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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 19:58:22 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 19:58:22 -0700 |
| commit | ad9eef0da50e86f32db6d3e9ecdeb5b4207154e7 (patch) | |
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You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: The Theory of the Relativity of Motion % +% % +% Author: Richard Chace Tolman % +% % +% Release Date: June 17, 2010 [EBook #32857] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK THE THEORY OF THE RELATIVITY *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{32857} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% fontenc: For boldface small-caps. Required. %% +%% %% +%% calc: Infix arithmetic. Required. %% +%% %% +%% ifthen: Logical conditionals. 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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: The Theory of the Relativity of Motion + +Author: Richard Chace Tolman + +Release Date: June 17, 2010 [EBook #32857] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE THEORY OF THE RELATIVITY *** +\end{PGtext} +\end{minipage} +\end{center} + + +%%%% Credits and transcriber's note %%%% +\clearpage +\thispagestyle{empty} + +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Berj Zamanian, Joshua +Hutchinson and the Online Distributed Proofreading Team +at http://www.pgdp.net (This file was produced from images +from the Cornell University Library: Historical Mathematics +Monographs collection.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter +\normalsize + + +%% -----File: 001.png---------- +\iffalse +Production Note + +Cornell University Library produced +this volume to replace the +irreparably deteriorated original. +It was scanned using Xerox software +and equipment at 600 dots +per inch resolution and compressed +prior to storage using +CCITT Group 4 compression. The +digital data were used to create +Cornell's replacement volume on +paper that meets the ANSI Standard +Z39.48-1984. The production +of this volume was supported in +part by the Commission on Preservation +and Access and the Xerox +Corporation. Digital file copyright +by Cornell University +Library 1992. +\fi +%% -----File: 002.png---------- +%[Blank Page] +%% -----File: 003.png---------- +\iffalse %[** TN: Cornell University Ex Libris page] + +%[Illustration: Cornell insignia] + +Cornell University Library +Ithaca, New York + +BOUGHT WITH THE INCOME OF THE +SAGE ENDOWMENT FUND +THE GIFT OF +HENRY W. SAGE + +1891 +\fi +%% -----File: 004.png---------- +%[Blank Page] +%% -----File: 005.png---Folio i------- +%% Title page +\begin{center} +\setlength{\TmpLen}{24pt}% +\LARGE\textbf{THE THEORY OF \\[\TmpLen] +THE RELATIVITY OF MOTION} \\[3\TmpLen] +\footnotesize BY \\[0.5\TmpLen] +\large RICHARD C. TOLMAN +\vfill + +\footnotesize UNIVERSITY OF CALIFORNIA PRESS \\ +BERKELEY \\[8pt] +1917 +\end{center} +%% -----File: 006.png---Folio ii------- +%% Verso +\clearpage +\null\vfill +\begin{center} +\scshape\tiny Press of \\ +The New Era Printing Company \\ +Lancaster, Pa +\end{center} +\vfill +%% -----File: 007.png---Folio iii------- +\clearpage +\null\vfill +\begin{center} +\footnotesize TO + +\large H. E. +\end{center} +\vfill +%% -----File: 008.png---Folio iv------- +%[Blank Page] +%% -----File: 009.png---Folio v------- +\cleardoublepage +%\pagestyle{fancy}**** +\phantomsection\pdfbookmark[0]{Table of Contents}{Contents} +\small +\tableofcontents +\normalsize + +\iffalse +%%%% Table of contents auto-generated; Scanned ToC commented out. %%%% +%[** TN: Heading below is printed by \contentsname] +THE THEORY OF THE RELATIVITY OF MOTION. +BY +RICHARD C. TOLMAN, PH.D. +TABLE OF CONTENTS. + +\textsc{Preface} 1 +\textsc{Chapter} I. Historical Development of Ideas as to the Nature of Space and +Time 5 +\textit{Part I}. The Space and Time of Galileo and Newton 5 +Newtonian Time 6 +Newtonian Space 7 +The Galileo Transformation Equations 9 +\textit{Part II}. The Space and Time of the Ether Theory 10 +Rise of the Ether Theory 10 +Idea of a Stationary Ether 12 +Ether in the Neighborhood of Moving Bodies 12 +Ether Entrained in Dielectrics 13 +The Lorentz Theory of a Stationary Ether 13 +\textit{Part III}. Rise of the Einstein Theory of Relativity 17 +The Michelson-Morley Experiment 17 +The Postulates of Einstein 18 +\textsc{Chapter} II. The Two Postulates of the Einstein Theory of Relativity 20 +The First Postulate of Relativity 20 +The Second Postulate of the Einstein Theory of Relativity 21 +Suggested Alternative to the Postulate of the Independence of the +Velocity of Light and the Velocity of the Source 23 +Evidence against Emission Theories of Light 24 +Different Forms of Emission Theory 25 +Further Postulates of the Theory of Relativity 27 +\textsc{Chapter} III. Some Elementary Deductions 28 +Measurements of Time in a Moving System 28 +Measurements of Length in a Moving System 30 +The Setting of Clocks in a Moving System 33 +The Composition of Velocities 35 +The Mass of a Moving Body 37 +The Relation between Mass and Energy 39 +\textsc{Chapter} IV. The Einstein Transformation Equations for Space and Time 42 +The Lorentz Transformation 42 +Deduction of the Fundamental Transformation Equations 43 +The Three Conditions to be Fulfilled 44 +The Transformation Equations 45 +Further Transformation Equations 47 +Transformation Equations for Velocity 47 +Transformation Equations for the Function $\dfrac{1}{\sqrt{1-\frac{u^2}{c^2}}}$ 47 +%% -----File: 010.png---Folio vi------- +Transformation Equations for Acceleration 48 +Chapter V. Kinematical Applications 49 +The Kinematical Shape of a Rigid Body 49 +The Kinematical Rate of a Clock 50 +The Idea of Simultaneity 51 +The Composition of Velocities 52 +The Case of Parallel Velocities 52 +Composition of Velocities in General 53 +Velocities Greater than that of Light 54 +Applications to Optical Problems 56 +The Doppler Effect 57 +The Aberration of Light 59 +Velocity of Light in Moving Media 60 +Group Velocity 61 +Chapter VI. The Dynamics of a Particle 62 +The Laws of Motion 62 +Difference between Newtonian and Relativity Mechanics 62 +The Mass of a Moving Particle 63 +Transverse Collision 63 +Mass the Same in all Directions 66 +Longitudinal Collision 67 +Collision of any Type 68 +Transformation Equations for Mass 72 +The Force Acting on a Moving Particle 73 +Transformation Equations for Force 73 +The Relation between Force and Acceleration 74 +Transverse and Longitudinal Acceleration 76 +The Force Exerted by a Moving Charge 77 +The Field around a Moving Charge 79 +Application to a Specific Problem 80 +Work 81 +Kinetic Energy 81 +Potential Energy 82 +The Relation between Mass and Energy 83 +Application to a Specific Problem 85 +Chapter VII. The Dynamics of a System of Particles 88 +On the Nature of a System of Particles 88 +The Conservation of Momentum 89 +The Equation of Angular Momentum 90 +The Function $T$ 92 +The Modified Lagrangian Function 93 +The Principle of Least Action 93 +Lagrange's Equations 95 +Equations of Motion in the Hamiltonian Form 96 +Value of the Function $T'$ 97 +The Principle of the Conservation of Energy 99 +On the Location of Energy in Space 100 +%% -----File: 011.png---Folio vii------- +\textsc{Chapter} VIII. The Chaotic Motion of a System of Particles 102 +The Equations of Motion 102 +Representation in Generalized Space 103 +Liouville's Theorem 103 +A System of Particles 104 +Probability of a Given Statistical State 105 +Equilibrium Relations 106 +The Energy as a Function of the Momentum 108 +The Distribution Law 109 +Polar Coördinates 110 +The Law of Equipartition 110 +Criterion for Equality of Temperature 112 +Pressure Exerted by a System of Particles 113 +The Relativity Expression for Temperature 114 +The Partition of Energy 117 +Partition of Energy for Zero Mass 117 +Approximate Partition for Particles of any Mass 118 +\textsc{Chapter} IX. The Principle of Relativity and the Principle of Least Action. 121 +The Principle of Least Action 121 +The Equations of Motion in the Lagrangian Form 122 +Introduction of the Principle of Relativity 124 +Relation between $\int W'dt'$ and $\int Wdt$ 124 +Relation between $H'$ and $H$ 127 +\textsc{Chapter} X. The Dynamics of Elastic Bodies 130 +On the Impossibility of Absolutely Rigid Bodies 130 +\textit{Part I}. Stress and Strain 130 +Definition of Strain 130 +Definition of Stress 132 +Transformation Equations for Strain 133 +Variation in the Strain 134 +\textit{Part II}. Introduction of the Principle of Least Action 137 +The Kinetic Potential for an Elastic Body 137 +Lagrange's Equations 138 +Transformation Equations for Stress 139 +Value of $E°$ 139 +The Equations of Motion in the Lagrangian Form 140 +Density of Momentum 142 +Density of Energy 142 +Summary of Results from the Principle of Least Action 142 +\textit{Part III}. Some Mathematical Relations 143 +The Unsymmetrical Stress Tensor $\mathrm{t}$ 143 +The Symmetrical Tensor $\mathrm{p}$ 145 +Relation between div $\mathrm{t}$ and $\mathrm{t}_n$ 146 +The Equations of Motion in the Eulerian Form 147 +\textit{Part IV}. Applications of the Results 148 +Relation between Energy and Momentum 148 +The Conservation of Momentum 149 +%% -----File: 012.png---Folio viii------- +The Conservation of Angular Momentum 150 +Relation between Angular Momentum and the Unsymmetrical +Stress Tensor 151 +The Right-Angled Lever 152 +Isolated Systems in a Steady State 154 +The Dynamics of a Particle 154 +Conclusion 154 +\textsc{Chapter} XI. The Dynamics of a Thermodynamic System 156 +The Generalized Coördinates and Forces 156 +Transformation Equation for Volume 156 +Transformation Equation for Entropy 157 +Introduction of the Principle of Least Action. The Kinetic +Potential 157 +The Lagrangian Equations 158 +Transformation Equation for Pressure 159 +Transformation Equation for Temperature 159 +The Equations of Motion for Quasistationary Adiabatic Acceleration +160 +The Energy of a Moving Thermodynamic System 161 +The Momentum of a Moving Thermodynamic System 161 +The Dynamics of a Hohlraum 162 +\textsc{Chapter} XII. Electromagnetic Theory 164 +The Form of the Kinetic Potential 164 +The Principle of Least Action 165 +The Partial Integrations 165 +Derivation of the Fundamental Equations of Electromagnetic +Theory 166 +The Transformation Equations for $\mathrm{e}$, $\mathrm{h}$ and $\rho$ 168 +The Invariance of Electric Charge 170 +The Relativity of Magnetic and Electric Fields 171 +Nature of Electromotive Force 172 +Derivation of the Fifth Fundamental Equation 172 +Difference between the Ether and the Relativity Theories of Electromagnetics +173 +Applications to Electromagnetic Theory 176 +The Electric and Magnetic Fields around a Moving Charge 176 +The Energy of a Moving Electromagnetic System 178 +Relation between Mass and Energy 180 +The Theory of Moving Dielectrics 181 +Relation between Field Equations for Material Media and +Electron Theory 182 +Transformation Equations for Moving Media 183 +Theory of the Wilson Experiment 186 +\textsc{Chapter} XIII. Four-Dimensional Analysis 188 +Idea of a Time Axis 188 +Non-Euclidean Character of the Space 189 +%% -----File: 013.png---Folio ix------- +Part I. Vector Analysis of the Non-Euclidean Four-Dimensional Manifold +191 +Space, Time and Singular Vectors 192 +Invariance of $x^2 + y^2 + z^2 - c^2t^2$ 192 +Inner Product of One-Vectors 193 +Non-Euclidean Angle 194 +Kinematical Interpretation of Angle in Terms of Velocity 194 +Vectors of Higher Dimensions 195 +Outer Products 195 +Inner Product of Vectors in General 198 +The Complement of a Vector 198 +The Vector Operator, $\Diamond$ or Quad 199 +Tensors 200 +The Rotation of Axes 201 +Interpretation of the Lorentz Transformation as a Rotation of +Axes 206 +Graphical Representation 208 +Part II. Applications of the Four-Dimensional Analysis 211 +Kinematics 211 +Extended Position 211 +Extended Velocity 212 +Extended Acceleration 213 +The Velocity of Light 214 +The Dynamics of a Particle 214 +Extended Momentum 214 +The Conservation Laws 215 +The Dynamics of an Elastic Body 216 +The Tensor of Extended Stress 216 +The Equation of Motion 216 +Electromagnetics 217 +Extended Current 218 +The Electromagnetic Vector $\vc{M}$ 217 +The Field Equations 217 +The Conservation of Electricity 218 +The Product $\vc{M} · \vc{q}$ 218 +The Extended Tensor of Electromagnetic Stress 219 +Combined Electrical and Mechanical Systems 221 +Appendix I. Symbols for Quantities 222 +Scalar Quantities 222 +Vector Quantities 223 +Appendix II. Vector Notation 224 +Three Dimensional Space 224 +Non-Euclidean Four Dimensional Space 225 +\fi +%%%% End of commented table of contents %%%% +%% +%% -----File: 014.png---Folio x------- +%[Blank Page] +%% -----File: 015.png---Folio 1------- +\mainmatter +\phantomsection\pdfbookmark[-1]{Main Matter}{Main Matter} + +\Preface + +Thirty or forty years ago, in the field of physical science, there +was a widespread feeling that the days of adventurous discovery had +passed forever, and the conservative physicist was only too happy to +devote his life to the measurement to the sixth decimal place of +quantities whose significance for physical theory was already an old +story. The passage of time, however, has completely upset such +bourgeois ideas as to the state of physical science, through the discovery +of some most extraordinary experimental facts and the development +of very fundamental theories for their explanation. + +On the experimental side, the intervening years have seen the +discovery of radioactivity, the exhaustive study of the conduction of +electricity through gases, the accompanying discoveries of cathode, +canal and X-rays, the isolation of the electron, the study of the +distribution of energy in the hohlraum, and the final failure of all +attempts to detect the earth's motion through the supposititious +ether. During this same time, the theoretical physicist has been +working hand in hand with the experimenter endeavoring to correlate +the facts already discovered and to point the way to further research. +The theoretical achievements, which have been found particularly +helpful in performing these functions of explanation and prediction, +have been the development of the modern theory of electrons, the +application of thermodynamic and statistical reasoning to the phenomena +of radiation, and the development of Einstein's brilliant +theory of the relativity of motion. + +It has been the endeavor of the following book to present an +introduction to this theory of relativity, which in the decade since +the publication of Einstein's first paper in 1905 (\textit{Annalen der Physik}) +has become a necessary part of the theoretical equipment of every +physicist. Even if we regard the Einstein theory of relativity merely +as a convenient tool for the prediction of electromagnetic and optical +phenomena, its importance to the physicist is very great, not only +because its introduction greatly simplifies the deduction of many +%% -----File: 016.png---Folio 2------- +theorems which were already familiar in the older theories based on a +stationary ether, but also because it leads simply and directly to correct +conclusions in the case of such experiments as those of Michelson +and Morley, Trouton and Noble, and Kaufman and Bucherer, which +can be made to agree with the idea of a stationary ether only by the +introduction of complicated and \textit{ad~hoc} assumptions. Regarded from +a more philosophical point of view, an acceptance of the Einstein +theory of relativity shows us the advisability of completely remodelling +some of our most fundamental ideas. In particular we shall now +do well to change our concepts of space and time in such a way as +to give up the old idea of their complete independence, a notion +which we have received as the inheritance of a long ancestral experience +with bodies moving with slow velocities, but which no longer proves +pragmatic when we deal with velocities approaching that of light. + +The method of treatment adopted in the following chapters is +to a considerable extent original, partly appearing here for the first +time and partly already published elsewhere.\footnote + {\textit{Philosophical Magazine}, vol.~18, p.~510 (1909); + \textit{Physical Review}, vol.~31, p.~26 (1910); + \textit{Phil.\ Mag.}, vol.~21, p.~296 (1911); + \textit{ibid}., vol.~22, p.~458 (1911); + \textit{ibid}., vol.~23, p.~375 (1912); + \textit{Phys.\ Rev.}, vol.~35, p.~136 (1912); + \textit{Phil.\ Mag.}, vol.~25, p.~150 (1913); + \textit{ibid}., vol.~28, p.~572 (1914); + \textit{ibid}., vol.~28, p.~583 (1914).} +\Chapref{III} follows +a method which was first developed by Lewis and Tolman,\footnote + {\textit{Phil.\ Mag.}, vol.~18, p.~510 (1909).} +and the +\Chapnumref[XIII]{last chapter} a method developed by Wilson and Lewis.\footnote + {\textit{Proceedings of the American Academy of Arts and Sciences}, + vol.~48, p.~389 (1912).} +The writer +must also express his special obligations to the works of Einstein, +Planck, Poincaré, Laue, Ishiwara and Laub. + +It is hoped that the mode of presentation is one that will be found +well adapted not only to introduce the study of relativity theory to +those previously unfamiliar with the subject but also to provide the +necessary methodological equipment for those who wish to pursue +the theory into its more complicated applications. + +After presenting, in the \Chapnumref[I]{first chapter}, a brief outline of the historical +development of ideas as to the nature of the space and time of science, +we consider, in \Chapref{II}, the two main postulates upon which the +theory of relativity rests and discuss the direct experimental evidence +for their truth. The \Chapnumref[III]{third chapter} then presents an elementary and +%% -----File: 017.png---Folio 3------- +non-mathematical deduction of a number of the most important +consequences of the postulates of relativity, and it is hoped that this +chapter will prove especially valuable to readers without unusual +mathematical equipment, since they will there be able to obtain a +real grasp of such important new ideas as the change of mass with +velocity, the non-additivity of velocities, and the relation of mass +and energy, without encountering any mathematics beyond the +elements of analysis and geometry. + +In \Chapref{IV} we commence the more analytical treatment of +the theory of relativity by obtaining from the two postulates of +relativity Einstein's transformation equations for space and time as +well as transformation equations for velocities, accelerations, and +for an important function of the velocity. \Chapref{V} presents +various kinematical applications of the theory of relativity following +quite closely Einstein's original method of development. In particular +we may call attention to the ease with which we may handle +the optics of moving media by the methods of the theory of relativity +as compared with the difficulty of treatment on the basis of the ether +theory. + +In Chapters \Chapnumref{VI},~\Chapnumref{VII} and~\Chapnumref{VIII} we develop and apply a theory of +the dynamics of a particle which is based on the Einstein transformation +equations for space and time, Newton's three laws of motion, +and the principle of the conservation of mass. + +We then examine, in \Chapref{IX}, the relation between the theory +of relativity and the principle of least action, and find it possible to +introduce the requirements of relativity theory at the very start into +this basic principle for physical science. We point out that we +might indeed have used this adapted form of the principle of least +action, for developing the dynamics of a particle, and then proceed +in Chapters \Chapnumref{X},~\Chapnumref{XI} and~\Chapnumref{XII} to develop the dynamics of an elastic +body, the dynamics of a thermodynamic system, and the dynamics +of an electromagnetic system, all on the basis of our adapted form +of the principle of least action. + +Finally, in \Chapref{XIII}, we consider a four-dimensional method +of expressing and treating the results of relativity theory. This +chapter contains, in Part~I, an epitome of some of the more important +methods in four-dimensional vector analysis and it is hoped that it +%% -----File: 018.png---Folio 4------- +can also be used in connection with the earlier parts of the book as a +convenient reference for those who are not familiar with ordinary +three-dimensional vector analysis. + +In the present book, the writer has confined his considerations to +cases in which there is a \emph{uniform} relative velocity between systems of +coördinates. In the future it may be possible greatly to extend the +applications of the theory of relativity by considering accelerated +systems of coördinates, and in this connection Einstein's latest work +on the relation between gravity and acceleration is of great interest. +It does not seem wise, however, at the present time to include such +considerations in a book which intends to present a survey of accepted +theory. + +The author will feel amply repaid for the work involved in the +preparation of the book if, through his efforts, some of the younger +American physicists can be helped to obtain a real knowledge of the +important work of Einstein. He is also glad to have this opportunity +to add his testimony to the growing conviction that the conceptual +space and time of science are not God-given and unalterable, but are +rather in the nature of human constructs devised for use in the description +and correlation of scientific phenomena, and that these +spatial and temporal concepts should be altered whenever the discovery +of new facts makes such a change pragmatic. + +The writer wishes to express his indebtedness to Mr.~William~H. +Williams for assisting in the preparation of Chapter~I\@. %[** TN: Not a useful cross-reference] +%% -----File: 019.png---Folio 5------- + + +\Chapter{I}{Historical Development of Ideas as to the Nature of +Space and Time.} +\SetRunningHeads{Chapter One.}{Historical Development.} + +\Paragraph{1.} Since the year 1905, which marked the publication of Einstein's +momentous article on the theory of relativity, the development of +scientific thought has led to a complete revolution in accepted ideas +as to the nature of space and time, and this revolution has in turn +profoundly modified those dependent sciences, in particular mechanics +and electromagnetics, which make use of these two fundamental +concepts in their considerations. + +In the following pages it will be our endeavor to present a description +of these new notions as to the nature of space and time,\footnote + {Throughout this work by ``space'' and ``time'' we shall mean the \emph{conceptual} + space and time of science.} +and to give a partial account of the consequent modifications which +have been introduced into various fields of science. Before proceeding +to this task, however, we may well review those older ideas +as to space and time which until now appeared quite sufficient for +the correlation of scientific phenomena. We shall first consider the +space and time of Galileo and Newton which were employed in the +development of the classical mechanics, and then the space and time +of the ether theory of light. + + +\Section[I]{The Space and Time of Galileo and Newton.} + +\Paragraph{2.} The publication in 1687 of Newton's \textit{Principia} laid down so +satisfactory a foundation for further dynamical considerations, that +it seemed as though the ideas of Galileo and Newton as to the nature +of space and time, which were there employed, would certainly remain +forever suitable for the interpretation of natural phenomena. And +indeed upon this basis has been built the whole structure of classical +mechanics which, until our recent familiarity with very high velocities, +has been found completely satisfactory for an extremely large number +of very diverse dynamical considerations. +%% -----File: 020.png---Folio 6------- + +An examination of the fundamental laws of mechanics will show +how the concepts of space and time entered into the Newtonian +system of mechanics. Newton's laws of motion, from which the +whole of the classical mechanics could be derived, can best be stated +with the help of the equation +\[ +\vc{F} = \frac{d}{dt} (m\vc{u}). +\Tag{1} +\] +This equation defines the force~$\vc{F}$ acting on a particle as equal to the +rate of change in its momentum (\ie, the product of its mass~$m$ and +its velocity~$\vc{u}$), and the whole of Newton's laws of motion may be +summed up in the statement that in the case of two interacting particles +the forces which they mutually exert on each other are equal in +magnitude and opposite in direction. + +Since in Newtonian mechanics the mass of a particle is assumed +constant, equation~(1) may be more conveniently written +\[ +\vc{F} + = m \frac{d\vc{u}}{dt} + = m \frac{d}{dt} \left( \frac{d\vc{r}}{dt} \right), +\] +or +\[ +\begin{aligned} + F_x &= m \frac{d}{dt} \left( \frac{dx}{dt} \right),\\ + F_y &= m \frac{d}{dt} \left( \frac{dy}{dt} \right),\\ + F_z &= m \frac{d}{dt} \left( \frac{dz}{dt} \right), +\end{aligned} +\Tag{2} +\] +and this definition of force, together with the above-stated principle +of the equality of action and reaction, forms the starting-point for +the whole of classical mechanics. + +The necessary dependence of this mechanics upon the concepts +of space and time becomes quite evident on an examination of this +fundamental equation~(2), in which the expression for the force acting +on a particle is seen to contain both the variables $x$,~$y$, and~$z$, which +specify the position of the particle in \emph{space}, and the variable~$t$, which +specifies the \emph{time}. + +\Subsubsection{3}{Newtonian Time.} To attempt a definite statement as to the +%% -----File: 021.png---Folio 7------- +meaning of so fundamental and underlying a notion as that of time +is a task from which even philosophy may shrink. In a general +way, conceptual time may be thought of as a \emph{one-dimensional}, \emph{unidirectional}, +\emph{one-valued} continuum. This continuum is a sort of framework +in which the instants at which actual occurrences take place +find an ordered position. Distances from point to point in the +continuum, that is intervals of time, are measured by the periods of +certain continually recurring cyclic processes such as the daily rotation +of the earth. A unidirectional nature is imposed upon the time +continuum among other things by an acceptance of the second law +of thermodynamics, which requires that actual progression in time +shall be accompanied by an increase in the entropy of the material +world, and this same law requires that the continuum shall be one-valued +since it excludes the possibility that time ever returns upon +itself, either to commence a new cycle or to intersect its former path +even at a single point. + +In addition to these characteristics of the time continuum, which +have been in no way modified by the theory of relativity, the \emph{Newtonian +mechanics always assumed a complete independence of time and +the three-dimensional space continuum} which exists along with it. +In dynamical equations time entered as an \emph{entirely independent} variable +in no way connected with the variables whose specification +determines position in space. In the following pages, however, we +shall find that the theory of relativity requires a very definite interrelation +between time and space, and in the Einstein transformation +equations we shall see the exact way in which measurements of time +depend upon the choice of a set of variables for measuring position +in space. + +\Subsubsection{4}{Newtonian Space.} An exact description of the concept of +space is perhaps just as difficult as a description of the concept of time. +In a general way we think of space as a \emph{three-dimensional}, \emph{homogeneous}, +\emph{isotropic} continuum, and these ideas are common to the +conceptual spaces of Newton, Einstein, and the ether theory of light. +The space of Newton, however, differs on the one hand from that of +Einstein because of a tacit assumption of the complete independence +of space and time measurements; and differs on the other hand from +that of the ether theory of light by the fact that ``free'' space was +%% -----File: 022.png---Folio 8------- +assumed completely empty instead of filled with an all-pervading +quasi-material medium---the ether. A more definite idea of the particularly +important characteristics of the Newtonian concept of space +may be obtained by considering somewhat in detail the actual methods +of space measurement. + +Positions in space are in general measured with respect to some +arbitrarily fixed system of reference which must be threefold in +character corresponding to the three dimensions of space. In particular +we may make use of a set of Cartesian axes and determine, +for example, the position of a particle by specifying its three Cartesian +coördinates $x$,~$y$ and~$z$. + +In Newtonian mechanics the particular set of axes chosen for +specifying position in space has in general been determined in the +first instance by considerations of convenience. For example, it is +found by experience that, if we take as a reference system lines drawn +upon the surface of the earth, the equations of motion based on Newton's +laws give us a simple description of nearly all dynamical phenomena +which are merely terrestrial. When, however, we try to +interpret with these same axes the motion of the heavenly bodies, we +meet difficulties, and the problem is simplified, so far as planetary +motions are concerned, by taking a new reference system determined +by the sun and the fixed stars. But this system, in its turn, becomes +somewhat unsatisfactory when we take account of the observed +motions of the stars themselves, and it is finally convenient to take a +reference system relative to which the sun is moving with a velocity +of twelve miles per second in the direction of the constellation Hercules. +This system of axes is so chosen that the great majority of stars have +on the average no motion with respect to it, and the actual motion +of any particular star with respect to these coördinates is called the +peculiar motion of the star. + +Suppose, now, we have a number of such systems of axes in uniform +relative motion; we are confronted by the problem of finding +some method of transposing the description of a given kinematical +occurrence from the variables of one of these sets of axes to those of +another. For example, if we have chosen a system of axes~$S$ and +have found an equation in $x$,~$y$,~$z$, and~$t$ which accurately describes the +motion of a given point, what substitutions for the quantities involved +%% -----File: 023.png---Folio 9------- +can be made so that the new equation thereby obtained will again +correctly describe the same phenomena when we measure the displacements +of the point relative to a new system of reference~$S'$ +which is in uniform motion with respect to~$S$? The assumption of +Galileo and Newton that ``free'' space is entirely empty, and the +further tacit assumption of the complete independence of space and +time, led them to propose a very simple solution of the problem, and +the transformation equations which they used are generally called +the Galileo Transformation Equations to distinguish them from the +Einstein Transformation Equations which we shall later consider. + +\Subsubsection{5}{The Galileo Transformation Equations.} Consider two systems +of right-angled coördinates, $S$~and~$S'$, which are in relative motion in +the $X$~direction with the velocity~$V$; for convenience let the $X$~axes, +$OX$~and~$O'X'$, of the two systems coincide in direction, and for further +simplification let us take as our zero point for time measurements the +instant when the two origins $O$~and~$O'$ coincide. Consider now a +point which at the time~$t$ has the coördinates $x$,~$y$ and~$z$ measured in +system~$S$. Then, according to the space and time considerations of +Galileo and Newton, the coördinates of the point with reference to +system~$S'$ are given by the following transformation equations: +\begin{align*} +x' &= x-Vt, \Tag{3}\displaybreak[0] \\ +y' &= y, \Tag{4}\displaybreak[0] \\ +z' &= z, \Tag{5}\displaybreak[0] \\ +t' &= t. \Tag{6} +\end{align*} +These equations are fundamental for Newtonian mechanics, and may +appear to the casual observer to be self-evident and bound up with +necessary ideas as to the nature of space and time. Nevertheless, +the truth of the first and the last of these equations is absolutely +dependent on the unsupported assumption of the complete independence +of space and time measurements, and since in the Einstein +theory we shall find a very definite relation between space and time +measurements we shall be led to quite a different set of transformation +equations. Relations (3),~(4),~(5) and~(6) will be found, however, to +be the limiting form which the correct transformation equations assume +when the velocity between the systems~$V$ becomes small compared +%% -----File: 024.png---Folio 10------- +with that of light. Since until very recent times the human +race in its entire past history has been familiar only with velocities +that are small compared with that of light, it need not cause surprise +that the above equations, which are true merely at the limit, should +appear so self-evident. + +\Paragraph{6.} Before leaving the discussion of the space and time system of +Newton and Galileo we must call attention to an important characteristic +which it has in common with the system of Einstein but +which is not a feature of that assumed by the ether theory. If we +have two systems of axes such as those we have just been considering, +we may with equal right consider either one of them at rest and the +other moving past it. All we can say is that the two systems are in +relative motion; it is meaningless to speak of either one as in any +sense ``\textit{absolutely}'' at rest. The equation $x' = x - Vt$ which we +use in transforming the description of a kinematical event from the +variables of system $S$ to those of system $S'$ is perfectly symmetrical +with the equation $x = x' + Vt'$ which we should use for a transformation +in the reverse direction. Of all possible systems no particular +set of axes holds a unique position among the others. We +shall later find that this important principle of the relativity of motion +is permanently incorporated into our system of physical science as +the \textit{first postulate of relativity}. This principle, common both to the +space of Newton and to that of Einstein, is not characteristic of the +space assumed by the classical theory of light. The space of this +theory was supposed to be filled with a stationary medium, the +luminiferous ether, and a system of axes stationary with respect to +this ether would hold a unique position among the other systems +and be the one peculiarly adapted for use as the ultimate system of +reference for the measurement of motions. + +We may now briefly sketch the rise of the ether theory of light and +point out the permanent contribution which it has made to physical +science, a contribution which is now codified as the second postulate +of relativity. + + +\Section[II]{The Space and Time of the Ether Theory.} + +\Subsubsection{7}{Rise of the Ether Theory.} Twelve years before the appearance +of the \textit{Principia}, Römer, a Danish astronomer, observed that an +%% -----File: 025.png---Folio 11------- +eclipse of one of the satellites of Jupiter occurred some ten minutes +later than the time predicted for the event from the known period +of the satellite and the time of the preceding eclipse. He explained +this delay by the hypothesis that it took light twenty-two minutes +to travel across the earth's orbit. Previous to Römer's discovery, +light was generally supposed to travel with infinite velocity. Indeed +Galileo had endeavored to find the speed of light by direct experiments +over distances of a few miles and had failed to detect any lapse of +time between the emission of a light flash from a source and its observation +by a distant observer. Römer's hypothesis has been repeatedly +verified and the speed of light measured by different methods +with considerable exactness. The mean of the later determinations +is $2.9986 × 10^{10}$ cm.~per second. + +\Paragraph{8.} At the time of Römer's discovery there was much discussion +as to the nature of light. Newton's theory that it consisted of particles +or corpuscles thrown out by a luminous body was attacked by +Hooke and later by Huygens, who advanced the view that it was +something in the nature of wave motions in a supposed space-filling +medium or ether. By this theory Huygens was able to explain +reflection and refraction and the phenomena of color, but assuming +\emph{longitudinal} vibrations he was unable to account for polarization. +Diffraction had not yet been observed and Newton contested the +Hooke-Huygens theory chiefly on the grounds that it was contradicted +by the fact of rectilinear propagation and the formation of +shadows. The scientific prestige of Newton was so great that the +emission or corpuscular theory continued to hold its ground for a +hundred and fifty years. Even the masterly researches of Thomas +Young at the beginning of the nineteenth century were unable to +dislodge the old theory, and it was not until the French physicist, +Fresnel, about 1815, was independently led to an undulatory theory +and added to Young's arguments the weight of his more searching +mathematical analysis, that the balance began to turn. From this +time on the wave theory grew in power and for a period of eighty +years was not seriously questioned. This theory has for its essential +postulate the existence of an all-pervading medium, the ether, in +which wave disturbances can be set up and propagated. And the +physical properties of this medium became an enticing field of inquiry +and speculation. +%% -----File: 026.png---Folio 12------- + +\Subsubsection{9}{Idea of a Stationary Ether.} Of all the various properties with +which the physicist found it necessary to endow the ether, for us the +most important is the fact that it must apparently remain stationary, +unaffected by the motion of matter through it. This conclusion was +finally reached through several lines of investigation. We may first +consider whether the ether would be dragged along by the motion of +nearby masses of matter, and, second, whether the ether enclosed in a +moving medium such as water or glass would partake in the latter's +motion. + +\Subsubsection{10}{Ether in the Neighborhood of Moving Bodies.} About the +year 1725 the astronomer Bradley, in his efforts to measure the +parallax of certain fixed stars, discovered that the apparent position +of a star continually changes in such a way as to trace annually a +small ellipse in the sky, the apparent position always lying in the +plane determined by the line from the earth to the center of the +ellipse and by the direction of the earth's motion. On the corpuscular +theory of light this admits of ready explanation as Bradley himself +discovered, since we should expect the earth's motion to produce an +apparent change in the direction of the oncoming light, in just the +same way that the motion of a railway train makes the falling drops +of rain take a slanting path across the window pane. If $\DPtypo{\vc{c}}{c}$~be the +velocity of a light particle and $\DPtypo{\vc{v}}{v}$~the earth's velocity, the apparent or +relative velocity would be $\DPtypo{\vc{c - v}}{c - v}$ and the tangent of the angle of +aberration would be~$\dfrac{v}{c}$. + +Upon the wave theory, it is obvious that we should \emph{also} expect a +similar aberration of light, provided only that the ether shall be +quite stationary and unaffected by the motion of the earth through it, +and this is one of the important reasons that most ether theories have +assumed a \emph{stationary ether unaffected by the motion of neighboring +matter}.\footnote + {The most notable exception is the theory of Stokes, which did assume that + the ether moved along with the earth and then tried to account for aberration with + the help of a velocity potential, but this led to difficulties, as was shown by Lorentz.} + +In more recent years further experimental evidence for assuming +that the ether is not dragged along by the neighboring motion of +large masses of matter was found by Sir Oliver Lodge. His final +experiments were performed with a large rotating spheroid of iron +%% -----File: 027.png---Folio 13------- +with a narrow groove around its equator, which was made the path +for two rays of light, one travelling in the direction of rotation and +the other in the opposite direction. Since by interference methods +no difference could be detected in the velocities of the two rays, here +also the conclusion was reached that \emph{the ether was not appreciably +dragged along by the rotating metal}. + +\Subsubsection{11}{Ether Entrained in Dielectrics.} With regard to the action of +a moving medium on the ether which might be entrained within it, +experimental evidence and theoretical consideration here too finally +led to the supposition that the ether itself must remain perfectly +stationary. The earlier view first expressed by Fresnel, in a letter +written to Arago in 1818, was that the entrained ether did receive a +fraction of the total velocity of the moving medium. Fresnel gave +to this fraction the value~$\dfrac{\mu^2-1}{\mu^2}$, where $\mu$~is the index of refraction of +the substance forming the medium. On this supposition, Fresnel +was able to account for the fact that Arago's experiments upon the +reflection and refraction of stellar rays show no influence whatever +of the earth's motion, and for the fact that Airy found the same angle +of aberration with a telescope filled with water as with air. Moreover, +the later work of Fizeau and the accurate determinations of +Michelson and Morley on the velocity of light in a moving stream +of water did show that the speed was changed by an amount corresponding +to Fresnel's fraction. The fuller theoretical investigations +of Lorentz, however, did not lead scientists to look upon this increased +velocity of light in a moving medium as an evidence that the ether +is pulled along by the stream of water, and we may now briefly sketch +the developments which culminated in the Lorentz theory of a completely +stationary ether. + +\Subsubsection{12}{The Lorentz Theory of a Stationary Ether.} The considerations +of Lorentz as to the velocity of light in moving media became +possible only after it was evident that optics itself is a branch of the +wider science of electromagnetics, and it became possible to treat +transparent media as a special case of dielectrics in general. In 1873, +in his \textit{Treatise on Electricity and Magnetism}, Maxwell first advanced +the theory that electromagnetic phenomena also have their seat in +the luminiferous ether and further that light itself is merely an electromagnetic +%% -----File: 028.png---Folio 14------- +disturbance in that medium, and Maxwell's theory was +confirmed by the actual discovery of electromagnetic waves in 1888 +by Hertz. + +The attack upon the problem of the relative motion of matter and +ether was now renewed with great vigor both theoretically and experimentally +from the electromagnetic side. Maxwell in his treatise had +confined himself to phenomena in stationary media. Hertz, however, +extended Maxwell's considerations to moving matter on the assumption +that the entrained ether is carried bodily along by it. It is evident, +however, that in the field of optical theory such an assumption +could not be expected to account for the Fizeau experiment, which +had already been explained on the assumption that the ether receives +only a fraction of the velocity of the moving medium; while in the +field of electromagnetic theory it was found that Hertz's assumptions +would lead us to expect \emph{no} production of a magnetic field in the +neighborhood of a rotating electric condenser providing the plates of +the condenser and the dielectric move together with the same speed +and this was decisively disproved by the experiment of Eichenwald. +The conclusions of the Hertz theory were also out of agreement with +the important experiments of H.~A.~Wilson on moving dielectrics. +It remained for Lorentz to develop a general theory for moving +dielectrics which was consistent with the facts. + +The theory of Lorentz developed from that of Maxwell by the +addition of the idea of the \emph{electron}, as the atom of electricity, and his +treatment is often called the ``electron theory.'' This atomistic +conception of electricity was foreshadowed by Faraday's discovery +of the quantitative relations between the amount of electricity associated +with chemical reactions in electrolytes and the weight of +substance involved, a relation which indicates that the atoms act as +carriers of electricity and that the quantity of electricity carried by a +single particle, whatever its nature, is always some small multiple of a +definite quantum of electricity, the electron. Since Faraday's time, +the study of the phenomena accompanying the conduction of electricity +through gases, the study of radioactivity, and finally indeed +the isolation and exact measurement of these atoms of electrical +charge, have led us to a very definite knowledge of many of the +properties of the electron. +%% -----File: 029.png---Folio 15------- + +While the experimental physicists were at work obtaining this +more or less first-hand acquaintance with the electron, the theoretical +physicists and in particular Lorentz were increasingly successful in +explaining the electrical and optical properties of matter in general +on the basis of the behavior of the electrons which it contains, the +properties of conductors being accounted for by the presence of movable +electrons, either free as in the case of metals or combined with +atoms to form ions as in electrolytes, while the electrical and optical +properties of dielectrics were ascribed to the presence of electrons +more or less bound by quasi-elastic forces to positions of equilibrium. +This Lorentz electron theory of matter has been developed in great +mathematical detail by Lorentz and has been substantiated by numerous +quantitative experiments. Perhaps the greatest significance +of the Lorentz theory is that such properties of matter as electrical +conductivity, magnetic permeability and dielectric inductivity, which +occupied the position of rather accidental experimental constants in +Maxwell's original theory, are now explainable as the statistical result +of the behavior of the individual electrons. + +With regard now to our original question as to the behavior of +\emph{moving} optical and dielectric media, the Lorentz theory was found +capable of accounting quantitatively for all known phenomena, including +Airy's experiment on aberration, Arago's experiments on the +reflection and refraction of stellar rays, Fresnel's coefficient for the +velocity of light in moving media, and the electromagnetic experiments +upon moving dielectrics made by Röntgen, Eichenwald, H.~A.~Wilson, +and others. For us the particular significance of the Lorentz +method of explaining these phenomena is that he does \emph{not} assume, as +did Fresnel, that the ether is partially dragged along by moving +matter. His investigations show rather that the ether must remain +perfectly stationary, and that such phenomena as the changed velocity +of light in moving media are to be accounted for by the modifying +influence which the electrons in the moving matter have upon the +propagation of electromagnetic disturbances, rather than by a dragging +along of the ether itself. + +Although it would not be proper in this place to present the +mathematical details of Lorentz's treatment of moving media, we +may obtain a clearer idea of what is meant in the Lorentz theory by a +%% -----File: 030.png---Folio 16------- +stationary ether if we look for a moment at the five fundamental +equations upon which the theory rests. These familiar equations, of +which the first four are merely Maxwell's four field equations, modified +by the introduction of the idea of the electron, may be written +\begin{align*} +\curl \vc{h} + &= \frac{1}{c}\, \frac{\partial \vc{e}}{\partial t} + + \rho\, \frac{\vc{u}}{c},\\ +\curl \vc{e} + &= -\frac{1}{c}\, \frac{\partial \vc{h}}{\partial t},\\ +\divg \vc{e} &= \rho,\\ +\divg \vc{h} &= 0,\\ +\vc{f} &= \rho\left\{ + \vc{e} + \left[\frac{\vc{u}}{c} × \vc{h}\right]^*\right\} +\end{align*} +in which the letters have their usual significance. (See \Chapref{XII}\@.) +Now the whole of the Lorentz theory, including of course his treatment +of moving media, is derivable from these five equations, and +the fact that the idea of a stationary ether does lie at the basis of +his theory is most clearly shown by the first and last of these equations, +which contain the velocity~$\vc{u}$ with which the charge in question +is moving, and \emph{for Lorentz this velocity is to be measured with respect +to the assumed stationary ether}. + +We have devoted this space to the Lorentz theory, since his work +marks the culmination of the ether theory of light and electromagnetism, +and for us the particularly significant fact is that by this +line of attack science was \emph{inevitably led to the idea of an absolutely +immovable and stationary ether}. + +\Paragraph{13.} We have thus briefly traced the development of the ether +theory of light and electromagnetism. We have seen that the space +continuum assumed by this theory is not empty as was the space of +Newton and Galileo but is assumed filled with a stationary medium, +the ether, and in conclusion should further point out that the \emph{time +continuum} assumed by the ether theory was apparently the same as +that of Newton and Galileo, and in particular that the \emph{old ideas as to +the absolute independence of space and time were all retained}. +%% -----File: 031.png---Folio 17------- + + +\Section[III]{Rise of the Einstein Theory of Relativity.} + +\Subsubsection{14}{The Michelson-Morley Experiment.} In spite of all the brilliant +achievements of the theory of a stationary ether, we must now +call attention to an experiment, performed at the very time when +the success of the ether theory seemed most complete, whose result +was in direct contradiction to its predictions. This is the celebrated +Michelson-Morley experiment, and to the masterful interpretation of +its consequences at the hands of Einstein we owe the whole theory of +relativity, a theory which will nevermore permit us to assume that +space and time are independent. + +If the theory of a stationary ether were true we should find, contrary +to the expectations of Newton, that systems of coördinates in +relative motion are not symmetrical, a system of axes fixed relatively +to the ether would hold a unique position among all other systems +moving relative to it and would be peculiarly adapted for the measurement +of displacements and velocities. Bodies at rest with respect +to this system of axes fixed in the ether would be spoken of as ``absolutely'' +at rest and bodies in motion through the ether would be +said to have ``absolute'' motion. From the point of view of the +ether theory one of the most important physical problems would be +to determine the velocity of various bodies, for example that of the +earth, through the ether. + +Now the Michelson-Morley experiment was devised for the very +purpose of determining the relative motion of the earth and the ether. +The experiment consists essentially in a comparison of the velocities +of light parallel and perpendicular to the earth's motion in its orbit. +A ray of light from the source~$S$ falls on the half silvered mirror~$A$, +where it is divided into two rays, one of which travels to the mirror~$B$ +and the other to the mirror~$C$, where they are totally reflected. The +rays are recombined and produce a set of interference fringes at~$\DPtypo{0}{O}$. +(See \Figref{1}.) + +We may now think of the apparatus as set so that one of the +divided paths is parallel to the earth's motion and the other perpendicular +to it. On the basis of the stationary ether theory, the +velocity of the light with reference to the apparatus would evidently +be different over the two paths, and hence on rotating the apparatus +%% -----File: 032.png---Folio 18------- +through an angle of ninety degrees we should expect a shift in the +position of the fringes. Knowing the magnitude of the earth's +velocity in its orbit and the dimensions of the apparatus, it is quite +possible to calculate the magnitude of the expected shift, a quantity +\begin{figure}[hbt] + \begin{center} + \Fig{1} + \Input[3in]{032} + \end{center} +\end{figure} +entirely susceptible of experimental determination. Nevertheless the +most careful experiments made at different times of day and at +different seasons of the year entirely failed to show any such shift +at all. + +This result is in direct contradiction to the theory of a stationary +ether and could be reconciled with that theory only by very arbitrary +assumptions. Instead of making such assumptions, the Einstein +theory of relativity finds it preferable to return in part to the older +ideas of Newton and Galileo. + +\Subsubsection{15}{The Postulates of Einstein.} In fact, in accordance with the +results of this work of Michelson-Morley and other confirmatory +experiments, the Einstein theory takes as its \emph{first postulate} the idea +familiar to Newton of the relativity of all motion. It states that +there is nothing out in space in the nature of an ether or of a fixed +set of coördinates with regard to which motion can be measured, +that there is no such thing as absolute motion, and that all we can +speak of is the relative motion of one body with respect to another. +%% -----File: 033.png---Folio 19------- + +Although we thus see that the Einstein theory of relativity has +returned in part to the ideas of Newton and Galileo as to the nature +of space, it is not to be supposed that the ether theory of light and +electromagnetism has made no lasting contribution to physical science. +Quite on the contrary, not only must the ideas as to the periodic and +polarizable nature of the light disturbance, which were first appreciated +and understood with the help of the ether theory, always +remain incorporated in every optical theory, but in particular the +Einstein theory of relativity takes as the basis for its \emph{second postulate} +a principle that has long been familiar to the ether theory, namely +that the velocity of light is independent of the velocity of the source. +We shall see in following chapters that it is the combination of this +principle with the first postulate of relativity that leads to the whole +theory of relativity and to our new ideas as to the nature and interrelation +of space and time. +%% -----File: 034.png---Folio 20------- + + + +\Chapter{II}{The Two Postulates of the Einstein Theory of +Relativity.} +\SetRunningHeads{Chapter Two.}{The Two Postulates.} + +\Paragraph{16.} There are two general methods of evaluating the theoretical +development of any branch of science. One of these methods is to +test by direct experiment the fundamental postulates upon which +the theory rests. If these postulates are found to agree with the facts, +we may feel justified in assuming that the whole theoretical structure +is a valid one, providing false logic or unsuspected and incorrect +assumptions have not later crept in to vitiate the conclusions. The +other method of testing a theory is to develop its interlacing chain of +propositions and theorems and examine the results both for their +internal coherence and for their objective validity. If we find that +the conclusions drawn from the theory are neither self-contradictory +nor contradictory of each other, and furthermore that they agree +with the facts of the external world, we may again feel that our theory +has achieved a measure of success. In the present chapter we shall +present the two main postulates of the theory of relativity, and indicate +the direct experimental evidence in favor of their truth. In following +chapters we shall develop the consequences of these postulates, show +that the system of consequences stands the test of internal coherence, +and wherever possible compare the predictions of the theory with +experimental facts. + + +\Subsection{The First Postulate of Relativity.} + +\Paragraph{17.} The first postulate of relativity as originally stated by Newton +was that it is impossible to measure or detect absolute translatory +motion through space. No objections have ever been made to this +statement of the postulate in its original form. In the development +of the theory of relativity, the postulate has been modified to include +the impossibility of detecting translatory motion through any medium +or ether which might be assumed to pervade space. + +In support of the principle is the general fact that no effects due +to the motion of the earth or other body through the supposed ether +%% -----File: 035.png---Folio 21------- +have ever been observed. Of the many unsuccessful attempts to +detect the earth's motion through the ether we may call attention to +the experiments on the refraction of light made by Arago, Respighi, +Hoek, Ketteler and Mascart, the interference experiments of Ketteler +and Mascart, the work of Klinkerfuess and Haga on the position of +the absorption bands of sodium, the experiment of Nordmeyer on the +intensity of radiation, the experiments of Fizeau, Brace and Strasser +on the rotation of the plane of polarized light by transmission through +glass plates, the experiments of Mascart and of Rayleigh on the +rotation of the plane of polarized light in naturally active substances, +the electromagnetic experiments of Röntgen, Des Coudres, J.~Koenigsberger, +Trouton, Trouton and Noble, and Trouton and Rankine, and +finally the Michelson and Morley experiment, with the further work +of Morley and Miller. For details as to the nature of these experiments +the reader may refer to the original articles or to an excellent +discussion by Laub of the experimental basis of the theory of relativity.\footnote + {\textit{Jahrbuch der Radioaktivität}, vol.~7, p.~405 (1910).} + +In none of the above investigations was it possible to detect any +effect attributable to the earth's motion through the ether. Nevertheless +a number of these experiments \emph{are} in accord with the final +form given to the ether theory by Lorentz, especially since his work +satisfactorily accounts for the Fresnel coefficient for the changed +velocity of light in moving media. Others of the experiments mentioned, +however, could be made to accord with the Lorentz theory +only by very arbitrary assumptions, in particular those of Michelson +and Morley, Mascart and Rayleigh, and Trouton and Noble. For +the purposes of our discussion we shall accept the principle of the +relativity of motion as an experimental fact. + + +\Subsection{The Second Postulate of the Einstein Theory of Relativity.} + +\Paragraph{18.} The second postulate of relativity states that \emph{the velocity of +light in free space appears the same to all observers regardless of the +relative motion of the source of light and the observer}. This postulate +may be obtained by combining the first postulate of relativity with a +principle which has long been familiar to the ether theory of light. +This principle states that the velocity of light is unaffected by a +motion of the emitting source, in other words, that the velocity with +%% -----File: 036.png---Folio 22------- +which light travels past any observer is not increased by a motion +of the source of light towards the observer. The first postulate of +relativity adds the idea that a motion of the source of light towards +the observer is identical with a motion of the observer towards the +source. The second postulate of relativity is seen to be merely a +combination of these two principles, since it states that the velocity +of light in free space appears the same to all observers regardless \emph{both} +of the motion of the source of light and of the observer. + +\Paragraph{19.} It should be pointed out that the two principles whose combination +thus leads to the second postulate of Einstein have come +from very different sources. The first postulate of relativity practically +denies the existence of any stationary ether through which +the earth, for instance, might be moving. On the other hand, the +principle that the velocity of light is unaffected by a motion of the +source was originally derived from the idea that light is transmitted +by a stationary medium which does not partake in the motion of the +source. This combination of two principles, which from a historical +point of view seem somewhat contradictory in nature, has given to +the second postulate of relativity a very extraordinary content. +Indeed it should be particularly emphasized that the remarkable +conclusions as to the nature of space and time forced upon science +by the theory of relativity are the special product of the second +postulate of relativity. + +A simple example of the conclusions which can be drawn from +this postulate will make its extraordinary nature evident. +\begin{figure}[hbt] + \begin{center} + \Fig{2} + \Input{036} + \end{center} +\end{figure} + +$S$~is a source of light and $A$~and~$B$ two moving systems. $A$~is +moving \emph{towards} the source~$S$, and $B$~\emph{away} from it. Observers on the +systems mark off equal distances $aa'$~and~$bb'$ along the path of the light +and determine the time taken for light to pass from $a$~to~$a'$ and $b$~to~$b'$ +respectively. Contrary to what seem the simple conclusions of +common sense, the second postulate requires that the time taken +%% -----File: 037.png---Folio 23------- +for the light to pass from $a$~to~$a'$ shall measure the same as the time +for the light to go from $b$~to~$b'$. Hence if the second postulate of +relativity is correct it is not surprising that science is forced in general +to new ideas as to the nature of space and time, ideas which are in +direct opposition to the requirements of so-called common sense. + + +\Subsection{Suggested Alternative to the Postulate of the Independence of the +Velocity of Light and the Velocity of the Source.} + +\Paragraph{20.} Because of the extraordinary conclusions derived by combining +the principle of the relativity of motion with the postulate +that the velocity of light is independent of the velocity of its source, +a number of attempts have been made to develop so-called \emph{emission} +theories of relativity based on the principle of the relativity of motion +and the further postulate that the velocity of light and the velocity +of its source are additive. + +Before examining the available evidence for deciding between the +rival principles as to the velocity of light, we may point out that +this proposed postulate, of the additivity of the velocity of source +and light, would as a matter of fact lead to a very simple kind of +relativity theory without requiring any changes in our notions of +space and time. For if light or other electromagnetic disturbance +which is being emitted from a source did partake in the motion of +that source in such a way that the velocity of the source is added to +the velocity of emission, it is evident that a system consisting of the +source and its surrounding disturbances would act as a whole and +suffer no \emph{permanent} change in configuration if the velocity of the +source were changed. This result would of course be in direct agreement +with the idea of the relativity of motion which merely requires +that the physical properties of a system shall be independent of its +velocity through space. + +As a particular example of the simplicity of emission theories we +may show, for instance, how easily they would account for the negative +\begin{wrapfigure}{l}{2in}%[** TN: Width-dependent line break] + \Fig{3} + \Input[2in]{038} +\end{wrapfigure} +result of the Michelson-Morley experiment. If~$O$, \Figref{3}, is a +source of light and $A$~and~$B$ are mirrors placed a meter away from~$O$, the +Michelson-Morley experiment shows that the time taken for light to +travel to~$A$ and back is the same as for the light to travel to~$B$ and +back, in spite of the fact that the whole apparatus is moving through +space in the direction $O - B$, due to the earth's motion around the sun. +%% -----File: 038.png---Folio 24------- +The basic assumption of emission theories, however, would require +exactly this result, since it says that light travels out from~$O$ with a +constant velocity in all directions with +respect to~$O$, and not with respect to +some ether through which $O$~is supposed +to be moving. + +The problem now before us is to +decide between the two rival principles +as to the velocity of light, and we shall +find that the bulk of the evidence is all +in favor of the principle which has led +to the Einstein theory of relativity with +its complete revolution in our ideas as to space and time, and against +the principle which has led to the superficially simple emission theories +of relativity. + +\Subsubsection{21}{Evidence Against Emission Theories of Light.} All emission +theories agree in assuming that light from a moving source has a +velocity equal to the vector sum of the velocity of light from a stationary +source and the velocity of the source itself at the instant of +emission. And without first considering the special assumptions +which distinguish one emission theory from another we may first +present certain astronomical evidence which apparently stands in +contradiction to this basic assumption of all forms of emission +theory. This evidence was pointed out by Comstock\footnote + {\textit{Phys.\ Rev}., vol.~30, p.~291 (1910).} +and later by +de Sitter.\footnote + {\textit{Phys.\ Zeitschr}., vol.~14, pp.~429, 1267 (1913).} + +Consider the rotation of a binary star as it would appear to an +observer situated at a considerable distance from the star and in its +plane of rotation. (See \Figref{4}.) If an emission theory of light +be true, the velocity of light from the star in position~$A$ will be $c + u$, +where $u$~is the velocity of the star in its orbit, while in the position~$B$ +the velocity will be $c - u$. Hence the star will be observed to arrive +in position~$A$, $\dfrac{l}{c+u}$~seconds after the event has actually occurred, and +in position~$B$, $\dfrac{l}{c-u}$~seconds after the event has occurred. This will +%% -----File: 039.png---Folio 25------- +make the period of half rotation from $A$~to~$B$ appear to be +\[ +\Delta t - \frac{l}{c+u} + \frac{l}{c-u} = \Delta t + \frac{2ul}{c^2}, +\] +where $\Delta t$~is the actual time of a half rotation in the orbit, which for +\begin{figure}[hbt] + \begin{center} + \Fig{4} + \Input[3.25in]{039} + \end{center} +\end{figure} +simplicity may be taken as circular. On the other hand, the period +of the next half rotation from $B$ back to~$A$ would appear to be +\[ +\Delta t - \frac{2ul}{c^2}. +\] + +Now in the case of most spectroscopic binaries the quantity~$\dfrac{2ul}{c^2}$ +is not only of the same order of magnitude as~$\Delta t$ but oftentimes probably +even larger. Hence, if an emission theory of light were true, +we could hardly expect without correcting for the variable velocity +of light to find that these orbits obey Kepler's laws, as is actually +the case. This is certainly very strong evidence against any form +of emission theory. It may not be out of place, however, to state +briefly the different forms of emission theory which have been tried. + +\Subsubsection{22}{Different Forms of Emission Theory.} As we have seen, emission +theories all agree in assuming that light from a moving source +%% -----File: 040.png---Folio 26------- +has a velocity equal to the vector sum of the velocity of light from a +stationary source and the velocity of the source itself at the instant +of emission. Emission theories differ, however, in their assumptions +as to the velocity of light after its reflection from a mirror. The three +assumptions which up to this time have been particularly considered +are (1)~that the excited portion of the reflecting mirror acts as a new +source of light and that the reflected light has the same velocity~$c$ +with respect to the mirror as has original light with respect to its source; +(2)~that light reflected from a mirror acquires a component of velocity +equal to the velocity of the mirror image of the original source, and +hence has the velocity~$c$ with respect to this mirror image; and (3)~that +light retains throughout its whole path the component of velocity +which it obtained from its original moving source, and hence after +reflection spreads out with velocity~$c$ in a spherical form around a +center which moves with the same speed as the original source. + +Of these possible assumptions as to the velocity of reflected light, +the first seems to be the most natural and was early considered by the +author but shown to be incompatible, not only with an experiment +which he performed on the velocity of light from the two limbs of +the sun,\footnote + {\textit{Phys.\ Rev}., vol.~31, p.~26 (1910).} +but also with measurements of the Stark effect in canal +rays.\footnote + {\textit{Phys.\ Rev}., vol.~35, p.~136 (1912).} +The second assumption as to the velocity of light was made +by Stewart,\footnote + {\textit{Phys.\ Rev}., vol.~32, p.~418 (1911).} +but has also been shown\footnotemark[2] %[** TN: Repeated footnote here, below] +to be incompatible with +measurements of the Stark effect in canal rays. Making use of the +third assumption as to the velocity of reflected light, a somewhat +complete emission theory has been developed by Ritz,\footnote + {\textit{Ann.\ de chim.\ et phys}., vol.~13, p.~145 (1908); + \textit{Arch.\ de Génève} vol.~26, p.~232 + (1908); \textit{Scientia}, vol.\ 5 (1909).} +and unfortunately +optical experiments for deciding between the Einstein +and Ritz relativity theories have never been performed, although +such experiments are entirely possible of performance.\footnotemark[2] Against the +Ritz theory, however, we have of course the general astronomical +evidence of Comstock and de Sitter which we have already described +above. + +For the present, the observations described above, comprise the +whole of the direct experimental evidence against emission theories +%% -----File: 041.png---Folio 27------- +of light and in favor of the principle which has led to the second +postulate of the Einstein theory. One of the consequences of the +Einstein theory, however, has been the deduction of an expression +for the mass of a moving body which has been closely verified by the +Kaufmann-Bucherer experiment. Now it is very interesting to note, +that starting with what has thus become an \emph{experimental} expression +for the mass of a moving body, it is possible to work backwards to a +derivation of the second postulate of relativity. For the details of +the proof we must refer the reader to the original article.\footnote + {\textit{Phys.\ Rev}., vol.\ 31, p.\ 26 (1910).} + + +\Subsection{Further Postulates of the Theory of Relativity.} + +\Paragraph{23.} In the development of the theory of relativity to which we +shall now proceed we shall of course make use of many postulates. +The two which we have just considered, however, are the only ones, +so far as we are aware, which are essentially different from those +common to the usual theoretical developments of physical science. +In particular in our further work we shall assume without examination +all such general principles as the homogeneity and isotropism of the +space continuum, and the unidirectional, one-valued, one-dimensional +character of the time continuum. In our treatment of the dynamics +of a particle we shall also assume Newton's laws of motion, and the +principle of the conservation of mass, although we shall find, of course, +that the Einstein ideas as to the connection between space and time +will lead us to a non-Newtonian mechanics. We shall also make +extensive use of the principle of least action, which we shall find a +powerful principle in all the fields of dynamics. +%% -----File: 042.png---Folio 28------- + + +\Chapter{III}{Some Elementary Deductions.} +\SetRunningHeads{Chapter Three.}{Some Elementary Deductions.} + +\Paragraph{24.} In order gradually to familiarize the reader with the consequences +of the theory of relativity we shall now develop by very +elementary methods a few of the more important relations. In this +preliminary consideration we shall lay no stress on mathematical +elegance or logical exactness. It is believed, however, that the +chapter will present a substantially correct account of some of the +more important conclusions of the theory of relativity, in a form +which can be understood even by readers without mathematical +equipment. + + +\Subsection{Measurements of Time in a Moving System.} + +\Paragraph{25.} We may first derive from the postulates of relativity a relation +connecting measurements of time intervals as made by observers in +systems moving with different velocities. Consider a system~$S$ +(\Figref[Fig.]{5}) provided with a plane mirror~$a\, a$, and an observer~$A$, who +\begin{figure}[hbt] + \begin{center} + \Fig{5} + \Input[3.25in]{042} + \end{center} +\end{figure} +has a clock so that he can determine the time taken for a beam of +light to travel up to the mirror and back along the path~$A\, m\, A$. +Consider also another similar system~$S'$, provided with a mirror~$b\, b$, +and an observer~$B$, who also has a clock for measuring the time it +takes for light to go up to his mirror and back. System~$S'$ is moving +past~$S$ with the velocity~$V$, the direction of motion being parallel +to the mirrors $a\, a$~and~$b\, b$, the two systems being arranged, moreover, +%% -----File: 043.png---Folio 29------- +so that when they pass one another the two mirrors $a\,a$~and~$b\,b$ +will coincide, and the two observers $A$~and~$B$ will also come into +coincidence. + +$A$,~considering his system at rest and the other in motion, measures +the time taken for a beam of light to pass to his mirror and return, +over the path~$A\, m\, A$, and compares the time interval thus obtained +with that necessary for the performance of a similar experiment +by~$B$, in which the light has to pass over a longer path such as~$B\, n\, B'$, +shown in \Figref{6}, where $B\, B'$~is the distance through which the +\begin{figure}[hbt] + \begin{center} + \Fig{6} + \Input{043} + \end{center} +\end{figure} +observer~$B$ has moved during the time taken for the passage of the +light up to the mirror and back. + +Since, in accordance with the second postulate of relativity, the +velocity of light is independent of the velocity of its source, it is +evident that the ratio of these two time intervals will be proportional +to the ratio of the two paths $A\, m\, A$~and~$B\, n\, B'$, and this can easily +be calculated in terms of the velocity of light~$c$ and the velocity~$V$ +of the system~$S'$. + +From \Figref{6} we have +\[ +(A\, m)^2 = (p\, n)^2 = (B\, n)^2 - (B\, p)^2. +\] +Dividing by $(B\, n)^2$, +\[ +\frac{(A\, m)^2}{(B\, n)^2} = 1 - \frac{(B\, p)^2}{(B\, n)^2}. +\] +But the distance $B\, p$ is to $B\, n$ as $V$ is to~$c$, giving us +\[ +\frac{A\, m }{ B\, n} = \sqrt{1 - \frac{V^2}{c^2}}, +\] +%% -----File: 044.png---Folio 30------- +and hence $A$~will find, either by calculation or by direct measurement +if he has arranged clocks at $B$~and~$B'$, that it takes a longer time for +the performance of $B$'s~experiment than for the performance of his +own in the ratio $1: \sqrt{ 1 - \dfrac{V^2}{c^2}}$. + +It is evident from the first postulate of relativity, however, that +$B$~himself must find exactly the same length of time for the light to +pass up to his mirror and come back as did~$A$ in his experiment, +because the two systems are, as a matter of fact, entirely symmetrical +and we could with equal right consider $B$'s~system to be the one at +rest and $A$'s~the one in motion. + +\emph{We thus find that two observers, $A$~and~$B$, who are in relative motion +will not in general agree in their measurements of the time interval necessary +for a given event to take place}, the event in this particular case, +for example, having been the performance of $B$'s~experiment; indeed, +making use of the ratio obtained in a preceding paragraph, we may +go further and make the quantitative statement that measurements of +\emph{time intervals made with a moving clock must be multiplied by the quantity +$\dfrac{1}{\sqrt{ 1 - \smfrac{V^2}{c^2}}}$ in order to agree with measurements made with a stationary +system of clocks}. + +It is sometimes more convenient to state this principle in the +form: A stationary observer using a set of stationary clocks will +obtain a greater measurement in the ratio $1: \sqrt{ 1 - \dfrac{V^2}{c^2}}$ for a given +time interval than an observer who uses a clock moving with the +velocity~$V$. + + +\Subsection{Measurements of Length in a Moving System.} + +\Paragraph{26.} We may now extend our considerations, to obtain a relation +between measurements of \emph{length} made in stationary and moving +systems. + +As to measurements of length \emph{perpendicular} to the line of motion +of the two systems $S$~and~$S'$, a little consideration will make it at once +evident that both $A$~and~$B$ must obtain identical results. This is +true because the possibility is always present of making a direct comparison +%% -----File: 045.png---Folio 31------- +of the meter sticks which $A$~and~$B$ use for such measurements +by holding them perpendicular to the line of motion. When the +relative motion of the two systems brings such meter sticks into +juxtaposition, it is evident from the first postulate of relativity that +$A$'s~meter and $B$'s~meter must coincide in length. Any difference in +length could be due only to the different velocity of the two systems +through space, and such an occurrence is ruled out by our first postulate. +\emph{Hence measurements made with a moving meter stick held perpendicular +to its line of motion will agree with those made with stationary meter +sticks.} + +\Paragraph{27.} With regard to measurements of length \emph{parallel} to the line of +motion of the systems, the affair is much more complicated. Any +direct comparison of the lengths of meter sticks in the two systems +would be very difficult to carry out; the consideration, however, of a +simple experiment on the velocity of light parallel to the motion of +the systems will lead to the desired relation. + +Let us again consider two systems $S$~and~$S'$ (\Figref[fig.]{7}), $S'$~moving +past~$S$ with the velocity~$V$. +\begin{figure}[hbt] + \begin{center} + \Fig{7} + \Input[3.5in]{045} + \end{center} +\end{figure} + +$A$ and $B$ are observers on these systems provided with clocks and +meter sticks. The two observers lay off, each on his own system, +paths for measuring the velocity of light. $A$~lays off a distance of +one meter,~$A\, m$, so that he can measure the time for light to travel +to the mirror~$m$ and return, and $B$, using a meter stick which has +the same length as~$A$'s when they are both at rest, lays off the distance~$B\, n$. + +Each observer measures the length of time it takes for light to +travel to his mirror and return, and will evidently have to find the +same length of time, since the postulates of relativity require that the +velocity of light shall be the same for all observers. +%% -----File: 046.png---Folio 32------- + +Now the observer~$A$, taking himself as at rest, finds that $B$'s~light +travels over a path~$B\, n'\, B'$ (\Figref[fig.]{8}), where $n\, n'$~is the distance +\begin{figure}[hbt] + \begin{center} + \Fig{8} + \Input[3in]{046} + \end{center} +\end{figure} +through which the mirror~$n$ moves while the light is travelling up to +it, and $B\, B'$~is the distance through which the source travels before +the light gets back. It is easy to calculate the length of this path. + +We have +\[ +\frac{n\,n'}{B\,n'} = \frac{V}{c} +\] +and +\[ +\frac{B\,B'}{B\,n'\,B'} = \frac{V}{c}. +\] +Also, from the figure, +\begin{align*} +B\,n' &= B\,n + n\,n',\\ +B\,n'\,B' &= B\,n\,B + 2\,n\,n'- B\, B'. +\end{align*} +Combining, we obtain +\[ +\frac{B\,n'\,B'}{B\,n\,B} = \frac{1}{1 - \smfrac{V^2}{c^2}}. +\] + +Thus $A$ finds that the path traversed by $B$'s~light, instead of being +exactly two meters as was his own, will be longer in the ratio of +$1:\left(1 - \dfrac{V^2}{c^2}\right)$. For this reason $A$~is rather surprised that $B$~does +not report a longer time interval for the passage of the light than he +himself found. He remembers, however, that he has already found +that measurements of time made with a moving clock must be multiplied +by the quantity $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$ in order to agree with his own, and +sees that this will account for part of the discrepancy between the +expected and observed results. To account for the remaining discrepancy +the further conclusion is now obtained \emph{that measurements of +%% -----File: 047.png---Folio 33------- +length made with a moving meter stick, parallel to its motion, must be +multiplied by the quantity $\sqrt{1 - \dfrac{V^2}{c^2}}$ in order to agree with those made +in a stationary system}. + +In accordance with this principle, a stationary observer will +obtain a smaller measurement for the length of a moving body than +will an observer moving along with the object. This has been called +the Lorentz shortening, the shortening occurring in the ratio +\[ +\sqrt{1 - \frac{V^2}{c^2}}:1 +\] +in the line of motion. + + +\Subsection{The Setting of Clocks in a Moving System.} + +\Paragraph{28.} It will be noticed that in our considerations up to this point +we have considered cases where only a \emph{single} moving clock was needed +in performing the desired experiment, and this was done purposely, +since we shall find, not only that a given time interval measures +shorter on a moving clock than on a system of stationary clocks, +but that a system of moving clocks which have been set in synchronism +by an observer moving along with them will not be set in synchronism +for a stationary observer. + +Consider again two systems $S$~and~$S'$ in relative motion with the +velocity~$V$. An observer~$A$ on system~$S$ places two carefully compared +clocks, unit distance apart, in the line of motion, and has the +time on each clock read when a given point on the other system +passes it. An observer~$B$ on system~$S'$ performs a similar experiment. +The time interval obtained in the two sets of readings must be the +same, since the first postulate of relativity obviously requires that the +relative velocity of the two systems $V$~shall have the same value for +both observers. + +The observer~$A$, however, taking himself as at rest, and familiar +with the change in the measurements of length and time in the moving +system which have already been deduced, expects that the velocity +as measured by~$B$ will be greater than the value that he himself +obtains in the ratio $\dfrac{1}{1 - \smfrac{V^2}{c^2}}$, since any particular one of $B$'s~clocks +%% -----File: 048.png---Folio 34------- +gives a shorter value for a given time interval than his own, while +$B$'s~measurements of the length of a moving object are greater than +his own, each by the factor $\sqrt{1 - \dfrac{V^2}{c^2}}$. In order to explain the actual +result of $B$'s~experiment he now has to conclude that the clocks which +for $B$ are set synchronously are not set in synchronism for himself. + +From what has preceded it is easy to see that in the moving system, +from the point of view of the stationary observer, clocks must be set +further and further ahead as we proceed towards the rear of the +system, since otherwise $B$~would not obtain a great enough difference +in the readings of the clocks as they come opposite the given point +on the other system. Indeed, if two clocks are situated in the moving +system,~$S'$, one in front of the other by the distance $l'$, as measured +by~$B$, then for $A$ it will appear as though $B$~had set his rear clock ahead +by the amount~$\dfrac{l'V}{c^2}$. + +\Paragraph{29.} We have now obtained all the information which we shall +need in this chapter as to measurements of time and length in systems +moving with different velocities. We may point out, however, before +proceeding to the application of these considerations, that our choice +of $A$'s system as the one which we should call stationary was of course +entirely arbitrary and immaterial. We can at any time equally well +take $B$'s~system as the one to which we shall ultimately refer all our +measurements, and indeed all that we shall mean when we call one of +our systems stationary is that for reasons of convenience we have +picked out that particular system as the one with reference to which +we particularly wish to make our measurements. We may also +point out that of course $B$~has to subject $A$'s~measurements of time +and length to just the same multiplications by the factor $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$ +as did~$A$ in order to make them agree with his own. + +These conclusions as to measurements of space and time are of course +very startling when first encountered. The mere fact, however, that +they appear strange to so-called ``common sense'' need cause us +no difficulty, since the older ideas of space and time were obtained +from an ancestral experience which never included experiments with +%% -----File: 049.png---Folio 35------- +high relative velocities, and it is only when the ratio $\dfrac{V^2}{c^2}$ becomes +appreciable that we obtain unexpected results. To those scientists +who do not wish to give up their ``common sense'' ideas of space +and time we can merely say that if they accept the two postulates +of relativity then they will also have to accept the consequences +which can be deduced therefrom. The remarkable nature of these +consequences merely indicates the very imperfect nature of our older +conceptions of space and time. + + +\Subsection{The Composition of Velocities.} + +\Paragraph{30.} Our conclusions as to the setting of clocks make it possible +to obtain an important expression for the composition of velocities. +Suppose we have a system~$S$, which we shall take as stationary, and +on the system an observer~$A$. Moving past~$S$ with the velocity~$V$ +is another system~$S'$ with an observer~$B$, and finally moving past~$S'$ +in the same direction is a body whose velocity is~$u'$ as measured by +observer~$B$. What will be the velocity~$u$ of this body as measured +by~$A$? + +Our older ideas led us to believe in the simple additivity of velocities +and we should have calculated~$u$ in accordance with the simple +expression +\[ +u = V + u'. +\] +We must now allow, however, for the fact that $u'$~is measured with +clocks which to~$A$ appear to be set in a peculiar fashion and running +at a different rate from his own, and with meter sticks which give +longer measurements than those used in the stationary system. + +The determination of~$u'$ by observer~$B$ would be obtained by +measuring the time interval necessary for the body in question to +move a given distance~$l'$ along the system~$S'$. If $t'$~is the difference +in the respective clock readings when the body reaches the ends of +the line~$l'$, we have +\[ +u' = \frac{l'}{t'}. +\] +We have already seen, however, that the two clocks are for~$A$ set $\dfrac{l'V}{c^2}$~units +apart and hence for clocks set together the time interval would +%% -----File: 050.png---Folio 36------- +have measured $t' + \dfrac{l'V}{c^2}$. Furthermore these moving clocks give +time measurements which are shorter in the ratio $\sqrt{1 - \dfrac{V^2}{c^2}}:1$ than +those obtained by~$A$, so that for~$A$ the time interval for the body to +move from one end of~$l'$ to the other would measure +\[ +\frac{t' + \smfrac{l'V}{c^2}}{\sqrt{1 - \smfrac{V^2}{c^2}}}; +\] +furthermore, owing to the difference in measurements of length, this +line~$l'$ has for~$A$ the length $l'\sqrt{1 - \dfrac{V^2}{c^2}}$. Hence $A$~finds that the +body is moving past~$S'$ with the velocity, +\[ +\frac{\ l'\sqrt{1 - \smfrac{V^2}{c^2}}\ } + {\smfrac{t' + \smfrac{l'V}{c^2}}{\sqrt{1 - \smfrac{V^2}{c^2}}}} + = \frac{\smfrac{l'}{t'} \left(1 - \smfrac{V^2}{c^2}\right)} + {1 + \smfrac{l'}{t'}\, \smfrac{V}{c^2}} + = \frac{u' \left(1 - \smfrac{V^2}{c^2}\right)} + {1+ \smfrac{u'V}{c^2}}. +\] +This makes the total velocity of the body past~$S$ equal to the sum +\[ +u = V + \frac{u' \left(1 - \smfrac{V^2}{c^2}\right)}{1 + \smfrac{u'V}{c^2}}, +\] +or +\[ +u = \frac{V + u'}{1 + \smfrac{u'V}{c^2}}. +\] + +This new expression for the composition of velocities is extremely +important. When the velocities $u'$~and~$V$ are small compared with +the velocity of light~$c$, we observe that the formula reduces to the simple +additivity principle which we know by common experience to be true +%% -----File: 051.png---Folio 37------- +for all ordinary velocities. Until very recently the human race has +had practically no experience with high velocities and we now see +that for velocities in the neighborhood of that of light, the simple +additivity principle is nowhere near true. + +In particular it should be noticed that by the composition of +velocities which are themselves less than that of light we can never +obtain any velocity greater than that of light. As an extreme case, +suppose for example that the system~$S'$ were moving past $S$~itself +with the velocity of light (\ie, $V = c$) and that in the system~$S'$ a +particle should itself be given the velocity of light in the same direction +(\ie, $u' = c$); we find on substitution that the particle still has +only the velocity of light with respect to~$S$. We have +\[ +u = \frac{c + c}{1 + \smfrac{c^2}{c^2}} = \frac{2c}{2} = c. +\] + +By the consideration of such conclusions as these the reader will +appreciate more and more the necessity of abandoning his older +naïve ideas of space and time which are the inheritance of a long +human experience with physical systems in which only slow velocities +were encountered. + + +\Subsection{The Mass of a Moving Body.} + +\Paragraph{31.} We may now obtain an important relation for the mass of a +moving body. Consider again two similar systems, $S$~at rest and $S'$~moving +past with the velocity~$V$. The observer~$A$ on system~$S$ has a +sphere made from some rigid elastic material, having a mass of $m$~grams, +and the observer~$B$ on system~$S'$ is also provided with a similar +sphere. The two spheres are made so that they are exactly alike +when both are at rest; thus $B$'s~sphere, since it is at rest with respect +to him, looks to him just the same as the other sphere does to~$A$. +As the two systems pass each other (\Figref[fig.]{9}) each of these clever experimenters +rolls his sphere towards the other system with a velocity of +$u$~cm.~per second, so that they will just collide and rebound in a line +perpendicular to the direction of motion. Now, from the first postulate +of relativity, system~$S'$ appears to~$B$ just the same as system $S$~appears +to~$A$, and $B$'s~ball appears to him to go through the same +evolutions that $A$~finds for his ball. $A$~finds that his ball on collision +%% -----File: 052.png---Folio 38------- +undergoes the algebraic change of velocity~$2u$, $B$~finds the same change +in velocity~$2u$ for his ball. $B$~reports this fact to~$A$, and $A$~knowing +that $B$'s~measurements of length agree with his own in this transverse +\begin{figure}[hbt] + \begin{center} + \Fig{9} + \Input{052} + \end{center} +\end{figure} +direction, but that his clock gives time intervals that are shorter than +his own in the ratio $\sqrt{1 - \dfrac{V^2}{c^2}}:1$, calculates that the change in velocity +of $B$'s~ball must be~$2u\sqrt{1 - \dfrac{V^2}{c^2}}$. + +From the principle of the conservation of momentum, however, +$A$~knows that the change in momentum of $B$'s~ball must be the same +as that of his own and hence can write the equation +\[ +m_au = m_bu\sqrt{1 - \frac{V^2}{c^2}}, +\] +where $m_a$~is the mass of $A$'s~ball and $m_b$~is the mass of $B$'s~ball. Solving +we have +\[ +m_b = \frac{m_a}{\sqrt{1 - \smfrac{V^2}{c^2}}}. +\] + +In other words, $B$'s~ball, which had the same mass~$m_a$ as~$A$'s when +%% -----File: 053.png---Folio 39------- +both were at rest, is found to have the larger mass $\dfrac{m_a}{\sqrt{1 - \smfrac{v^2}{c^2}}}$ when +placed in a system that is moving with the velocity~$V$.\footnote + {In carrying out this experiment the transverse velocities of the balls should + be made negligibly small in comparison with the relative velocity of the systems~$V$.} + +The theory of relativity thus leads to the general expression +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{v^2}{c^2}}} +\] +for the mass of a body moving with the velocity~$u$ and having the +mass~$m_0$ when at rest. + +Since we have very few velocities comparable with that of light +it is obvious that the quantity $\sqrt{1 - \dfrac{v^2}{c^2}}$ seldom differs much from +unity, which makes the experimental verification of this expression +difficult. In the case of electrons, however, which are shot off from +radioactive substances, or indeed in the case of cathode rays produced +with high potentials, we do have particles moving with velocities +comparable to that of light, and the experimental work of Kaufmann, +Bucherer, Hupka and others in this field provides one of the most +striking triumphs of the theory of relativity. + + +\Subsection{The Relation Between Mass and Energy.} + +\Paragraph{32.} The theory of relativity has led to very important conclusions +as to the nature of mass and energy. In fact, we shall see that matter +and energy are apparently different names for the same fundamental +entity. + +When we set a body in motion it is evident from the previous +section that we increase both its mass as well as its energy. Now +we can show that there is a definite ratio between the amount of +energy that we give to the body and the amount of mass that we +give to it. + +If the force~$f$ acts on a particle which is free to move, its increase in +kinetic energy is evidently +\[ +\Delta E = \int f\, dl. +\] +But the force acting\DPtypo{, is}{ is,} by definition, equal to the rate of increase in +%% -----File: 054.png---Folio 40------- +the momentum of the particle +\[ +f = +\frac{d}{dt}(mu). +\] +Substituting we have +\[ +\Delta E + = \int \frac{d(mu)}{dt}\, dl + = \int \frac{dl}{dt}\, d(mu) + = \int u\, d(mu). +\] +We have, however, from the previous section, +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +which, solved for~$u$, gives us +\[ +u = c \sqrt{1 - \frac{{m_0}^2}{m^2}}. +\] +Substituting this value of~$u$ in our equation for~$\Delta E$ we obtain, after +simplification, +\[ +\Delta E = \int c^2\, dm = c^2\, \Delta m. +\] + +This says that the increase of the kinetic energy of the particle, +in ergs, is equal to the increase in mass, in grams, multiplied by the +square of the velocity of light. If now we bring the particle to rest +it will give up both its kinetic energy and its excess mass. Accepting +the principles of the conservation of mass and energy, we know, however, +that neither this energy nor the mass has been destroyed; they +have merely been passed on to other bodies. There is, moreover, +every reason to believe that this mass and energy, which were associated +together when the body was in motion and left the body when +it was brought to rest, still remain always associated together. For +example, if the body should be brought to rest by setting another +body into motion, it is of course a necessary consequence of our considerations +that the kinetic energy and the excess mass both pass +on together to the new body which is set in motion. A similar conclusion +would be true if the body is brought to rest by frictional forces, +since the heat produced by the friction means an increase in the kinetic +energies of ultimate particles. +%% -----File: 055.png---Folio 41------- + +In general we shall find it pragmatic to consider that matter and +energy are merely different names for the same fundamental entity. +One gram of matter is equal to $10^{21}$~ergs of energy. +\[ +c^2 = (2.9986 × 10^{10})^2 = \text{approx.\ }10^{21}. +\] + +This apparently extraordinary conclusion is in reality one which +produces the greatest simplification in science. Not to mention +numerous special applications where this principle is useful, we may +call attention to the fact that the great laws of the conservation of +mass and of energy have now become identical. We may also point +out that those opposing camps of philosophic materialists who defend +matter on the one hand or energy on the other as the fundamental +entity of the universe may now forever cease their unimportant bickerings. +%% -----File: 056.png---Folio 42------- + + +\Chapter{IV}{The Einstein Transformation Equations for Space +and Time.} +\SetRunningHeads{Chapter Four.}{Transformation Equations for Space and Time.} + +\Subsection{The Lorentz Transformation.} + +\Paragraph{33.} We may now proceed to a systematic study of the consequences +of the theory of relativity. + +The fundamental problem that first arises in considering +spatial and temporal measurements is that of transforming the +description of a given kinematical occurrence from the variables of +one system of coördinates to those of another system which is in +motion relative to the first. + +Consider two systems of right-angled Cartesian coördinates $S$~and~$S'$ +(\Figref[fig.]{10}) in relative motion in the $X$~direction with the velocity~$V$. +\begin{figure}[hbt] + \begin{center} + \Fig{10} + \Input{056} + \end{center} +\end{figure} +The \emph{position} of any given point in space can be determined by specifying +its coördinates $x$,~$y$, and~$z$ with respect to system~$S$ or its coördinates +$x'$,~$y'$ and~$z'$ with respect to system~$S'$. Furthermore, for the +purpose of determining the \emph{time} at which any event takes place, we +may think of each system of coördinates as provided with a whole +series of clocks placed at convenient intervals throughout the system, +the clocks of each series being set and regulated\footnote + {We may think of the clocks as being set in any of the ways that are usual + in practice. Perhaps the simplest is to consider the clocks as mechanisms which + have been found to ``keep time'' when they are all together where they can be + examined by one individual observer. The assumption can then be made, in accordance + with our ideas of the homogeneity of space, that they will continue to + ``keep time'' after they have been distributed throughout the system.} +by observers in the +%% -----File: 057.png---Folio 43------- +corresponding system. The time at which the event in question +takes place may be denoted by~$t$ if determined by the clocks belonging +to system~$S$ and by~$t'$ if determined by the clocks of system~$S'$. + +For convenience the two systems $S$~and~$S'$ are chosen so that the +axes $OX$~and~$O'X'$ lie in the same line, and for further simplification +we choose, as our starting-point for time measurements, $t$~and~$t'$ both +equal to zero when the two origins come into coincidence. + +The specific problem now before us is as follows: If a given kinematical +occurrence has been observed and described in terms of the +variables $x'$,~$y'$,~$z'$ and~$t'$, what substitutions must we make for the +values of these variables in order to obtain a correct description of the +\emph{same} kinematical event in terms of the variables $x$,~$y$,~$z$ and~$t$? In +other words, we want to obtain a set of transformation equations +from the variables of system~$S'$ to those of system~$S$. The equations +which we shall present were first obtained by Lorentz, and the process +of changing from one set of variables to the other has generally been +called the Lorentz transformation. The significance of these equations +from the point of view of the theory of relativity was first appreciated +by Einstein. + + +\Subsection{Deduction of the Fundamental Transformation Equations.} + +\Paragraph{34.} It is evident that these transformation equations are going +to depend on the relative velocity $V$ of the two systems, so that we +may write for them the expressions +\begin{align*} +x' &= F_1(V, x, y, z, t), \displaybreak[0] \\ +y' &= F_2(V, x, y, z, t), \displaybreak[0] \\ +z' &= F_3(V, x, y, z, t), \displaybreak[0] \\ +t' &= F_4(V, x, y, z, t), +\end{align*} +where $F_1$,~$F_2$,~etc., are the unknown functions whose form we wish +to determine. + +It is possible at the outset, however, greatly to simplify these +relations. If we accept the idea of the homogeneity of space it is +evident that any other line parallel to~$OXX'$ might just as well have +been chosen as our line of $X$-axes, and hence our two equations for +$x'$~and~$t'$ must be independent of $y$~and~$z$. Moreover, as to the equations +%% -----File: 058.png---Folio 44------- +for $y'$~and~$z'$ it is at once evident that the only possible solutions +are $y' = y$ and $z' = z$. This is obvious because a meter stick held +in the system~$S'$ perpendicular to the line of relative motion,~$OX'$, +of the system can be directly compared with meter sticks held similarly +in system~$S$, and in accordance with the first postulate of relativity +they must agree in length in order that the systems may be entirely +symmetrical. We may now rewrite our transformation equations +in the simplified form +\begin{align*} +x' &= F_1(V, t, x), \\ +y' &= y, \\ +z' &= z, \\ +t' &= F_2(V, t, x), +\end{align*} +and have only two functions, $F_1$~and~$F_2$, whose form has to be determined. + +To complete the solution of the problem we may make use of three +further conditions which must govern the transformation equations. + +\Subsubsection{35}{Three Conditions to be Fulfilled.} In the first place, when the +velocity~$V$ between the systems is small, it is evident that the transformation +equations must reduce to the form that they had in Newtonian +mechanics, since we know both from measurements and from +everyday experience that the Newtonian concepts of space and time +are correct as long as we deal with slow velocities. Hence the limiting +form of the equations as $V$~approaches zero will be (cf.~\Chapref{I}, +equations \DPchg{3--4--5--6}{(3),~(4), (5),~(6)}) +\begin{align*} +x' &= x - Vt,\\ +y' &= y, \\ +z' &= z, \\ +t' &= t. +\end{align*} + +\Paragraph{36.} A second condition is imposed upon the form of the functions +$F_1$~and~$F_2$ by the first postulate of relativity, which requires that the +two systems $S$~and~$S'$ shall be entirely symmetrical. Hence the +transformation equations for changing from the variables of system~$S$ +to those of system~$S'$ must be of exactly the same form as those used +in the reverse transformation, containing, however, $-V$~wherever +$+V$~occurs in the latter equations. Expressing this requirement in +%% -----File: 059.png---Folio 45------- +mathematical form, we may write as true equations +\begin{align*} +x &= F_1(-V, t', x'), \\ +t &= F_2(-V, t', x'), +\end{align*} +where $F_1$~and~$F_2$ must have the same form as above. + +\Paragraph{37.} A final condition is imposed upon the form of $F_1$~and~$F_2$ by +the second postulate of relativity, which states that the velocity of a +beam of light appears the same to all observers regardless of the +motion of the source of light or of the observer. Hence our transformation +equations must be of such a form that a given beam of +light has the same velocity,~$c$, when measured in the variables of either +system. Let us suppose, for example, that at the instant $t = t' = 0$, +when the two origins come into coincidence, a light impulse is started +from the common point occupied by $O$~and~$O'$. Then, measured in +the coördinates of either system, the optical disturbance which is +generated must spread out from the origin in a spherical form with +the velocity~$c$. Hence, using the variables of system~$S$, the coördinates +of any point on the surface of the disturbance will be given by the +expression +\[ +x^2 + y^2 + z^2 = c^2t^2, +\Tag{7} +\] +while using the variables of system~$S'$ we should have the similar +expression +\[ +x'^2 + y'^2 + z'^2 = c^2t'^2. +\Tag{8} +\] +Thus we have a particular kinematical occurrence, the spreading out +of a light disturbance, whose description is known in the variables +of either system, and our transformation equations must be of such +a form that their substitution will change equation (8) to equation (7). +In other words, the expression $x^2 + y^2 + z^2 - c^2t^2$ is to be an invariant +for the Lorentz transformation. + +\Subsubsection{38}{The Transformation Equations.} The three sets of conditions +which, as we have seen in the last three paragraphs, are imposed upon +the form of $F_1$~and~$F_2$ are sufficient to determine the solution of the +problem. The natural method of solution is obviously that of trial, +%% -----File: 060.png---Folio 46------- +and we may suggest the solution: +\begin{align*} +x' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, (x - Vt) + = \kappa(x - Vt), \Tag{9}\\ +y' &= y, \Tag{10}\\ +z' &= z, \Tag{11}\\ +t' &= \frac{1}{\sqrt{1- \smfrac{V^2}{c^2}}} \left(t - \frac{V}{c^2}\, x\right) + = \kappa \left(t - \frac{V}{c^2}\, x\right), \Tag{12} +\end{align*} +where we have placed~$\kappa$ to represent the important and continually +recurring quantity $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$. + +It will be found as a matter of fact by examination that these\rule{0em}{1.8em} +solutions do fit all three requirements which we have stated. Thus, +when $V$~becomes small compared with the velocity of light,~$c$, the +equations do reduce to those of Galileo and Newton. Secondly, if +the equations are solved for the unprimed quantities in terms of the +primed, the resulting expressions have an unchanged form except for +the introduction of~$-V$ in place of~$+V$, thus fulfilling the requirements +of symmetry imposed by the first postulate of relativity. And +finally, if we substitute the expressions for $x'$,~$y'$,~$z'$ and~$t'$ in the polynomial +$x'^2 + y'^2 + z'^2 = c^2t'^2$, we shall obtain the expression $x^2 + y^2 ++ z^2 - c^2t^2$ and have thus secured the invariance of $x^2 + y^2 + z^2 - c^2t^2$ +which is required by the second postulate of relativity. + +We may further point out that the whole series of possible Lorentz +transformations form a group such that the result of two successive +transformations could itself be represented by a single transformation +provided we picked out suitable magnitudes and directions for the +velocities between the various systems. + +It is also to be noted that the transformation becomes imaginary +for cases where $V > c$, and we shall find that this agrees with ideas +obtained in other ways as to the speed of light being an upper limit +for the magnitude of all velocities. +%% -----File: 061.png---Folio 47------- + + +\Subsection{Further Transformation Equations.} + +\Paragraph{39.} Before making any applications of our equations we shall find +it desirable to obtain by simple substitutions and differentiations a +series of further transformation equations which will be of great value +in our future work. + +By the simple differentiation of equation~(12) we can obtain +\[ +\frac{dt'}{dt} = \kappa\left(1 - \frac{\dot{x}V}{c^2}\right), +\Tag{13} +\] +where we have put~$\dot{x}$ for~$\dfrac{dx}{dt}$. + +\Subsubsection{40}{Transformation Equations for Velocity.} By differentiation of +the equations for $x'$,~$y'$ and~$z'$, nos.\ (9),~(10) and~(11), and substitution +of the value just found for~$\dfrac{dt'}{dt}$ we may obtain the following transformation +equations for velocity: +\begin{alignat*}{3} +\dot{x}' &= \frac{\dot{x} - V}{1 - \smfrac{\dot{x}V}{c^2}} + &&\qquad\text{or}\qquad& + u'_x &= \frac{u_x - V}{1 - \smfrac{u_xV}{c^2}}, \Tag{14} \\ +% +\dot{y}' &= \frac{\dot{y}\kappa^{-1}}{1 - \smfrac{\dot{y}V}{c^2}} &&& + u'_y &= \frac{u_y\kappa^{-1}}{1 - \smfrac{u_xV}{c^2}}, \Tag{15} \\ +% +\dot{z}' &= \frac{\dot{z}\kappa^{-1}}{1 - \smfrac{\dot{z}V}{c^2}} &&& + u'_z &= \frac{u_z\kappa^{-1}}{1 - \smfrac{u_xV}{c^2}}, \Tag{16} +\end{alignat*} +where the placing of a dot has the familiar significance of differentiation +with respect to time, $\dfrac{dx}{dt}$~being represented by~$\dot{x}$ and $\dfrac{dx'}{dt'}$ by~$\dot{x}'$. + +The significance of these equations for the transformation of +velocities is as follows: If for an observer in system~$S$ a point appears +to be moving with the uniform velocity $(\dot{x}, \dot{y}, \dot{z})$ its velocity $(\dot{x}', \dot{y}', \dot{z}')$, +as measured by an observer in system~$S'$, is given by these expressions +(14),~(15) and~(16). + +\Subsubsection{41}{Transformation Equations for the Function $\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}$.} These +%% -----File: 062.png---Folio 48------- +three transformation equations for the velocity components of a point\DPtypo{,}{} +permit us to obtain a further transformation equation for an important +function of the velocity which we shall find continually recurring in +our later work. This is the function $\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, where we have indicated +the total velocity of the point by~$u$, according to the expression +$u^2 = \dot{x}^2 + \dot{y}^2 + \dot{z}^2$. By the substitution of equations (14),~(15) and~(16) +we obtain the transformation equation +\[ +\frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = \frac{\left(1 - \smfrac{u_xV}{c^2}\right) \kappa} + {\sqrt{1 - \smfrac{u^2}{c^2}}}. +\Tag{17} +\] + +\Subsubsection{42}{Transformation Equations for Acceleration.} By further differentiating +equations (14),~(15) and~(16) and simplifying, we easily +obtain three new equations for transforming measurements of \emph{acceleration} +from system $S'$~to~$S$, viz.: +\begin{align*} +\ddot{x}' &= \left(1 - + \frac{\dot{x}V}{c^2}\right)^{-3}\kappa^{-3}\ddot{x}, +\Tag{18} \\ +% +\ddot{y}' &= \left(1 - \frac{\dot{x}V}{c^2}\right)^{-2} \kappa^{-2}\ddot{y} + + \dot{y}\, \frac{V}{c^2} + \left(1 - \frac{\dot{x}V}{c^2}\right)^{-3} \kappa^{-2}\ddot{x}, +\Tag{19} \\ +% +\ddot{z}' &= \left(1 - \frac{\dot{x}V}{c^2}\right)^{-2} \kappa^{-2}\ddot{z} + + \dot{z} \frac{V}{c^2} + \left(1 - \frac{\dot{x}V}{c^2}\right)^{-3} \kappa^{-2}\ddot{x}, +\Tag{20} +\intertext{or} +{\dot{u}_x}' &= \left(1 - \frac{u_xV}{c^2}\right)^{-3} \kappa^{-3}\ddot{u}_x, +\Tag{18} \\ +% +{\dot{u}_y}' &= \left(1 - \frac{u_xV}{c^2}\right)^{-2} \kappa^{-2}\ddot{u}_y + + u_y\, \frac{V}{c^2} + \left(1 - \frac{u_xV}{c^2}\right)^{-3} \kappa^{-2}\dot{u}_x, +\Tag{19} \\ +% +{\dot{u}_z}' &= \left(1 - \frac{u_xV}{c^2}\right)^{-2} \kappa^{-2}\ddot{u}_z + + u_z\, \frac{V}{c^2} + \left(1 - \frac{u_xV}{c^2}\right)^{-3} \kappa^{-2}\dot{u}_x. +\Tag{20} +\end{align*} +%% -----File: 063.png---Folio 49------- + + +\Chapter{V}{Kinematical Applications.} +\SetRunningHeads{Chapter Five.}{Kinematical Applications.} + +\Paragraph{43.} The various transformation equations for spatial and temporal +measurements which were derived in the \Chapnumref[IV]{previous chapter} may now be +used for the treatment of a number of kinematical problems. In +particular it will be shown in the latter part of the chapter that a +number of optical problems can be handled with extraordinary facility +by the methods now at our disposal. + + +\Subsection{The Kinematical Shape of a Rigid Body.} + +\Paragraph{44.} We may first point out that the conclusions of relativity theory +lead us to quite new ideas as to what is meant by the shape of a rigid +body. We shall find that the shape of a rigid body will depend entirely +upon the relative motion of the body and the observer who is making +measurements on it. + +Consider a rigid body which is at rest with respect to system~$S'$. +Let $x_1'$,~$y_1'$,~$z_1'$ and $x_2'$,~$y_2'$,~$z_2'$ be the coördinates in system~$S'$ of two +points in the body. The coördinates of the same points as measured +in system~$S$ can be found from transformation equations (9),~(10) +and~(11), and by subtraction we can obtain the following expressions +\begin{gather*} +(x_2 - x_1) = \sqrt{1 - \frac{V^2}{c^2}}\, ({x_2}' - {x_1}'), +\Tag{21} \\ +(y_2 - y_1) = (y_2' - y_1'), +\Tag{22} \\ +(z_2 - \DPtypo{y_2}{z_1}) = (z_2' - z_1'), +\Tag{23} +\end{gather*} +connecting the distances between the pair of points as viewed in the +two systems. In making this subtraction terms containing~$t$ have +been cancelled out since we are interested in the \emph{simultaneous} positions +of the points. In accordance with these equations we may distinguish +then between the \emph{geometrical shape} of a body, which is the shape that +it has when measured on a system of axes which are at rest relative +to it, and its \emph{kinematical shape}, which is given by the coördinates which +%% -----File: 064.png---Folio 50------- +express the \emph{simultaneous} positions of its various points when it is in +motion with respect to the axes of reference. We see that the kinematical +shape of a rigid body differs from its geometrical shape by a +shortening of all its dimensions in the line of motion in the ratio +$\sqrt{1 - \dfrac{V^2}{c^2}}:1$; thus a sphere, for example, becomes a Heaviside ellipsoid. + +In order to avoid incorrectness of speech we must be very careful +not to give the idea that the kinematical shape of a body is in +any sense either more or less real than its geometrical shape. We +must merely learn to realize that the shape of a body is entirely dependent +on the particular set of coördinates chosen for the making +of space measurements. + + +\Subsection{The Kinematical Rate of a Clock.} + +\Paragraph{45.} Just as we have seen that the shape of a body depends upon +our choice of a system of coördinates, so we shall find that the rate of +a given clock depends upon the relative motion of the clock and its +observer. Consider a clock or any mechanism which is performing +a periodic action. Let the clock be at rest with respect to system~$S'$ +and let a given period commence at~${t_1}'$ and end at~${t_2}'$, the length of +the interval thus being $\Delta t' = {t_2}' - {t_1}'$. + +From transformation equation~(12) we may obtain +\begin{align*} +t_1' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}} + \left(t_1 - \frac{V}{c^2}\, x_1\right), \\ +t_2' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}} + \left(t_2 - \frac{V}{c^2}\, x_2\right), +\end{align*} +and by subtraction, since $x_2 - x_1$ is obviously equal to~$Vt$, we have +\begin{align*}%[** TN: Not aligned in orig.] +t_2 - t_1 &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, ({t_2}' - {t_1}'), \\ +\Delta t &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, \Delta t'. +\end{align*} +%% -----File: 065.png---Folio 51------- +Thus an observer who is moving past a clock finds a longer period for +the clock in the ratio $1 : \sqrt{1 - \dfrac{V^2}{c^2}}$ than an observer who is stationary +with respect to it. Suppose, for example, we have a particle which +is turning alternately red and blue. For an observer who is moving +past the particle the periods for which it remains a given color measure +longer in the ratio $1 : \sqrt{1 -\dfrac{V^2}{c^2}}$ than they do to an observer who is +stationary with respect to the particle. + +\Paragraph{46.} A possible opportunity for testing this interesting conclusion +of the theory of relativity is presented by the phenomena of canal +rays. We may regard the atoms which are moving in these rays as +little clocks, the frequency of the light which they emit corresponding +to the period of the clock. If now we should make spectroscopic +observations on canal rays of high velocity, the frequency of the +emitted light ought to be less than that of light from stationary atoms +of the same kind if our considerations are correct. It would of course +be necessary to view the canal rays at right angles to their direction +of motion, to prevent a confusion of the expected shift in the spectrum +with that produced by the ordinary Doppler effect (see \Secref{54}). + + +\Subsection{The Idea of Simultaneity.} + +\Paragraph{47.} We may now also point out that the idea of the \emph{absolute} simultaneity +of two events must henceforth be given up. Suppose, for +example, an observer in the system~$S$ is interested in two events +which take place simultaneously at the time~$t$. Suppose one of these +events occurs at a point having the $X$~coördinate~$x_1$ and the other +at a point having the coördinate~$x_2$; then by transformation equation~(12) +it is evident that to an observer in system~$S'$, which is moving +relative to~$S$ with the velocity~$V$, the two events would take place +respectively at the times +\begin{align*} +{t_1}' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}} + \left(t - \frac{V}{c^2}\, x_1\right) \\ +\intertext{and} +{t_2}' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}} + \left(t - \frac{V}{c^2}\, x_2\right) +\end{align*} +%% -----File: 066.png---Folio 52------- +or the difference in time between the occurrence of the events would +appear to this other observer to be +\[ +{t_2}' - {t_1}' + = \frac{V}{c^2\, \sqrt{1 - \smfrac{V^2}{c^2}}}\, (x_1 - x_2). +\Tag{25} +\] + + +\Subsection{The Composition of Velocities.} + +\Subsubsection{48}{The Case of Parallel Velocities.} We may now present one of +the most important characteristics of Einstein's space and time, +which can be best appreciated by considering transformation equation~(14), +or more simply its analogue for the transformation in the reverse +direction +\[ +u_x = \frac{{u_x}' + V}{1 + \smfrac{{u_x}'V}{c^2}}. +\Tag{26} +\] + +Consider now the significance of the above equation. If we +have a particle which is moving in the $X$~direction with the velocity~$u_x'$ +as measured in system~$S'$, its velocity~$u_x$ with respect to system~$S$ +is to be obtained by adding the relative velocity of the two systems~$V$ +\emph{and dividing the sum of the two velocities by} $1 + \dfrac{{u_x}'V}{c^2}$. Thus we see +that we must completely throw overboard our old naïve ideas of the +direct additivity of velocities. Of course, in the case of very slow +velocities, when $u_x'$~and~$V$ are both small compared with the velocity +of light, the quantity~$\dfrac{{u_x}'V}{c^2}$ is very nearly zero and the direct addition +of velocities is a close approximation to the truth. In the case of +velocities, however, which are in the neighborhood of the speed of +light, the direct addition of velocities would lead to extremely erroneous +results. + +\Paragraph{49.} In particular it should be noticed that by the composition of +velocities which are themselves less than that of light we can never +obtain any velocity greater than that of light. Suppose, for example, +that the system~$S'$ were moving past~$S$ with the velocity of light +(\ie, $V = c$), and that in the system~$S'$ a particle should itself be +given the velocity of light in the $X$~direction (\ie, $u_x' = c$); we find +on substitution that the particle still has only the velocity of light +%% -----File: 067.png---Folio 53------- +with respect to~$S$. We have +\[ +u_x = \frac{c + c}{1 + \smfrac{c^2}{c^2}} = \frac{2c}{2} = c. +\] + +If the relative velocity between the systems should be one half +the velocity of light,~$\dfrac{c}{2}$, and an experimenter on~$S'$ should shoot off a +particle in the $X$~direction with half the velocity of light, the total +velocity with respect to~$S$ would be +\[ +u_x = \frac{\frac{1}{2}c + \frac{1}{2}c}{1 + \smfrac{\frac{1}{4}c^2}{c^2}} + = \frac{4}{5}\, c. +\] + +\Subsubsection{50}{Composition of Velocities in General.} In the case of particles +which have components of velocity in other than the $X$~direction it +is obvious that our transformation equations will here also provide +methods of calculation to supersede the simple addition of velocities. +If we place +\begin{align*} +u^2 &= {u_x}^2 + {u_y}^2 + {u_z}^2 ,\\ +{u'}^2 &= {{u_x}'}^2 + {{u_y}'}^2 + {{u_z}'}^2 , +\end{align*} +we may obtain by the substitution of equations (14),~(15) and~(16) +\[ +u = \frac{\left({u'}^2 + V^2 + 2u'V \cos\alpha + - \smfrac{{u'}^2V^2 \sin^2\alpha}{c^2}\right)^{1/2}} + {1 + \smfrac{u'V \cos\alpha}{c^2}}, +\Tag{27} +\] +where $\alpha$ is the angle in the system~$S'$ between the $X'$~axis and the +velocity of the particle~$u'$. For the particular case that $V$~and~$u'$ +are in the same direction, the equation obviously reduces to the +simpler form +\[ +u = \frac{u' + V}{1 + \smfrac{u'V}{c^2}}, +\] +which we have already considered. + +\Paragraph{51.} We may also call attention at this point to an interesting characteristic +of the equations for the transformation of velocities. It will +%% -----File: 068.png---Folio 54------- +be noted from an examination of these equations that if to any observer +a particle appears to have a constant velocity, \ie, to be +unacted on by any force, it will also appear to have a \emph{uniform} although +of course different velocity to any observer who is himself in uniform +motion with respect to the first. An examination, however, of the +transformation equations for acceleration (18),~(19),~(20) will show +that here a different state of affairs is true, since it will be seen that a +point which has \emph{uniform acceleration} $(\ddot{x}, \ddot{y}, \ddot{z})$ with respect to an observer +in system~$S$ will not in general have a uniform acceleration in +another system~$S'$, since the acceleration in system~$S'$ depends not +only on the constant acceleration but also on the velocity in system~$S$, +which is necessarily varying. + + +\Subsection{Velocities Greater than that of Light.} + +\Paragraph{52.} In the preceding section we have called attention to the fact +that the mere composition of velocities which are not themselves +greater than that of light will never lead to a speed that is greater +than that of light. The question naturally arises whether velocities +which are greater than that of light could ever possibly be obtained +in any way. + +This problem can be attacked in an extremely interesting manner. +Consider two points $A$~and~$B$ on the $X$~axis of the system~$S$, and +suppose that some impulse originates at~$A$, travels to~$B$ with the +velocity~$u$ and at~$B$ produces some observable phenomenon, the starting +of the impulse at~$A$ and the resulting phenomenon at~$B$ thus +being connected by the relation of \emph{cause and effect}. + +The time elapsing between the cause and its effect as measured +in the units of system~$S$ will evidently be +\[ +\Delta t = t_B - t_A = \frac{x_B - x_A}{u}, +\Tag{28} +\] +where $x_A$~and~$x_B$ are the coördinates of the two points $A$~and~$B$. + +Now in another system~$S'$, which has the velocity~$V$ with respect +to~$S$, the time elapsing between cause and effect would evidently be +\[ +\Delta t' = {t'}_B - {t'}_A + = \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\left(t_B - \frac{V}{c^2}\, x_B\right) + - \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\left(t_A - \frac{V}{c^2}\, x_A\right), +\] +%% -----File: 069.png---Folio 55------- +where we have substituted for $t'_B$~and~$t'_A$ in accordance with equation~(12). +Simplifying and introducing equation~(28) we obtain +\[ +\Delta t' + = \frac{1 - \smfrac{uV}{ c^2}} + {\sqrt{1 - \smfrac{V^2}{c^2}}}\, \Delta t. +\Tag{29} +\] +Let us suppose now that there are no limits to the possible magnitude +of the velocities $u$~and~$V$, and in particular that the causal impulse +can travel from~$A$ to~$B$ with a velocity~$u$ greater than that of +light. It is evident that we could then take a velocity~$u$ great enough +so that $\dfrac{uV}{c^2}$~would be greater than unity and $\Delta t'$~would become negative. +In other words, for an observer in system~$S'$ the effect which +occurs at~$B$ would \emph{precede} in time its cause which originates at~$A$. +Such a condition of affairs might not be a logical impossibility; nevertheless +its extraordinary nature might incline us to believe that no +causal impulse can travel with a velocity greater than that of light. + +We may point out in passing, however, that in the case of kinematic +occurrences in which there is no causal connection there is no +reason for supposing that the velocity must be less than that of light. +Consider, for example, a set of blocks arranged side by side in a long +row. For each block there could be an \emph{independent} time mechanism +like an alarm clock which would go off at just the right instant so +that the blocks would fall down one after another along the line. +The velocity with which the phenomenon would travel along the +line of blocks could be arranged to have any value. In fact, the +blocks could evidently all be fixed to fall just at the same instant, +which would correspond to an infinite velocity. It is to be noticed +here, however, that there is no causal connection between the falling +of one block and that of the next, and no transfer of energy. + +%[** TN: ToC entry reads "Applications to Optical Problems"] +\Subsection{Application of the Principles of Kinematics to Certain Optical Problems.} + +\Paragraph{53.} Let us now apply our kinematical considerations to some +problems in the field of optics. We may consider a beam of light +as a periodic electromagnetic disturbance which is propagated through +a vacuum with the velocity~$c$. At any point in the path of a beam of +%% -----File: 070.png---Folio 56------- +light the intensity of the electric and magnetic fields will be undergoing +periodic changes in magnitude. Since the intensities of both the +electric and the magnetic fields vary together, the statement of a +single vector is sufficient to determine the instantaneous condition +at any point in the path of a beam of light. It is customary to call +this vector (which might be either the strength of the electric or of +the magnetic field) the light vector. + +For the case of a simple plane wave (\ie, a beam of monochromatic +light from a distant source) the light vector at any point in the path +of the light may be put proportional to +\[ +\sin\omega \left(t - \frac{lx + my + nz}{c}\right), +\Tag{30} +\] +where $x$,~$y$ and~$z$ are the coördinates of the point under observation, +$t$~is the time, $l$,~$m$ and~$n$ are the cosines of the angles $\alpha$,~$\beta$ and~$\gamma$ which +determine the direction of the beam of light with reference to our +system, and $\omega$~is a constant which determines the period of the light. + +If now this same beam of light were examined by an observer in +system~$S'$ which is moving past the original system in the $X$~direction +with the velocity~$V$, we could write the light vector proportional to +\[ +\sin\omega' \left(t' - \frac{l'x' + m'y' + n'z'}{c}\right), +\Tag{31} +\] +It is not difficult to show that the transformation equations which +we have already developed must lead to the following relations between +the measurements in the two systems\footnote + {Methods for deriving the relation between the accented and unaccented + quantities will be obvious to the reader. For example, consider the relation between + $\omega$~and~$\omega'$. At the origin of coördinates $x = y = z = 0$ in system~$S$, we shall have + in accordance with expression~(30) the light vector proportional to $\sin \omega t$, and hence + similarly at the point~$O'$, which is the origin of coördinates in system~$S'$, we shall + have the light vector proportional to $\sin \omega' t'$. But the point~$O'$ as observed from + system~$S$ moves with the velocity~$V$ along the $X$\DPchg{-}{~}axis and at any instant has the + position $x = Vt$; hence substituting in expression~(30) we have the light vector at + the point~$O'$ as measured in system~$S$ proportional to + \[ + \sin\omega t \left(1 - l\, \frac{V}{c}\right), + \Tag{36} + \] + while as measured in system~$S'$ the intensity is proportional to + \[ + \sin\omega' t'. + \Tag{37} + \] + We have already obtained, however, a transformation equation for~$t'$, namely, + \[ + t' = \kappa \left(t - \frac{V}{c^2}\, x\right), + \] + and further may place $x = Vt$. Making these substitutions and comparing expressions + (36)~and~(37) we see that we must have the relation + \[ + \omega' = \omega \kappa \left(1 - l\, \frac{V}{c}\right). + \] + Methods of obtaining the relation between the cosines $l$,~$m$ and~$n$ and the corresponding + cosines $l'$,~$m'$, and~$n'$ as measured in system~$S'$ may be left to the reader.} +%% -----File: 071.png---Folio 57------- +\begin{align*}%[* TN: Aligning; centered in original] +\omega' &= \omega\kappa \left(1 - l\, \smfrac{V}{c}\right), \Tag{32} \\ +l' &= \frac{l - \smfrac{V}{c}}{1 - l\, \smfrac{V}{c}}, \Tag{33} \\ +m' &= \frac{m}{\kappa\left(1 - l\smfrac{V}{c}\right)}, \Tag{34} \\ +n' &= \frac{n}{\kappa\left(1 - l\smfrac{V}{c}\right)}. \Tag{35} +\end{align*} + +With the help of these equations we may now treat some important +optical problems. + +\Subsubsection{54}{The Doppler Effect.} At the origin of coördinates, $x = y = z += 0$, in system~$S$ we shall evidently have from expression~(30) the +light vector proportional to $\sin \omega t$. That means that the vector +becomes zero whenever $\omega t = 2N \pi$, where $N$~is any integer; in other +words, the period of the light is $p = \dfrac{2\pi}{\omega}$ or the frequency +\[ +\nu = \frac{\omega}{2\pi}. +\] +Similarly the frequency of the light as measured by an observer in +system~$S'$ would be +\[ +\nu' = \frac{\omega'}{2\pi}. +\] +%% -----File: 072.png---Folio 58------- +Combining these two equations and substituting the equation connecting +$\omega$~and~$\omega'$ we have +\[ +\nu = \frac{\nu'}{\kappa \left(1 - l\smfrac{V}{c}\right)}. +\] +This is the relation between the frequencies of a given beam of light +as it appears to observers who are in relative motion. + +If we consider a source of light at rest with respect to system~$S'$ +and at a considerable distance from the observer in system~$S$, we +may substitute for~$\nu'$ the frequency of the source itself,~$\nu_0$, and for~$l$ +we may write~$\cos\phi$, where $\phi$~is the angle between the line connecting +source and observer and the direction of motion of the source, leading +to the expression +\[ +\nu = \frac{\nu_0}{\kappa \left(1 - \cos\phi\, \smfrac{V}{c}\right)}. +\Tag{38} +\] + +This is the most general equation for the \emph{Doppler effect}. When +the source of light is moving directly in the line connecting source +and observer, we have $\cos\phi = 1$, and the equation reduces to +\[ +\nu = \frac{\nu_0}{\kappa \left(1 - \smfrac{V}{c}\right)}, +\Tag{39} +\] +which except for second order terms is identical with the older expressions +for the Doppler effect, and hence agrees with experimental +determinations. + +We must also observe, however, that even when the source of +light moves at right angles to the line connecting source and observer +there still remains a second-order effect on the observed frequency, +in contradiction to the predictions of older theories. We have in this +case $\cos\phi = 0$, +\[ +\nu = \nu_0\, \sqrt{1 - \frac{V^2}{c^2}}. +\Tag{40} +\] +This is the change in frequency which we have already considered +when we discussed the rate of a moving clock. The possibilities of +%% -----File: 073.png---Folio 59------- +direct experimental verification should not be overlooked (see \Secref[section]{46}). + +\Subsubsection{55}{The Aberration of Light.} Returning now to our transformation +equations, we see that equation~(33) provides an expression for +calculating the \emph{aberration of light}. Let us consider that the source +of light is stationary with respect to system~$S$, and let there be an +observer situated at the origin of \DPchg{coordinates}{coördinates} of system~$S'$ and thus +moving past the source with the velocity~$V$ in the $X$~direction. Let $\phi$~be +the angle between the $X$\DPchg{-}{~}axis and the line connecting source of +light and observer and let $\phi'$~be the same angle as it appears to the +moving observer; then we can obviously substitute in equation~(33), +$\cos\phi = l$, $\cos\phi' = l'$, giving us +\[ +\cos\phi' = \frac{\cos\phi - \smfrac{V}{c}}{1 - \cos\phi\, \smfrac{V}{c}}. +\Tag{41} +\] +This is a general equation for the aberration of light. + +For the particular case that the direction of the beam of light is +perpendicular to the motion of the observer we have $\cos\phi = 0$ +\[ +\cos\phi' = - \frac{V}{c}, +\Tag{42} +\] +which, except for second-order differences, is identical with the familiar +expression which makes the tangent of the angle of aberration numerically +equal to~$V/c$.\DPnote{** Slant fractions start here} The experimental verification of the formula +by astronomical measurements is familiar. + +\Subsubsection{56}{Velocity of Light in Moving Media.} It is also possible to treat +very simply by kinematic methods the problem of the velocity of +light in moving media. We shall confine ourselves to the particular +case of a beam of light in a medium which is itself moving parallel +to the light. + +Let the medium be moving with the velocity~$V$ in the $X$~direction, +and let us consider the system of coördinates~$S'$ as stationary with +respect to the medium. Now since the medium appears to be stationary +with respect to observers in~$S'$ it is evident that the velocity +of the light with respect to~$S'$ will be~$c/\mu$, where $\mu$~is index of refraction +%% -----File: 074.png---Folio 60------- +for the medium. If now we use our equation~(26) for the addition of +velocities we shall obtain for the velocity of light, as measured by +observers in~$S$, +\[ +u = \frac{\smfrac{c}{\mu} + V}{1 + \smfrac{V\, \smfrac{c}{\mu}}{c^2}}. +\Tag{43} +\] +Carrying out the division and neglecting terms of higher order we +obtain +\[ +u = \frac{c}{\mu} + \left(\frac{\mu^2 - 1}{\mu^2}\right) V. +\Tag{44} +\] +The equation thus obtained is identical with that of Fresnel, the +quantity $\left(\dfrac{\mu^2 - 1}{\mu^2}\right)$ being the well-known Fresnel coefficient. The +empirical verification of this equation by the experiments of Fizeau +and of Michelson and Morley is too well known to need further +mention. + +For the case of a dispersive medium we should obviously have to +substitute in equation~(44) the value of~$\mu$ corresponding to the particular +frequency,~$\nu'$, which the light has in system~$S'$. It should be +noticed in this connection that the frequencies $\nu'$~and~$\nu$ which the +light has respectively in system~$S$ and system~$S'$, although nearly +enough the same for the practical use of equation~(44), are in reality +connected by an expression which can easily be shown (see \Secref[section]{54}) +to have the form +\[ +\nu' = \kappa \left(1 - \frac{V}{c}\right)\nu. +\Tag{45} +\] + +\Subsubsection{57}{Group Velocity.} In an entirely similar way we may treat the +problem of group velocity and obtain the equation +\[ +G = \frac{G' + V}{1 + \smfrac{G'V}{c^2}}, +\Tag{46} +\] +where $G'$ is the group velocity as it appears to an observer who is +%% -----File: 075.png---Folio 61------- +stationary with respect to the medium. $G'$~is, of course, an experimental +quantity, connected with frequency and the properties of the +medium, in a way to be determined by experiments on the stationary +medium. + +In conclusion we wish to call particular attention to the extraordinary +simplicity of this method of handling the optics of moving +media as compared with those that had to be employed before the +introduction of the principle of relativity. +%% -----File: 076.png---Folio 62------- + + +\Chapter{VI}{The Dynamics of a Particle.} +\SetRunningHeads{Chapter Six.}{Dynamics of a Particle.} + +\Paragraph{58.} In this chapter and the two following, we shall present a +system of ``relativity mechanics'' based on Newton's three laws of +motion, the Einstein transformation equations for space and time, +and the principle of the conservation of mass. + + +\Subsection{The Laws of Motion.} + +Newton's laws of motion may be stated in the following form: + +I\@. Every particle continues in its state of rest or of uniform motion +in a straight line, unless it is acted upon by an external force. + +II\@. The rate of change of the momentum of the particle is equal +to the force acting and is in the same direction. + +III\@. For the action of every force there is an equal force acting +in the opposite direction. + +Of these laws the first two merely serve to define the concept of +force, and their content may be expressed in mathematical form by +the following equation of definition +\[ +\vc{F} + = \frac{d}{dt}(m\vc{u}) + = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u}, +\Tag{47} +\] +where $\vc{F}$ is the force acting on a particle of mass~$m$ which has the +velocity~$\vc{u}$, and hence the momentum~$m\vc{u}$. + +Quite different in its nature from the first two laws, which merely +give us a definition of force, the third law states a very definite physical +postulate, since it requires for every change in the momentum of a +body an equal and opposite change in the momentum of some other +body. The truth of this postulate will of course be tested by comparing +with experiment the results of the theory of mechanics which +we base upon its assumption. + + +\Subsection{Difference between Newtonian and Relativity Mechanics.} + +\Paragraph{59.} Before proceeding we may point out the particular difference +between the older Newtonian mechanics, which were based on the +laws of motion and the \emph{Galilean} transformation equations for space +%% -----File: 077.png---Folio 63------- +and time, and our new system of relativity mechanics based on +those same laws of motion and the \emph{Einstein} transformation equations. + +In the older mechanics there was no reason for supposing that the +mass of a body varied in any way with its velocity, and hence force +could be defined interchangeably as the rate of change of momentum +or as mass times acceleration, since the two were identical. In relativity +mechanics, however, we shall be forced to conclude that the +mass of a body increases in a perfectly definite way with its velocity, +and hence in our new mechanics we must define force as equal to the +total rate of change of momentum +\[ +\frac{d(m\vc{u})}{dt} + = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u} +\] +instead of merely as mass times acceleration $m\, \dfrac{d\vc{u}}{dt}$. If we should try +to define force in ``relativity mechanics'' as merely equal to mass +times acceleration, we should find that the application of Newton's +third law of motion would then lead to very peculiar results, which +would make the mass of a body different in different directions and +force us to give up the idea of the conservation of mass. + + +\Subsection{The Mass of a Moving Particle.} + +\Paragraph{60.} In \Secref{31} we have already obtained in an elementary way +an expression for the mass of a moving particle, by considering a +collision between elastic particles and calculating how the resulting +changes in velocity would appear to different observers who are +themselves in relative motion. Since we now have at our command +general formulæ for the transformation of velocities, we are now in +a position to handle this problem much more generally, and \emph{in particular +to show that the expression obtained for the mass of a moving particle +is entirely independent of the consideration of any particular type of +collision}. + +\Subsubsection{61}{Transverse Collision.} Let us first treat the case of a so-called +``transverse'' collision. Consider a system of coördinates and two +\begin{wrapfigure}{l}{3in}%[** TN: Width-dependent break] + \Fig{11} + \Input[3in]{078} +\end{wrapfigure} +exactly similar elastic particles, each having the mass~$m_0$ when at +rest, one moving in the $X$~direction with the velocity~$+u$ and the +other with the velocity~$-u$. (See \Figref{11}.) Besides the large +components of velocity $+u$~and~$-u$ which they have in the $X$~direction +%% -----File: 078.png---Folio 64------- +let them also have small components of velocity in the $Y$~direction, +$+v$~and~$-v$. The experiment is so arranged that the particles +will just undergo a glancing collision as they pass each other and +rebound with components +of velocity in the $Y$~direction +of the same magnitude,~$v$, +which they originally had, +but in the reverse direction. +(It is evident from the symmetry of the arrangement that the experiment +would actually occur as we have stated.) + +We shall now be interested in the way this experiment would appear +to an observer who is in motion in the $X$ direction with the velocity~$V$ +relative to our original system of coördinates. + +From equation~(14) for the transformation of velocities, it can +be seen that this \emph{new observer} would find for the $X$~component velocities +of the two particles the values +\[ +u_1 = \frac{ u - V}{1 - \smfrac{uV}{c^2}} \qquad\text{and}\qquad +u_2 = \frac{-u - V}{1 + \smfrac{uV}{c^2}} +\Tag{48} +\] +and from equation~(15) for the $Y$~component velocities would find the +values +\[ +v_1 = \pm \frac{v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}} +\qquad\text{and}\qquad +v_2 = \mp \frac{v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}}, +\Tag{49} +\] +the signs depending on whether the velocities are measured before or +after the collision. + +Now from Newton's third law of motion (\ie, the principle of +the equality of action and reaction) it is evident that on collision +the two particles must undergo the same numerical change in momentum. + +For the experiment that we have chosen the only change in momentum +is in the $Y$~direction, and the observer whose measurements +we are considering finds that one particle undergoes the total change +%% -----File: 079.png---Folio 65------- +in velocity +\begin{align*} +2v_1 &= \frac{2v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}} \\ +\intertext{and that the other particle undergoes the change in velocity} +2v_2 &= \frac{2v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 + \smfrac{uV}{c^2}}. +\end{align*} + +Since these changes in the velocities of the particles are not equal, +it is evident that their masses must also be unequal if the principle +of the equality of action and reaction is true for all observers, as we +have assumed. This difference in the mass of the particles, each of +which has the mass~$m_0$ when at rest, arises from the fact that the mass +of a particle is a function of its velocity and for the observer in question +the two particles are not moving with the same velocity. + +Using the symbols $m_1$~and~$m_2$ for the masses of the particles, we +may now write as a mathematical expression of the requirements of +the third law of motion +\[ +\frac{2m_1v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}} = +\frac{2m_2v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 + \smfrac{uV}{c^2}}. +\] + +Simplifying, we obtain by direct algebraic transformation +%[** TN: Setting innermost denominator fractions textstyle for clarity] +\[ +\frac{m_1}{m_2} + = \frac{1 - \smfrac{uV}{c^2}}{1 + \smfrac{uV}{c^2}} + = \frac{\sqrt{ + 1 - \smfrac{\Biggl(\smfrac{-u - V}{1 + \tfrac{uV}{c^2}}\Biggr)^2}{c^2}}} + {\sqrt{ + 1 - \smfrac{\Biggl(\smfrac{ u - V}{1 - \tfrac{uV}{c^2}}\Biggr)^2}{c^2}}}, +\] +%% -----File: 080.png---Folio 66------- +which on the substitution of equations~(48) gives us +\[ +\frac{m_1}{m_2} + = \frac{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}} + {\sqrt{1 - \smfrac{{u_1}^2}{c^2}}}. +\Tag{50} +\] + +This equation thus shows that the mass of a particle moving with +the velocity~$u$\footnote + {For simplicity of calculation we consider the case where the components of + velocity in the $Y$~direction are small enough to be negligible in their effect on the + mass of the particles compared with the large components of velocity $u_1$~and~$u_2$ in + the $X$~direction.} +is inversely proportional to $\sqrt{1 - \dfrac{u^2}{c^2}}$, and, denoting +the mass of the particle at rest by~$m_0$, we may write as a \emph{general expression +for the mass of a moving particle} +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}. +\Tag{51} +\] + +\Subsubsection{62}{Mass the Same in All Directions.} The method of derivation +that we have just used to obtain this expression for the mass of a +moving particle is based on the consideration of a so-called ``transverse +collision,'' and in fact the expression obtained has often been +spoken of as that for the \emph{transverse mass} of a moving particle, while +a different expression, $\dfrac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}$, has been used for the so-called +\emph{longitudinal mass} of the particle. These expressions $\dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ and +$\dfrac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}$ are, as a matter of fact, the values of the electric force +necessary to give a charged particle unit acceleration respectively +at right angles and in the same direction as its original velocity, and +hence such expressions would be proper for the mass of a moving particle +if we should define force as mass times acceleration. As already +%% -----File: 081.png---Folio 67------- +stated, however, it has seemed preferable to retain, for force, Newton's +original definition which makes it equal to the rate of change of +momentum, and we shall presently see that this more suitable definition +is in perfect accord with the idea that the mass of a particle is +the same in all directions. + +Aside from the unnecessary complexity which would be introduced, +the particular reason making it unfortunate to have different +expressions for mass in different directions is that under such conditions +it would be impossible to retain or interpret the principle of +the conservation of mass. And we shall now proceed to show that +by introducing the principle of the conservation of mass, the consideration +of a ``longitudinal collision'' will also lead to exactly the +same expression, $\dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, for the mass of a moving particle as we +have already obtained from the consideration of a transverse collision. + +\Subsubsection{63}{Longitudinal Collision.} Consider a system of coördinates and +two elastic particles moving in the $X$~direction with the velocities +$+u$~and~$-u$ so that a ``longitudinal'' (\ie, head-on) collision will +occur. Let the particles be exactly alike, each of them having the +mass~$m_0$ when at rest. On collision the particles will evidently come +to rest, and then under the action of the elastic forces developed start +up and move back over their original paths with the respective velocities +$-u$~and~$+u$ of the same magnitude as before. + +Let us now consider how this collision would appear to an observer +who is moving past the original system of coördinates with the velocity~$V$ +in the $X$~direction. Let $u_1$~and~$u_2$ be the velocities of the particles +as they appear to this new observer before the collision has taken +place. Then, from our formula for the transformation of velocities~(14), +it is evident that we shall have +\[ +u_1 = \frac{ u - V}{1 - \smfrac{uV}{c^2}}\qquad\text{and}\qquad +u_2 = \frac{-u - V}{1 + \smfrac{uV}{c^2}}. +\Tag{52} +\] + +Since these velocities $u_1$~and~$u_2$ are not of the same magnitude, +the two particles which have the same mass when at rest do not have +the same mass for this observer. Let us call the masses before collision +$m_1$~and~$m_2$. +%% -----File: 082.png---Folio 68------- + +Now during the collision the velocities of the particles will all the +time be changing, but from the principle of the conservation of mass +the sum of the two masses must all the time be equal to $m_1 + m_2$. +When in the course of the collision the particles have come to relative +rest, they will be moving past our observer with the velocity~$-V$, +and their momentum will be $-(m_1 + m_2)V$. But, from the principle +of the equality of action and reaction, it is evident that this momentum +must be equal to the original momentum before collision occurred. +This gives us the equation $-(m_1 + m_2)V = m_1 u_1 + m_2 u_2$. Substituting +our values~(52) for $u_1$~and~$u_2$ we have +\[ +\frac{m_1}{\left(1 - \smfrac{uV}{c^2}\right)} = +\frac{m_2}{\left(1 + \smfrac{uV}{c^2}\right)}, +\] +and by direct algebraic transformation, as in the previous proof, +this can be shown to be identical with +\[ +\frac{m_1}{m_2} + = \frac{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}} + {\sqrt{1 - \smfrac{{u_1}^2}{c^2}}}, +\] +leading to the same expression that we obtained before for the mass +of a moving particle, viz.: +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}. +\] + +\Subsubsection{64}{Collision of Any Type.} We have derived this formula for the +mass of a moving particle first from the consideration of a transverse +and then of a longitudinal collision between particles which are elastic +and have the same mass when at rest. It seems to be desirable to +show, however, that the consideration of any type of collision between +particles of any mass leads to the same formula for the mass of a +moving particle. + +For the mass~$m$ of a particle moving with the velocity~$u$ let us +write the equation $m = m_0 F(u^2)$, where $F(\:)$~is the function whose +form we wish to determine. The mass is written as a function of +%% -----File: 083.png---Folio 69------- +the square of the velocity, since from the homogeneity of space the +mass will be independent of the direction of the velocity, and the +mass is made proportional to the mass at rest, since a moving body +may evidently be thought of as divided into parts without change in +mass. It may be further remarked that the form of the function~$F(\:)$ +must be such that its value approaches unity as the variable +approaches zero. + +Let us now consider two particles having respectively the masses +$m_0$~and~$n_0$ when at rest, moving with the velocities $u$~and~$w$ before +collision, and with the velocities $U$~and~$W$ after a collision has taken +place. + +From the principle of the conservation of mass we have +\begin{multline*} +m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2) + +n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2) \\ += m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2) ++ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2), +\Tag{53} +\end{multline*} +and from the principle of the equality of action and reaction (\ie, +Newton's third law of motion) +\begin{multline*} +m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2)u_x + +n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2)w_x \\ += m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2)U_x ++ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2)W_x, +\Tag{54} +\end{multline*} +\begin{multline*} +m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2)u_y + +n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2)w_y \\ += m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2)U_y ++ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2)W_y, +\Tag{55} +\end{multline*} +\begin{multline*} +m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2)u_z + +n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2)w_z \\ += m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2)U_z ++ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2)W_z. +\Tag{56} +\end{multline*} + +These velocities, $u_x$,~$u_y$,~$u_x$, $w_x$,~$w_y$,~$w_z$, $U_x$,~etc., are measured, of +course, with respect to some definite system of ``space-time'' coördinates. +An observer moving past this system of coördinates with the +velocity~$V$ in the $X$~direction would find for the corresponding component +velocities the values +\[ +\frac{u_x - V}{1 - \smfrac{u_xV}{c^2}},\quad +\frac{\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}}\, u_y,\quad +\frac{\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}}\, u_z,\quad +\frac{w_x - V}{1 - \smfrac{w_xV}{c^2}},\quad\text{etc.}, +\] +as given by our transformation equations for velocity \DPchg{(14, 15, 16)}{(14),~(15),~(16)}. +%% -----File: 084.png---Folio 70------- + +Since the law of the conservation of mass and Newton's third +law of motion must also hold for the measurements of the new observer, +we may write the following new relations corresponding to +equations \DPchg{53~to~56}{(53)~to~(56)}: + +{\footnotesize% +\[ +\begin{aligned} +m_0 F&\left\{ + \left(\frac{u_x - V}{1 - \smfrac{u_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_y}{1 - \smfrac{u_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_z}{1 - \smfrac{u_x V}{c^2}}\right)^2 +\right\} \\ ++ n_0F&\left\{ +\left(\frac{w_x - V}{1 - \smfrac{w_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_y}{1 - \smfrac{w_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_z}{1 - \smfrac{w_x V}{c^2}}\right)^2 +\right\} \\ += m_0F&\left\{ +\left(\frac{U_x - V}{1 - \smfrac{U_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_y}{1 - \smfrac{U_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_z}{1 - \smfrac{U_x V}{c^2}}\right)^2 +\right\} \\ ++ n_0F&\left\{ +\left(\frac{W_x - V}{1 - \smfrac{W_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_y}{1 - \smfrac{W_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_z}{1 - \smfrac{W_x V}{c^2}}\right)^2 +\right\}, +\end{aligned} +\Tag{53\textit{a}} +\]}% +\[ +\begin{aligned} +&m_0F\{u_x\cdots\}\, \frac{u_x - V}{1 - \smfrac{u_xV}{c^2}} + +n_0F\{w_x\cdots\}\, \frac{w_x - V}{1 - \smfrac{w_xV}{c^2}} \\ +&\qquad= +m_0F\{U_x\cdots\}\, \frac{U_x - V}{1 - \smfrac{U_xV}{c^2}} + +n_0F\{W_x\cdots\}\, \frac{W_x - V}{1 - \smfrac{W_xV}{c^2}}, +\end{aligned} +\Tag{54\textit{a}} +\] +{\small% +\[ +\begin{aligned} +&m_0F\{u_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_y}{1 - \smfrac{u_xV}{c^2}} + +n_0F\{w_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_y}{1 - \smfrac{w_xV}{c^2}} \\ +&\qquad= +m_0F\{U_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_y}{1 - \smfrac{U_xV}{c^2}} + +n_0F\{W_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_y}{1 - \smfrac{W_xV}{c^2}}, +\end{aligned} +\Tag{55\textit{a}} +\]}% +%% -----File: 085.png---Folio 71------- +\[ +\begin{aligned} +&m_0F{u_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_x}{1 - \smfrac{u_xV}{c^2}} + +n_0F{w_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_x}{1 - \smfrac{w_xV}{c^2}} \\ +&\qquad= +m_0F{U_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_x}{1 - \smfrac{U_xV}{c^2}} + +n_0F{W_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_x}{1 - \smfrac{W_xV}{c^2}}. +\end{aligned} +\Tag{56\textit{a}} +\] + +It is evident that these equations \DPchg{(53\textit{a}--56\textit{a})}{(53\textit{a})--(56\textit{a})} must be true no +matter what the velocity between the new observer and the original +system of coördinates, that is, true for all values of~$V$. The velocities +$u_x$,~$u_y$,~$u_z$, $w_x$,~etc., are, however, perfectly definite quantities, measured +with reference to a definite system of coördinates and entirely independent +of~$V$. If these equations are to be true for perfectly definite +values of $u_x$,~$u_y$,~$u_z$, $w_x$,~etc., and for all values of~$V$, it is evident that +the function~$F(\:\,)$ must be of such a form that the equations are +identities in~$V$. As a matter of fact, it is found by trial that $V$~can +be cancelled from all the equations if we make $F(\:\,)$ of the form +$\dfrac{1}{\sqrt{1 - \smfrac{(\:)}{c^2}}}$; and we see that the expected relation is a solution of the +equations, although perhaps not necessarily a unique solution. + +Before proceeding to use our formula for the mass of a moving +particle for the further development of our system of mechanics, +we may call attention in passing to the fact that the experiments of +Kaufmann, Bucherer, and Hupka have in reality shown that the mass +of the electron increases with its velocity according to the formula +which we have just obtained. We shall consider the dynamics of the +electron more in detail in the chapter devoted to \Chapnumref[XII]{electromagnetic +theory}. We wish to point out now, however, that in this derivation +we have made no reference to any electrical charge which might be +carried by the particle whose mass is to be determined. Hence we +may reject the possibility of explaining the Kaufmann experiment +by assuming that the charge of the electron decreases with its velocity, +since the increase in mass is alone sufficient to account for the results +of the measurement. +%% -----File: 086.png---Folio 72------- + + +\Subsection{Transformation Equations for Mass.} + +\Paragraph{65.} Since the velocity of a particle depends on the particular +system of coördinates chosen for the measurement, it is evident that +the mass of the particle will also depend on our reference system of +coördinates. For the further development of our system of dynamics, +we shall find it desirable to obtain transformation equations for mass +similar to those already obtained for velocity, acceleration, etc. + +We have +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +where the velocity~$u$ is measured with respect to some definite system +of coördinates,~$S$. Similarly with respect to a system of coördinates~$S'$ +which is moving relatively to~$S$ with the velocity~$V$ in the $X$~direction +we shall have +\[ +m' = \frac{m_0}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}. +\] +We have already obtained, however, a transformation equation~(17) +for the function of the velocity occurring in these equations and on +substitution we obtain the desired transformation equation +\[ +m' = \left(1 - \frac{u_x V}{c^2}\right) \kappa m, +\Tag{57} +\] +where $\kappa$ has the customary significance +\[ +\kappa = \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}. +\] + +By differentiation of~(57) with respect to the time and simplification, +we obtain the following transformation equation for the +\emph{rate at which the mass of a particle is changing} owing to change in +velocity +\[ +\dot{m}' = \dot{m} - \frac{mV}{c^2} + \left(1 - \frac{u_xV}{c^2}\right)^{-1} \frac{du_x}{dt}. +\Tag{58} +\] +%% -----File: 087.png---Folio 73------- + +%[** TN: ToC entry reads "The Force Acting on a Moving Particle"] +\Subsection{Equation for the Force Acting on a Moving Particle.} + +\Paragraph{66.} We are now in a position to return to our development of the +dynamics of a particle. In the first place, the equation which we +have now obtained for the mass of a moving particle will permit +us to rewrite the original equation by which we defined force, in a +number of ways which will be useful for future reference. + +We have our equation of definition~(47) +\[ +\vc{F} = \frac{d}{dt}(m\vc{u}) + = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u}, +\] +which, on substitution of the expression for~$m$, gives us +\[ +\vc{F} + = \frac{d}{dt} \Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{u}\Biggr] + = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt} + + \frac{d}{dt} \Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr] \vc{u} +\Tag{59} +\] +or, carrying out the indicated differentiation, +\[ +\vc{F} + = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt} + + \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, + \frac{u}{c^2}\, \frac{du}{dt}\, \vc{u}. +\Tag{60} +\] + + +\Subsection{Transformation Equations for Force.} + +\Paragraph{67.} We are also in position to obtain transformation equations for +force. We have +\[ +\vc{F} = \frac{d}{dt}(m\vc{u}) = m\vc{\dot{u}} + \dot{m}\vc{u} +\] +or +\begin{align*} +F_x &= m\dot{u}_x + \dot{m}u_x, \\ +F_y &= m\dot{u}_y + \dot{m}u_y, \\ +F_z &= m\dot{u}_z + \dot{m}u_z. +\end{align*} +We have transformation equations, however, for all the quantities +on the right-hand side of these equations. For the velocities we +have equations (14),~(15) and~(16), for the accelerations (18),~(19) +and~(20), for mass, equation~(57) and for rate of change of mass, +equation~(58). Substituting above we obtain as our \emph{transformation +%% -----File: 088.png---Folio 74------- +equations for force} +\begin{align*} +F_x' &= \frac{F_x - \dot{m}V}{1 - \smfrac{u_xV}{c^2}} + = F_x - \frac{u_y V}{c^2 - u_x V}\, F_y + - \frac{u_z V}{c^2 - u_x V}\, F_z, \Tag{61} \\ +F_y' &= \frac{\kappa^{-1}}{1 - \smfrac{u_x V}{c^2}}\, F_y, \Tag{62}\\ +F_z' &= \frac{\kappa^{-1}}{1 - \smfrac{u_x V}{c^2}}\, F_z. \Tag{63} +\end{align*} + +We may now consider a few applications of the principles governing +the dynamics of a particle. + + +\Subsection{The Relation between Force and Acceleration.} + +\Paragraph{68.} If we examine our equation~(59) for the force acting on a +particle +\[ +F = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt} + + \frac{d}{dt} \Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr] \vc{u}, +\Tag{59} +\] +we see that the force is equal to the sum of two vectors, one of which +is in the direction of the acceleration $\dfrac{d\vc{u}}{dt}$ and the other in the direction +of +\begin{wrapfigure}[17]{l}{2.5in}%[** TN: Width-dependent break] + \Fig{12} + \Input[2.5in]{088} +\end{wrapfigure} +the existing velocity~$\vc{u}$, so that \emph{in general force and the acceleration +it produces are not in the same direction}. +We shall find it interesting +to see, however, that if the force +which does produce acceleration in +a given direction be resolved perpendicular +and parallel to the acceleration, +the two components will +be connected by a definite relation. + +Consider a particle (\Figref[fig.]{12}) in +plane space moving with the velocity +\[ +\vc{u} = {u_x}\vc{i} + {u_y}\vc{j}. +\] +%% -----File: 089.png---Folio 75------- +Let it be accelerated in the $X$~direction by the action of the component +forces $F_x$~and~$F_y$. + +From our general equation~(59) for the force acting on a particle +we have for these component forces +\begin{align*} +F_x &= \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{du_x}{dt} + + \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]u_x, +\Tag{64} \\ +F_y &= \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{du_y}{dt} + + \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]u_y. +\Tag{65} +\end{align*} + +Introducing the condition that all the acceleration is to be in the $Y$~direction, +which makes $\dfrac{du_x}{dt} = 0$, and further noting that $u^2 = u_x^2 + u_y^2$, +by the division of equation~(64) by~(65), we obtain +\begin{align*} +\frac{F_x}{F_y} &= \frac{u_x u_y}{c^2 - {u_x}^2}, \\ +F_x &= \frac{u_x u_y}{c^2 - {u_x}^2}\, F_y. +\Tag{66} +\end{align*} + +\emph{Hence, in order to accelerate a particle in a given direction, we may +apply any force~$F_y$ in the desired direction, but must at the same time +apply at right angles another force~$F_x$ whose magnitude is given by +equation~\upshape{(66)}.} + +Although at first sight this state of affairs might seem rather +unexpected, a simple qualitative consideration will show the necessity +of a component of force perpendicular to the desired acceleration. +Refer again to \Figref{12}; since the particle is being accelerated in the $Y$~direction, +its total velocity and hence its mass are increasing. This +increasing mass is accompanied by increasing momentum in the $X$~direction +even when the velocity in that direction remains constant. +The component force~$F_x$ is necessary for the production of this increase +in $X$-momentum. + +In a later paragraph we shall show an application of equation~(66) +in electrical theory. +%% -----File: 090.png---Folio 76------- + + +\Subsection{Transverse and Longitudinal Acceleration.} + +\Paragraph{69.} An examination of equation~(66) shows that there are two +special cases in which the component force~$F_x$ disappears and the +force and acceleration are in the same direction. $F_x$~will disappear +when either $u_x$~or~$u_y$ is equal to zero, so that force and acceleration +will be in the same direction when the force acts exactly at right +angles to the line of motion of the particle, or in the direction of the +motion (or of course also when $u_x$~and~$u_y$ are both equal to zero and +the particle is at rest). It is instructive to obtain simplified expressions +for force for these two cases of transverse and longitudinal +acceleration. + +Let us again examine our equation~(60) for the force acting on a +particle +\[ +\vc{F} + = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt} + + \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, + \frac{u}{c^2}\, \frac{du}{dt} \vc{u}. +\Tag{60}%[** TN: [sic] Repeated equation] +\] + +For the case of a \emph{transverse acceleration} there is no component of +force in the direction of the velocity~$\vc{u}$ and the second term of the +equation is equal to zero, giving us +\[ +\vc{F} = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt}. +\Tag{67} +\] + +For the case of \emph{longitudinal acceleration}, the velocity~$\vc{u}$ and the +acceleration~$\dfrac{d\vc{u}}{dt}$ are in the same direction, so that we may rewrite the +second term of~(60), giving us +\[ +\vc{F} + = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt} + + \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, + \frac{u^2}{c^2}\, \frac{d\vc{u}}{dt}, +\] +and on simplification this becomes +\[ +\vc{F} + = \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, \frac{d\vc{u}}{dt}. +\Tag{68} +\] +%% -----File: 091.png---Folio 77------- +An examination of this expression shows the reason why $\dfrac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}$ +is sometimes spoken of as the expression for the \emph{longitudinal mass} of a +particle. + + +\Subsection{The Force Exerted by a Moving Charge.} + +\Paragraph{70.} In a \Chapnumref[XII]{later chapter} we shall present a consistent development +of the fundamentals of electromagnetic theory based on the Einstein +transformation equations for space and time and the four field equations. +At this point, however, it may not be amiss to point out that +the principles of mechanics themselves may sometimes be employed +to obtain a simple and direct solution of electrical problems. + +Suppose, for example, we wish to calculate the force with which a +\emph{point charge in uniform motion} acts on any other point charge. We +can solve this problem by considering a system of coördinates which +move with the same velocity as the charge itself. An observer +making use of the new system of coördinates could evidently calculate +the force exerted by the charge in question by Coulomb's familiar +inverse square law for static charges, and the magnitude of the force +as measured in the original system of coördinates can then be determined +from our transformation equations for force. Let us proceed +to the specific solution of the problem. + +Consider a system of coördinates~$S$, and a charge~$e$ in uniform +motion along the $X$~axis with the velocity~$V$. We desire to know +the force acting at the time~$t$ on any other charge~$e_1$ which has any +desired coördinates $x$,~$y$, and~$z$ and any desired velocity $u_x$,~$u_y$ and~$u_z$. + +Assume a system of coördinates,~$S'$, moving with the same velocity +as the charge~$e$ which is taken coincident with the origin. To an +observer moving with the system~$S'$, the charge~$e$ appears to be +always at rest and surrounded by a pure electrostatic field. Hence +in system~$S'$ the force with which $e$~acts on~$e_1$ will be, in accordance +with Coulomb's law\footnote + {It should be noted that in its original form Coulomb's law merely stated + that the force between two stationary charges was proportional to the product of + the charges and inversely to the distance between them. In the present derivation + we have extended this law to apply to the instantaneous force exerted by a stationary + charge upon any other charge. + + The fact that a charge of electricity appears the same to observers in all systems + is obviously also necessary for the setting up of equations (69),~(70),~(71). That + such is the case, however, is an evident consequence of the atomic nature of electricity. + The charge~$e$ would appear of the same magnitude to observers both in + system~$S$ and system~$S'$, since they would both count the same number of electrons + on the charge. (See \Secref{157}.)} +\[ +\vc{F'} = \frac{e e_1 \vc{r'}}{{r'}^3} +\] +%% -----File: 092.png---Folio 78------- +or +\begin{align*} +F_x' &= \frac{ee_1x'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \Tag{69} \\ +F_y' &= \frac{ee_1x'}{({y'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \Tag{70} \\ +F_z' &= \frac{ee_1x'}{({z'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \Tag{71} +\end{align*} +where $x'$,~$y'$, and~$z'$ are the coördinates of the charge~$e_1$ at the time~$t'$. +For simplicity let us consider the force at the time $t' = 0$; then from +transformation equations (9),~(10), (11),~(12) we shall have +\[ +x' = \kappa^{-1} x,\qquad y' = y, \qquad z'= z. +\] +Substituting in (69),~(70),~(71) and also using our transformation +equations for force (61),~(62),~(63), we obtain the following equations +for the force acting on~$e_1$, as it appears to an observer in system~$S$: +\begin{align*}%[** TN: Aligning on "="s] +F_x &= \frac{ee_1x'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}} + \left(x + \frac{V}{c^2}\, \kappa^2(yu_y + zu_z)\right), +\Tag{72} \\ +F_y &= \frac{ee_1\left(1 - \smfrac{u_xV}{c^2}\right) \kappa y} + {({\kappa^{-2}x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, +\Tag{73} \\ +F_z &= \frac{ee_1\left(1 - \smfrac{u_xV}{c^2}\right) \kappa z} + {({\kappa^{-2}x'}^2 + {y'}^2 + {z'}^2)^{3/2}}. +\Tag{74} +\end{align*} + +These equations give the force acting on~$e_1$ at the time~$t$. From +transformation equation~(12) we have $t = \dfrac{V}{c^2}\, x$, since $t' = 0$. At this +time the charge~$e$, which is moving with the uniform velocity~$V$ along +%% -----File: 093.png---Folio 79------- +the $X$~axis, will evidently have the position +\[ +x_e = \frac{V^2}{c^2}\, x,\qquad +y_e = 0, \qquad +z_e = 0. +\] + +For convenience we may now refer our results to a system of +coördinates whose origin coincides with the position of the charge~$e$ +at the instant under consideration. If $X$,~$Y$ and~$Z$ are the coördinates +of~$e_1$ with respect to this new system, we shall evidently have +the relations +\begin{gather*} +X = x - \frac{V^2 }{c^2}\, x = \kappa^{-2} x,\qquad Y = y,\qquad Z = z,\\ +U_x = u_x, \qquad U_y = u_y, \qquad U_z = u_z. +\end{gather*} +Substituting into (72),~(73),~(74) we obtain +\begin{align*}%[** TN: Aligning on "="s] +F_x &= \frac{ee_1}{s^3} \left(1 - \frac{V^2}{c^2}\right) + \left(X + \frac{V}{c^2}\, (YU_y + ZU_z)\right), +\Tag{75} \\ +F_y &= \frac{ee_1}{s^3} \left(1 - \frac{V^2}{c^2}\right) + \left(1 - \frac{U_xV}{c^2})\right) Y, +\Tag{76} \\ +F_z &= \frac{ee_1}{s^3} \left(1 - \frac{V^2}{c^2}\right) + \left(1 - \frac{U_xV}{c^2})\right) Z, +\Tag{77} +\end{align*} +where for simplicity we have placed +\[ +s = \sqrt{X^2 + \left(1 - \frac{V^2}{c^2}\right)(Y^2 + Z^2)}. +\] + +These are the same equations which would be obtained by substituting +the well-known formulæ for the strength of the electric and +magnetic field around a moving point charge into the fifth fundamental +equation of the Maxwell-Lorentz theory, $\vc{f} = \rho \left(\vc{e} + \dfrac{1}{c}\, [\vc{u} × \vc{h}]^*\right)$. +They are really obtained in this way more easily, however, and are +seen to come directly from Coulomb's law. + +%[** TN: Unnumbered, but has a ToC entry] +\Subsubsection{}{The Field around a Moving Charge.} Evidently we may also use +these considerations to obtain an expression for the electric field +produced by a moving charge~$e$, if we consider the particular case +that the charge~$e_1$ is stationary (\ie, $U_x = U_y = U_z = 0$) and equal +%% -----File: 094.png---Folio 80------- +to unity. Making these substitutions in (75),~(76),~(77) we obtain +the well-known expression for the electrical field in the neighborhood +of a moving point charge +\[ +\vc{F} = e = \frac{\vc{e}}{s^3} \left(1 - \frac{V^2}{c^2}\right)\vc{r}, +\Tag{78} +\] +where +\[ +\vc{r} = X\vc{i} + Y\vc{j} + Z\vc{k}. +\] + +\Subsubsection{71}{Application to a Specific Problem.} Equations (75), (76), (77) +can also be applied in the solution of a +rather interesting specific problem. + +Consider a charge~$e$ constrained to +move in the $X$~direction with the velocity~$V$ +and at the instant under consideration +let it coincide with the origin +of a system of stationary coördinates +$YeX$ (\Figref[fig.]{13}). Suppose now a second +charge~$e_1$, situated at the point $X = 0$, +$Y = Y$ and moving in the $X$~direction +with the same velocity~$V$ as the charge~$e$, +and also having a component velocity +in the $Y$~direction~$U_y$. Let us +%[** TN: Move down past page break; width-dependent line break] +\begin{wrapfigure}{l}{2.25in} + \Fig{13} + \Input[2.25in]{094} +\end{wrapfigure} +predict +the nature of its motion under the influence +of the charge~$e$, it being otherwise +unconstrained. + +From the simple qualitative considerations placed at our disposal +by the theory of relativity, it seems evident that the charge~$e_1$ ought +merely to increase its component of velocity in the $Y$~direction and +retain unchanged its component in the $X$~direction, since from the +point of view of an observer moving along with~$e$ the phenomenon is +merely one of ordinary \emph{electrostatic} repulsion. + +Let us see whether our equations for the force exerted by a moving +charge actually lead to this result. By making the obvious substitutions +in equations (75)~and~(76) we obtain for the component +forces on~$e_1$ +\begin{align*}%[** TN: Aligning on "="s] +F_x &= \frac{e e_1}{s^3}\left(1 - \frac{V^2}{c^2}\right) + \frac{V}{c^2}\, Y U_y, +\Tag{79} \\ +F_x &= \frac{e e_1}{s^3}\left(1 - \frac{V^2}{c^2}\right)^2 Y. +\Tag{80} +\end{align*} +%% -----File: 095.png---Folio 81------- + +Now under the action of the component force~$F_x$ we might at +first sight expect the charge~$e_1$ to obtain an acceleration in the $X$~direction, +in contradiction to the simple qualitative prediction that +we have just made on the basis of the theory of relativity. We +remember, however, that equation~(66) prescribes a definite ratio +between the component forces $F_x$~and~$F_y$ if the acceleration is to be +in the $Y$~direction, and dividing~(79) by~(80) we actually obtain the +necessary relation +\[ +\frac{F_x}{F_y} = \frac{V U_y}{c^2 - V^2}. +\] + +Other applications of the new principles of dynamics to electrical, +magnetic and gravitational problems will be evident to the reader. + + +\Subsection{Work.} + +\Paragraph{72.} Before proceeding with the further development of our theory +of dynamics we shall find it desirable to define the quantities work, +kinetic, and potential energy. + +We have already obtained an expression for the force acting on a +particle and shall define the work done on the particle as the integral +of the force times the distance through which the particle is displaced. +Thus +\[ +W = \int \vc{F} · d\vc{r}, +\Tag{81} +\] +where $\vc{r}$ is the radius vector determining the position of the particle. + + +\Subsection{Kinetic Energy.} + +\Paragraph{73.} When a particle is brought from a state of rest to the velocity~$\vc{u}$ +by the action of an unbalanced force~$\vc{F}$, we shall define its kinetic +energy as numerically equal to the work done in producing the velocity. +Thus +\[ +K = W = \int \vc{F} · d\vc{r}. +\] + +Since, however, the kinetic energy of a particle turns out to be +entirely independent of the particular choice of forces used in producing +the final velocity, it is much more useful to have an expression +for kinetic energy in terms of the mass and velocity of the particle. + +We have +\[ +K = \int \vc{F} · d\vc{r} + = \int \vc{F} · \frac{d\vc{r}}{dt}\, dt + = \int \vc{F} · \vc{u}\, dt. +\] +%% -----File: 096.png---Folio 82------- +Substituting the value of~$\vc{F}$ given by the equation of definition~(47) +we obtain +\begin{align*} +K &= \int m\, \frac{d\vc{u}}{dt} · \vc{u}\, dt + + \int \frac{dm}{dt}\, \vc{u} · \vc{u}dt \\ + &= \int m\, \vc{u} · d\vc{u} + \int \vc{u} · \vc{u}\, dm \\ + &= \int mu\, du + \int u^2\, dm. +\end{align*} +Introducing the expression~(51) for the mass of a moving particle +$m = \dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, we obtain +\[ +K = \int m_0\, \frac{u}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, du + + \int\frac{m_0}{c^2}\, \frac{u^3}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, du +\] +and on integrating and evaluating the constant of integration by +placing the kinetic energy equal to zero when the velocity is zero, +we easily obtain the desired expression for the kinetic energy of a +particle: +\begin{align*} +K &= m_0 c^2 \Biggl[\frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} - 1\Biggr], +\Tag{82} \\ + &= c^2(m - m_0). +\Tag{83} +\end{align*} + +It should be noticed, as was stated above, that the kinetic energy +of a particle \emph{does} depend merely on its mass and final velocity and is +entirely independent of the particular choice of forces which happened +to be used in producing the state of motion. + +It will also be noticed, on expansion into a series, that our expression~(82) +for the kinetic energy of a particle approaches at low +velocities the form familiar in the older Newtonian mechanics, +\[ +K = \tfrac{1}{2} m_0 u^2. +\] + + +\Subsection{Potential Energy.} + +\Paragraph{74.} When a moving particle is brought to rest by the action of a +%% -----File: 097.png---Folio 83------- +\emph{conservative}\footnote + {A conservative force is one such that any work done by displacing a system + against it would be completely regained if the motion of the system should be reversed. + + Since we believe that the forces which act on the ultimate particles and constituents + of matter are in reality all of them conservative, we shall accept the general + principle of the conservation of energy just as in Newtonian mechanics. (For a + logical deduction of the principle of the conservation of energy in a system of particles, + see the next chapter, \Secref[section]{89}.)} %[** TN: Not a useful chapter cross-ref] +force we say that its kinetic energy has been transformed +into potential energy. The increase in the potential energy +of the particle is equal to the kinetic energy which has been destroyed +and hence equal to the work done by the particle against the force, +giving us the equation +\[ +\Delta U = -W = -\int \vc{F} · d\vc{r}. +\Tag{84} +\] + + +\Subsection{The Relation between Mass and Energy.} + +\Paragraph{75.} We may now consider a very important relation between the +mass and energy of a particle, which was first pointed out in our +chapter on ``\Chapnumref[III]{Some Elementary Deductions}.'' + +When an isolated particle is set in motion, both its mass and +energy are increased. For the increase in mass we may write +\[ +\Delta m = m - m_0, +\] +and for the increase in energy we have the expression for kinetic energy +given in equation~(83), giving us +\[ +\Delta E = c^2(m-m_0), +\] +or, combining with the previous equation, +\[ +\Delta E = c^2 \Delta m. +\Tag{85} +\] + +Thus the increase in the kinetic energy of a particle always bears +the same definite ratio (the square of the velocity of light) to its +increase in mass. Furthermore, when a moving particle is brought +to rest and thus loses both its kinetic energy and its extra (``kinetic'') +mass, there seems to be every reason for believing that this mass +and energy which are associated together when the particle is in +motion and leave the particle when it is brought to rest will still +remain always associated together. For example, if the particle is +brought to rest by collision with another particle, it is an evident +%% -----File: 098.png---Folio 84------- +consequence of our considerations that the energy and the mass +corresponding to it do remain associated together since they are both +passed on to the new particle. On the other hand, if the particle +is brought to rest by the action of a conservative force, say for example +that exerted by an elastic spring, the kinetic energy which leaves the +particle will be transformed into the potential energy of the stretched +spring, and since the mass which has undoubtedly left the particle +must still be in existence, we shall believe that this mass is now associated +with the potential energy of the stretched spring. + +\Paragraph{76.} Such considerations have led us to believe that matter and +energy may be best regarded as different names for the same fundamental +entity: \emph{matter}, the name which has been applied when we +have been interested in the property of mass or inertia possessed +by the entity, and \emph{energy}, the name applied when we have been +interested in the part taken by the entity in the production of motion +and other changes in the physical universe. We shall find these +ideas as to the relations between matter, energy and mass very fruitful +in the simplification of physical reasoning, not only because it +identifies the two laws of the conservation of mass and the conservation +of energy, but also for its frequent application in the solution +of specific problems. + +\Paragraph{77.} We must call attention to the great difference in size between +the two units, the gram and the erg, both of which are used for the +measurement of the one fundamental entity, call it matter or energy +as we please. Equation~(85) gives us the relation +\[ +E = c^2 m, +\Tag{86} +\] +where $E$~is expressed in ergs and $m$~in grams; hence, taking the velocity +of light as $3 × 10^{10}$~centimeters per second, we shall have +\[ +1\text{ gram} = 9 × 10^{20}\text{ ergs}. +\Tag{87} +\] +The enormous number of ergs necessary for increasing the mass of +a system to the amount of a single gram makes it evident that experimental +proofs of the relation between mass and energy will be hard to +find, and outside of the experimental work on electrons at high velocities, +already mentioned in \Secref{64} and the well-known relations +%% -----File: 099.png---Folio 85------- +between the energy and momentum of a beam of light, such evidence +has not yet been forthcoming. + +As to the possibility of obtaining further direct experimental +evidence of the relation between mass and energy, we certainly cannot +look towards thermal experiments with any degree of confidence, +since even on cooling a body down to the absolute zero of temperature +it loses but an inappreciable fraction of its mass at ordinary temperatures.\footnote + {It should be noticed that our theory points to the presence of enormous + stores of interatomic energy which are still left in substances cooled to the absolute + zero.} +In the case of some radioactive processes, however, we may +find a transfer of energy large enough to bring about measurable +differences in mass. And making use of this point of view we might +account for the lack of exact relations between the atomic weights of +the successive products of radioactive decomposition.\footnote + {See, for example, Comstock, \textit{Philosophical Magazine}, vol.~15, p.~1 (1908).} + +\Subsubsection{78}{Application to a Specific Problem.} We may show an interesting +application of our ideas as to the relation between mass and +energy, in the treatment of a specific problem. Consider, just as in +\Secref{63}, two elastic particles both of which have the mass~$m_0$ at +rest, one moving in the $X$~direction with the velocity~$+u$ and the +other with the velocity~$-u$, in such a way that a head-on collision +between the particles will occur and they will rebound over their +original paths with the respective velocities $-u$~and~$+u$ of the +same magnitude as before. + +Let us now consider how this collision would appear to an observer +who is moving past the original system of coördinates with the velocity~$V$ +in the $X$~direction. To this new observer the particles will be +moving before the collision with the respective velocities +\[ +u_1 = \frac{ u - V}{1 - \smfrac{uV}{c^2}}\qquad\text{and}\qquad +u_2 = \frac{-u - V}{1 - \smfrac{uV}{c^2}}, +\Tag{88} +\] +as given by equation~(14) for the transformation of velocities. Furthermore, +when in the course of the collision the particles have come +to relative rest they will obviously be moving past our observer with +the velocity~$-V$. +%% -----File: 100.png---Folio 86------- + +Let us see what the masses of the particles will be both before and +during the collision. Before the collision, the mass of the first particle +will be +\[ +\frac{m_0}{\sqrt{1 - \smfrac{{u_1}^2}{c^2}}} = +\frac{m_0}{\sqrt{1 - \smfrac{\Biggl[\smfrac{ u - V}{1 - \tfrac{uV}{c^2}}\Biggr]^2}{c^2}}} = +\frac{m_0 \left(1 - \smfrac{uV}{c^2}\right)} + {\sqrt{\left(1 - \smfrac{V^2}{c^2}\right)\left(1 - \smfrac{u^2}{c^2}\right)}} +\] +and the mass of the second particle will be +\[ +\frac{m_0}{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}} = +\frac{m_0}{\sqrt{1 - \smfrac{\Biggl[\smfrac{-u - V}{1 + \tfrac{uV}{c^2}}\Biggr]^2}{c^2}}} = +\frac{m_0 \left(1 + \smfrac{uV}{c^2}\right)} + {\sqrt{\left(1 - \smfrac{V^2}{c^2}\right)\left(1 - \smfrac{u^2}{c^2}\right)}}. +\] +Adding these two expressions, we obtain for the sum of the masses of +the two particles before collision, +\[ +\frac{2m_0}{\sqrt{\left(1 - \smfrac{V^2}{c^2}\right) + \left(1 - \smfrac{u^2}{c^2}\right)}}. +\] + +Now during the collision, when the two particles have come to +relative rest, they will evidently both be moving past our observer +with the velocity~$-V$ and hence the sum of their masses at the +instant of relative rest would appear to be +\[ +\frac{2m_0}{\sqrt{1 - \smfrac{V^2}{c^2}}}, +\] +a quantity which is smaller than that which we have just found for +the sum of the two masses before the collision occurred. This apparent +discrepancy between the total mass of the system before and during +the collision, is removed, however, if we realize that when the particles +%% -----File: 101.png---Folio 87------- +have come to relative rest an amount of potential energy of +elastic deformation has been produced, which is just sufficient to restore +them to their original velocities, and the mass corresponding to +this potential energy will evidently be just sufficient to make the +total mass of the system the same as before collision. + +In the following chapter on the dynamics of a system of particles +we shall make further use of our ideas as to the mass corresponding +to potential energy. +%% -----File: 102.png---Folio 88------- + + +\Chapter{VII}{The Dynamics of a System of Particles.} +\SetRunningHeads{Chapter Seven.}{Dynamics of a System of Particles.} + +\Paragraph{79.} In the \Chapnumref[VI]{preceding chapter} we discussed the laws of motion +of a particle. With the help of those laws we shall now derive some +useful general dynamical principles which describe the motions of a +system of particles, and in the \Chapnumref[VIII]{following chapter} shall consider an +application of some of these principles to the kinetic theory of gases. + +The general dynamical principles which we shall present in this +chapter will be similar \emph{in form} to principles which are already familiar +in the classical Newtonian mechanics. Thus we shall deduce principles +corresponding to the principles of the conservation of momentum, +of the conservation of moment of momentum, of least action and of +\textit{vis~viva}, as well as the equations of motion in the Lagrangian and +Hamiltonian (canonical) forms. For cases where the velocities of all +the particles involved are slow compared with that of light, we shall +find, moreover, that our principles become identical in content, as +well as in form, with the corresponding principles of the classical +mechanics. Where high velocities are involved, however, our new +principles will differ from those of Newtonian mechanics. In particular +we shall find among other differences that in the case of high +velocities it will no longer be possible to define the Lagrangian function +as the difference between the kinetic and potential energies of the +system, nor to define the generalized momenta used in the Hamiltonian +equations as the partial differential of the kinetic energy with +respect to the generalized velocity. + + +\Subsection{On the Nature of a System of Particles.} + +\Paragraph{80.} Our purpose in this chapter is to treat dynamical systems +consisting of a finite number of particles, each obeying the equation +of motion which we have already written in the forms, +\begin{gather*} +\vc{F} + = \frac{d}{dt}(m\vc{u}) + = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u}, +\Tag{47} \displaybreak[0] \\ +\vc{F} + = \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c}}}\,\vc{u}\Biggr] + = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c}}}\, \frac{d\vc{u}}{dt} + + \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c}}}\Biggr] \vc{u}. +\Tag{59} +\end{gather*} +%% -----File: 103.png---Folio 89------- + +It is not to be supposed, however, that the total mass of such a +system can be taken as located solely in these particles. It is evident +rather, since potential energy has mass, that there will in general be +mass distributed more or less continuously throughout the space in +the neighborhood of the particles. Indeed we have shown at the +end of the preceding chapter\DPnote{** TN: Not a useful cross-ref} (\Secref{78}) that unless we take account +of the mass corresponding to potential energy we can not maintain +the principle of the conservation of mass, and we should also find it +impossible to retain the principle of the conservation of momentum +unless we included the momentum corresponding to potential energy. + +For a continuous distribution of mass we may write for the force +acting at any point on the material in a small volume,~$\delta V$, +\[ +\vc{f}\, \delta V = \frac{d}{dt} (\vc{g}\, \delta V), +\Tag{47\textit{A}} +\] +where $\vc{f}$ is the force per unit volume and $\vc{g}$~is the density of momentum. +This equation is of course merely an equation of definition for the +intensity of force at a point. We shall assume, however, that Newton's +third law, that is, the principle of the equality of action and +reaction, holds for forces of this type as well as for those acting on +particles. In later chapters we shall investigate the way in which $\vc{g}$~depends +on velocity, state of strain, etc., but for the purposes of this +chapter we shall not need any further information as to the nature +of the distributed momentum. + +Let us proceed to the solution of our specific problems. + + +\Subsection{The Conservation of Momentum.} + +\Paragraph{81.} We may first show from Newton's third law of motion that +the momentum of an isolated system of particles remains constant. + +Considering a system of particles of masses $m_1$,~$m_2$, $m_3$,~etc., we +may write in accordance with equation~\DPtypo{47}{(47)}, +\[ +\begin{aligned} +\vc{F}_1 + \vc{I}_1 &= \frac{d}{dt} (m_1 \vc{u}_1), \\ +\vc{F}_2 + \vc{I}_2 &= \frac{d}{dt} (m_2 \vc{u}_2), \\ +\text{etc.,}\quad & +\end{aligned} +\Tag{89} +\] +%% -----File: 104.png---Folio 90------- +where $\vc{F}_1$,~$\vc{F}_2$,~etc., are the external forces impressed on the individual +particles from outside the system and $\vc{I}_1$,~$\vc{I}_2$,~etc., are the internal +forces arising from mutual reactions within the interior of the system. +Considering the distributed mass in the system, we may also write, +in accordance with~(47\textit{A}) the further equation +\[ +(\vc{f} + \vc{i})\, \delta V = \frac{d}{dt}(\vc{g}\, \delta V), +\Tag{90} +\] +where $\vc{f}$~and~$\vc{i}$ are respectively the external and internal forces acting +\emph{per unit volume} of the distributed mass. Integrating throughout the +whole volume of the system~$V$ we have +\[ +\int (\vc{f} + \vc{i})\, dV = \frac{d\vc{G}}{dt}, +\Tag{91} +\] +where $\vc{G}$ is the total distributed momentum in the system. Adding +this to our previous equations~(89) for the forces acting on the individual +particles, we have +\[ +%[** TN: \textstyle \sum in original] +\Sum \vc{F}_1 + \Sum \vc{I}_1 + \int \vc{f}\, dV + \int \vc{i}\, dV + = \frac{d}{dt} \Sum m_1 u_1 + \frac{d\vc{G}}{dt}. +\] + +But from Newton's third law of motion (\ie, the principle of the +equality of action and reaction) it is evident that the sum of the +internal forces, $\Sum \vc{I}_1 + \int \vc{i}\, dV$, which arise from mutual reactions within +the system must be equal to zero, which leads to the desired equation +of momentum +\[ +\Sum \vc{F}_1 + \int \vc{f}\, dv = \frac{d}{dt}(\Sum m_1 u_1 + \vc{G}). +\Tag{92} +\] + +In words this equation states that at any given instant the vector +sum of the external forces acting on the system is equal to the rate +at which the total momentum of the system is changing. + +For the particular case of an isolated system there are no external +forces and our equation becomes a statement of the principle of the +\emph{conservation of momentum}. + + +\Subsection{The Equation of Angular Momentum.} + +\Paragraph{82.} We may next obtain an equation for the moment of momentum +of a system about a point. +%% -----File: 105.png---Folio 91------- +Consider a particle of mass~$m_1$ and velocity~$u_1$. Let $\vc{r}_1$~be the +radius vector from any given point of reference to the particle. Then +for the moment of momentum of the particle about the point we may +write +\[ +\vc{M}_1 = \vc{r}_1 × m_1\vc{u}_1, +\] +and summing up for all the particles of the system we may write +\[ +\Sum \vc{M}_1 = \Sum (\vc{r}_1 × m_1\vc{u}_1). +\Tag{93} +\] +Similarly, for the moment of momentum of the \emph{distributed mass} we +may write +\[ +\vc{M}_{\text{dist.}} = \int (\vc{r} × \vc{g})\, dV, +\Tag{94} +\] +where $\vc{r}$ is the radius vector from our chosen point of reference to a +point in space where the density of momentum is~$\vc{g}$ and the integration +is to be taken throughout the whole volume,~$V$, of the system. + +Adding these two equations (93)~and~(94), we obtain for the total +amount of momentum of the system about our chosen point +\[ +\vc{M} = \Sum(\vc{r}_1 × m_1\vc{u}_1) + \int (\vc{r} × \vc{g})\, dV; +\] +and differentiating with respect to the time we have, for the rate of +change of the moment of momentum, +\begin{multline*} +\frac{d\vc{M}}{dt} + = \Sum \left\{\vc{r}_1 × \frac{d}{dt}(m_1\vc{u}_1)\right\} + + \Sum \left(\frac{d\vc{r}_1}{dt} × m_1\vc{u}_1\right) \\ + + \int \left(\vc{r} × \frac{d\vc{g}}{dt} \right) dV + + \int \left(\frac{d\vc{r}}{dt} × \vc{g} \right) dV; +\end{multline*} +or, making the substitutions given by equations (89)~and~(90), and +writing $\dfrac{d\vc{r}_1}{dt} = \vc{u}_1$, etc.\DPtypo{}{,} we have +\begin{multline*} +\frac{d\vc{M}}{dt} + = \Sum (\vc{r}_1 × \vc{F}_1) + \Sum (\vc{r}_1 × \vc{I}_1) + + \Sum (\vc{u}_1 × m_1\vc{u}_1) \\ + + \int (\vc{r} × \vc{f})\, dV + \int (\vc{r} × \vc{i})\, dV + + \int (\vc{u} × \vc{g})\, dV. +\end{multline*} +To simplify this equation we may note that the third term is equal to +zero because it contains the outer product of a vector by itself. Furthermore, +if we accept the principle of the equality of action and +%% -----File: 106.png---Folio 92------- +reaction, together with the further requirement that forces are not +only equal and opposite but that their points of application be in the +same straight line, we may put the moment of all the internal forces +equal to zero and thus eliminate the second and fifth terms. We +obtain as the equation of angular momentum +\[ +\frac{d\vc{M}}{dt} = \Sum(\vc{r}_1 × \vc{F}_1) + + \int (\vc{r} × \vc{f})\, dV + \int (\vc{u} × \vc{g})\, dV. +\Tag{95} +\] + +We may call attention to the inclusion in this equation of the +interesting term $\int(\vc{u} × \vc{g})\, dV$. If density of momentum and velocity +should always be in the same direction this term would vanish, since +the outer product of a vector by itself is equal to zero. In our consideration +of the ``Dynamics of Elastic Bodies,'' however, we shall +find bodies with a component of momentum at right angles to their +direction of motion and hence must include this term in a general +treatment. For a completely isolated system it can be shown, however, +that this term vanishes along with the external forces and we +then have the principle of the \emph{conservation of moment of momentum.} + + +\Subsection{The Function $T$.} + +\Paragraph{83.} We may now proceed to the definition of a function which +will be needed in our treatment of the principle of least action. + +One of the most valuable properties of the Newtonian expression, +$\frac{1}{2}m_0u^2$, for kinetic energy was the fact that its derivative with respect +to velocity is evidently the Newtonian expression for momentum,~$m_0u$. +It is not true, however, that the derivative of our new expression +for kinetic energy (see \Secref{73}), $m_0c^2 \Biggl[\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} - 1\Biggr]$, with respect +to velocity is equal to momentum, and for that reason in our non-Newtonian +mechanics we shall find it desirable to define a new function,~$T$, +by the equation, +\[ +T = m_0c^2\left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right). +\Tag{96} +\] + +For slow velocities (\ie, small values of~$u$) this reduces to the +Newtonian expression for kinetic energy and at all velocities we have +%% -----File: 107.png---Folio 93------- +the relation, +\[ +\frac{dT}{du} + = -m_0 c^2\, \frac{d}{du} \sqrt{1 - \frac{u^2}{c^2}} + = \frac{m_0u}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = mu, +\Tag{97} +\] +showing that the differential of~$T$ with respect to velocity is momentum. + +For a system of particles we shall define~$T$ as the summation of +the values for the individual particles: +\[ +T = \Sum m_0 c^2 \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right). +\Tag{98} +\] + + +\Subsection{The Modified Lagrangian Function.} + +\Paragraph{84.} In the older mechanics the Lagrangian function for a system +of particles was defined as the difference between the kinetic and +potential energies of the system. The value of the definition rested, +however, on the fact that the differential of the kinetic energy with +respect to velocity was equal to momentum, so that we shall now +find it advisable to define the Lagrangian function with the help of +our new function~$T$ in accordance with the equation +\[ +L = T - U. +\Tag{99} +\] + + +\Subsection{The Principle of Least Action.} + +\Paragraph{85.} We are now in a position to derive a principle corresponding +to that of least action in the older mechanics. Consider the path +by which our dynamical system actually moves from state~(1) to +state~(2). The motion of any particle in the system of mass $m$ will +be governed by the equation +\[ +\vc{F} = \frac{d}{dt} (m\vc{u}). +\Tag{100} +\] + +Let us now compare the actual path by which the system moves +from state~(1) to state~(2) with a slightly displaced path in which the +laws of motion are not obeyed, and let the displacement of the particle +at the instant in question be~$\delta \vc{r}$. + +Let us take the inner product of both sides of equation~(100) with~$\delta \vc{r}$; +%% -----File: 108.png---Folio 94------- +we have +\begin{gather*} +\begin{aligned} +\vc{F} ·\delta\vc{r} + &= \frac{d}{dt}(m \vc{u}) · \delta \vc{r} \\ + &= \frac{d}{dt}(m\vc{u} · \delta\vc{r}) + - m\vc{u} · \frac{d\, \delta\vc{r}}{dt} \\ + &= \frac{d}{dt}(m\vc{u} · \delta\vc{r}) - m\vc{u} · \delta\vc{u}) +\end{aligned} \\ +(m\vc{u} · \delta\vc{u} + \vc{F} · \delta\vc{r})\, dt + = d(m\vc{u} · \delta\vc{r}). +\end{gather*} + +Summing up for all the particles of the system and integrating +between the limits $t_1$~and~$t_2$, we have +\[ +\int_{t_1}^{t_2} \left(\Sum m\vc{u}· \delta\vc{u} + \Sum \vc{F} · \delta\vc{r}\right) dt + = \left[\Sum m\vc{u} · \delta\vc{r} \right]_{t_1}^{t_2}. +\] +Since $t_1$~and~$t_2$ are the times when the actual and displaced motions +coincide, we have at these times $\delta\vc{r} = 0$; furthermore we also have +$\vc{u} · \delta\vc{u} = u\, \delta u$, so that we may write +\[ +\int_{t_1}^{t_2}\left(\Sum mu\, \delta u + \vc{F} · \delta\vc{r}\right) dt = 0. +\] +With the help of equation~(97), however, we see that $\Sum mu\, \delta u = \delta T$, +giving us +\[ +\int_{t_1}^{t_2} (\delta T + \vc{F} · \delta r)\, dt = 0. +\Tag{101} +\] +\emph{If the forces~$F$ are conservative}, we may write $\vc{F} · \delta r = -\delta U$, where +$\delta U$~is the difference between the potential energies of the displaced +and the actual configurations. This gives us +\[ +\delta \int_{t_1}^{t_2} (T - U)\, dt = 0 +\] +or +\[ +\delta \int_{t_1}^{t_2} L\, dt = 0, +\Tag{102} +\] +which is the modified principle of least action. The principle evidently +requires that for the actual path by which the system goes +%% -----File: 109.png---Folio 95------- +from state~(1) to state~(2), the quantity $\ds\int_{t_1}^{t_2} L\, dt$ shall be a minimum (or +maximum). + + +\Subsection{Lagrange's Equations.} + +\Paragraph{86.} We may now derive the Lagrangian equations of motion from +the above principle of least action. Let us suppose that the position +of each particle of the system under consideration is completely determined +by $n$~\emph{independent} generalized coördinates $\phi_{1}$,~$\phi_{2}$, $\phi_{3} \cdots \phi_{n}$ and +hence that $L$~is some function of $\phi_{1}$,~$\phi_{2}$, $\phi_{3} \cdots \phi_{n}$, $\dot{\phi}_{1}$,~$\dot{\phi}_{2}$, $\dot{\phi}_{3} \cdots \dot{\phi}_{n}$, +where for simplicity we have put $\dot{\phi}_{1} = \dfrac{d\phi_1}{dt}$, $\dot{\phi}_{2} = \dfrac{d\phi_2}{dt}$,~etc. + +%%%% Use of "1" as a subscript in the original starts here %%%% +From equation~(102) we have +\[ +\int_{t_1}^{t_2} (\delta L)\, dt = \int_{t_1}^{t_2} \left( + \Sum_1^n \frac{\partial L}{\partial\phi_{\1}}\, \delta\phi_{\1} + + \Sum_1^n \frac{\partial L}{\partial\dot{\phi}_{\1}}\, \delta\dot{\phi}_{\1} + \right)dt = 0. +\Tag{103} +\] +But +\[ +\delta\dot{\phi}_{\1} = \frac{d}{dt}(\delta\phi_{\1})\DPchg{}{,} +\] +which gives us +\begin{align*} +\int_{t_1}^{t_2} \frac{\partial L}{\partial\dot{\phi}_{\1}}\, + \delta\dot{\phi}_{\1}\, dt + &= \int_{t_1}^{t_2} \frac{\partial L}{\partial\dot{\phi}_{\1}}\, + \frac{d}{dt}(\delta\phi_{\1})\, dt \\ + &= \left[\frac{\partial L}{\partial\dot{\phi}_{\1}}\, \delta\phi_{\1}\right]_{t_1}^{t_2} + - \int_{t_1}^{t_2} \delta\phi_{\1}\, + \frac{d}{dt} \left(\frac{\partial L}{\partial\dot{\phi}_{\1}}\right) dt +\end{align*} +or, since at times $t_{1}$~and~$t_{2}$, $\delta \phi_{\1}$~is zero, the first term in this expression +disappears and on substituting in equation~(103) we obtain +\[ +\int_{t_1}^{t2} \left[\Sum_{1}^{n} \delta \phi_{\1} + \left\{ \frac{\partial L}{\partial\phi_{\1}} + - \frac{d}{dt} \left(\frac{\partial L}{\partial\dot{\phi}_{\1}}\right) + \right\} \right] dt = 0. +\] +Since, however, the limits $t_{1}$~and~$t_{2}$ are entirely at our disposal we must +have at every instant +\[ +\Sum_{1}^{n} \delta \phi_{\1} + \left\{ \frac{\partial L}{\partial\phi_{\1}} + - \frac{d}{dt} \left(\frac{\partial L}{\partial\dot{\phi}_{\1}}\right) + \right\} = 0. +\] +Finally, moreover, since the $\phi$'s~are independent parameters, we can +assign perfectly arbitrary values to $\delta\phi_{1}$,~$\delta\phi_{2}$,~etc., and hence must have +%% -----File: 110.png---Folio 96------- +the series of equations +\[ +\begin{aligned} +&\frac{d}{dt} \left( \frac{\partial L}{\partial\dot{\phi}_1} \right) + - \frac{\partial L}{\partial\phi_1} = 0, \\ +&\frac{d}{dt} \left( \frac{\partial L}{\partial\dot{\phi}_2} \right) + - \frac{\partial L}{\partial\phi_2} = 0, \\ +&\text{etc.} +\end{aligned} +\Tag{104} +\] +These correspond to Lagrange's equations in the older mechanics, +differing only in the definition of~$L$. + + +\Subsection{Equations of Motion in the Hamiltonian Form.} + +\Paragraph{87.} We shall also find it desirable to obtain equations of motion +in the Hamiltonian or canonical form. + +Let us define the \emph{generalized momentum}~$\psi_{\1}$ corresponding to the +coördinate~$\phi_{\1}$ by the equation, +\[ +\psi_{\1} = \frac{\partial T}{\partial\dot{\phi}_{\1}}. +\Tag{105} +\] + +It should be noted that the generalized momentum is not as in +ordinary mechanics the derivative of the kinetic energy with respect +to the generalized velocity but approaches that value at low velocities. + +Consider now a function~$T'$ defined by the equation +\[ +T' = \psi_1\dot{\phi}_1 + \psi_{2}\dot{\phi}_2 + \cdots - T. +\Tag{106} +\] +Differentiating we have +\begin{align*} +dT' &= \psi_1\, d\dot{\phi}_1 + \psi_2\, d\dot{\phi}_2 + \cdots \\ + &\quad+ \dot{\phi}_1\, d\psi_{1} + \dot{\phi}_2\, d\psi_{2} + \cdots \\ + &\quad- \frac{\partial T}{\partial\phi_1}\, d\phi_1 + - \frac{\partial T}{\partial\phi_2}\, d\phi_2 - \cdots \\ + &\quad- \frac{\partial T}{\partial\dot{\phi}_1}\, d\dot{\phi}_1 + - \frac{\partial T}{\partial\dot{\phi}_2}\, d\dot{\phi}_2 - \cdots, +\end{align*} +and this, by the introduction of~(105), becomes +\[ +dT' = \dot{\phi}_1\, d\psi_1 + \dot{\phi}_2\, d\psi_{2} + \cdots + - \frac{\partial T}{\partial\phi_1}\, d\phi_1 + - \frac{\partial T}{\partial\phi_2}\, d\phi_2 - \cdots. +\Tag{107} +\] +%% -----File: 111.png---Folio 97------- +Examining this equation we have +\begin{align*} +\frac{\partial T'}{\partial\phi_{\1}} + &= - \frac{\partial T}{\partial\phi_{\1}}, +\Tag{108} \\ +\frac{\partial T'}{\partial\psi_{\1}} + & = \dot{\phi}_{\1}. +\Tag{109} +\end{align*} +In Lagrange's equations we have +\[ +\frac{d}{dt}\left\{ \frac{\partial}{\partial\dot{\phi}_{\1}}(T - U)\right\} + - \frac{\partial}{\partial\phi_{\1}}(T - U) = 0. +\] +But since $U$ is independent of~$\psi_{\1}$ we may write +\[ +\frac{\partial(T - U)}{\partial\dot{\phi}_{\1}} + = \frac{\partial T}{\partial\dot{\phi}_{\1}} = \psi_{\1}, +\] +and furthermore by~(108), +\[ +\frac{\partial T}{\partial\phi_{\1}} = -\frac{\partial T'}{\partial\phi_{\1}}. +\] +Substituting these two expressions in Lagrange's equations we obtain +\[ +\frac{d\psi_{\1}}{dt} = -\frac{\partial(T' + U)}{\partial\phi_{\1}} +\] +or, writing $T' + U = E$, we have +\[ +\frac{d\psi_{\1}}{dt} = -\frac{\partial E}{\partial\phi_{\1}} +\Tag{110} +\] +and since $U$~is independent of~$\psi_{\1}$ we may rewrite equation~(109) in +the form +\[ +\frac{d\phi_{\1}}{dt} = \frac{\partial E}{\partial\psi_{\1}}. +\Tag{111} +\] + +The set of equations corresponding to (110)~and~(111) for all the +coördinates $\phi_{1}$,~$\phi_{2}$, $\phi_{3}, \cdots \phi_{n}$ and the momenta $\psi_{1}$,~$\psi_{2}$, $\psi_{3}, \cdots \psi_{n}$ are +the desired equations of motion in the canonical form. + +\Subsubsection{88}{Value of the Function $T'$.} We have given the symbol~$E$ to +the quantity $T' + U$, since $T'$~actually turns out to be identical with +%% -----File: 112.png---Folio 98------- +the expression by which we defined kinetic energy, thus making +$E = T' + U$ the sum of the kinetic and potential energies of the +system. + +To show that $T'$~is equal to~$K$, the kinetic energy, we have by the +equation of definition~(106) +\begin{align*} +T' &= \phi_1\psi_1 + \phi_2\psi_2 + \cdots - T, \\ + &= \phi_1\, \frac{\partial T}{\partial\dot{\phi}_1} + + \phi_2\, \frac{\partial T}{\partial\dot{\phi}_2} + \cdots - T. +\end{align*} +But $T$ by definition, equation~(98), is +\begin{align*} +T &= \Sum c^2m_0 \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right), \\ +\intertext{which gives us} +\frac{\partial T}{\partial\dot{\phi}_{\1}} + &= \Sum m_0 \left(1 - \frac{u^2}{c^2}\right)^{-1/2} + u\, \frac{\partial u}{\partial\dot{\phi}_{\1}} \\ + &= \Sum mu\, \frac{\partial u}{\partial\dot{\phi}_{\1}} +\end{align*} +and substituting we obtain +\[ +\begin{aligned} +T' &= \dot{\phi}_1 \Sum mu\, \frac{\partial u}{\partial\dot{\phi}_1} + + \dot{\phi}_2 \Sum mu\, \frac{\partial u}{\partial\dot{\phi}_2} + + \cdots - T \\ + &= \Sum mu \left\{ + \dot{\phi}_1\, \frac{\partial u}{\partial\dot{\phi}_1} + + \dot{\phi}_2\, \frac{\partial u}{\partial\dot{\phi}_2} + + \cdots \right\} - T. +\end{aligned} +\Tag{112} +\] +We can show, however, that the term in parenthesis is equal to~$u$. +If the \DPchg{coordinates}{coördinates} $x$,~$y$,~$z$ determine the position of the particle in +question, we have, +\begin{align*} +x &= f(\phi_1\phi_2\phi_3 \cdots \phi_n), \\ +\dot{x} = \frac{dx}{dt} + &= \dot{\phi}_1\, \frac{\partial f(\:)}{\partial\phi_1} + + \dot{\phi}_2\, \frac{\partial f(\:)}{\partial\phi_2} + + \dot{\phi}_3\, \frac{\partial f(\:)}{\partial\phi_3} + \cdots +\end{align*} +and differentiating with respect to the~$\dot{\phi}$'s, we obtain, +\[ +\frac{\partial\dot{x}}{\partial\dot{\phi}_1} + = \frac{\partial f(\:)}{\partial\phi_1} + = \frac{\partial x}{\partial\phi_1}, \quad +\frac{\partial\dot{x}}{\partial\dot{\phi}_2} + = \frac{\partial x}{\partial\phi_2}, \quad +\frac{\partial\dot{x}}{\partial\dot{\phi}_3} + = \frac{\partial x}{\partial\phi_3}, \quad \text{etc.}\DPtypo{,}{} +\] +%% -----File: 113.png---Folio 99------- +Similarly +\begin{alignat*}{3} +\frac{\partial\dot{y}}{\partial\dot{\phi}_1} + &= \frac{\partial y}{\partial\phi_1}, +&\qquad +\frac{\partial\dot{y}}{\partial\dot{\phi}_2} + &= \frac{\partial y}{\partial\phi_2}, &\qquad \text{etc.}, \\ +\frac{\partial\dot{z}}{\partial\dot{\phi}_1} + &= \frac{\partial z}{\partial\phi_1}, +&\qquad +\frac{\partial\dot{z}}{\partial\dot{\phi}_2} + &= \frac{\partial z}{\partial\phi_2}, &\qquad \text{etc.}, +\end{alignat*} +Let us write now +\begin{align*} +u &= \sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}, \\ +\frac{\partial u}{\partial\dot{\phi}_{\1}} + &= \frac{1}{\sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}} + \left(\dot{x}\, \frac{\partial\dot{x}}{\partial\dot{\phi}_{\1}} + + \dot{y}\, \frac{\partial\dot{y}}{\partial\dot{\phi}_{\1}} + + \dot{z}\, \frac{\partial\dot{z}}{\partial\dot{\phi}_{\1}}\right), +\end{align*} +or making the substitutions for $\dfrac{\partial\dot{x}}{\partial\dot{\phi}_{\1}}$, $\dfrac{\partial\dot{y}}{\partial\dot{\phi}_{\1}}$, etc., given above, we have, +\[ +\frac{\partial u}{\partial\dot{\phi}_{\1}} + = \frac{1}{u} + \left(\dot{x}\, \frac{\partial x}{\partial\phi_{\1}} + + \dot{y}\, \frac{\partial y}{\partial\phi_{\1}} + + \dot{z}\, \frac{\partial z}{\partial\phi_{\1}}\right). +\] +%%%% Use of "1" as a subscript in the original ends here %%%% +Substituting now in~(112) we shall obtain, +{\footnotesize% +\begin{align*} +T'& = \Sum mu +\begin{aligned}[t] +\Biggl\{\frac{\dot{x}}{u} + \left(\phi_1\, \frac{\partial x}{\partial\phi_1} + + \phi_2\, \frac{\partial x}{\partial\phi_2} + \cdots \right) + &+ \frac{\dot{y}}{u} + \left(\phi_1\, \frac{\partial y}{\partial\phi_1} + + \phi_2\, \frac{\partial y}{\partial\phi_2} + \cdots \right) \\ + &+ \frac{\dot{z}}{u} + \left(\phi_1\, \frac{\partial z}{\partial\phi_1} + + \phi_2\, \frac{\partial z}{\partial\phi_2} + \cdots \right) + \Biggr\} - T +\end{aligned} \\ + &= \Sum mu^2 - T +\end{align*}}% +or, introducing the value of~$T$ given by equation~(98), we have +\begin{align*} +T' &= \Sum \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} + \left\{u^2 - c^2 \sqrt{1 - \frac{u^2}{c^2}} + + c^2 \left(1 - \frac{u^2}{c^2}\right)\right\} \\ + &= \Sum c^2(m - m_0), +\end{align*} +which is the expression~(83) for kinetic energy. + +Hence we see that the Hamiltonian function $E = T' + U$ is the +sum of the kinetic and potential energies of the system as in Newtonian +mechanics. + + +\Subsection{The Principle of the Conservation of Energy.} + +\Paragraph{89.} We may now make use of our equations of motion in the +canonical form to show that the total energy of a system of interacting +%% -----File: 114.png---Folio 100------- +particles remains constant. If such were not the case it is obvious +that our definitions of potential and kinetic energy would not be +very useful. + +Since $E = T' + U$ is a function of $\phi_1$,~$\phi_2$, $\phi_3, \cdots$ $\psi_1$,~$\psi_2$, $\psi_3, \cdots$, we +may write +\begin{align*} +\frac{dE}{dt} + &= \frac{\partial E}{\partial\phi_1}\, \dot{\phi}_1 + + \frac{\partial E}{\partial\phi_2}\, \dot{\phi}_2 + \cdots \\ + &\quad + + \frac{\partial E}{\partial\psi_1}\, \dot{\psi}_1 + + \frac{\partial E}{\partial\psi_2}\, \dot{\psi}_2 + \cdots. +\end{align*} +Substituting the values of $\dfrac{\partial E}{\partial\phi_1}$, $\dfrac{\partial E}{\partial\psi_1}$, etc., given by the canonical +equations of motion (110)~and~(111), we have +\begin{align*} +\frac{dE}{dt} + &= -\dot{\psi}_1\dot{\phi}_1 - \dot{\psi}_2\dot{\phi}_2 - \cdots \\ + &\quad + + \dot{\psi}_1\dot{\phi}_1 + \dot{\psi}_2\dot{\phi}_2 + \cdots \\ + &= 0, +\end{align*} +which gives us the desired proof that just as in the older Newtonian +mechanics the total energy of an isolated system of particles is a +conservative quantity. + + +\Subsection{On the Location of Energy in Space.} + +\Paragraph{90.} This proof of the conservation of energy in a system of interacting +particles justifies us in the belief that the concept of energy +will not fail to retain in the newer mechanics the position of great +importance which it gradually acquired in the older systems of physical +theory. Indeed, our newer considerations have augmented the +important rôle of energy by adding to its properties the attribute of +mass or inertia, and thus leading to the further belief that matter +and energy are in reality different names for the same fundamental +entity. + +The importance of this entity, energy, makes it very interesting +to consider the possibility of ascribing a definite location in space to +any given quantity of energy. In the older mechanics we had a +hazy notion that the kinetic energy of a moving body was probably +located in some way in the moving body itself, and possibly a vague +%% -----File: 115.png---Folio 101------- +idea that the potential energy of a raised weight might be located in +the space between the weight and the earth. Our discovery of the +relation between mass and energy has made it possible, however, to +give a much more definite, although not a complete, answer to inquiries +of this kind. + +In our discussions of the dynamics of a particle (Chapter~VI, %[** TN: Not a useful cross-reference] +\Secref{61}) we saw that an acceptance of Newton's principle of the +equality of action and reaction forced us to ascribe an increased mass +to a moving particle over that which it has at rest. This increase in +the mass of the moving particle is necessarily located either in the +particle itself or distributed in the surrounding space in such a way +that its center of mass always coincides with the position of the +particle, and since the kinetic energy of the particle is the energy +corresponding to this increased mass we may say that \emph{the kinetic energy +of a moving particle is so distributed in space that its center of mass +always coincides with the position of the particle}. + +If now we consider the transformation of kinetic energy into +potential energy we can also draw somewhat definite conclusions as to +the location of potential energy. By the principle of the conservation +of mass we shall be able to say that the mass of any potential +energy formed is just equal to the ``kinetic'' mass which has disappeared, +and by the principle of the conservation of momentum we +can say that the velocity of this potential energy is just that necessary +to keep the total momentum of the system constant. Such considerations +will often permit us to reach a good idea as to the location +of potential energy. + +Consider, for example, a pair of similar attracting particles which +are moving apart from each other with the velocities $+u$~and~$-u$ +and are gradually coming to rest under the action of their mutual +attraction, their kinetic energy thus being gradually changed into +potential energy. Since the total momentum of the system must +always remain zero, we may think of the potential energy which is +formed as left stationary in the space between the two particles. +%% -----File: 116.png---Folio 102------- + + +\Chapter{VIII}{The Chaotic Motion of a System of Particles.} +\SetRunningHeads{Chapter Eight.}{Chaotic Motion of a System of Particles.} + +The discussions of the \Chapnumref[VII]{previous chapter} have placed at our disposal +generalized equations of motion for a system of particles similar in +form to those familiar in the classical mechanics, and differing only +in the definition of the Lagrangian function. With the help of these +equations it is possible to carry out investigations parallel to those +already developed in the classical mechanics, and in the present +chapter we shall discuss the chaotic motion of a system of particles. +This problem has received much attention in the classical mechanics +because of the close relations between the theoretical behavior of +such an ideal system of particles and the actual behavior of a monatomic +gas. We shall find no more difficulty in handling the problem +than was experienced in the older mechanics, and our results will of +course reduce to those of Newtonian mechanics in the case of slow +velocities. Thus we shall find a distribution law for momenta which +reduces to that of Maxwell for slow velocities, and an equipartition +law for the average value of a function which at low velocities becomes +identical with the kinetic energy of the particles. + +\Subsubsection{91}{The Equations of Motion.} It has been shown that the Hamiltonian +equations of motion +\[ +\begin{aligned} +&\frac{\partial E}{\partial\phi_1} = -\frac{d\psi_1}{dt} = -\dot{\psi}_1, \\ +&\frac{\partial E}{\partial\psi_1} = \frac{d\phi_1}{dt} = \dot{\phi}_1, \\ +&\text{etc.}, +\end{aligned} +\Tag{113} +\] +will hold in relativity mechanics provided we define the generalized +momenta $\psi_1$,~$\psi_2$,~etc., \emph{not} as the differential of the kinetic energy +with respect to the generalized velocities $\dot{\phi}_1$,~$\dot{\phi}_2$,~etc., but as the differential +with respect to $\dot{\phi}_1$,~$\dot{\phi}_2$,~etc., of a function +\[ +T = \Sum m_0c^2 \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right), +\] +%% -----File: 117.png---Folio 103------- +where $m_0$~is the mass of a particle having the velocity~$u$ and the summation~$\Sum$ +extends over all the particles of the system. + +\Subsubsection{92}{Representation in Generalized Space.} Consider now a system +defined by the $n$~generalized coördinates $\phi_1$,~$\phi_2$, $\phi_3, \cdots, \phi_n$, and the +corresponding momenta $\psi_1$,~$\psi_2$, $\psi_3, \cdots, \psi_n$. Employing the methods +so successfully used by Jeans,\footnote + {Jeans, \textit{The Dynamical Theory of Gases}, Cambridge, 1916.} +we may think of the state of the +system at any instant as determined by the position of a point plotted +in a $2n$-dimensional space. Suppose now we had a large number of +systems of the same structure but differing in state, then for each +system we should have at any instant a corresponding point in our +$2n$-dimensional space, and as the systems changed their state, in the +manner required by the laws of motion, the points would describe +stream lines in this space. + +\Subsubsection{93}{Liouville's Theorem.} Suppose now that the points were +originally distributed in the generalized space with the uniform +density~$\rho$. Then it can be shown by familiar methods that, just as +in the classical mechanics, the density of distribution remains uniform. + +Take, for example, some particular cubical element of our generalized +space $d\phi_1\, d\phi_2\, d\phi_3 \dots d\psi_1\, d\psi_2\, d\psi_3\dots$. The density of distribution +will evidently remain uniform if the number of points +entering any such cube per second is equal to the number leaving. +Consider now the two parallel bounding surfaces of the cube which +are perpendicular to the $\phi_1$~axis, one cutting the axis at the point~$\phi_1$ +and the other at the point~$\phi_1 + d\phi_1$. The area of each of these +surfaces is $d\phi_2\, d\phi_3\dots d\psi_1\, d\psi_2\, d\psi_3\dots$, and hence, if $\dot{\phi}_1$~is the component +of velocity which the points have parallel to the $\phi_1$~axis, and $\dfrac{\partial\dot{\phi}_1}{\partial\phi_1}$~is +the rate at which this component is changing as we move along the +axis, we may obviously write the following expression for the difference +between the number of points leaving and entering per second +through these two parallel surfaces +\[ +\rho\left[\left(\frac{\partial\dot{\phi}_1}{\partial\phi_1}\right) d\phi_1\right] +d\phi_2\, d\phi_3\, \dots d\psi_1\, d\psi_2\, d\psi_3 \cdots + = \rho\, \frac{\partial\dot{\phi}_1}{\partial\phi_1}\, dV. +\] + +Finally, considering all the pairs of parallel bounding surfaces, we +%% -----File: 118.png---Folio 104------- +find for the total decrease per second in the contents of the element +\[ +\rho\left( + \frac{\partial\dot{\phi}_1}{\partial\phi_1} + + \frac{\partial\dot{\phi}_2}{\partial\phi_2} + + \frac{\partial\dot{\phi}_3}{\partial\phi_3} + \cdots + + \frac{\partial\dot{\psi}_1}{\partial\psi_1} + + \frac{\partial\dot{\psi}_2}{\partial\psi_2} + + \frac{\partial\dot{\psi}_3}{\partial\psi_3} + \cdots\right) dV. +\] +But the motions of the points are necessarily governed by the Hamiltonian +equations~(113) given above, and these obviously lead to the +relations +\begin{align*} +& \frac{\partial\dot{\phi}_1}{\partial\phi_1} ++ \frac{\partial\dot{\psi}_1}{\partial\psi_1} = 0, \\ +& \frac{\partial\dot{\phi}_2}{\partial\phi_2} ++ \frac{\partial\dot{\psi}_2}{\partial\psi_2} = 0\DPtypo{.}{,} \\ +& \text{etc.} +\end{align*} +So that our expression for the change per second in the number of +points in the cube becomes equal to zero, the necessary requirement +for preserving uniform density. + +This maintenance of a uniform distribution means that there is +no tendency for the points to crowd into any particular region of the +generalized space, and hence if we start some one system going and +plot its state in our generalized space, we may \emph{assume} that, after an +indefinite lapse of time, the point is equally likely to be in any one of +the little elements~$dV$. \emph{In other words, the different states of a system, +which we can specify by stating the region $d\phi_1\, d\phi_2\, d\phi_3 \dots d\psi_1\, d\psi_2\, d\psi_3 \dots$ +in which the values of the \DPchg{coordinates}{coördinates} and momenta of the system fall, +are all equally likely to occur.}\footnote + {The criterion here used for determining whether or not the states are equally + liable to occur is obviously a necessary requirement, although it is not so evident + that it is a sufficient requirement for equal probability.} + +\Subsubsection{94}{A System of Particles.} Consider now a system containing $N_a$~particles +which have the mass~$m_a$ when at rest, $N_b$~particles which +have the mass~$m_b$, $N_c$~particles which have the mass~$m_c$, etc. If at +any given instant we specify the particular differential element +$dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$ which contains the coördinates $x$,~$y$,~$z$, and the +corresponding momenta $\psi_x$,~$\psi_y$,~$\psi_z$ for \emph{each} particle, we shall thereby +completely determine what Planck\footnote + {Planck, \textit{Wärmestrahlung}, Leipzig, 1913.} +has well called the \emph{microscopic} +state of the system, and by the previous paragraph any microscopic +%% -----File: 119.png---Folio 105------- +state of the system in which we thus specify the six-dimensional +position of each particle is just as likely to occur as any other microscopic +state. + +It must be noticed, however, that many of the possible microscopic +states which are determined by specifying the six-dimensional +position of each individual particle are in reality completely identical, +since if all the particles having a given mass~$m_a$ are alike among themselves, +it makes no difference which particular one of the various +available identical particles we pick out to put into a specified range +$dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$. + +For this reason we shall usually be interested in specifying the +\emph{statistical} state\footnote + {What we have here defined as the \emph{statistical} state is what Planck calls the + \emph{macroscopic} state of the system. The word macroscopic is unfortunate, however, in + implying a less minute observation as to the size of the elements $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$ +in which the representative points are found.} +of the system, for which purpose we shall merely +state the number of particles of a given kind which have coördinates +falling in a given range $dx\, dy\,dz\, d\psi_x\, d\psi_y\, d\psi_z$. We see that corresponding +to any given statistical state there will be in general a +large number of microscopic states. + +\Subsubsection{95}{Probability of a Given Statistical State.} We shall now be +particularly interested in the probability that the system of particles +will actually be in some specified \emph{statistical} state, and since Liouville's +theorem has justified our belief that all \emph{microscopic} states are +equally likely to occur, we see that the probability of a given statistical +state will be proportional to the number of microscopic states +which correspond to it. + +For the system under consideration let a particular statistical +state be specified by stating that ${N_a}'$,~${N_a}''$, ${N_a}''', \cdots$, ${N_b}'$,~${N_b}''$, ${N_b}''', \cdots$,~etc., are the number of particles of the corresponding masses +$m_a$,~$m_b$,~etc., which fall in the specified elementary regions $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$, Nos.\ $1a$,~$2a$, $3a, \cdots$, $1b$,~$2b$, $3b, \cdots$,~etc. By familiar +methods of calculation it is evident that the number of arrangements +by which the particular distribution of particles can be effected, +that is, in other words, the number of microscopic states,~$W$, which +correspond to the given statistical state, is given by the expression +\[ +%[** TN: Modernized factorial notation] +W = \frac{N_a!\, N_b!\, N_c! \cdots} + {{N_a}'!\, {N_a}''!\, {N_a}'''! \cdots + {N_b}'!\, {N_b}''!\, {N_b}'''! \cdots} +\] +%% -----File: 120.png---Folio 106------- +and this number~$W$ is proportional to the probability that the system +will be found in the particular statistical state considered. + +If now we assume that each of the regions +\[ +dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z,\ +\text{Nos.}\ 1a,\ 2a,\ 3a,\ \cdots,\ 1b,\ 2b,\ 3b,\ \cdots\ \text{etc.} +\] +is great enough to contain a large number of particles,\footnote + {The idea of successive orders of infinitesimals which permit the differential + region $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$, to contain a large number of particles is a familiar one in + mathematics.} +we may +apply the Stirling formula +\[ +N! = \sqrt{2\pi\, N} \left(\frac{N}{\epsilon}\right)^N +\] +for evaluating $N_a!$,~$N_b!$,~etc., and omitting negligible terms, shall +obtain for~$\log W$ the result +\begin{align*} +\log W &= -N_a \left( + \frac{{N_a}'}{N_a} \log\frac{{N_a}'}{N_a} + + \frac{{N_a}''}{N_a} \log\frac{{N_a}''}{N_a} + + \frac{{N_a}'''}{N_a} \log\frac{{N_a}'''}{N_a} + \cdots\right) \\ + &\quad -N_b\left( + \frac{{N_b}'}{N_b} \log\frac{{N_b}'}{N_b} + + \frac{{N_b}''}{N_b} \log\frac{{N_b}''}{N_b} + + \frac{{N_b}'''}{N_b} \log\frac{{N_b}'''}{N_b} + \cdots\right),\\ + &\quad\text{etc.} +\end{align*} + +For simplicity let us denote the ratios $\dfrac{{N_a}'}{N_a}$, $\dfrac{{N_a}''}{N_a}$,~etc., by the +symbols ${w_a}'$, ${w_a}''$,~etc. These quantities ${w_a}'$, ${w_a}''$,~etc., are evidently +the probabilities, in the case of this particular statistical state, +that any given particle~$m_a$ will be found in the respective regions +Nos.\ $1a$,~$2a$,~etc. + +We may now write +\[ +\log W = -N_a\Sum w_a\log w_a - N_b\Sum w_b\log w_b -{}, \text{ etc.}, +\] +where the summation extends over all the regions Nos.\ $1a$,~$2a$, $\cdots +1b$, $2b$,~etc. + +\Subsubsection{96}{Equilibrium Relations.} Let us now suppose that the system +of particles is contained in an enclosed space and has the definite +energy content~$E$. Let us find the most probable distribution of the +particles. For this the necessary condition will be +\begin{multline*} +\delta\log W = -N_a\Sum (\log w_a + 1)\, \delta w_a \\ + -N_b\Sum (\log w_b + 1)\, \delta w_b \cdots = 0. +\Tag{114} +\end{multline*} +In carrying out our variation, however, the number of particles of +%% -----File: 121.png---Folio 107------- +each kind must remain constant so that we have the added relations +\[ +\Sum \delta w_a=0, \qquad +\Sum \delta w_b=0, \qquad \text{etc.} +\Tag{115} +\] +Finally, since the energy is to have a definite value~$E$, it must also +remain constant in the variation, which will provide still a further +relation. Since the energy of a particle will be a definite function of +its position and momentum,\footnote + {We thus exclude from our considerations systems in which the potential energy + depends appreciably on the \emph{relative} positions of the independent particles.} +let us write the energy of the system +in the form +\[ +E = N_a \Sum w_a E_a + N_b \Sum w_b E_b + \cdots, +\] +where $E_a$ is the energy of a particle in the region $1a$,~etc. + +Since in carrying out our variation the energy is to remain constant, +we have the relation +\[ +E = N_a \Sum E_a\, \delta w_a + + N_b \Sum E_b\, \delta w_b + \cdots = 0. +\Tag{116} +\] + +Solving the simultaneous equations (114),~(115),~(116) by familiar +methods we obtain +\begin{align*} +&\log w_a + 1 + \lambda E_a + \mu_b = 0, \\ +&\log w_b + 1 + \lambda E_b + \mu_b = 0, \\ +&\text{etc.}, +\end{align*} +where $\lambda$,~$\mu_a$, $\mu_b$,~etc., are undetermined constants. (It should be +specially noticed that $\lambda$~is the same constant in each of the series of +equations.) + +Transforming we have +\[ +\begin{aligned} +& w_a = \alpha_a\, e^{-hE_a}, \\ +& w_b = \alpha_b\, e^{-hE_b}, \\ +& \text{etc.}, +\end{aligned} +\Tag{117} +\] +as the expressions which determine the chance that a given particle +of mass $m_a$,~$m_b$,~etc., will fall in a given region $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$, +when we have the distribution of maximum probability. It should +be noticed that~$h$, which corresponds to the~$\lambda$ of the preceding equations, +is the same constant in all of the equations, while $\alpha_a$,~$\alpha_b$,~etc., +are different constants, depending on the mass of the particles $m_a$,~$m_b$,~etc. +%% -----File: 122.png---Folio 108------- + +\Subsubsection{97}{The Energy as a Function of the Momentum.} $E_a$,~$E_b$,~etc., +are of course functions of $x$,~$y$,~$z$, $\psi_x$,~$\psi_y$,~$\psi_z$. Let us now obtain an +expression for~$E_a$ in terms of these quantities. If there is no external +field of force acting, the energy of a particle~$E_a$ will be independent +of $x$,~$y$, and~$z$, and will be determined entirely by its velocity and +mass. In accordance with the theory of relativity we shall have\footnote + {This expression is that for the total energy of the particle, including that internal energy~$m_0 c^2$ + which, according to relativity theory, the particle has when it is at rest. (See \Secref{75}.) + It would be just as correct to substitute for~$E_a$ in equation~(117) the value of the kinetic energy + $m_a c^2 \Biggl(\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}- 1 \Biggr)$ + instead of the total energy $\dfrac{m_ac^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, + since the two differ merely by a constant~$m_a c^2$ which would be taken care of by assigning a suitable value to~$\alpha_a$.} +\[ +E_a = \frac{m_ac^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\Tag{118} +\] +where $m_a$ is the mass of the particle at rest. + +Let us now express $E_a$ as a function of $\psi_x$,~$\psi_y$,~$\psi_z$. + +We have from our equations (105)~and~(98), which were used for +defining momentum +\begin{align*} +\psi_x + &= \frac{\partial}{\partial\dot{x}}\, m_a + \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right)\displaybreak[0] \\ + &= \frac{\partial}{\partial\dot{x}}\, m_a + \left(1 - \sqrt{1 - \frac{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}{c^2}}\right)\displaybreak[0] \\ + &= \frac{m_0\dot{x}}{\sqrt{1 - \smfrac{u^2}{c^2}}}. +\end{align*} +Constructing the similar expressions for $\psi_y$~and~$\psi_z$ we may write the +relation +\[ +\psi^2 = \psi_x^2 + \psi_y^2 + \psi_z^2 + = \frac{m^2_a (\dot{x}^2 + \dot{y}^2 + \dot{z}^2)}{1 - \smfrac{u^2}{c^2}} + = \frac{m^2_au^2}{1 - \smfrac{u^2}{c^2}}, +\Tag{119} +\] +which also defines~$\psi^2$. +%% -----File: 123.png---Folio 109------- + +By simple transformations and the introduction of equation~(118) +we obtain the desired relation +\[ +E_a = c\sqrt{\psi^2 + {m_a}^2c^2}. +\Tag{120} +\] + +\Subsubsection{98}{The Distribution Law.} We may now rewrite equations~(117) +in the form +\[ +\begin{aligned} +& w_a = \alpha_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}, \\ +& w_b = \alpha_b\, e^{-hc\sqrt{\psi^2 + {m_b}^2c^2}}, \\ +& \text{etc.} +\end{aligned} +\Tag{121} +\] + +These expressions determine the probability that a given particle +of mass $m_a$,~$m_b$,~etc.\DPtypo{}{,} will fall in a given region $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$, and +correspond to Maxwell's distribution law in ordinary mechanics. We +see that these probabilities are independent of the position $x$,~$y$,~$z$\footnote + {This is true only when, as assumed, no external field of force is acting.} +but dependent on the momentum. + +$\alpha_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}$ is the probability that a given particle will fall in a +particular six-dimensional cube of volume $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$. Let us +now introduce, for convenience, a new quantity $a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}$ which +will be the probability per \emph{unit} volume that a given particle will have +the six dimensional location in question, the constants $\alpha_a$~and~$a_a$ +standing in the same ratio as the volumes $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$ and unity. + +We may then write +\begin{alignat*}{2} +w_a &= \alpha_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}} & + &= a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, + dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z \\ +w_b &= \alpha_b\, e^{-hc\sqrt{\psi^2 + {m_b}^2c^2}} & + &= a_b\, e^{-hc\sqrt{\psi^2 + {m_b}^2c^2}}\, + dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z +\end{alignat*} +etc. + +Since every particle must have components of momentum lying +between minus and plus infinity, and lie somewhere in the whole +volume~$V$ occupied by the mixture, we have the relation +\[ +V \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} + a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, d\psi_x\, d\psi_y\, d\psi_z = 1. +\Tag{122} +\] + +It is further evident that the average value of any quantity~$A$ +which depends on the momentum of the particles is given by the +%% -----File: 124.png---Folio 110------- +expression +\[ +[A]_{\text{av.}} + = V \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} + a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\,A \, d\psi_x\, d\psi_y\, d\psi_z, +\Tag{123} +\] +where $A$ is some function of $\psi_x$,~$\psi_y$, and~$\psi_z$. + +\Subsubsection{99}{Polar Coördinates.} We may express relations corresponding +to (122)~and~(123) more simply if we make use of polar coördinates. +Consider instead of the elementary volume $d\psi_x\, d\psi_y\, d\psi_z$ the volume +$\psi^2\sin\theta\, d\theta\, d\phi\, d\psi$ expressed in polar coördinates, where +\[ +\psi^2 = {\psi_x}^2 + {\psi_y}^2 + {\psi_z}^2. +\] + +The probability that a particle~$m_a$ will fall in the region +\[%[** TN: Displaying to avoid bad line break] +dx\, dy\, dz\, \psi^2 \sin\theta\, d\theta\, d\phi\, d\psi +\] +will be +\[ +a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, + dx\, dy\, dz\, \psi^2 \sin\theta\, d\theta\, d\phi\, d\psi, +\] +and since each particle must fall somewhere in the space $x\:y\:z\: \psi_x\: \psi_y\: \psi_z$ +we shall have corresponding to~(122) the relation +\[ +\begin{gathered} +V \int_{0}^{\pi}\int_{0}^{2\pi}\int_{0}^{\infty} + a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, \psi^2 \sin\theta\, + d\theta\, d\phi\, d\psi = 1, \\ +4\pi V \int_{0}^{\infty} a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, \psi^2\, d\psi = 1. +\end{gathered} +\Tag{124} +\] +Corresponding to equation~(123), we also see that the average value +of any quantity~$A$, which is dependent on the momentum of the +molecules of mass~$m_a$, will be given by the expression +\[ +[A]_{\text{av.}} + = 4\pi V \int_{0}^{\infty} a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, + A\, \psi^2\, d\psi. +\Tag{125} +\] + +\Subsubsection{100}{The Law of Equipartition.} We may now obtain a law which +corresponds to that of the equipartition of \textit{vis~viva} in the classical +mechanics. Considering equation~(124) let us integrate by parts, we +obtain +\begin{multline*} +\left[ 4\pi V a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, \frac{\psi^3}{3}\right]_{\psi=0}^{\psi=\infty} \\ +-4\pi V\int_0^{\infty} \frac{\psi^3}{3}\, a_a\, + e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\,(-hc)\, + \frac{\psi}{\sqrt{\psi^2 + {m_a}^2c^2}}\, d\psi = 1. +\end{multline*} +%% -----File: 125.png---Folio 111------- +Substituting the limits into the first term we find that it becomes +zero and may write +\[ +4\pi V\int_0^{\infty} a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, + \frac{\psi^2c}{\sqrt{\psi^2 + {m_a}^2c^2}}\, \psi^2\, d\psi = \frac{3}{h}. +\] + +But by equation~(125) the left-hand side of this relation is the +average value of $\dfrac{\psi^2c}{\sqrt{\psi^2 + {m_a}^2c^2}}$ for the particles of mass~$m_a$. We have +\[ +\left[ \frac{\psi^2c}{\sqrt{\psi^2 + {m_a}^2c^2}}\right]_{\text{av.}} + = \frac{3}{h}. +\] +Introducing equation~(119) which defines~$\psi^2$, we may transform this +expression into +\[ +\Biggl[\frac{m_au^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggl]_{\text{av.}} + = \frac{3}{h}. +\Tag{126} +\] + +Since we have shown that $h$~is independent of the mass of the +particles, \emph{we see that the average value of $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is the same for particles +of all different masses}. This is the principle in relativity mechanics +that corresponds to the law of the equipartition of \textit{vis~viva} in the +classical mechanics. Indeed, for low velocities the above expression +reduces to~$m_0 u^2$, the \textit{vis~viva} of Newtonian mechanics, a fact which +affords an illustration of the general principle that the laws of Newtonian +mechanics are always the limiting form assumed at low velocities +by the more exact formulations of relativity mechanics. + +We may now call attention in passing to the fact that this quantity +$\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, whose value is the same for particles of different masses, is +not the relativity expression for kinetic energy, which is given rather +by the formula $c^2\Biggl[\dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} - m_0\Biggr]$. So that in relativity mechanics +%% -----File: 126.png---Folio 112------- +the principle of the equipartition of energy is merely an approximation. +We shall later return to this subject. + +\Subsubsection{101}{Criterion for Equality of Temperature.} For a system of particles +of masses $m_a$,~$m_b$,~etc., enclosed in the volume~$V$, and having the +definite energy content~$E$, we have shown that +\[ +4\pi V\, a_a\, e^{-hc\sqrt{\psi^2+{m_a}^2c^2}}\, \psi^2\, d\psi +\] +and +\[ +4\pi V\, a_b\, e^{-hc\sqrt{\psi^2+{m_b}^2c^2}}\, \psi^2\, d\psi +\] +are the respective probabilities that given particles of mass~$m_a$ or +mass~$m_b$ will have momenta between $\psi$~and~$\psi + d\psi$. Suppose now +we consider a differently arranged system in which we have $N_a$~particles +of mass~$m_a$ by themselves in a space of volume~$V_a$ and $N_b$~particles +of mass~$m_b$ in a contiguous space of volume~$V_b$, separated +from~$V_a$ by a partition which permits a transfer of energy, and let +the total energy of the double system be, as before, a definite quantity~$E$ +(the energy content of the partition being taken as negligible). +Then, by reasoning entirely similar to that just employed, we can +obviously show that +\[ +4\pi V_a\, a_a\, e^{-hc\sqrt{\psi^2+{m_a}^2c^2}}\, \psi^2\, d\psi +\] +and +\[ +4\pi V_b\, a_b\, e^{-hc\sqrt{\psi^2+{m_b}^2c^2}}\, \psi^2\, d\psi +\] +are now the respective probabilities that given particles of mass~$m_a$ +or mass~$m_b$ will have momenta between $\psi$~and~$\psi + d\psi$, the only +changes in the expressions being the substitution of the volumes +$V_a$~and~$V_b$ in the place of the one volume~$V$. Furthermore, this +distribution law will evidently lead as before to the equality of the +average values of +\[ +\frac{m_au^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\qquad \text{and}\qquad +\frac{m_bu^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}. +\] +Since, however, the spaces containing the two kinds of particles are in +thermal contact, their temperature is the same. Hence we find that +\emph{the equality of the average values of $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is the necessary condition for +equality of temperature}. +%% -----File: 127.png---Folio 113------- + +\emph{The above distribution law also leads to the important corollary that +for any given system of particles at a definite temperature the momenta +and hence the total energy content is independent of the volume.} + +We may now proceed to the derivation of relations which will +permit us to show that the important quantity $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is directly +proportional to the temperature as measured on the absolute thermodynamic +temperature scale. + +\Subsubsection{102}{Pressure Exerted by a System of Particles.} We first need +to obtain an expression for the pressure exerted by a system of $N$~particles +enclosed in the volume~$V$. Consider an element of surface~$dS$ +perpendicular to the $X$~axis, and let the pressure acting on it be~$p$. +The total force which the element~$dS$ exerts on the particles that +impinge will be~$p\, dS$, and this will be equal to the rate of change of +the momenta in the $X$~direction of these particles.\footnote + {The system is considered dilute enough for the mutual attractions of the + particles to be negligible in their effect on the external pressure.} + +Now by equation~(122) the total number of particles having +momenta between $\psi_x$~and~$\psi_x + d\psi_x$ in the \emph{positive} direction is +\[ +NV \int_{\psi_x}^{\psi_x + d\psi_x}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} + a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, d\psi_x\, d\psi_y\, d\psi_z. +\] +But $\dot{x}\, dS$ gives us the volume which contains the number of particles +having momenta between $\psi_x$~and~$\psi_x + d\psi_x$ which will reach~$dS$ in a +second. Hence the number of such particles which impinge per +second will be +\[ +NV\, \frac{\dot{x}\, dS}{V} + \int_{\psi_x}^{\psi_x + d\psi_x}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} + a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, d\psi_x\, d\psi_y\, d\psi_z. +\] +and their change in momentum, allowing for the effect of the rebound, +will be +\[ +2N\, dS \int_{\psi_x}^{\psi_x + d\psi_x}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} + a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi_x\, \dot{x}\, d\psi_x\, d\psi_y\, d\psi_z. +\] +Finally, the total change in momentum per second for all particles +can be found by integrating for all possible positive values of~$\psi_x$. +%% -----File: 128.png---Folio 114------- +Equating this to the total force~$p\, dS$ we have +\[ +p\, dS = 2N\, dS + \int_{0}^{\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} + a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi_x\, \dot{x}\, d\psi_x\, d\psi_y\, d\psi_z. +\] +Cancelling~$dS$, multiplying both sides of the equation by the volume~$V$, +changing the limits of integration and substituting $\dfrac{m_0\dot{x}}{\sqrt{1 + \smfrac{u^2}{c^2}}}$ for~$\psi_x$, +we have +\[ +pV = NV + \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} + a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, + \frac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, d\psi_x\, d\psi_y\, d\psi_z. +\] +But this by equation~(123) reduces to +\[ +pV = N \Biggl[\frac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}} +\] +or, since +\[ +\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = \frac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}} + + \frac{m_0\dot{y}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}} + + \frac{m_0\dot{z}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +we have from symmetry +\[ +pV = \frac{N}{3}\Biggl[\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Bigg]_{\text{av.}}. +\Tag{127} +\] +Since at a given temperature we have seen that the term in parenthesis +is independent of the volume and the nature of the particles, we see +that the laws of Boyle and Avogadro hold also in relativity mechanics +for a system of particles. + +For slow velocities equation~(127) reduces to the familiar expression +$pV = \dfrac{N}{3}\, (m_0u^2)_{\text{av.}}$. + +\Subsubsection{103}{The Relativity Expression for Temperature.} We are now in +a position to derive the relativity expression for temperature. The +thermodynamic scale of temperature may be defined in terms of the +efficiency of a heat engine. Consider a four-step cycle performed +with a working substance contained in a cylinder provided with a +piston. In the first step let the substance expand isothermally and +%% -----File: 129.png---Folio 115------- +reversibly, absorbing the heat~$Q_2$ from a reservoir at temperature~$T_2$; +in the second step cool the cylinder down at constant volume to~$T_1$; +in the third step compress to the original volume, giving out the +heat~$Q_1$ at temperature~$T_1$, and in the fourth step heat to the original +temperature. Now if the working substance is of such a nature that +the heat given out in the second step could be used for the \emph{reversible} +heating of the cylinder in the fourth step, we may define the absolute +temperatures $T_2$~and~$T_1$ by the relation $\dfrac{T_2}{T_1} = \dfrac{Q_2}{Q_1}$.\footnote + {We have used this cycle for defining the thermodynamic temperature scale + instead of the familiar Carnot cycle, since it avoids the necessity of obtaining an + expression for the relation between pressure and volume in an adiabatic expansion.} + +Consider now such a cycle performed on a cylinder which contains +one of our systems of particles. Since we have shown (\Secref{101}) +that at a definite temperature the energy content of such a +system is independent of the volume, it is evident that our working +substance fulfils the requirement that the heat given out in the second +step shall be sufficient for the reversible heating in the last step. +Hence, in accordance with the thermodynamic scale, we may measure +the temperatures of the two heat reservoirs by the relation $\dfrac{T_2}{T_1} = \dfrac{Q_2}{Q_1}$ +and may proceed to obtain expressions for $Q_2$~and~$Q_1$. + +In order to obtain these expressions we may again make use of the +principle that the energy content at a definite temperature is independent +of the volume. This being true, we see that $Q_2$~and~$Q_1$ +must be equal to the work done in the changes of volume that take +place respectively at $T_2$~and~$T_1$, and we may write the relations +\begin{align*} +Q_2 &= \int_V^{V'} p\, dV\quad \text{(at $T_2$)}, \\ +Q_1 &= \int_V^{V'} p\, dV\quad \text{(at $T_1$)}. +\end{align*} +But equation~(127) provides an expression for~$p$ in terms of~$V$, leading +on integration to the relations +\begin{align*} +Q_2 &= \frac{N}{3}\Biggl[ + \frac{m_0{u_2}^2}{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}} + \Biggr]_{\text{av.}} \log\frac{V'}{V}, \\ +%% -----File: 130.png---Folio 116------- +Q_1 &= \frac{N}{3}\Biggl[ + \frac{m_0{u_1}^2}{\sqrt{1 - \smfrac{{u_1}^2}{c^2}}} + \Biggl]_{\text{av.}} \log\frac{V'}{V}, +\end{align*} +which gives us on division +\[ +\frac{T_2}{T_1} = \frac{Q_2}{Q_1} + = \frac{\Biggl[\smfrac{m_0{u_2}^2}{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}}\Biggr]_{\text{av.}}} + {\Biggl[\smfrac{m_0{u_1}^2}{\sqrt{1 - \smfrac{{u_1}^2}{c^2}}}\Biggr]_{\text{av.}}}. +\] + +\emph{We see that the absolute temperature measured on the thermodynamic +scale is proportional to the average value of $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$.} + +We may finally express our temperature in the same units customarily +employed by comparing equation~(127) +\[ +pV = \frac{N}{3}\Biggl[\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}}, +\] +with the ordinary form of the gas law +\[ +pV = nRT, +\] +where $n$~is the number of mols of gas present. + +We evidently obtain +\[ +\begin{aligned} +nRT &= \frac{N}{3} \Biggl[\frac{m_0 u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}},\\ +T &= \frac{N}{3nR}\Biggl[\frac{m_0 u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}} + = \frac{1}{3k} \Biggl[\frac{m_0 u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}}, +\end{aligned} +\Tag{128} +\] +where the quantity $\dfrac{nR}{N}$, which may be called the gas constant for a +single molecule, has been denoted, as is customary, by the letter~$k$. +%% -----File: 131.png---Folio 117------- +Remembering the relation $\Biggl[\dfrac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}} = \dfrac{3}{h}$, we have +\[ +kT = \frac{1}{h}. +\Tag{129} +\] + +\Subsubsection{104}{The Partition of Energy.} We have seen that our new equipartition +law precludes the possibility of an exact equipartition of +energy. It becomes very important to see what the average energy +of a particle of a given mass does become at any temperature. + +Equation~(125) provides a general expression for the average value +of any property of the particles. For the average value of the energy +$c\sqrt{\psi^2 + {m_0}^2c^2}$ of particles of mass~$m_0$ (see equation~120) we shall have +\[ +[E]_{\text{av.}} + = 4\pi V \int_0^{\infty} a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, + c\, \sqrt{\psi^2 + {m_0}^2c^2}\, \psi^2\, d\psi. +\] +The unknown constant~$a$ may be eliminated with the help of the +relation~(124) +\[ +4\pi V \int_0^{\infty} a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi^2\, d\psi = 1 +\] +and for~$h$ we may substitute the value given by~(129), which gives us +the desired equation +\[ +[E]_{\text{av.}} + = \frac{\ds\int_0^{\infty} e^{-(c/kT)\sqrt{\psi^2 + {m_0}^2c^2}}\, c\, \sqrt{\psi^2 + {m_0}^2c^2}\, \psi^2\, d\psi} + {\ds\int_0^{\infty} e^{-(c/kT)\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi^2\, d\psi}. +\Tag{130} +\] + +\Subsubsection{105}{Partition of Energy for Zero Mass.} Unfortunately, no general +method for the evaluation of this expression seems to be available. +For the particular case that the mass~$m_0$ of the particles approaches +zero compared to the momentum, the expression reduces to +\[ +[E]_{\text{av.}} + = \frac{c\ds\int_0^{\infty} e^{-(c\psi /kT)}\, \psi^3\, d\psi} + { \ds\int_0^{\infty} e^{-(c\psi /kT)}\, \psi^2\, d\psi} +\] +%% -----File: 132.png---Folio 118------- +in terms of integrals whose values are known. Evaluating, we obtain +\[ +[E]_{\text{av.}} = 3kT. +\] +For the total energy of $N$ such particles we obtain +\[ +E = 3NkT, +\] +and introducing the relation $k = \dfrac{nR}{N}$ by which we defined~$k$ we have +\[ +E = 3nRT +\Tag{131} +\] +as the expression for the energy of $n$~mols of particles if their value of~$m_0$ +is small compared with their momentum. + +It is instructive to compare this with the ordinary expression of +Newtonian mechanics +\[ +E = \frac{3}{2}\, nRT, +\] +which undoubtedly holds when the masses are so large and the velocities +so small that no appreciable deviations from the laws of Newtonian +mechanics are to be expected. We see that for particles of +very small mass the average kinetic energy at any temperature is +twice as large as that for large particles at the same temperature. +It is also interesting to note that in accordance with equation~(131) +a mol of particles which approach zero mass at the absolute zero, +would have a mass of +\[ +\frac{3 × 8.31 × 10^{7} × 300}{10^{21}} = 7.47 × 10^{-11} +\] +grams at room temperature ($300°$~absolute). This suggests a field +of fascinating if profitless speculation. + +%[** TN: ToC entry reads "Approximate Partition for Particles of any Mass"] +\Subsubsection{106}{Approximate Partition of Energy for Particles of any Desired +Mass.} For particles of any desired mass we may obtain an approximate +idea of the relation between energy and temperature by expanding +the expression for kinetic energy into a series. For the average +kinetic energy of a particle we have +\[ +[K]_{\text{av.}} + = c^2\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} - m_0\Biggr]_{\text{av.}}. +\] +%% -----File: 133.png---Folio 119------- +Expanding into a series we obtain for the total kinetic energy of $N$~particles +\[ +K = Nm_0\left( + \frac{1}{2}\, \vc{u}^2 + \frac{3}{8}\, \frac{\vc{u}^4}{c^2} + + \frac{15}{48}\, \frac{\vc{u}^6}{c^4} + + \frac{105}{384}\, \frac{\vc{u}^8}{c^6} + \cdots\right), +\Tag{132} +\] +where $\vc{u}^2$, $\vc{u}^4$,~etc., are the average values of $u^2$,~$u^4$,~etc., for the individual +particles. + +To determine approximately how the value of~$K$ varies with the +temperature we may also expand our expression~(128) for temperature, +\[ +T = \frac{1}{3k}\Biggl[\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{Av.}}, +\] +into a series; we obtain +{\small% +\[ +\frac{3}{2}\, kNT = \frac{3}{2}\, nRT + = Nm_0 \left( + \frac{1}{2}\, \vc{u}^2 + \frac{1}{4}\, \frac{\vc{u}^4}{c^2} + + \frac{3}{16}\, \frac{\vc{u}^6}{c^4} + + \frac{15}{96}\, \frac{\vc{u}^8}{c^6} + \cdots\right). +\Tag{133} +\]}% +Combining expressions (132)~and~(133) by subtraction and transposition, +we obtain +\[ +K = \frac{3}{2}\, nRT + + Nm_0 \left( + \frac{1}{8}\, \frac{\vc{u}^4}{c^2} + + \frac{1}{8}\, \frac{\vc{u}^6}{c^4} + + \frac{15}{128}\, \frac{\vc{u}^8}{c^6} + \cdots\right). +\Tag{134} +\] +For the case of velocities low enough so that $\vc{u}^4$~and higher powers +can be neglected, this reduces to the familiar expression of Newtonian +mechanics,~$K = \dfrac{3}{2}\, nRT$. + +In case we neglect in expression~(134) powers higher than~$\vc{u}^4$ we +have the approximate relation +\[ +\frac{Nm_0 \vc{u}^4}{8c^2} + = \frac{1}{2Nm_0c^2} \left(\frac{Nm_0\vc{u}^2}{2}\right)^2, +\] +the left-hand term really being the larger, since the average square of a +quantity is greater than the square of its average. Since $\left(\dfrac{Nm_0\vc{u}^2}{2}\right)^2$ +is approximately equal to $\left(\dfrac{3}{2}\, nRT\right)^2$, we may write the approximation +%% -----File: 134.png---Folio 120------- +\[ +K = \frac{3}{2}\, nRT + \frac{1}{2Nm_0c^2} \left(\frac{3}{2}\, nRT\right)^2, +\] +or, noting that $N m_0 = M$, the total mass of the system at the absolute +zero, we have +\[ +K = \frac{3}{2}\, nRT + \frac{9}{8}\, \frac{n^2R^2}{Mc^2}\, T^2. +\] +If we use the erg as our unit of energy, $R$~will be~$8.31 × 10^7$; expressing +velocities in centimeters per second, $c^2$~will be~$10^{21}$, and $M$~will be the +mass of the system in grams. + +For one mol of a monatomic gas we should have in ergs +\[ +K = 12.4 × 10^7T + \frac{7.77}{M}\, 10^{-6}\, T^2. +\] + +In the case of the electron $M$~may be taken as approximately +$1/1800$. At room temperature the second term of our equation would +be entirely negligible, being only $3.5 × 10^{-6}$~per cent of the first, and +still be only $3.5 × 10^{-4}$~per cent in a fixed star having a temperature of +$30,000°$. Hence at all ordinary temperatures we may expect the +law of the equipartition of energy to be substantially exact for particles +of mass as small as the electron. + +Our purpose in carrying through the calculations of this chapter +has been to show that a very important and interesting problem in +the classical mechanics can be handled just as easily in the newer +mechanics, and also to point out the nature of the modifications in +existing theory which will have to be introduced if the later developments +of physics should force us to consider equilibrium relations for +particles of mass much smaller than that of the electron. + +We may also call attention to the fact that we have here considered +a system whose equations of motion agree with the principles +of dynamics and yet do not lead to the equipartition of energy. This +is of particular interest at a time when many scientists have thought +that the failure of equipartition in the hohlraum stood in necessary +conflict with the principles of dynamics. +%% -----File: 135.png---Folio 121------- + + +\Chapter{IX}{The Principle of Relativity and the Principle of +Least Action.} +\SetRunningHeads{Chapter Nine.}{Relativity and the Principle of Least Action.} + +It has been shown by the work of Helmholtz, J.~J. Thomson, +Planck and others that the principle of least action is applicable in +the most diverse fields of physical science, and is perhaps the most +general dynamical principle at our disposal. Indeed, for any system +whose future behavior is determined by the instantaneous values of a +number of \DPchg{coordinates}{coördinates} and their time rate of change, it seems possible +to throw the equations describing the behavior of the system into +the form prescribed by the principle of least action. This generality +of the principle of least action makes it very desirable to develop the +relation between it and the principle of relativity, and we shall obtain +in this way the most important and most general method for deriving +the consequences of the theory of relativity. We have already +developed in \Chapref{VII} the particular application of the principle +of least action in the case of a system of particles, and with the help +of the more general development which we are about to present, we +shall be able to apply the principle of relativity to the theories of +elasticity, of thermodynamics and of electricity and magnetism. + +\Subsubsection{107}{The Principle of Least Action.} For our purposes the principle +of least action may be most simply stated by the equation +\[ +\int_{t_1}^{t_2}(\delta H + W)\, dt = 0. +\Tag{135} +\] +This equation applies to any system whose behavior is determined +by the values of a number of independent coördinates $\phi_1\phi_2\phi_3\cdots$ +and their rate of change with the time $\dot{\phi}_1\dot{\phi}_2\dot{\phi}_3\cdots$, and the equation +describes the path by which the system travels from its configuration +at any time~$t_1$ to its configuration at any subsequent time~$t_2$. + +$H$~is the so-called kinetic potential of the system and is a function +of the coördinates and their generalized velocities: +\[ +H = F(\phi_1\phi_2\phi_3\cdots \dot{\phi}_1\dot{\phi}_2\dot{\phi}_3\cdots). +\Tag{136} +\] +%% -----File: 136.png---Folio 122------- +$\delta H$~is the variation of~$H$ at any instant corresponding to a slightly +displaced path by which the system might travel from the same +initial to the same final state in the same time interval, and $W$~is the +external work corresponding to the variation~$\delta$ which would be done +on the system by the external forces if at the instant in question the +system should be displaced from its actual configuration to its configuration +on the displaced path. Thus +\[ +W = \Phi_1\, \delta\phi_1 + + \Phi_2\, \delta\phi_2 + + \Phi_3\, \delta\phi_3 + \cdots, +\Tag{137} +\] +where $\Phi_1$, $\Phi_2$,~etc., are the so-called generalized external forces which +act in such a direction as to increase the values of the corresponding +coördinates. + +The form of the function which determines the kinetic potential~$H$ +depends on the particular nature of the system to which the principle +of least action is being applied, and it is one of the chief tasks of +general physics to discover the form of the function in the various +fields of mechanical, electrical and thermodynamic investigation. +As soon as we have found out experimentally what the form of~$H$ is +for any particular field of investigation, the principle of least action, +as expressed by equation~(135), becomes the basic equation for the +mathematical development of the field in question, a development +which can then be carried out by well-known methods. + +The special task for the theory of relativity will be to find a general +relation applicable to any kind of a system, which shall connect the +value of the kinetic potential~$H$ as measured with respect to a set of +coördinates~$S$ with its value~$H'$ as measured with reference to another +set of coördinates~$S'$ which is in motion relative to~$S$. This relation +will of course be of such a nature as to agree with the principle of the +relativity of motion, and in this way we shall introduce the principle +of relativity at the very start into the fundamental equation for all +fields of dynamics. + +Before proceeding to the solution of that problem we may put +the principle of least action into another form which is sometimes +more convenient, by obtaining the equations for the motion of a +system in the so-called Lagrangian form. + +\Subsubsection{108}{The Equations of Motion in the Lagrangian Form.} To obtain +the equations of motion in the Lagrangian form we may evidently +%% -----File: 137.png---Folio 123------- +rewrite our fundamental equation~(135) in the form +\[ +\begin{aligned} +\int_{t_1}^{t_2} \biggl( + \frac{\partial H}{\partial\phi_1}\, \delta\phi_1 + + \frac{\partial H}{\partial\phi_2}\, \delta\phi_2 + \cdots + &+ \frac{\partial H}{\partial\dot{\phi}_1}\, \delta\dot{\phi}_1 + + \frac{\partial H}{\partial\dot{\phi}_2}\, \delta\dot{\phi}_2 + \cdots \\ + &+ \Phi_1\, \delta\phi_1 + \Phi_2\, \delta\phi_2 + \cdots\biggr) dt = 0 +\end{aligned} +\Tag{138} +\] + +We have now, however, +\[ +\delta\dot\phi_1 = \frac{d}{dt}(\delta\phi_1), \qquad +\delta\dot\phi_2 = \frac{d}{dt}(\delta\phi_2), \qquad \text{etc.,} +\] +which gives us +\begin{align*} +\int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_1}\, \delta\dot\phi_1\, dt + &= \int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_1}\, + \frac{d}{dt}(\delta\phi_1)\, dt \\ + &= \left[\frac{\partial H}{\partial\dot\phi_1}\, \delta\phi_1\right]_{t_1}^{t_2} + - \int_{t_1}^{t_2} \delta\phi_1\, + \frac{d}{dt} \left(\frac{\partial H}{\partial\dot\phi_1}\right) dt, +\end{align*} +or, since $\delta\phi_1$, $\delta\phi_2$,~etc., are by hypothesis zero at times $t_1$~and~$t_2$, we +obtain +\begin{align*} +&\int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_1}\, \delta\dot\phi_1 + = - \int_{t_1}^{t_2} \frac{d}{dt} + \left(\frac{\partial H}{\partial\dot\phi_1}\right) \delta\phi_1\, dt, \\ +&\int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_2}\, \delta\dot\phi_2 + = - \int_{t_1}^{t_2} \frac{d}{dt} + \left(\frac{\partial H}{\partial\dot\phi_2}\right) \delta\phi_2\, dt, \\ +&\text{etc.} +\end{align*} +On substituting these expressions in~(138) we obtain +\begin{multline*} + \int_{t_1}^{t_2} \left[ + \left(\frac{\partial H}{\partial\phi_1} + - \frac{d}{dt} \left(\frac{\partial H}{\partial\dot\phi_1}\right) + + \Phi_1\right) \right. \delta\phi_1 \\ + \left. + \left(\frac{\partial H}{\partial\phi_2} + - \frac{d}{dt} \left(\frac{\partial H}{\partial\dot\phi_2}\right) + + \Phi_2\right) \delta\phi_2 + \cdots \right] dt = 0\DPtypo{}{,} +\end{multline*} +and since the variations of $\phi_1$, $\phi_2$,~etc., are entirely independent and +the limits of integration $t_1$~and~$t_2$ are entirely at our disposal, this +equation will be true only when each of the following equations is +true. And these are the equations of motion in the desired Lagrangian +%% -----File: 138.png---Folio 124------- +form, +\[ +\begin{aligned} +&\frac{d}{dt}\, \frac{\partial H}{\partial\dot{\phi}_1} + - \frac{\partial H}{\partial \phi_1} = \Phi_1, \\ +&\frac{d}{dt}\, \frac{\partial H}{\partial\dot{\phi}_2} + - \frac{\partial H}{\partial \phi_2} = \Phi_2, \\ +&\text{etc.} +\end{aligned} +\Tag{139} +\] + +In these equations $H$ is the kinetic potential of a system whose +state is determined by the generalized coördinates $\phi_1$,~$\phi_2$,~etc., and +their time derivatives $\dot{\phi}_1$,~$\dot{\phi}_2$~etc., where $\Phi_1$,~$\Phi_2$,~etc., are the generalized +external forces acting on the system in such a sense as to tend +to \emph{increase} the values of the corresponding generalized coördinates. + +\Subsubsection{109}{Introduction of the Principle of Relativity.} Let us now investigate +the relation between our dynamical principle and the principle +of the relativity of motion. To do this we must derive an equation +for transforming the kinetic potential~$H$ for a given system +from one set of \DPchg{coordinates}{coördinates} to another. In other words, if $S$~and~$S'$ +are two sets of reference axes, $S'$~moving past~$S$ in the $X$\DPchg{-}{~}direction +with the velocity~$V$, what will be the relation between $H$~and~$H'$, +the values for the kinetic potential of a given system as measured +with reference to $S$~and~$S'$? + +It is evident from the theory of relativity that our fundamental +equation~(135) must hold for the behavior of a given system using +either set of \DPchg{coordinates}{coördinates} $S$~or~$S'$, so that both of the equations +\[ +\int_{t_1}^{t_2} (\delta H + W)\, dt = 0\qquad\text{and}\qquad +\int_{{t_1}'}^{{t_2}'} (\delta H' + W')\, dt' = 0\DPtypo{}{,} \Tag{140} +\] +or +\[ +\int_{t_1}^{t_2} (\delta H + W)\, dt + = \int_{{t_1}'}^{{t_2}'} (\delta H' + W')\, dt' = 0\DPtypo{}{,} +\] +must hold for a given process, where it will be necessary, of course, +to choose the limits of integration $t_1$~and~$t_2$, ${t_1}'$~and~${t_2}'$ wide enough +apart so that for both sets of coördinates the varied motion will be +completed within the time interval. Since we shall find it possible +now to show that in general $\ds\int W\, dt = \int W'\, dt'$, we shall be able to +obtain from the above equations a simple relation between $H$~and~$H'$. + +%[** TN: Bold symbols in original] +\Subsubsection{110}{Relation between $\int W\, dt$ and $\int W'\, dt'$.} To obtain the desired +%% -----File: 139.png---Folio 125------- +proof we must call attention in the first place to the fact that all +kinds of force which can act at a given point must be governed by +the same transformation equations when changing from system~$S$ to +system~$S'$. This arises because when two forces of a different nature +are of such a magnitude as to exactly balance each other and produce +no acceleration for measurements made with one set of coördinates +they must evidently do so for any set of coördinates (see Chapter~IV, %[** TN: Not a useful cross-reference] +\Secref{42}). Since we have already found transformation equations +for the force acting at a point, in our consideration of the dynamics +of a particle, we may now use these expressions in general for the +evaluation~$\int W'\, dt'$. + +$W'$ is the work which would be done by the external forces if at +any instant~$t'$ we should displace our system from its actual configuration +to the simultaneous configuration on the displaced path. +Hence it is evident that $\int W'\, dt\DPtypo{}{'}$~will be equal to a sum of terms of the +type +\[ +%[** TN: Subscripts y and z misprinted (not as subscripts) in original] +\int ({F_x}'\, \delta x' + {F_y}'\, \delta y' + {F_z}'\, \delta z')\, dt', +\] +where ${F_x}'$,~${F_y}'$,~${F_z}'$, is the force acting at a given point of the system +and $\delta x'$,~$\delta y'$,~$\delta z'$ are the displacements necessary to reach the corresponding +point on the displaced path, all these quantities being +measured with respect to~$S'$. + +Into this expression we may substitute, however, in accordance +with equations (61),~(62),~(63) and~(13), the values +\[ +\begin{aligned} +{F_x}' &= F_x - \frac{\dot{y}V}{c^2}\, \frac{1}{1 - \smfrac{\dot{x}V}{c^2}}\, F_y + - \frac{\dot{z}V}{c^2}\, \frac{1}{1 - \smfrac{\dot{x}V}{c^2}}\, F_z, \\ +{F_y}' &= \frac{F_y\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}, \\ +{F_z}' &= \frac{F_z\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}, \\ +dt' &= \kappa \left(1 - \frac{\dot{x}V}{c^2}\right) dt. +\end{aligned} +\Tag{141} +\] +%% -----File: 140.png---Folio 126------- + +We may also make substitutions for $\delta x'$,~$\delta y'$ and~$\delta z'$ in terms of +$\delta x$,~$\delta y$ and~$\delta z$, but to obtain transformation equations for these quantities +is somewhat complicated owing to the fact that positions on the +actual and displaced path, which are simultaneous when measured +with respect to~$S'$, will not be simultaneous with respect to~$S$. We +have denoted by~$t'$ the time in system~$S'$ when the point on the \emph{actual} +path has the position $x'$,~$y'$,~$z'$ and simultaneously the point on the +\emph{displaced} path has the position $(x'+ \delta x')$, $(y' + \delta y')$, $(z' + \delta z')$, +when measured in system~$S'$, or by our fundamental transformation +equations (9),~(10) and~(11) the positions $\kappa (x' + Vt')$,~$y'$,~$z'$ and +$\kappa \bigl([x' + \delta x'] + Vt'\bigr)$, $(y'+ \delta y')$, $(z'+\delta z')$ when measured in system~$S$. +If now we denote by $t_A$~and~$t_D$ the corresponding times in system~$S$ +we shall have, by our fundamental transformation equation~(12), +\begin{align*} +t_A &= \kappa \left(t' + \frac{Vx'}{c^2}\right), \\ +t_D &= \kappa \left(t' + \frac{V}{c^2}[x' + \delta x']\right), +\end{align*} +and we see that in system~$S$ the point has reached the displaced +position at a time later than that of the actual position by the amount +\[ +t_D - t_A = \frac{\kappa V}{c^2}\, \delta x', +\] +and, since during this time-interval the displaced point would have +moved, neglecting higher-order terms, the distances +\[ +\dot{x}\, \frac{\kappa V}{c^2}\, \delta x', \qquad +\dot{y}\, \frac{\kappa V}{c^2}\, \delta x', \qquad +\dot{z}\, \frac{\kappa V}{c^2}\, \delta x', +\] +these quantities must be subtracted from the coördinates of the +displaced point in order to obtain a position on the displaced path +which will be simultaneous with~$t_A$ as measured in system~$S$. We +obtain for the simultaneous position on the displaced path +\begin{gather*} +\kappa \bigl([x' + \delta x'] + Vt'\bigr) + - \kappa\, \frac{\dot{x}V}{c^2}\, \delta x', \qquad +y' + \delta y' - \kappa\, \frac{\dot{x}V}{c^2}\, x', \\ +z' + \delta z' - \kappa\, \frac{\dot{z}V}{c^2}\, \delta x', +\end{gather*} +%% -----File: 141.png---Folio 127------- +and for the corresponding position on the actual path +\[ +\kappa (x' + Vt'), \quad y', \quad z', +\] +and obtain by subtraction +\[ +\begin{aligned} +\delta x &= \kappa \left(1 - \frac{\dot{x}V}{c^2}\right) \delta x', \\ +\delta y &= \delta y' - \kappa \frac{\dot{y}V}{c^2}\, \delta x', \\ +\delta z &= \delta z' - \kappa \frac{\dot{z}V}{c^2}\, \delta x'. +\end{aligned} +\Tag{142} +\] +Substituting now these equations, together with the other transformation +equations~(141), in our expression we obtain +\[ +\begin{aligned} +\int ({F_x}'\, \delta x' &+ {F_y}'\, \delta y' + {F_z}'\, \delta z')\, dt' \\ + &= \int \Biggl(\Biggl[ + F_x - \frac{\dot{y}V}{c^2}\, \frac{F_y}{1 - \smfrac{\dot{x}V}{c^2}} + - \frac{\dot{z}V}{c^2}\, \frac{F_z}{1 - \smfrac{\dot{x}V}{c^2}} + \Biggr] +%[** TN: \rlap hack to get equation number centered] + \rlap{$\ds \frac{\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}\, \delta x$} \\ + &+ \frac{\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}\, F_y \Biggl[ + \delta y + \frac{\dot{y}V/c^2}{1 - \smfrac{\dot{x}V}{c^2}}\, \delta x + \Biggr] \\ + &+ \frac{\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}\, F_z \Biggl[ + \delta z + \frac{\dot{z}V/c^2}{1 - \smfrac{\dot{x}V}{c^2}}\, \delta x + \Biggr]\Biggr) \kappa \left(1 - \frac{\dot{x}V}{c^2}\right) \\ + &= \int(F_x\, \delta x + F_y\, \delta y + F_z\, \delta z)\, dt'. +\end{aligned} +\Tag{143} +\] +We thus see that we must always have the general equality +\[ +\int W'\, dt' = \int W\, dt. +\Tag{144} +\] + +\Subsubsection{111}{Relation between $H'$~and~$H$.} Introducing this equation into +our earlier expression~(140) we obtain as a general relation between +$H'$~and~$H$ +\[ +\int \delta H'\, dt' = \int \delta H\, dt. +\Tag{145} +\] + +Restricting ourselves to systems of such a nature that we can +%% -----File: 142.png---Folio 128------- +assign them a definite velocity $u = \dot{x}\vc{i} + \dot{y}\vc{j} + \dot{z}\vc{k}$, we can rewrite +this expression in the following form, where by $H_D$~and~$H_A$ we denote +the values of the kinetic potential respectively on the displaced and +actual paths +\begin{align*} +\int \delta H'\, dt' = \int {H_D}'\, dt' + &- \int {H_A}'\, dt' + = \int {H_D}' \kappa + \left(1 - \frac{(\dot{x} + \delta \dot{x})V}{c^2}\right) dt \\ + &- \int {H_A}' \kappa + \left(1 - \frac{\dot{x}V}{c^2}\right) dt + = \int H_D\, dt - \int H_A\, dt, +\end{align*} +and hence obtain for such systems the simple expression +\[ +H' = \frac{H}{\kappa \left(1 - \smfrac{\dot{x}V}{c^2}\right)}. +\] +Noting the relation between $\sqrt{1 - \dfrac{{u'}^2}{c^2}}$ and $\sqrt{1 - \dfrac{u^2}{c^2}}$ given in equation~(17), +this can be rewritten +\[ +\frac{H'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}} + = \frac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\Tag{146} +\] +and this is the expression which we shall find most useful for our +future development of the consequences of the theory of relativity. +Expressing the requirement of the equation in words we may say +that the theory of relativity requires an invariance of $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ in the +Lorentz transformation. + +\Paragraph{112.} As indicated above, the use of this equation is obviously +restricted to systems moving with some perfectly definite velocity~$\vc{u}$. +Systems satisfying this condition would include particles, infinitesimal +portions of continuous systems, and larger systems in a steady state. + +\Paragraph{113.} Our general method of procedure in different fields of investigation +will now be to examine the expression for kinetic potential +which is known to hold for the field in question, provided the velocities +involved are low and by making slight alterations when necessary, +%% -----File: 143.png---Folio 129------- +see if this expression can be made to agree with the requirements of +equation~(146) without changing its value for low velocities. Thus +it is well known, for example, that, in the case of low velocities, for a +single particle acted on by external forces the kinetic potential may +be taken as the kinetic energy~$\frac{1}{2}m_0u^2$. For relativity mechanics, as +will be seen from the developments of \Chapref{VII}, we may take for +the kinetic potential, $-m_0c^2 \sqrt{1 - \dfrac{u^2}{c^2}}$, an expression which, except for +an additive constant, becomes identical with~$\frac{1}{2}m_0u^2$ at low velocities, +and which at all velocities agrees with equation~(146). +%% -----File: 144.png---Folio 130------- + + +\Chapter{X}{The Dynamics of Elastic Bodies.} +\SetRunningHeads{Chapter Ten.}{Dynamics of Elastic Bodies.} + +We shall now treat with the help of the principle of least action +the rather complicated problem of the dynamics of continuous elastic +media. Our considerations will \emph{extend} the appreciation of the intimate +relation between mass and energy which we found in our treatment +of the dynamics of a particle. We shall also be able to show +that the dynamics of a particle may be regarded as a special case +of the dynamics of a continuous elastic medium, and to apply our +considerations to a number of other important problems. + +\Subsubsection{114}{On the Impossibility of Absolutely Rigid Bodies.} In the +older treatises on mechanics, after considering the dynamics of a +particle it was customary to proceed to a discussion of the dynamics +of rigid bodies. These rigid bodies were endowed with definite and +\DPtypo{nu}{un}changeable size and shape and hence were assigned five degrees +of freedom, since it was necessary to state the values of five variables +completely to specify their position in space. As pointed out by +Laue, however, our newer ideas as to the velocity of light as a limiting +value will no longer permit us to conceive of a continuous body as +having only a finite number of degrees of freedom. This is evident +since it is obvious that we could start disturbances simultaneously +at an indefinite number of points in a continuous body, and as these +disturbances cannot spread with infinite velocity it will be necessary +to give the values of an infinite number of variables in order completely +to specify the succeeding states of the system. For our newer +mechanics the nearest approach to an absolutely rigid body would +of course be one in which disturbances are transmitted with the +velocity of light. Since, then, the theory of relativity does not +permit rigid bodies we may proceed at once to the general theory of +deformable bodies. + + +\Section[I]{Stress and Strain.} + +\Subsubsection{115}{Definition of Strain.} In the more familiar developments of +the theory of elasticity it is customary to limit the considerations to +%% -----File: 145.png---Folio 131------- +the case of strains small enough so that higher powers of the displacements +can be neglected, and this introduces considerable simplification +into a science which under any circumstances is necessarily +one of great complication. Unfortunately for our purposes, we +cannot in general introduce such a simplification if we wish to apply +the theory of relativity, since in consequence of the Lorentz shortening +a body which appears unstrained to one observer may appear tremendously +compressed or elongated to an observer moving with a +different velocity. The best that we can do will be arbitrarily to +choose our state of zero deformation such that the strains will be +small when measured in the particular system of coördinates $S$ in +which we are specially interested. + +A theory of strains of any magnitude was first attempted by +Saint-Venant and has been amplified and excellently presented by +Love in his \textit{Treatise on the Theory of Elasticity}, Appendix to Chapter~I. +In accordance with this theory, the strain at any point in a body is +completely determined by six component strains which can be defined +by the following equations, wherein $(u, v, w)$~is the displacement of a +point having the unstrained position $(x, y, z)$: +%[** TN: Setting as two groups, both numbered (148), to permit a page break] +\begin{align*} +&\begin{aligned} +\epsilon_{xx} &= \frac{\partial u}{\partial x} + + \tfrac{1}{2} \left\{ + \left(\frac{\partial u}{\partial x}\right)^2 + + \left(\frac{\partial v}{\partial x}\right)^2 + + \left(\frac{\partial w}{\partial x}\right)^2 \right\}, \\ +% +\epsilon_{yy} &= \frac{\partial y}{\partial v} + + \tfrac{1}{2} \left\{ + \left(\frac{\partial u}{\partial y}\right)^2 + + \left(\frac{\partial v}{\partial y}\right)^2 + + \left(\frac{\partial w}{\partial y}\right)^2 \right\}, \\ +% +\epsilon_{zz} &= \frac{\partial w}{\partial z} + + \tfrac{1}{2} \left\{ + \left(\frac{\partial u}{\partial z}\right)^2 + + \left(\frac{\partial v}{\partial z}\right)^2 + + \left(\frac{\partial w}{\partial z}\right)^2 \right\}, +\end{aligned} +\Tag{148} +\displaybreak[0] \\ +% +&\begin{aligned} +\epsilon_{yz} &= \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z} + + \frac{\partial u}{\partial y}\, \frac{\partial u}{\partial z} + + \frac{\partial v}{\partial y}\, \frac{\partial v}{\partial z} + + \frac{\partial w}{\partial y}\, \frac{\partial w}{\partial z}, \\ +% +\epsilon_{xz} &= \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} + + \frac{\partial u}{\partial x}\, \frac{\partial u}{\partial z} + + \frac{\partial v}{\partial x}\, \frac{\partial v}{\partial z} + + \frac{\partial w}{\partial x}\, \frac{\partial w}{\partial z}, \\ +% +\epsilon_{xy} &= \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} + + \frac{\partial u}{\partial x}\, \frac{\partial u}{\partial y} + + \frac{\partial v}{\partial x}\, \frac{\partial v}{\partial y} + + \frac{\partial w}{\partial x}\, \frac{\partial w}{\partial y}. +\end{aligned} +\Tag{148} +\end{align*} + +It will be seen that these expressions for strain reduce to those +familiar in the theory of small strains if such second-order quantities as +$\left(\dfrac{\partial u}{\partial x}\right)^2$ or $\dfrac{\partial u}{\partial y}\,\dfrac{\partial u}{\partial z}$ can be neglected. +%% -----File: 146.png---Folio 132------- + +\Paragraph{116.} A physical significance for these strain components will be +obtained if we note that it can be shown from geometrical considerations +that lines which are originally parallel to the axes have, when +strained, the elongations +\[ +\begin{aligned} +e_x &= \sqrt{1 + 2\epsilon_{xx}} - 1, \\ +e_y &= \sqrt{1 + 2\epsilon_{yy}} - 1, \\ +e_z &= \sqrt{1 + 2\epsilon_{zz}} - 1, +\end{aligned} +\Tag{149} +\] +and that the angles between lines originally parallel to the axes are +given in the strained condition by the expressions +\[ +\begin{aligned} +\cos \theta_{yz} + &= \frac{\epsilon_{yz}} + {\sqrt{1 + 2\epsilon_{yy}}\, \sqrt{1 + 2\epsilon_{zz}}}, \\ +\cos \theta_{xz} + &= \frac{\epsilon_{xz}} + {\sqrt{1 + 2\epsilon_{xx}}\, \sqrt{1 + 2\epsilon_{zz}}}, \\ +\cos \theta_{xy} + &= \frac{\epsilon_{xy}} + {\sqrt{1 + 2\epsilon_{xx}}\, \sqrt{1 + 2\epsilon_{yy}}}, +\end{aligned} +\Tag{150} +\] + +Geometrical considerations are also sufficient to show that in +case the strain is a simple elongation of amount~$e$ the following equation +will be true: +\[ +\frac{\epsilon_{xx}}{l^2} = +\frac{\epsilon_{yy}}{m^2} = +\frac{\epsilon_{zz}}{n^2} = +\frac{\epsilon_{yz}}{2mn} = +\frac{\epsilon_{xz}}{2ln} = +\frac{\epsilon_{xy}}{2lm} = e + \tfrac{1}{2}e^2, +\Tag{151} +\] +where $l$,~$m$,~$n$ are the cosines which determine the direction of the +elongation. + +\Subsubsection{117}{Definition of Stress.} We have just considered the expressions +for the strain at a given point in an elastic medium; we may +now define stress in terms of the work done in changing from one +state of strain to another. Considering the material contained in +\emph{unit volume when the body is unstrained}, we may write, for the work +done by this material on its surroundings when a change in strain +takes place, +%% -----File: 147.png---Folio 133------- +\[ +\begin{aligned} +\delta W = -\delta E + = t_{xx}\, \delta\epsilon_{xx} + + t_{yy}\, \delta\epsilon_{yy} + &+ t_{zz}\, \delta\epsilon_{zz} \\ + &+ t_{yz}\, \delta\epsilon_{yz} + + t_{xz}\, \delta\epsilon_{xz} + + t_{xy}\, \delta\epsilon_{xy}, +\end{aligned} +\Tag{152} +\] +and this equation serves to define the stresses $t_{xx}$,~$t_{yy}$,~etc. In case +the strain varies from point to point we must consider of course the +work done \textit{per}~unit volume of the unstrained material. In case the +strains are small it will be noticed that the stresses thus defined are +identical with those used in the familiar theories of elasticity. + +\Subsubsection{118}{Transformation Equations for Strain.} We must now prepare +for the introduction of the theory of relativity into our considerations, +by determining the way the strain at a given point~$P$ appears to observers +moving with different velocities. Let the point~$P$ in question +be moving with the velocity $\vc{u} = x\vc{i} + y\vc{j} + z\vc{k}$ as measured in system~$S$. +Since the state of zero deformation from which to measure +strains can be chosen perfectly arbitrarily, let us for convenience +take the strain as zero as measured in system~$S$, giving us +\[ +\epsilon_{xx} = +\epsilon_{yy} = +\epsilon_{zz} = +\epsilon_{yz} = +\epsilon_{xz} = +\epsilon_{xy} = 0. +\Tag{153} +\] +What now will be the strains as measured by an observer moving +along with the point~$P$ in question? Let us call the system of coördinates +used by this observer~$S°$. It is evident now from our considerations +as to the shape of moving systems presented in \Chapref{V} that +in system~$S°$ the material in the neighborhood of the point in question +will appear to have been elongated in the direction of motion in the +ratio of $1 : \sqrt{1 - \dfrac{u^2}{c^2}}$. Hence in system~$S°$ the strain will be an elongation +\[ +e = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} - 1 +\Tag{154} +\] +in the line determined by the direction cosines +\[ +l = \frac{\dot{x}}{u},\qquad +m = \frac{\dot{y}}{u},\qquad +\DPtypo{u}{n} + = \frac{\dot{z}}{u}. +\Tag{155} +\] + +We may now calculate from this elongation the components of +strain by using equation~(151). We obtain +%% -----File: 148.png---Folio 134------- +{\small%[** TN: Setting on two lines, not six] +\[ +\begin{aligned} +%[** TN: \llap coaxes equation to the left without crowding the tag] +\llap{$\epsilon°$}_{xx} + &= \frac{\dot{x}^2}{2c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],& +{\epsilon°}_{yy} + &= \frac{\dot{y}^2}{2c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],& +{\epsilon°}_{zz} + &= \frac{\dot{z}^2}{2c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],\\ +% +\llap{$\epsilon°$}_{yz}%[** See above] + &= \ \frac{\dot{y}\dot{z}}{c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],& +{\epsilon°}_{xz} + &= \ \frac{\dot{x}\dot{z}}{c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],& +{\epsilon°}_{xy} + &= \ \frac{\dot{x}\dot{y}}{c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr], +\end{aligned} +\Tag{156} +\]}% +and these are the desired equations for the strains at the point~$P$, +the accent~$°$ indicating that they are measured with reference to a +system of coördinates~$S°$ moving along with the point itself. + +\Subsubsection{119}{Variation in the Strain.} We shall be particularly interested +in the variation in the strain as measured in~$S°$ when the velocity +experiences a small variation~$\delta\vc{u}$, the strains remaining zero as measured +in~$S$. For the sake of simplicity let us choose our coördinates +in such a way that the $X$\DPchg{-}{~}axis is parallel to the original velocity, so +that our change in velocity will be from $\vc{u} = \dot{x}\vc{i}$ to +\[ +\vc{u} + \delta\vc{u} + = (\dot{x} + \delta\dot{x})\, \vc{i} + + \delta\dot{y}\, \vc{j} + \delta\dot{z}\, \vc{k}. +\] +Taking $\delta\vc{u}$~small enough so that higher orders can be neglected, and +noting that $\dot{y} = \dot{z} = 0$, we shall then have, from equations~(156), +%% -----File: 149.png---Folio 135------- +\[ +\begin{aligned} +\delta{\epsilon°}_{xx} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta\dot{x},& +\delta{\epsilon°}_{yy} &= 0, \\ +\delta{\epsilon°}_{zz} &= 0, & +\delta{\epsilon°}_{yz} &= 0, \\ +\delta{\epsilon°}_{xz} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta\dot{z},&\qquad +\delta{\epsilon°}_{xy} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta\dot{y}. +\end{aligned} +\Tag{157} +\] + +We shall also be interested in the variation in the strain as measured +in~$S°$ produced by a variation in the strain as measured in~$S$. Considering +again for simplicity that the $X$\DPchg{-}{~}axis is parallel to the motion +of the point, we must calculate the variation produced in ${\epsilon°}_{xx}$,~${\epsilon°}_{yy}$,~etc., +by changing the values of $\epsilon_{xx}$,~$\epsilon_{yy}$,~etc., from zero to $\delta\epsilon_{xx}$,~$\delta\epsilon_{yy}$,~etc. + +The variation~$\delta\epsilon_{xx}$ will produce a variation in~${\epsilon°}_{xx}$ whose amount +can be calculated as follows: By equations~(149) a line which has unit +length and is parallel to the $X$\DPchg{-}{~}axis in the unstrained condition will +have when strained the length $\sqrt{1 + 2\epsilon_{xx}}$ when measured in system~$S$ +and $\sqrt{1 + 2{\epsilon°}_{xx}}$ when measured in system~$S°$. Since the strain in +system~$S$ is small, the line remains sensibly parallel to the $X$\DPchg{-}{~}axis, +which is also the direction of motion, and these quantities will be +connected in accordance with the Lorentz shortening by the equation +\[ +\sqrt{1 + 2\epsilon_{xx}} + = \sqrt{1 - \frac{u^2}{c^2}}\, + \sqrt{1 + 2{\epsilon°}_{xx}}. +\Tag{158} +\] +Carrying out now our variation~$\delta\epsilon_{xx}$, neglecting~$\epsilon_{xx}$ in comparison +with larger quantities and noting that except for second order quantities, +\[ +\sqrt{1 + 2{\epsilon°}_{xx}} = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} +\Tag{159} +\] +we obtain +\[ +\delta{\epsilon°}_{xx} + = \frac{\delta\epsilon_{xx}}{\left(1 - \smfrac{u^2}{c^2}\right)}. +\Tag{160} +\] + +Since the variations $\delta\epsilon_{yy}$,~$\delta\epsilon_{zz}$,~$\delta\epsilon_{yz}$ affect only lines which are at +right angles to the direction of motion, we may evidently write +\[ +\delta{\epsilon°}_{yy} = \delta\epsilon_{yy}, \qquad +\delta{\epsilon°}_{zz} = \delta\epsilon_{zz}, \qquad +\delta{\epsilon°}_{yz} = \delta\epsilon_{yz}. +\Tag{161} +\] +%% -----File: 150.png---Folio 136------- +To calculate $\delta{\epsilon°}_{xz}$ we may note that in accordance with equations~(150) +we must have +\begin{align*} +\cos \theta_{xz} + &= \frac{\epsilon_{xz}} + {\sqrt{1 + 2\epsilon_{xx}}\, \sqrt{1+2\epsilon_{zz}}},\\ +\cos {\theta°}_{xz} + &= \frac{{\epsilon°}_{xz}} + {\sqrt{1 + 2{\epsilon°}_{xx}}\, \sqrt{1 + 2{\epsilon°}_{zz}}}, +\end{align*} +where $\theta_{xz}$~is the angle between lines which in the unstrained condition +are parallel to the $X$~and~$Z$ axes respectively. In accordance with +the Lorentz shortening, however, we shall have +\[ +\cos \theta_{xz} = \sqrt{1 - \frac{u^2}{c^2}} \cos {\theta°}_{xz}. +\] +Introducing this relation, remembering that $\epsilon_{xx} = {\epsilon°}_{zz} = 0$, and +noting equation~(159), we obtain +\begin{align*} +\delta {\epsilon°}_{xz} + &= \frac{\delta\epsilon_{xz}}{\left(1 - \smfrac{u^2}{x^2}\right)}, +\Tag{162} \\ +\intertext{and similarly} +\delta {\epsilon°}_{xy} + &= \frac{\delta\epsilon_{xy}}{\left(1 - \smfrac{u^2}{x^2}\right)}. +\Tag{163} +\end{align*} + +We may now combine these equations (160),~(161),~(162) and~(163) +with those for the variation in strain with velocity and obtain +the final set which we desire: +\[ +\begin{aligned} +\delta {\epsilon°}_{xx} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta \dot{x} + + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \delta\epsilon_{xx}, \\ +\delta {\epsilon°}_{yy} &= \delta\epsilon_{yy}, \\ +\delta {\epsilon°}_{zz} &= \delta\epsilon_{zz}, \\ +\delta {\epsilon°}_{yz} &= \delta\epsilon_{yz}, \\ +\delta {\epsilon°}_{xz} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta \dot{z} + + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \delta\epsilon_{xz}, \\ +\delta {\epsilon°}_{xy} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta \dot{y} + + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \delta\epsilon_{xy}. +\end{aligned} +\Tag{164} +\] +%% -----File: 151.png---Folio 137------- + +These equations give the variation in the strain measured in +system~$S°$ at a point~$P$ moving in the $X$~direction with velocity~$u$, +provided the strains are negligibly small as measured in~$S$. + + +\Section[II]{Introduction of the Principle of Least Action.} + +\Subsubsection{120}{The Kinetic Potential for an Elastic Body.} We are now in +a position to develop the mechanics of an elastic body with the help +of the principle of least action. In Newtonian mechanics, as is well +known, the kinetic potential for unit volume of material at a given +point~$P$ in an elastic body may be put equal to the density of kinetic +energy minus the density of potential energy, and it is obvious that +our choice for kinetic potential must reduce to that value at low +velocities. Our choice of an expression for kinetic potential is furthermore +limited by the fundamental transformation equation for kinetic +potential which we found in the last chapter +\[ +\frac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = \frac{H'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}. +\Tag{146} +\] + +Taking these requirements into consideration, we may write for +the kinetic potential per unit volume of the material at a point~$P$ +moving with the velocity~$\vc{u}$ the expression +\[ +H = -E° \sqrt{1 - \frac{u^2}{c^2}}, +\] +where $E°$~is the energy as measured in system~$S°$ of the amount of +material which in the unstrained condition (\ie, as measured in +system~$S$) is contained in unit volume. + +The above expression obviously satisfies our fundamental transformation +equation~(146) and at low velocities reduces in accordance +with the requirements of Newtonian mechanics to +\[ +H = \tfrac{1}{2} m° u^2 - E°, +\] +provided we introduce the substitution made familiar by our previous +work, $m° = \dfrac{E°}{c^2}$. +%% -----File: 152.png---Folio 138------- + +\Subsubsection{121}{Lagrange's Equations.} Making use of this expression for the +kinetic potential in an elastic body, we may now obtain the equations +of motion and stress for an elastic body by substituting into Lagrange's +equations~(139) Chapter~IX\@. %[** TN: Not a useful cross-reference.] + +Considering the material at the point~$P$ contained in unit volume +in the unstrained condition, we may choose as our generalized coördinates +the six component strains $\epsilon_{xx}$,~$\epsilon_{yy}$,~etc., with the corresponding +stresses $-t_{xx}$,~$-t_{yy}$,~etc., as generalized forces, and the +three coördinates $x$,~$y$,~$z$ which give the position of the point with the +corresponding forces $F_x$,~$F_y$ and~$F_z$. + +It is evident that the kinetic potential will be independent of +the time derivatives of the strains, and if we consider cases in which +$E°$~is independent of position, the kinetic potential will also be independent +of the absolute magnitudes of the coördinates $x$,~$y$ and~$z$. +Substituting in Lagrange's equations~(139), we then obtain +\[ +\left. +\begin{aligned} +-\frac{\partial}{\partial \epsilon_{xx}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{xx}, \\ +-\frac{\partial}{\partial \epsilon_{yy}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{yy}, \\ +-\frac{\partial}{\partial \epsilon_{zz}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{zz}, \\ +-\frac{\partial}{\partial \epsilon_{yz}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{yz}, \\ +-\frac{\partial}{\partial \epsilon_{xz}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{xz}, \\ +-\frac{\partial}{\partial \epsilon_{xy}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{xy}, +\end{aligned} +\right\} +\Tag{165} +\] +\[ +\left. +\begin{aligned} +\frac{d}{dt}\, \frac{\partial}{\partial \dot{x}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_x, \\ +\frac{d}{dt}\, \frac{\partial}{\partial \dot{y}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_y, \\ +\frac{d}{dt}\, \frac{\partial}{\partial \dot{z}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_z. +\end{aligned} +\right\} +\Tag{166} +\] +%% -----File: 153.png---Folio 139------- + +We may simplify these equations, however; by performing the +indicated differentiations and making suitable substitutions, we have +\[ +\frac{\partial {E°}_{xx}}{\partial \epsilon_{xx}} + = \frac{\partial {E°}_{xx}}{\partial {\epsilon°}_{xx}}\, + \frac{\partial {\epsilon°}_{xx}}{\partial \epsilon_{xx}}. +\] +But in accordance with equation~(152) we may write +\[ +\frac{\partial {E°}_{xx}}{\partial {\epsilon°}_{xx}} = -{t°}_{xx} +\] +and from equations~(164) we may put +\[ +\frac{\partial {\epsilon°}_{xx}}{\partial \epsilon_{xx}} + = \frac{1}{1 - \smfrac{u^2}{c^2}}. +\] +Making the substitutions in the first of the Lagrangian equations we +obtain +\[ +t_{xx} = -\frac{\partial}{\partial\epsilon_{xx}} + \left(E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) + = {t°}_{xx} \frac{1}{1 - \smfrac{u^2}{c^2}} \sqrt{1 - \frac{u^2}{c^2}} + = \frac{{t°}_{xx}}{\sqrt{1 - \frac{u^2}{c^2}}}. +\] + +\Subsubsection{122}{Transformation Equations for Stress.} Similar substitutions +can be made in all the equations of stress, and we obtain as our set +of transformation equations +\[ +\begin{aligned} +t_{xx} &= \frac{{t°}_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, & +t_{yy} &= \sqrt{1 - \smfrac{u^2}{c^2}}\, {t°}_{yy}, & +t_{zz} &= \sqrt{1 - \smfrac{u^2}{c^2}}\, {t°}_{zz}, \\ +t_{yx} &= \sqrt{1 - \smfrac{u^2}{c^2}}\, {t°}_{yx}, & +t_{xz} &= \frac{{t°}_{xz}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, & +t_{xy} &= \frac{{t°}_{xy}}{\sqrt{1 - \smfrac{u^2}{c^2}}}. +\end{aligned} +\Tag{167} +\] + +%[** TN: Bold symbol in original] +\Subsubsection{123}{Value of $E^\circ$.} With the help of these transformation equations +for stress we may calculate the value of~$E°$, the energy content, as +measured in system~$S°$, of material which in the unstrained condition +is contained in unit volume. + +Consider unit volume of the material in the unstrained condition +and call its energy content~$w°°$. Give it now the velocity $u = \dot{x}$, +keeping its state of strain unchanged in system~$S$. Since the \emph{strain} +%% -----File: 154.png---Folio 140------- +is not changing in system~$S$, the stresses $t_{xx}$,~etc., will also be constant +in system~$S$. In system~$S°$, however, the component strain will +change in accordance with equations~(156) from zero to +\[ +{\epsilon°}_{xx} + = \frac{\dot{x}^2}{2c^2}\, \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}, +\] +and the corresponding stress will be given at any instant by the +expression just derived, +\[ +{t°}_{xx} = t_{xx} \sqrt{1 - \frac{u^2}{c^2}}, +\] +$t_{xx}$ being, as we have just seen, a constant. We may then write for~$E°$ +the expression +\[ +E° = w°° - t_{xx} \int_0^w \sqrt{1 - \frac{u^2}{c^2}}\, + d\Biggl[\frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, + \frac{\dot{x}}{2c^2}\Biggr]. +\] +Noting that $u = \dot{x}$ we obtain on integration, +\[ +E° = w°° + t_{xx} - \frac{t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}} +\Tag{168} +\] +as the desired expression for the energy as measured in system~$S°$ +contained in the material which in system~$S$ is unstrained and has +unit volume. + +\Subsubsection{124}{The Equations of Motion in the Lagrangian Form.} We are +now in a position to simplify the three Lagrangian equations~(166) +for $F_x$,~$F_y$ and~$F_z$. Carrying out the indicated differentiation we have +\[ +F_x = \frac{d}{dt}\, \frac{\partial}{\partial \dot{x}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\right) + = \frac{d}{dt} \Biggl[ + \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2} + - \sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial\dot{x}} + \Biggr], +\] +and introducing the value of~$E°$ given by equation~(168) we obtain +\[ +F_x = \frac{d}{dt} \Biggl[ + \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}\Biggr]. +\Tag{169} +\] +%% -----File: 155.png---Folio 141------- +Simple calculations will also give us values for $F_y$~and~$F_z$. We have +from~(166) +\[ +F_y = \frac{d}{dt}\, \frac{\partial}{\partial \dot{y}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\right) + = \frac{d}{dt} \Biggl[ + \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{y}}{c^2} + - \sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial\dot{y}} + \Biggr]. +\] +But since we have adapted our considerations to cases in which the +direction of motion is along the $X$\DPchg{-}{~}axis, we have $\dot{y} = 0$; furthermore +we may substitute, in accordance with equations (152),~(157) and~(167), +\[ +\frac{\partial E°}{\partial \dot{y}} + = \frac{\partial E°}{\partial {\epsilon°}_{xy}}\, + \frac{\partial {\epsilon°}_{xy}}{\partial \dot{y}} + = -{t°}_{xy}\, \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, + \frac{\dot{x}}{c^2} + = \frac{-t_{xy}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2}. +\] +We thus obtain as our three equations of motion +\[ +\begin{aligned} +F_x &= \frac{d}{dt}\Biggl[ + \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2} + \Biggr], \\ +F_y &= \frac{d}{dt} \left(t_{xy}\, \frac{\dot{x}}{c^2}\right), \\ +F_z &= \frac{d}{dt} \left(t_{xz}\, \frac{\dot{x}}{c^2}\right). +\end{aligned} +\Tag{170} +\] +In these equations the quantities $F_x$,~$F_y$ and~$F_z$ are the components +of force acting on a particular system, namely that quantity of material +which at the instant in question has unit volume. Since the volume +of this material will in general be changing, $F_x$,~$F_y$ and~$F_z$ do not give +us the force per unit volume as usually defined. If we represent, +however, by $f_x$,~$f_y$ and~$f_z$ the components of force per unit volume, +we may rewrite these equations in the form +\[ +\begin{aligned} +F_x\, \delta V &= \frac{d}{dt}\Biggl[ + \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}\, \delta V\Biggr], \\ +F_y\, \delta V &= \frac{d}{dt}\left(t_{xy}\, \frac{\dot{x}}{c^2}\, \delta V\right),\\ +F_z\, \delta V &= \frac{d}{dt}\left(t_{xz}\, \frac{\dot{x}}{c^2}\, \delta V\right), +\end{aligned} + \Tag{171} +\] +%% -----File: 156.png---Folio 142------- +where by $\delta V$ we mean a small element of volume at the point in +question. + +\Subsubsection{125}{Density of Momentum.} Since we customarily define force as +equal to the time rate of change of momentum, we may now write for +the density of momentum~$\vc{g}$ at a point in an elastic body which is +moving in the $X$~direction with the velocity $u = \dot{x}$ +\[ +\vc{g}_x = \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}, \qquad +\vc{g}_y = t_{xy}\, \frac{\dot{x}}{c^2}, \qquad +\vc{g}_y = t_{xy}\, \frac{\dot{x}}{c^2}. +\Tag{172} +\] + +It is interesting to point out that there are components of momentum +in the $Y$~and~$Z$ directions in spite of the fact that the material +at the point in question is moving in the $X$~direction. We shall +later see the important significance of this discovery. + +\Subsubsection{126}{Density of Energy.} It will be remembered that the forces +whose equations we have just obtained are those acting on unit +volume of the material as measured in system~$S$, and hence we are +now in a position to calculate the energy density of our material. +Let us start out with unit volume of our material at rest, with the +energy content~$w°°$ and determine the work necessary to give it the +velocity $u = \dot{x}$ without change in stress or strain. Since the only +component of force which suffers displacement is~$F_x$, we have +\[ +\begin{aligned} +%[** TN: Commas present in original, arguably serve a grammatical purpose] +w &= w°° + \int_0^u \frac{d}{dt} \Biggl[ + \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}\Biggr] \dot{x}\, dt, \\ + &= w°° + (w°° + t_{xx}) + \int_0^u \dot{x}\, d\Biggl[ + \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2}\Biggr], \\ + &= \Biggl\{\frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}} + - t_{xx}\Biggr\} +\end{aligned} +\Tag{173} +\] +as an expression for the energy density of the elastic material. + +\Subsubsection{127}{Summary of Results Obtained from the Principle of Least +Action.} We may now tabulate for future reference the results obtained +from the principle of least action. +%% -----File: 157.png---Folio 143------- + +At a given point in an elastic medium which is moving in the $X$~direction +with the velocity $u = \dot{x}$, we have for the components of +stress +\[ +\begin{aligned} +t_{xx} &= \frac{{t°}_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, & +t_{yy} &= \sqrt{1 - \frac{u^2}{c^2}}\, {t°}_{yy}, & +t_{zz} &= \sqrt{1 - \frac{u^2}{c^2}}\, {t°}_{zz}, \\ +t_{yz} &= \sqrt{1 - \frac{u^2}{c^2}}\, {t°}_{yz}, & +t_{xz} &= \frac{{t°}_{xz}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, & +t_{xy} &= \frac{{t°}_{xz}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\end{aligned} +\Tag{167} +\] +For the density of energy at the point in question we have +\[ +w = \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}} - t_{xx}. +\Tag{173} +\] +For the density of momentum we have +\[ +\vc{g}_{x} + = \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}, \qquad +\vc{g}_y = t_{xy}\, \frac{\dot{x}}{c^2}, \qquad +\vc{g}_z = t_{xz}\, \frac{\dot{x}}{c^2}. +\Tag{172} +\] + + +\Section[III]{Some Mathematical Relations.} + +Before proceeding to the applications of these results which we +have obtained from the principle of least action, we shall find it desirable +to present a number of mathematical relations which will +later prove useful. + +\Subsubsection{128}{The Unsymmetrical Stress Tensor $\vc{t}$.} We have defined the +components of stress acting at a point by equation~(152) +\[ +\delta W + = t_{xx}\, \delta\epsilon_{xx} + + t_{yy}\, \delta\epsilon_{yy} + + t_{zz}\, \delta\epsilon_{zz} + + t_{yz}\, \delta\epsilon_{yz} + + t_{xz}\, \delta\epsilon_{xz} + + t_{xy}\, \delta\epsilon_{xy}, +\] +where $\delta W$~is the work which accompanies a change in strain and is +performed on the surroundings by the amount of material which was +contained in unit volume in the unstrained state. Since for convenience +we have taken as our state of zero strain the condition of +the body as measured in system~$S$, it is evident that the components +$t_{xx}$,~$t_{yy}$,~etc., may be taken as the forces acting on the faces of a unit +cube of material at the point in question, the first letter of the subscript +%% -----File: 158.png---Folio 144------- +indicating the direction of the force and the second subscript +the direction of the normal to the face in question. + +Interpreting the components of stress in this fashion, we may +now add three further components and obtain a complete tensor +\[ +\vc{t} = \left\{ +\begin{matrix} +t_{xx} & t_{xy} & t_{xz} \\ +t_{yx} & t_{yy} & t_{yz} \\ +t_{zx} & t_{zy} & t_{zz} +\end{matrix} +\right. +\Tag{174} +\] + +The three new components $t_{yx}$,~$t_{zx}$,~$t_{zy}$ are forces acting on the +unit cube, in the directions and on the faces indicated by the subscripts. +A knowledge of their value was not necessary for our developments +of the consequences of the principle of least action, since it was +possible to obtain an expression for the work accompanying a change +in strain without their introduction. We shall find them quite important +for our later considerations, however, and may proceed to +determine their value. + +$t_{yz}$ is the force acting in the $Y$~direction tangentially to a face of +the cube perpendicular to the $X$\DPchg{-}{~}axis, and measured with a system +of coördinates~$S$. Using a system of \DPchg{coordinates}{coördinates}~$S°$ which is stationary +with respect to the point in question, we should obtain, for the measurement +of this force, +\[ +{t°}_{yx} = \frac{t_{yx}}{\sqrt{1 - \smfrac{u^2}{c^2}}} +\] +in accordance with our transformation equation for force~(62), Chapter~VI\@. %[** TN: Not a useful cross-reference] +Similarly we shall have the relation +\[ +{t°}_{xy} = t_{xy}. +\] +In accordance with the elementary theory of elasticity, however, the +forces ${t°}_{yx}$~and~${t°}_{xy}$ which are measured by an observer moving with +the body will be connected by the relation +\[ +{t°}_{xy} = \frac{{t°}_{yx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +${t°}_{xy}$ being larger than~${t°}_{yx}$ in the ratio of the areas of face upon which +they act. Combining these three equations, and using similar methods +%% -----File: 159.png---Folio 145------- +for the other quantities, we can obtain the desired relations +\[ +t_{yx} = \left(1 - \frac{u^2}{c^2}\right) t_{xy}, \qquad +t_{zx} = \left(1 - \frac{u^2}{c^2}\right) t_{xz}, \qquad +t_{zy} = t_{yz}. +\Tag{175} +\] +We see that $\vc{t}$~is an unsymmetrical tensor. + +\Subsubsection{129}{The Symmetrical Tensor $\vc{p}$.} Besides this unsymmetrical tensor~$\vc{t}$ +we shall find it desirable to define a further tensor~$\vc{p}$ by the +equation +\[ +\vc{p} = \vc{t} + \vc{gu}. +\Tag{176} +\] + +We shall call $\vc{gu}$ the tensor product of $\vc{g}$~and~$\vc{u}$ and may indicate +tensor products in general by a simple juxtaposition of vectors. $\vc{gu}$~is +itself a tensor with components as indicated below: +\[ +\vc{gu} = \left\{ +\begin{matrix} +g_xu_x & g_xu_y & g_xu_z, \\ +g_yu_x & g_yu_y & g_yu_z, \\ +g_zu_x & g_zu_y & g_zu_z. +\end{matrix} +\right. +\Tag{177} +\] + +Unlike $\vc{t}$, $\vc{p}$~will be a symmetrical tensor, since we may show, by +substitution of the values for $\vc{g}$~and~$\vc{u}$ already obtained, that +\[ +p_{yx} = p_{xy}, \qquad +p_{zx} = p_{xz}, \qquad +p_{zy} = p_{yz}. +\Tag{178} +\] +Consider for example the value of~$p_{yx}$; we have from our definition +\[ +p_{yx} = t_{yx} + g_{y}u_{x}, +\] +and by equations (175)~and~(172) we have +\[ +t_{xy} = \left(1 - \frac{u^2}{c^2}\right) t_{xy},\qquad +g_y = t_{xy}\, \frac{u_x}{c^2}, +\] +and hence by substitution obtain +\[ +p_{yx} = t_{xy}. +\] +We also have, however, by definition +\[ +p_{xy} = t_{xy} + g_xu_y, +\] +and since for the case we are considering $u_y = 0$, we arrive at the +equality +\[ +p_{xy} = p_{yx}. +\] +The other equalities may be shown in a similar way. +%% -----File: 160.png---Folio 146------- + +\Subsubsection{130}{Relation between $\divg\vc{t}$~and~$\vc{t}_n$.} At a given point~$P$ in our +elastic body we shall define the divergence of the tensor~$\vc{t}$ by the equation +\[ +\begin{aligned} +\divg\vc{t} + &= \left(\frac{\partial t_{xx}}{\partial x} + + \frac{\partial t_{xy}}{\partial y} + + \frac{\partial t_{xz}}{\partial z}\right) \vc{i} \\ + &+ \left(\frac{\partial t_{yx}}{\partial x} + + \frac{\partial t_{yy}}{\partial y} + + \frac{\partial t_{yz}}{\partial z}\right) \vc{j} \\ + &+ \left(\frac{\partial t_{zx}}{\partial x} + + \frac{\partial t_{zy}}{\partial y} + + \frac{\partial t_{zz}}{\partial z}\right) \vc{k}, +\end{aligned} + \Tag{179} +\] +where $\vc{i}$,~$\vc{j}$ and~$\vc{k}$ are unit vectors parallel to the axes, $\divg\vc{t}$~thus being +an ordinary vector. It will be seen that $\divg\vc{t}$~is the elastic force +acting per unit volume of material at the point~$P$. + +Considering an element of surface~$dS$, we shall define a further +vector~$\vc{t}_n$ by the equation +\[ +\begin{aligned} +\vc{t}_n + &= (t_{xx}\cos\alpha + t_{xy}\cos\beta + t_{xz}\cos\gamma)\, \vc{i} \\ + &+ (t_{yx}\cos\alpha + t_{yy}\cos\beta + t_{yz}\cos\gamma)\, \vc{j} \\ + &+ (t_{zx}\cos\alpha + t_{zy}\cos\beta + t_{zz}\cos\gamma)\, \vc{k}, +\end{aligned} +\Tag{180} +\] +where $\cos \alpha$,~$\cos \beta$ and~$\cos \gamma$ are the direction cosines of the inward-pointing +normal to the element of surface~$dS$. + +Considering now a definite volume~$V$ enclosed by the surface~$S$ +it is evident that $\divg\vc{t}$~and~$\vc{t}_n$ will be connected by the relation +\[ +-\int \divg\vc{t}\, dV = \int_0 \vc{t}_n\, dS, +\Tag{181} +\] +where the symbol~$0$ indicates that the integration is to be taken over +the whole surface which encloses the volume~$V$. This equation is +of course merely a direct application of Gauss's formula, which states +in general the equality +{\small%[** TN: Not breaking] +\[ +-\int \left( + \frac{\partial P}{\partial x} + + \frac{\partial Q}{\partial y} + + \frac{\partial R}{\partial z}\right) dV + = \int_0 (P\cos \alpha + Q\cos \beta + R\cos \gamma)\, dS, +\Tag{182} +\]}% +where $P$,~$Q$ and~$R$ may be any functions of $x$,~$y$ and~$z$. +%% -----File: 161.png---Folio 147------- + +We shall also find use for a further relation between $\divg\vc{t}$~and~$\vc{t}_n$. +Consider a given point of reference~$O$, and let $\vc{r}$~be the radius vector +to any point~$P$ in the elastic body; we can then show with the help +of Gauss's Formula~(182) that +\begin{multline*} +-\int (\vc{r} × \divg\vc{t})\, dV = \int_0 (\vc{r} × \vc{t}_n)\, dS \\ +-\int \bigl[(t_{yz}-t_{zy})\vc{jk} + + (t_{xz}-t_{zx})\vc{ik} + + (t_{xy}-t_{yx})\vc{ij}\bigr]\, dV, +\end{multline*} +where $×$~signifies as usual the outer product. Taking account of +equations (172)~and~(175) this can be rewritten +\[ +-\int (\vc{r} × \divg\vc{t})\, dV + = \int_0 (\vc{r} × \vc{t}_n)\, dS - \int(\vc{u} × \vc{g})\, dV. +\Tag{183} +\] + +\Subsubsection{131}{The Equations of Motion in the Eulerian Form.} We saw in +\DPchg{sections (\Secnumref{124})~and~(\Secnumref{125})}{Sections \Secnumref{124}~and~\Secnumref{125}} that the equations of motion in the Lagrangian +form might be written +\[ +\vc{f}\, \delta V = \frac{d}{dt} (\vc{g}\, \delta V), +\] +where $\vc{f}$~is the density of force acting at any point and $\vc{g}$~is the density +of momentum. + +Provided that there are no external forces acting and $\vc{f}$~is produced +solely by the elastic forces, our definition of the divergence of a +tensor will now permit us to put +\[ +\vc{f} = - \divg\vc{t}, +\] +and write for our equation of motion +\[ +(-\divg\vc{t})\, \delta V + = \frac{d}{dt} (\vc{g}\, \delta V) + = \delta V\, \frac{d\vc{g}}{dt} + \vc{g}\, \frac{d(\delta V)}{dt}. +\] +Expressing $\dfrac{d\vc{g}}{dt}$ in terms of partial differentials, and putting +\[ +\frac{d(\delta V)}{dt} = \delta V \divg\vc{u} +\] +we obtain +\[ +-\divg \vc{t} = \left(\frac{\partial \vc{g}}{\partial t} + + u_x\, \frac{\partial \vc{g}}{\partial x} + + u_y\, \frac{\partial \vc{g}}{\partial y} + + u_z\, \frac{\partial \vc{g}}{\partial z}\right) + + \vc{g} \divg\vc{u}. +\] +%% -----File: 162.png---Folio 148------- +Our symmetrical tensor~$\vc{p}$, however, was defined by the equation~(176) +\[ +\vc{p} = \vc{t} + \vc{gu}, +\] +and hence we may now write our equations of motion in the very +beautiful Eulerian form +\[ +-\divg\vc{p} = \frac{\partial \vc{g}}{\partial t}. +\Tag{184} +\] + +We shall find this simple form for the equations of motion very +interesting in connection with our considerations in the last chapter. + + +\Section[IV]{Applications of the Results.} + +We may now use the results which we have obtained from the +principle of least action to elucidate various problems concerning +the behavior of elastic bodies. + +\Subsubsection{132}{Relation between Energy and Momentum.} In our work on +the dynamics of a particle we found that the mass of a particle was +equal to its energy divided by the square of the velocity of light, and +hence have come to expect in general a necessary relation between +the existence of momentum in any particular direction and the transfer +of energy in that same direction. We find, however, in the case +of elastically stressed bodies a somewhat more complicated state of +affairs than in the case of particles, since besides the energy which is +transported bodily by the motion of the medium an additional quantity +of energy may be transferred through the medium by the action +of the forces which hold it in its state of strain. Thus, for example, +in the case of a longitudinally compressed rod moving parallel to its +length, the forces holding it in its state of longitudinal compression +will be doing work at the rear end of the rod and delivering an equal +quantity of energy at the front end, and this additional transfer of +energy must be included in the calculation of the momentum of the +bar. + +As a matter of fact, an examination of the expressions for momentum +which we obtained from the principle of least action will show +the justice of these considerations. For the density of momentum +in the $X$~direction we obtained the expression +\[ +g_x = (w + t_{xx})\, \frac{\dot{x}}{c^2}, +\] +%% -----File: 163.png---Folio 149------- +and we see that in order to calculate the momentum in the $X$~direction +we must consider not merely the energy~$w$ which is being bodily +carried along in that direction with the velocity~$\dot{x}$, but also must take +into account the additional flow of energy which arises from the +stress~$t_{xx}$. As we have already seen in \Secref{128}, this stress~$t_{xx}$ can +be thought of as resulting from forces which act on the front and +rear faces of a centimeter cube of our material. Since the cube is +moving with the velocity~$\dot{x}$, the force on the rear face will do the +work $t_{xx}\dot{x}$~per second and this will be given up at the forward face. +We thus have an additional density of energy-flow in the $X$~direction +of the magnitude~$t_{xx}\dot{x}$ and hence a corresponding density of momentum~$\dfrac{t_{xx}\dot{x}}{c^2}$. + +Similar considerations explain the interesting occurrence of components +of momentum in the $Y$~and~$Z$ directions, +\[ +g_y = t_{xy}\, \frac{\dot{x}}{c^2},\qquad +g_z = t_{xz}\, \frac{\dot{x}}{c^2}, +\] +in spite of the fact that the material involved is moving in the $X$~direction. +The stress~$t_{xy}$, for example, can be thought of as resulting +from forces which act tangentially in the $X$~direction on the pair of +faces of our unit cube which are perpendicular to the $Y$~axis. Since +the cube is moving in the $X$~direction with the velocity~$\dot{x}$, we shall +have the work~$t_{xy}\dot{x}$, done at one surface per second and transferred to +the other, and the resulting flow of energy in the $X$~direction is accompanied +by the corresponding momentum~$\dfrac{t_{xy}\dot{x}}{c^2}$. + +\Subsubsection{133}{The Conservation of Momentum.} It is evident from our +previous discussions that we may write the equation of motion for +an elastic medium in the form +\[ +\vc{f}\, \delta V = \frac{d(\vc{g}\, \delta V)}{dt}, +\] +where $\vc{g}$~is the density of momentum at any given point and $\vc{f}$~is the +force acting per unit volume of material. We have already obtained, +from the principle of least action, expressions~(172) which permit +the calculation of~$\vc{g}$ in terms of the energy density, stress and velocity +at the point in question, and our present problem is to discuss somewhat +further the nature of the force~$\vc{f}$. +%% -----File: 164.png---Folio 150------- + +We shall find it convenient to analyze the total force per unit +volume of material~$\vc{f}$ into those external forces~$\vc{f}_{\text{ext.}}$ like gravity, which +are produced by agencies outside of the elastic body and the internal +force~$\vc{f}_{\text{int.}}$ which arises from the elastic interaction of the parts of the +strained body itself. It is evident from the way in which we have +defined the divergence of a tensor~(179) that for this latter we may +write +\[ +\vc{f}_{\text{int.}} = -\divg\vc{t}. +\Tag{185} +\] +Our equation of motion then becomes +\[ +(\vc{f}_{\text{ext.}} - \divg\vc{t})\, \delta V + = \frac{d(\vc{g}\, \delta V)}{dt}, +\Tag{186} +\] +or, integrating over the total volume of the elastic body, +\[ +\int \vc{f}_{\text{ext.}}\, dV - \int \divg\vc{t}\, dV + = \frac{d}{dt} \int \vc{g}\, dV + = \frac{d\vc{G}}{dt}, +\Tag{187} +\] +where $\vc{G}$ is the total momentum of the body. With the help of the +purely analytical relation~(181) we may transform the above equation +into +\[ +\int \vc{f}_{\text{ext.}}\, dV + \int \vc{t}_n\, dS = \frac{d\vc{G}}{dt}, +\Tag{188} +\] +where $\vc{t}_n$~is defined in accordance with~(180) so that the integral +$\ds\int_{0} \vc{t}_n\, dS$ becomes the force exerted by the surroundings on the surface +of the elastic body. + +In the case of an isolated system both $\vc{f}_{\text{ext.}}$~and~$\vc{t}_n$ would evidently +be equal to zero and we have the principle of the conservation of +momentum. + +\Subsubsection{134}{The Conservation of Angular Momentum.} Consider the +%[** TN: O and P in next line are boldface in the original] +radius vector~$\vc{r}$ from a point of reference~$O$ to any point~$P$ in an elastic +body; then the angular momentum of the body about~$O$ will be +\[ +\vc{M} = \int (\vc{r} × \vc{g})\, dV, +\] +and its rate of change will be +\[ +\frac{d\vc{M}}{dt} + = \int \left(\vc{r} × \frac{d\vc{g}}{dt}\right) dV + + \int \left(\frac{d\vc{r}}{dt} × \vc{g}\right) dV. +\Tag{189} +\] +%% -----File: 165.png---Folio 151------- +Substituting equation~(186), this may be written +\[ +\frac{d\vc{M}}{dt} + = \int (\vc{r} × \vc{f}_{\text{ext.}})\, dV + - \int (\vc{r} × \divg\vc{t})\, dV + \int (\vc{u} × \vc{g})\, dV, +\] +or, introducing the purely mathematical relation~(183) we have, +\[ +\frac{d\vc{M}}{dt} + = \int (\vc{r} × \vc{f}_{\text{ext.}})\, dV + + \int_{0} (\vc{r} × \vc{t}_n)\, dS. +\Tag{190} +\] +We see from this equation that the rate of change of the angular +momentum of an elastic body is equal to the moment of the external +forces acting on the body plus the moment of the surface forces. + +In the case of an isolated system this reduces to the important +principle of the conservation of angular momentum. + +\Subsubsection{135}{Relation between Angular Momentum and the Unsymmetrical +Stress Tensor.} The fact that at a point in a strained elastic medium +there may be components of momentum at right angles to the motion +of the point itself, leads to the interesting conclusion that even in a +state of steady motion the angular momentum of a strained body +will in general be changing. + +This is evident from equation~(189), in the preceding section, +which may be written +\[ +\frac{d\vc{M}}{dt} + = \int \left(\vc{r} × \frac{d\vc{g}}{dt}\right) dV + + \int(\vc{u} × \vc{g})\, dV. +\Tag{191} +\] +In the older mechanics velocity~$\vc{u}$ and momentum~$\vc{g}$ were always in +the same direction so that the last term of this equation became zero. +%[** TN: Awkward grammar/repeated verb in the original.] +In our newer mechanics, however, we have found~(172) components +of momentum at right angles to the velocity and \emph{hence even for a body +moving in a straight line with unchanging stresses and velocity we find +that the angular momentum is increasing at the rate +\[ +\frac{d\vc{M}}{dt} = \int (\vc{u} × \vc{g})\, dV, +\Tag{192} +\] +and in order to maintain the body in its state of uniform motion we +must apply external forces with a turning moment of this same amount}. + +The presence of this increasing angular momentum in a strained +body arises from the unsymmetrical nature of the stress tensor, the integral +$\int (\vc{u} × \vc{g})\, dV$ being as a matter of fact exactly equal to the integral +%% -----File: 166.png---Folio 152------- +over the same volume of the turning moments of the unsymmetrical +components of the stress. Thus, for example, if we have a body moving +in the $X$~direction with the velocity $\vc{u} = \dot{x}\vc{i}$ we can easily see from +equations (172)~and~(175) the truth of the equality +\[ +(\vc{u} × \vc{g}) + = \bigl[(t_{yz} - t_{zy})\, \vc{jk} + + (t_{xz} - t_{zx})\, \vc{ik} + + (t_{xy} - t_{yx})\, \vc{ij}\bigr]. +\] + +\Subsubsection{136}{The Right-Angled Lever.} An interesting example of the +\begin{wrapfigure}{l}{2.125in} + \Fig{14} + \Input[2in]{166} +\end{wrapfigure} +principle that in general a turning +moment is needed for the uniform +translatory motion of a strained body +is seen in the apparently paradoxical +case of the right-angled lever. + +Consider the right-angled lever +shown in \Figref{14}. This lever is stationary +with respect to a system of +coördinates~$S°$. Referred to~$S°$ the +two lever arms are equal in length: +\[ +{l_1}° = {l_2}°, +\] +and the lever is in equilibrium under the action of the equal forces +\[ +{F_1}° = {F_2}°. +\] + +Let us now consider the equilibrium as it appears, using a system +of coördinates~$S$ with reference to which the lever is moving in $X$~direction +with the velocity~$V$. Referred to this new system of coördinates +the length~$l_1$ of the arm which lies in the $Y$~direction will be +the same as in system~$S°$, giving us +\[ +l_1 = {l_1}°. +\] +But for the other arm which lies in the direction of motion we shall +have, in accordance with the Lorentz shortening, +\[ +l_2 = {l_2}° \sqrt{1 - \frac{V^2}{c^2}}. +\] +For the forces $F_1$~and~$F_2$ we shall have, in accordance with our equations +%% -----File: 167.png---Folio 153------- +for the transformation of force (61)~and~(62), +\begin{align*} +F_1 &= {F_1}°, \\ +F_2 &= {F_2}° \sqrt{1 - \frac{V^2}{c^2}}. +\end{align*} +We thus obtain for the moment of the forces around the pivot~$B$ +\[ +F_1l_1 - F_2l_2 + = {F_1}° {l_1}° + - {F_2}° {l_2}° \left(1 - \frac{V^2}{c^2}\right) + = {F_1}°{l_1}°\, \frac{V^2}{c^2}, + = F_1l_1\, \frac{V^2}{c^2}, +\] +and are led to the remarkable conclusion that such a moving lever +will be in equilibrium only if the external forces have a definite turning +moment of the magnitude given above. + +The explanation of this apparent paradox is obvious, however, +in the light of our previous discussion. In spite of the fact that the +lever is in uniform motion in a straight line, its angular momentum +is continually increasing owing to the fact that it is elastically strained, +and it can be shown by carrying out the integration indicated in +equation~(192) that the rate of change of angular momentum is as a +matter of fact just equal to the turning moment $F_1l_1\, \dfrac{V^2}{c^2}$. + +This necessity for a turning moment $F_1l_1\, \dfrac{V^2}{c^2}$ can also be shown +directly from a consideration of the energy flow in the lever. Since +the force~$F_1$ is doing the work $F_1V$~per second at the point~$A$, a stream +of energy of this amount is continually flowing through the lever +from~$A$ to the pivot~$B$. In accordance with our ideas as to the relation +between energy and mass, this new energy which enters at~$A$ each +second has the mass~$\dfrac{F_1V}{c^2}$, and hence each second the angular momentum +of the system around the point~$B$ is increased by the amount +\[ +\frac{F_1V}{c^2}\, Vl_1 = F_1l_1\, \frac{V^2}{c^2}. +\] +We have already found, however, exactly this same expression for +the moment of the forces around the pivot~$B$ and hence see that they +are of just the magnitude necessary to keep the lever from turning, +thus solving completely our apparent paradox. +%% -----File: 168.png---Folio 154------- + +\Subsubsection{137}{Isolated Systems in a Steady State.} Our considerations have +shown that the density of momentum is equal to the density of energy +flow divided by the square of the velocity of light. If we have a +system which is in a steady internal state, and is either isolated or +merely subjected to an external pressure with no components of force +tangential to the bounding surface, it is evident that the resultant +flow of energy for the whole body must be in the direction of motion, +and hence for these systems momentum and velocity will be in the +same direction without the complications introduced by a transverse +energy flow. + +Thus for an \emph{isolated} system in a steady \emph{internal} state we may +write, +\[ +\vc{G} = \frac{E}{c^2}\, \vc{u} + = \frac{\smfrac{E°}{c^2}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{u}. +\Tag{193} +\] + +\Subsubsection{138}{The Dynamics of a Particle.} It is important to note that +particles are interesting examples of systems in which there will +obviously be no transverse component of energy flow since their +infinitesimal size precludes the action of tangential surface forces. +We thus see that the dynamics of a particle may be regarded as a +special case of the more general dynamics which we have developed +in this chapter, the equation of motion for a particle being +\[ +\vc{F} = \frac{d}{dt} \left[ + \frac{\smfrac{E°}{c^2}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,\vc{u} + \right] + = \frac{d}{dt} \Biggl[ + \frac{m°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\vc{u}\Biggr], +\] +in agreement with the work of \Chapref{VI}. + +\Subsubsection{139}{Conclusion.} We may now point out in conclusion the chief +results of this chapter. With the help of Einstein's equations for +spatial and temporal considerations, we have developed a set of +transformation equations for the strain in an elastic body. Using the +components of strain and velocity as generalized coördinates, we then +introduced the principle of least action, choosing a form of function +%% -----File: 169.png---Folio 155------- +for kinetic potential which agrees at low velocities with the choice +made in the older theories of elasticity and at all velocities agrees +with the requirements of the principle of relativity. Using the +Lagrangian equations, we were then able to develop all that is necessary +for a complete theory of elasticity. + +The most important consequence of these considerations is an +extension in our ideas as to the relation between momentum and +energy. We find that the density of momentum in any direction +must be placed equal to the total density of energy flow in that same +direction divided by the square of the velocity of light; and we find +that we must include in our density of energy flow that transferred +through the elastic body by the forces which hold it in its state of +strain and suffer displacement as the body moves. This involves in +general a flow of energy and hence momentum at right angles to the +motion of the body itself. + +At present we have no experiments of sufficient accuracy so that +we can investigate the differences between this new theory of elasticity +and the older ones, and hence of course have found no experimental +contradiction to the new theory. It will be seen, however, from the +expressions for momentum that even at low velocities the consequences +of this new theory will become important as soon as we +run across elastic systems in which very large stresses are involved. +It is also important to show that a theory of elasticity can be developed +which agrees with the requirements of the theory of relativity. +In fairness, it must, however, be pointed out in conclusion that since +our expression for kinetic potential was not absolutely uniquely determined +there may also be other theories of elasticity which will agree +with the principle of relativity and with all the facts as now known. +%% -----File: 170.png---Folio 156------- + + +\Chapter{XI}{The Dynamics of a Thermodynamic System.} +\SetRunningHeads{Chapter Eleven.}{Dynamics of a Thermodynamic System.} + +We may now use our conclusions as to the relation between the +principle of least action and the theory of relativity to obtain information +as to the behavior of thermodynamic systems in motion. + +\Subsubsection{140}{The Generalized Coördinates and Forces.} Let us consider a +thermodynamic system whose state is defined by the \emph{generalized +coördinates} volume~$v$, entropy~$S$ and the values of $x$,~$y$ and~$z$ which +determine its position. Corresponding to these coördinates we shall +have the generalized external forces, the negative of the pressure,~$-p$, +temperature,~$T$, and the components of force, $F_x$,~$F_y$ and~$F_z$. +These generalized coördinates and forces are related to the energy +change~$\delta E$ accompanying a small displacement~$\delta$, in accordance with +the equation +\[ +\delta E = -\delta W + = -p\, \delta v + T\, \delta S + + F_x\, \delta x + F_y\, \delta y + F_z\, \delta z. +\Tag{194} +\] + +\Subsubsection{141}{Transformation Equation for Volume.} Before we can apply +the principle of least action we shall need to have transformation +equations for the generalized coördinates, volume and entropy. + +In accordance with the Lorentz shortening, we may write the +following expression for the volume~$v$ of the system in terms of~$v°$ as +measured with a set of axes~$S°$ with respect to which the system is +stationary: +\[ +v = v° \sqrt{1 - \frac{u^2}{c^2}} + = v° \sqrt{1 - \frac{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}{c^2}}, +\] +where $u$ is the velocity of the system. + +By differentiation we may obtain expressions which we shall find +useful, +\begin{align*} +\frac{\partial v°}{\partial v} + &= \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\Tag{195}\displaybreak[0] \\ +\frac{\partial v°}{\partial \dot{x}} + &= \frac{v}{\left(1 - \smfrac{u^2}{c^2}\right)^{\frac{3}{2}}}\, + \frac{\dot{x}}{c^2} + = \frac{v°}{\left(1 - \smfrac{u^2}{c^2}\right)}\, + \frac{\dot{x}}{c^2}. +\Tag{196} +\end{align*} +%% -----File: 171.png---Folio 157------- + +\Subsubsection{142}{Transformation Equation for Entropy.} As for the entropy +of a thermodynamic system, this is a quantity which must appear +the same to all observers regardless of their motion. This invariance +of entropy is a direct consequence of the close relation between the +entropy of a system in a given state and the probability of that state. +Let us write, in accordance with the Boltzmann-Planck ideas as to +the interdependence of these quantities, +\[ +S = k\log W, +\] +where $S$ is the entropy of the system in the state in question, $k$~is a +universal constant, and $W$~the probability of having a microscopic +arrangement of molecules or other elementary constituent parts which +corresponds to the desired thermodynamic state. Since this probability +is evidently independent of the relative motion of the observer +and the system we see that the entropy of a system~$S$ must be an +invariant and may write +\[ +S = S°. +\Tag{197} +\] + +\Subsubsection{143}{Introduction of the Principle of Least Action. The Kinetic +Potential.} We are now in a position to introduce the principle of +least action into our considerations by choosing a form of function +for the kinetic potential which will agree at low velocities with the +familiar principles of thermodynamics and will agree at all velocities +with the requirements of the theory of relativity. + +If we use volume and entropy as our generalized coördinates, these +conditions are met by taking for kinetic potential the expression +\[ +H = -E° \sqrt{1 - \frac{u^2}{c^2}}. +\Tag{198} +\] + +This expression agrees with the requirements of the theory of +relativity that $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ shall be an invariant (see \Secref{111}) and +at low velocities reduces to $H = -E$, which with our choice of +coördinates is the familiar form for the kinetic potential of a thermodynamic +system. +%% -----File: 172.png---Folio 158------- + +It should be noted that this expression for the kinetic potential +of a thermodynamic system applies of course only provided we pick +out volume~$v$ and entropy~$S$ as generalized coördinates. If, following +Helmholtz, we should think it more rational to take $v$ as one coördinate +and a quantity~$\theta$ whose time derivative is equal to temperature, +$\dot{\theta} = T$, as the other coördinate, we should obtain of course a different +expression for the kinetic potential; in fact should have under those +circumstances +\[ +H = (E° - T° S°) \sqrt{1 - \frac{u^2}{c^2}}. +\] +Using this value of kinetic potential, however, with the corresponding +coördinates we should obtain results exactly the same as those which +we are now going to work out with the help of the other set of coördinates. + +\Subsubsection{144}{The Lagrangian Equations.} Having chosen a form for the +kinetic potential we may now substitute into the Lagrangian equations~(139) +and obtain the desired information with regard to the +behavior of thermodynamic systems. + +Since we shall consider cases in which the energy of the system is +independent of the position in space, the kinetic potential will be +independent of the coördinates $x$,~$y$ and~$z$, depending only on their +time derivatives. Noting also that the kinetic potential is independent +of the time derivatives of volume and entropy, we shall +obtain the Lagrangian equations in the simple form +\[ +\begin{aligned} +-\frac{\partial}{\partial v} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -p, \\ +-\frac{\partial}{\partial S} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= T, \\ +\frac{d}{dt}\, \frac{\partial}{\partial\dot{x}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_x, \\ +\frac{d}{dt}\, \frac{\partial}{\partial\dot{y}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_y, \\ +\frac{d}{dt}\, \frac{\partial}{\partial\dot{z}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_z. +\end{aligned} +\Tag{199} +\] +%% -----File: 173.png---Folio 159------- + +\Subsubsection{145}{Transformation Equation for Pressure.} We may use the first +of these equations to show that the pressure is a quantity which +appears the same to all observers regardless of their relative motion. +We have +\[ +p = \frac{\partial}{\partial v} \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\right) + = -\sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial v} + = -\sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial v°}\, + \frac{\partial v°}{\partial v}. +\] +But, in accordance with equation~(194), $p° = -\dfrac{\partial E°}{\partial v°}$, and in accordance +with equation~(195), +\[ +\frac{\partial v°}{\partial v} = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +which gives us the desired relation +\[ +p = p°. +\Tag{200} +\] + +Defining pressure as force per unit area, this result will be seen +to be identical with that which is obtained from the transformation +equations for force and area which result from our earliest considerations. + +\Subsubsection{146}{Transformation Equation for Temperature.} The second of +the Lagrangian equations~(199) will provide us information as to +measurements of temperature made by observers moving with different +velocities. We have +\[ +T = \frac{\partial}{\partial S} + \left(E° \sqrt{1 - \frac{u^2}{c^2}}\right) + = \sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial S°}\, + \frac{\partial S°}{\partial S}. +\] +But, in accordance with equation~(194), $\dfrac{\partial E°}{\partial S°} = T°$ and in accordance +with~(197) $\dfrac{\partial S°}{\partial S} = 1$. We obtain as our transformation equation, +\[ +T = T° \sqrt{1 - \frac{u^2}{c^2}}, +\Tag{201} +\] +and see that the quantity $\dfrac{T}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is an invariant for the Lorentz +transformation\DPchg{}{.} +%% -----File: 174.png---Folio 160------- + +\Subsubsection{147}{The Equations of Motion for Quasistationary Adiabatic +Acceleration.} Let us now turn our attention to the last three of the +Lagrangian equations. These are the equations for the motion of a +thermodynamic system under the action of external force. It is +evident, however, that these equations will necessarily apply only +to cases of quasistationary acceleration, since our development of +the principle of least action gave us an equation for kinetic potential +which was true only for systems of infinitesimal extent or large systems +in a steady internal state. It is also evident that we must confine our +considerations to cases of adiabatic acceleration, since otherwise the +value of~$E°$ which occurs in the expression for kinetic potential might +be varying in a perfectly unknown manner. + +The Lagrangian equations for force may be advantageously transformed. +We have +\begin{align*} +F_x &= \frac{d}{dt}\, \frac{\partial}{\partial \dot{x}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) + = \frac{d}{dt} \Biggl[ + \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2} + - \sqrt{1 - \frac{u^2}{c^2}}\, + \frac{\partial E°}{\partial \dot{x}}\Biggr] \\ + &= \frac{d}{dt}\Biggl\{ + \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2} + - \sqrt{1 - \frac{u^2}{c^2}} + \left(\frac{\partial E°}{\partial v°}\, + \frac{\partial v°}{\partial \dot{x}} + + \frac{\partial E°}{\partial S°}\, + \frac{\partial S°}{\partial \dot{x}}\right)\Biggr\}. +\end{align*} +But by equations (194),~(196) and~(197) we have +\[ +\frac{\partial E°}{\partial v°} = -p°, \qquad +\frac{\partial v°}{\partial \dot{x}} + = \frac{v°}{\left(1 - \smfrac{u^2}{c^2}\right)}\, + \frac{\dot{x}}{c^2}, \qquad\text{and}\qquad +\frac{\partial S°}{\partial \dot{x}} = 0. +\] +We obtain +\[ +F_x = \frac{d}{dt}\Biggl\{ + \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}\Biggr\}. +\Tag{202} +\] + +Similar equations may be obtained for the components of force in +the $Y$~and~$Z$ directions and these combined to give the vector equation +\[ +\vc{F} = \frac{d}{dt} \Biggl\{ + \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}\Biggr\}. +\Tag{203} +\] +%% -----File: 175.png---Folio 161------- + +This is the fundamental equation of motion for the dynamics of a +thermodynamic system. + +\Subsubsection{148}{The Energy of a Moving Thermodynamic System.} We may +use this equation to obtain an expression for the energy of a moving +thermodynamic system. If we adiabatically accelerate a thermodynamic +system in the direction of its motion, its energy will increase +both because of the work done by the force +\[ +\vc{F} = \frac{d}{dt} \Biggl\{ + \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}\Biggr\} +\] +which produces the acceleration and because of the work done by the +pressure $p = p°$ which acts on a volume which is continually diminishing +as the velocity~$u$ increases, in accordance with the expression +$v = v° \sqrt{1 - \dfrac{u^2}{c^2}}$. Hence we may write for the total energy +\begin{align*} +E &= E° + \int_0^u \frac{d}{dt}\Biggl\{ + \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}\Biggr\} \vc{u}\, dt + + p° v° \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\:\right)\DPchg{}{,} \\ +E &= \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}} + - p° v° \sqrt{1 - \frac{u^2}{c^2}} + = \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}} - pv. +\Tag{204} +\end{align*} + +\Subsubsection{149}{The Momentum of a Moving Thermodynamic System.} We +may compare this expression for the energy of a thermodynamic +system with the following expression for momentum which is evident +from the equation~(203) for force: +\[ +\vc{G} = \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}. +\Tag{205} +\] + +We find again, as in our treatment of \Chapnumref[X]{elastic bodies} presented +in the last chapter, that the momentum of a moving system may be +calculated by taking the \emph{total} flow of energy in the desired direction +%% -----File: 176.png---Folio 162------- +and dividing by~$c^2$. Thus, comparing equations (204)~and~(205), +we have +\[ +\vc{G} = \frac{E}{c^2}\, \vc{u} + \frac{pv}{c^2}\, \vc{u}, +\Tag{206} +\] +where the term $\dfrac{E}{c^2}\, \vc{u}$ takes care of the energy transported bodily along +by the system and the term $\dfrac{pv}{c^2}\, \vc{u}$ takes care of the energy transferred +in the $\vc{u}$~direction by the action of the external pressure on the rear +and front end of the moving system. + +\Subsubsection{150}{The Dynamics of a Hohlraum.} As an application of our considerations +we may consider the dynamics of a hohlraum, since a +hohlraum in thermodynamic equilibrium is of course merely a special +example of the general dynamics which we have just developed. The +simplicity of the hohlraum and its importance from a theoretical +point of view make it interesting to obtain by the present method the +same expression for momentum that can be obtained directly but +with less ease of calculation from electromagnetic considerations. + +As is well known from the work of Stefan and Boltzmann, the +energy content~$E°$ and pressure~$p°$ of a hohlraum at rest and in thermodynamic +equilibrium are completely determined by the temperature~$T°$ +and volume~$v°$ in accordance with the equations +\begin{align*} +E° &= av° {T°}^4, \\ +p° &= \frac{a}{3}\, {T°}^4, +\end{align*} +where $a$~is the so-called Stefan's constant. + +Substituting these values of $E°$~and~$p°$ in the equation for the +motion of a thermodynamic system~(203), we obtain +\[ +\vc{F} = \frac{d}{dt}\Biggl[ + \frac{4}{3}\, \frac{av° {T°}^4}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}\Biggr] + = \frac{d}{dt}\Biggl[ + \frac{4}{3}\, \frac{avT^4}{\left(1 - \smfrac{u^2}{c^2}\right)^3}\, + \frac{\vc{u}}{c^2}\Biggr] +\Tag{207} +\] +as the equation for the quasistationary adiabatic acceleration of a +%% -----File: 177.png---Folio 163------- +hohlraum. In view of this equation we may write for the momentum +of a hohlraum the expression +\[ +\vc{G} = \frac{4}{3}\, \frac{av°{T°}^4}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}. +\Tag{208} +\] + +It is a fact of significance that our dynamics leads to a result for +the momentum of a hohlraum which had been adopted on the ground +of electromagnetic considerations even without the express introduction +of relativity theory. +%% -----File: 178.png---Folio 164------- + + +\Chapter{XII}{Electromagnetic Theory.} +\SetRunningHeads{Chapter Twelve.}{Electromagnetic Theory.} + +The Einstein theory of relativity proves to be of the greatest +significance for electromagnetics. On the one hand, the new electromagnetic +theory based on the first postulate of relativity obviously +accounts in a direct and straightforward manner for the results of the +Michelson-Morley experiment and other unsuccessful attempts to +detect an ether drift, and on the other hand also accounts just as +simply for the phenomena of moving dielectrics as did the older +theory of a stationary ether. Furthermore, the theory of relativity +provides considerably simplified methods for deriving a great many +theorems which were already known on the basis of the ether theory, +and gives us in general a clarified insight into the nature of electromagnetic +action. + +\Subsubsection{151}{The Form of the Kinetic Potential.} In \Chapref{IX} we investigated +the general relation between the principle of least action +and the theory of the relativity of motion. We saw that the development +of any branch of dynamics would agree with the requirements +of relativity provided only that the kinetic potential~$H$ has such a form +that the quantity $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is an invariant for the Lorentz transformation. +Making use of this discovery we have seen the possibility +of developing the dynamics of a particle, the dynamics of an elastic +body, and the dynamics of a thermodynamic system, all of them in +forms which agree with the theory of relativity by merely introducing +slight modifications into the older expressions for kinetic potential in +such a way as to obtain the necessary invariance for $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$. +In the case of electrodynamics, however, on account of the closely +interwoven historical development of the theories of electricity and +relativity, we shall not find it necessary to introduce any modification +%% -----File: 179.png---Folio 165------- +in the form of the kinetic potential, but may take for~$H$ the following +expression, which is known to lead to the familiar equations of the +Lorentz electron theory +\[ +H = \int dV \left\{\frac{\vc{e}^2}{2} + \frac{\curl \vc{\phi})^2}{2} + - \vc{\phi} · \left(\frac{\dot{\vc{e}}}{c} + \rho\, \frac{\vc{u}}{c}\right) +\right\}, +\Tag{209} +\] +where the integration is to extend over the whole volume of the +system~$V$, $\vc{e}$~is the intensity of the electric field at the point in question, +$\vc{\phi}$~is the value of the vector potential, $\rho$~the density of charge and $\vc{u}$~its +velocity.\footnote + {Strictly speaking this expression for kinetic potential is not quite correct, + since kinetic potential must have the dimensions of energy. To complete the equation + and give all the terms their correct dimensions, we could multiply the first term + by the dielectric inductivity of free space~$\epsilon$, and the last two terms by the magnetic + permeability~$\mu$. Since, however, $\epsilon$~and~$\mu$ have the numerical value unity with the + usual choice of units, we shall not be led into error in our particular considerations + if we omit these factors.} + +Let us now show that the expression which we have chosen for +kinetic potential does lead to the familiar equations of the electron +theory. + +\Subsubsection{152}{The Principle of Least Action.} If now we denote by~$\vc{f}$ the +force per unit volume of material exerted by the electromagnetic +action it is evident that we may write in accordance with the principle +of least action~(135) +\[ +\int dt\, dV \left[\delta \left\{ + \frac{\vc{e}^2}{2} + \frac{(\curl \vc{\phi})^2}{2} + - \vc{\phi} · \left(\frac{\dot{\vc{e}}}{c} + \rho\, \frac{\vc{u}}{c}\right) + \right\} + \vc{f}· \delta\vc{r} \right] = 0, +\Tag{210} +\] +where $\delta\vc{r}$ is the variation in the radius vector to the particle under +consideration, and where the integration is to be taken over the +whole volume occupied by the system and between two instants of +time $t_1$~and~$t_2$ at which the actual and displaced configurations of the +system coincide. + +\Subsubsection{153}{The Partial Integrations.} In order to simplify this equation, +we shall need to make use of two results which can be obtained by +partial integrations with respect to time and space respectively. + +Thus we may write +\[ +\int_{t_1}^{t_2} dt\, (a\, \dot{\delta b}) + = \int_{t_1}^{t_2} a\, d(\delta b) + = [a\, \delta b]_{t_1}^{t_2} + - \int_{t_1}^{t_2} dt \left(\frac{da}{dt}\, \delta b\right), +\] +%% -----File: 180.png---Folio 166------- +or, since the displaced and actual motions coincide at $t_1$~and~$t_2$, +\[ +\int dt\, (a\, \dot{\delta b}) + = -\int dt \left(\frac{da}{dt}\, \delta b\right)\DPtypo{}{.} +\Tag{211} +\] +We may also write +\[ +\int dV \left(a\, \frac{db}{dx}\right) + = \int dy\, dz\, (a\, db) + = \int dy\, dz\, [ab]_{x=-\infty}^{x=+\infty} + - \int dV \left(b\, \frac{da}{dx}\right), +\] +or, since we are to carry out our integrations over the whole volume +occupied by the system, we shall take our functions as zero at the +limits of integration and may write +\[ +\int dV \left(a\, \frac{db}{dx}\right) + = -\int dV \left(b\, \frac{da}{dx}\right). +\Tag{212} +\] +Since similar considerations apply to derivatives with respect to the +other variables $y$~and~$z$, we can also obtain +\begin{gather*} +\int dV\, a \divg\vc{b} = -\int dV\, \vc{b} · \grad a, +\Tag{213} \\ +\int dV\, \vc{a} · \curl\vc{b} = \int dV\, \vc{b} · \curl\vc{a}. +\Tag{214} +\end{gather*} + +\Subsubsection{154}{Derivation of the Fundamental Equations of Electromagnetic +Theory.} {\stretchyspace% +Making use of these purely mathematical relationships we +are now in a position to develop our fundamental equation~(210). +Carrying out the indicated variation, noting that $\delta \vc{u} = \dfrac{d(\delta\vc{r})}{dt}$ and +making use of (211)~and~(214) we easily obtain} +\[ +\begin{aligned} +\int dt\, dV \Biggl[ + \left\{\vc{e} + \frac{1}{c}\, \frac{\partial\vc{\phi}}{\partial t}\right\} + · \delta\vc{e} + &+ \left\{\curl\curl\vc{\phi} - \left(\frac{\dot{\vc{e}}}{c} + + \rho\, \frac{\vc{u}}{c}\right) \right\} · \delta\vc{\phi} \\ + &\qquad\qquad + - \frac{\vc{\phi}}{c} · \delta(\rho\vc{u}) + \vc{f}· \delta\vc{r}\Biggr] + = 0. +\end{aligned} +\Tag{215} +\] + +In developing the consequences of this equation, it should be +noted, however, that the variations are not all of them independent; +thus, since we shall define the density of charge by the equation +\[ +\rho = \divg\vc{e}, +\Tag{216} +\] +it is evident that it will be necessary to preserve the truth of this +equation in any variation that we carry out. This can evidently be +%% -----File: 181.png---Folio 167------- +done if we add to our equation~(215) the expression +\[ +\int dt\, dV\, \psi[\delta\rho - \divg\delta\vc{e}] = 0, +\] +where $\psi$~is an undetermined scalar multiplier. We then obtain with +the help of~(213) +{\small% +%[** TN: Re-breaking] +\[ +\begin{aligned} +&\int dt\, dV \Biggl[ + \left\{\vc{e} + \frac{1}{c}\, \frac{\partial\vc{\phi}}{\partial t} + + \grad\psi \right\} · \delta\vc{e} \\ ++& \left\{\curl\curl\vc{\phi} - \left(\frac{\dot{\vc{e}}}{c} + + \rho\, \frac{\vc{u}}{c}\right)\right\} · \delta\vc{\phi} + - \frac{\vc{\phi}}{c} · \delta(\rho\vc{u}) + \psi\, \delta\rho + + \vc{f} · \delta\vc{r}\Biggr] = 0, +\end{aligned} +\Tag{217} +\]}% +and may now treat the variations $\delta \vc{e}$~and~$\delta\vc{\phi}$ as entirely independent +of the others; we must then have the following equations true +\begin{gather*} +\vc{e} = -\frac{1}{c}\, \frac{\partial \vc{\phi}}{\partial t} - \grad \psi, +\Tag{218} \\ +\curl\curl\vc{\phi} = \frac{\dot{\vc{e}}}{c} + \frac{\rho\vc{u}}{c}, +\Tag{219} +\end{gather*} +and have thus derived from the principle of least action the fundamental +equations of modern electron theory. We may put these in +their familiar form by defining the magnetic field strength~$\vc{h}$ by the +equation +\[ +\vc{h} = \curl\vc{\phi}\DPtypo{}{.} +\Tag{220} +\] +We then obtain from~(219) +\begin{align*}%[** TN: Next four equations not aligned in original] +\curl\vc{h} &= \frac{1}{c}\, \frac{\partial\vc{e}}{\partial t} + + \rho\, \frac{\vc{u}}{c}, +\Tag{221} \\ +\intertext{and, noting the mathematical identity $\curl\grad\psi = 0$, we obtain +from (218)} +\curl\vc{e} &= -\frac{1}{c}\, \frac{\partial\vc{h}}{\partial t}. +\Tag{222} \\ +\intertext{We have furthermore by definition~(216)} +\divg\vc{e} &= \rho, +\Tag{223} \\ +\intertext{and noting equation~(220) may write the mathematical identity} +\divg\vc{h} &= 0. +\Tag{224} +\end{align*} +%% -----File: 182.png---Folio 168------- + +These four equations~\DPchg{(221--4)}{(221)--(224)} are the familiar expressions which +have been made the foundation of modern electron theory. They +differ from Maxwell's original four field equations only by the introduction +in (221)~and~(223) of terms which arise from the density of +charge~$\rho$ of the electrons, and reduce to Maxwell's set in free space. + +\Paragraph{155.} We have not yet made use of the last three terms in the +fundamental equation~(217) which results from the principle of least +action. As a matter of fact, it can be shown that these terms can be +transformed into the expression +\[ +\int dt\, dV \left[ + \frac{\rho}{c}\, \frac{\partial\vc{\phi}}{\partial t} + - \frac{\rho}{c}\, [\vc{u} × \curl\vc{\phi}]^* + + \rho \grad\psi + \vc{f}\right] · \delta\vc{r}, +\Tag{225} +\] +and hence lead to the familiar fifth fundamental equation of modern +electron theory, +\begin{align*} +\vc{f} &= \rho \left\{-\frac{\partial\vc{\phi}}{c\partial t} + - \grad\psi + \left[\frac{\vc{u}}{c} × \curl\vc{\Phi}\right]^*\right\}, \\ +\vc{f} &= \rho \left\{\vc{e} + \left[\frac{\vc{u}}{c} × \vc{h}\right]^*\right\}. +\Tag{226} +\end{align*} +The transformation of the last three terms of~(217) into the form +given above~(225) is a complicated one and it has not seemed necessary +to present it here since in a later paragraph we shall show the +possibility of deriving the fifth fundamental equation of the electron +theory~(226) by combining the four field equations~\DPchg{(221--4)}{(221)--(224)} with the +transformation equations for force already obtained from the principle +of relativity. The reader may carry out the transformation himself, +however, if he makes use of the partial integrations which we have +already obtained, notes that in accordance with the principle of the +conservation of electricity we must have $\delta\rho = - \divg\rho\, \delta\vc{r}$ and notes +that $\delta\vc{u} = \dfrac{d(\delta\vc{r})}{dt}$, where the differentiation $\smash{\dfrac{d}{dt}}\rule{0pt}{12pt}$ indicates that we are +following some particular particle in its motion, while the differentiation +$\dfrac{\partial}{\partial t}$ occurring in $\dfrac{\partial\vc{\phi}}{\partial t}$ indicates that we intend the rate of change +at some particular stationary point. + +\Subsubsection{156}{The Transformation Equations for $\vc{e}$,~$\vc{h}$ and~$\rho$.} We have thus +shown the possibility of deriving the fundamental equations of modern +%% -----File: 183.png---Folio 169------- +electron theory from the principle of least action. We now wish to +introduce the theory of relativity into our discussions by presenting +a set of equations for transforming measurements of $\vc{e}$,~$\vc{h}$ and~$\rho$ from +one set of space-time coördinates~$S$ to another set~$S'$ moving past~$S$ +in the $X$\DPchg{-}{~}direction with the velocity~$V$. This set of equations is as +follows: +\begin{gather*} %[** TN: Set equation groups on one line each] +\begin{alignat*}{3} +{e_x}' &= e_x, \qquad & +{e_y}' &= \kappa \left(e_y - \frac{V}{c}h_z\right),\qquad & +{e_z}' &= \kappa \left(e_z + \frac{V}{c}h_y\right), \Tag{227}\displaybreak[0] \\ +{h_x}' &= h_x, & +{h_y}' &= \kappa \left(h_y + \frac{V}{c}e_z\right), & +{h_z}' &= \kappa \left(h_z - \frac{V}{c}e_y\right), \Tag{228} +\end{alignat*} \displaybreak[0] \\ +\rho' = \rho\kappa \left(1 - \frac{u_zV}{c^2}\right), \Tag{229} +\end{gather*} +where $\kappa$ has its customary significance $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$. +\bigskip%[** Explicit space] + +As a matter of fact, this set of transformation equations fulfills +all the requirements imposed by the theory of relativity. Thus, in +the first place, it will be seen, on development, that these equations +are themselves perfectly symmetrical with respect to the primed and +unprimed quantities except for the necessary change from $+V$~to~$-V$. +In the second place, it will be found that the substitution of +these equations into our five fundamental equations for electromagnetic +theory \DPchg{(221--2--3--4--6)}{(221), (222), (223), (224), (226)} will successfully transform them +into an entirely similar set with primed quantities replacing the +unprimed ones. And finally it can be shown that these equations +agree with the general requirement derived in \Chapref{IX} that the +%% -----File: 184.png---Folio 170------- +quantity $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ shall be an invariant for the Lorentz transformation. + +To demonstrate this important invariance of $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ we may +point out that by introducing equations (220),~(221) and~(214), our +original expression for kinetic potential +\[ +H = \int dV \left\{ + \frac{\vc{e}^2}{2} + \frac{(\curl\vc{\phi})^2}{2} + - \vc{\phi} · \left(\frac{\dot{\vc{e}}}{c} + \rho\, \frac{\vc{u}}{c}\right) + \right\} +\] +can easily be shown equal to +\[ +\int dV \left(\frac{\vc{e}^2}{2} - \frac{\vc{h}^2}{2}\right), +\Tag{230} +\] +and, noting that our fundamental equations for space and time provide +us with the relation +\[ +\frac{dV}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = \frac{dV'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}, +\] +we can easily show that our transformation equations for $\vc{e}$~and~$\vc{h}$ do +lead to the equality +\[ +\frac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = \frac{H'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}. +\] + +We thus know that our development of the fundamental equations +for electromagnetic theory from the principle of least action is indeed +in complete accordance with the theory of relativity, since it conforms +with the general requirement which was found in \Chapref{IX} to be +imposed by the theory of relativity on all dynamical considerations. + +\Subsubsection{157}{The Invariance of Electric Charge.} As to the significance of +the transformation equations which we have presented for $\vc{e}$,~$\vc{h}$ and~$\rho$, +we may first show, in accordance with the last of these equations, +that a given electric charge will appear the same to all observers no +matter what their relative motion. +%% -----File: 185.png---Folio 171------- + +To demonstrate this we merely have to point out that, by introducing +equation~(17), we may write our transformation equation +for~$\rho$~(229) in the form +\[ +\frac{\rho'}{\rho} + = \frac{\sqrt{1 - \smfrac{u^2}{c^2}}}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}, +\] +which shows at once that the two measurements of density of charge +made by $O$~and~$O'$ are in exactly the same ratio as the corresponding +measurements for the Lorentz shortening of the charged body, so +that the total charge will evidently measure the same for the two +observers. + +We might express this invariance of electric charge by writing the +equation +\[ +Q' = Q. +\Tag{231} +\] + +It should be noted in passing that this result is in entire accord +with the whole modern development of electrical theory, which lays +increasing stress on the fundamentality and indivisibility of the +electron as the natural unit quantity of electricity. On this basis +the most direct method of determining the charge on an electrified +body would be to count the number of electrons present and this +number must obviously appear the same both to observer~$O$ and +observer~$O'$.\footnote + {A similar invariance of electric charge has been made fundamental in the + author's development of the theory of similitude (\ie, the theory of the relativity + of size). See for example \textit{Phys.\ Rev}., vol.~3, p.~244 (1914).} + +\Subsubsection{158}{The Relativity of Magnetic and Electric Fields.} As to the +significance of equations (227)~and~(228) for transforming the values +of the electric and magnetic field strengths from one system to another, +we see that at a given point in space we may distinguish between the +electric vector $\vc{e} = e_x\, \vc{i} + e_y\, \vc{j} + e_z\, \vc{k}$ as measured by our original +observer~$O$ and the vector $\vc{e}' = {e_x}'\, \vc{i} + {e_y}'\, \vc{j} + {e_z}'\, \vc{k}$ as measured in +units of his own system by an observer~$O'$ who is moving past~$O$ with +the velocity~$V$ in the $X$\DPchg{-}{~}direction. Thus if $O$~finds in an unvarying +electromagnetic field that $Q\vc{e}$~is the force on a small test charge~$Q$ +which is stationary with respect to his system, $O'$~will find experimentally +%% -----File: 186.png---Folio 172------- +for a similar test charge that moves along with him a value +for the force~$Q\vc{e}'$, where $\vc{e}'$~can be calculated from with the help of +these equations~(227). Similar remarks would apply to the forces +which would act on magnetic poles. + +These considerations show us that we should now use caution in +speaking of a pure electrostatic or pure magnetic field, since the +description of an electromagnetic field is determined by the particular +choice of coördinates with reference to which the field is measured. + +\Subsubsection{159}{Nature of Electromotive Force.} We also see that the ``electromotive'' +force which acts on a charge moving through a magnetic +field finds its interpretation as an ``electric'' force provided we make +use of a system of coördinates which are themselves stationary with +respect to the charge. Such considerations throw light on such questions, +for example, as to the seat of the ``electromotive'' forces in +``homopolar'' electric dynamos where there is relative motion of a +conductor and a magnetic field. + + +\Subsection{Derivation of the Fifth Fundamental Equation.} + +\Paragraph{160.} We may now make use of this fact that the forces acting on +a moving charge of electricity may be treated as purely electrostatic, +by using a set of coördinates which are themselves moving along with +the charge, to derive the fifth fundamental equation of electromagnetic +theory. + +Consider an electromagnetic field having the values $\vc{e}$~and~$\vc{h}$ for +the electric and magnetic field strengths at some particular point. +What will be the value of the electromagnetic force~$\vc{f}$ acting per +unit volume on a charge of density~$\rho$ which is passing through the +point in question with the velocity~$\vc{u}$? + +To solve the problem take a system of coördinates~$S'$ which itself +moves with the same velocity as the charge, for convenience letting +the $X$\DPchg{-}{~}axis coincide with the direction of the motion of the charge. +Since the charge of electricity is stationary with respect to this system, +the force acting on it as measured in units of this system will be by +definition equal to the product of the charge by the strength of the +electric field as it appears to an observer in this system, so that we may +write +\[ +\vc{F} = Q'\vc{e}', +\] +%% -----File: 187.png---Folio 173------- +or +\[ +{F_x}' = Q'{e_x}', \qquad +{F_y}' = Q'{e_y}', \qquad +{F_z}' = Q'{e_z}'. +\] +For the components of the electrical field ${e_x}'$,~${e_y}'$,~${e_z}'$, we have just +obtained the transformation equations~(227), while in our earlier +dynamical considerations in \Chapref{VI} we obtained transformation +equations (61),~(62), and~(63) for the components of force. Substituting +above and bearing in mind that $u_x = V$, $u_y = u_z = 0$, and +that $Q' = Q$, we obtain on simplification +\begin{align*} +F_x &= Q e_x, \\ +F_y &= Q \left(e_y - \frac{u_x}{c}h_z\right), \\ +F_z &= Q \left(e_z - \frac{u_x}{c}h_y\right), +\end{align*} +which in vectorial form gives us the equation +\[ +\vc{F} = Q \left(\vc{e} - \frac{1}{c}\, [\vc{u} × \vc{h}]^*\right) +\] +or for the force per unit volume +\[ +\vc{f} = \rho \left(\vc{e} + \frac{1}{c}\, [\vc{u} × \vc{h}]^*\right). +\Tag{226} +\] + +This is the well-known fifth fundamental equation of the Maxwell-Lorentz +theory of electromagnetism. We have already indicated the +method by which it could be derived from the principle of least action. +This derivation, however, from the transformation equations, provided +by the theory of relativity, is particularly simple and attractive. + + +\Subsection{Difference between the Ether and the Relativity Theories of Electromagnetism.} + +\Paragraph{161.} In spite of the fact that we have now found five equations +which can be used as a basis for electromagnetic theory which agree +with the requirements of relativity and also have exactly the same +form as the five fundamental equations used by Lorentz in building +up the stationary ether theory, it must not be supposed that the +relativity and ether theories of electromagnetism are identical. Although +the older equations have exactly the same form as the ones +which we shall henceforth use, they have a different interpretation, +since our equations are true for measurements made with the help +of any non-accelerated set of coördinates, while the equations of +%% -----File: 188.png---Folio 174------- +Lorentz were, in the first instance, supposed to be true only for measurements +which were referred to a set of coördinates which were +stationary with respect to the assumed luminiferous ether. Suppose, +for example, we desire to calculate with the help of equation~(226), +\[ +\vc{t} = \rho \left(\vc{e} + \frac{1}{\vc{c}}\, [\vc{u} × \vc{h}]^*\right), +\] +the force acting on a charged body which is moving with the velocity~$\vc{u}$; +we must note that for the stationary ether theory, $\vc{u}$~must be the +velocity of the charged body through the ether, while for us $\vc{u}$~may be +taken as the velocity past any set of unaccelerated coördinates, provided +$\vc{e}$~and~$\vc{h}$ are measured with reference to the same set of coördinates. +It will be readily seen that such an extension in the meaning +of the fundamental equations is an important simplification. + +\Paragraph{162.} A word about the development from the theory of a stationary +ether to our present theory will not be out of place. When it was +found that the theory of a stationary ether led to incorrect conclusions +in the case of the Michelson-Morley experiment, the hypothesis +was advanced by Lorentz and Fitzgerald that the failure of that +experiment to show any motion through the ether was due to a contraction +of the apparatus in the direction of its motion through the +ether in the ratio $1 : \sqrt{1 - \dfrac{u^2}{c^2}}$. Lorentz then showed that if all systems +should be thus contracted in the line of their motion through the +ether, and observers moving with such system make use of suitably +contracted meter sticks and clocks adjusted to give what Lorentz +called the ``local time,'' their measurements of electromagnetic +phenomena could be described by a set of equations which have +nearly the same form as the original four field equations which would +be used by a stationary observer. It will be seen that Lorentz was +thus making important progress towards our present idea of the complete +relativity of motion. The final step could not be taken, however, +without abandoning our older ideas of space and time and giving up +the Galilean transformation equations as the basis of kinematics. +It was Einstein who, with clearness and boldness of vision, pointed +out that the failure of the Michelson-Morley experiment, and all +other attempts to detect motion through the ether, is not due to a +%% -----File: 189.png---Folio 175------- +fortuitous compensation of effects but is the expression of an important +general principle, and the new transformation equations for kinematics +to which he was led have not only provided the basis for an \emph{exact} +transformation of the field equations but have so completely revolutionized +our ideas of space and time that hardly a branch of science +remains unaffected. + +\Paragraph{163.} With regard to the present status of the ether in scientific +theory, it must be definitely stated that this concept has certainly +lost both its fundamentality and the greater part of its usefulness, +and this has been brought about by a gradual process which has only +found its culmination in the work of Einstein. Since the earliest +days of the luminiferous ether, the attempts of science to increase the +substantiality of this medium have met with little success. Thus +we have had solid elastic ethers of most extreme tenuity, and ethers +with a density of a thousand tons per cubic millimeter; we have had +quasi-material tubes of force and lines of force; we have had vibratory +gyrostatic ethers and perfect gases of zero atomic weight; but after +every debauch of model-making, science has recognized anew that a +correct mathematical description of the actual phenomena of light +propagation is superior to any of these sublimated material media. +Already for Lorentz the ether had been reduced to the bare function +of providing a stationary system of reference for the measurement of +positions and velocities, and now even this function has been taken +from it by the work of Einstein, which has shown that any unaccelerated +system of reference is just as good as any other. + +To give up the notion of an ether will be very hard for many +physicists, in particular since the phenomena of the interference and +polarization of light are so easily correlated with familiar experience +with wave motions in material elastic media. Consideration will +show us, however, that by giving up the ether we have done nothing +to destroy the periodic or polarizable nature of a light disturbance. +When a plane polarized beam of light is passing through a given +point in space we merely find that the electric and magnetic fields at +that point lie on perpendiculars to the direction of propagation and +undergo regular periodic changes in magnitude. There is no need of +going beyond these actual experimental facts and introducing any +hypothetical medium. It is just as simple, indeed simpler, to say +%% -----File: 190.png---Folio 176------- +that the electric or magnetic field has a certain intensity at a given +point in space as to speak of a complicated sort of strain at a given +point in an assumed ether. + + +\Subsection{Applications to Electromagnetic Theory.} + +\Paragraph{164.} The significant fact that the fundamental equations of the +new electromagnetic theory have the same form as those of Lorentz +makes it of course possible to retain in the structure of modern electrical +theory nearly all the results of his important researches, care +being taken to give his mathematical equations an interpretation in +accordance with the fundamental ideas of the theory of relativity. It +is, however, entirely beyond our present scope to make any presentation +of electromagnetic theory as a whole, and in the following paragraphs +we shall confine ourselves to the proof of a few theorems which +can be handled with special ease and directness by the methods introduced +by the theory of relativity. + +\Subsubsection{165}{The Electric and Magnetic Fields around a Moving Charge.} +Our transformation equations for the electromagnetic field make it +very easy to derive expressions for the field around a point charge in +uniform motion. Consider a point charge~$Q$ moving with the velocity~$V$. +For convenience consider a system of reference~$S$ such that $Q$~is +moving along the $X$\DPchg{-}{~}axis and at the instant in question, $t=0$, let the +charge coincide with the origin of coördinates~$O$. We desire now to +calculate the values of electric field~$\vc{e}$ and the magnetic field~$\vc{h}$ at any +point in space $x$,~$y$,~$z$. + +Consider another system of reference,~$S'$, which moves along with +the same velocity as the charge~$Q$, the origin of coördinates~$O'$\DPchg{,}{} and +the charge always coinciding in position. Since the charge is stationary +with respect to their new system of reference, we shall have +the electric field at any point $x'$,~$y'$,~$z'$ in this system given by the +equations +\begin{align*} +{e_x}' &= \frac{Qx'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \\ +{e_y}' &= \frac{Qy'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \\ +{e_z}' &= \frac{Qz'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, +\end{align*} +%% -----File: 191.png---Folio 177------- +while the magnetic field will obviously be zero for measurements made +in system~$S'$, giving us +\[%[** TN: Setting on one line] +{h_x}' = 0, \qquad {h_y}' = 0, \qquad {h_z}' = 0. +\] +Introducing our transformation equations (9),~(10) and~(11) for $x'$,~$y'$ +and~$z'$ and our transformation equations (227)~and~(228) for the +electric and magnetic fields and substituting $t=0$, we obtain for the +values of $\vc{e}$~and~$\vc{h}$ in system~$S$ at the instant when the charge passes +through the point~$O$, +\begin{align*} +e_x &= \frac{Q\kappa x}{(\kappa^2x^2 + y^2 + z^2)^{3/2}} + = \frac{Q \left(1 - \smfrac{V^2}{c^2}\right) x} + {\left[x^2 + \left(1 - \smfrac{V^2}{c^2}\right)(y^2 + z^2)\right]^{3/2}}, +\displaybreak[0] \\ +e_y &= \frac{Q\kappa y}{(\kappa^2x^2 + y^2 + z^2)^{3/2}} + = \frac{Q \left(1 - \smfrac{V^2}{c^2}\right) y} + {\left[x^2 + \left(1 - \smfrac{V^2}{c^2}\right)(y^2 + z^2)\right]^{3/2}}, +\displaybreak[0] \\ +e_z &= \frac{Q\kappa z}{(\kappa^2x^2 + y^2 + z^2)^{3/2}} + = \frac{Q \left(1 - \smfrac{V^2}{c^2}\right) z} + {\left[x^2 + \left(1 - \smfrac{V^2}{c^2}\right)(y^2 + z^2)\right]^{3/2}}, +\displaybreak[0] \\ +h_x &= 0, \\ +h_y &= -\frac{V}{c}\, e_z,\\ +h_z &= \frac{V}{c}\, e_y, +\end{align*} +or, putting $s$ for the important quantity $\sqrt{x^2 + \left(1 - \dfrac{V^2}{c^2}\right)(y^2 + z^2)}$ +and writing the equations in the vectorial form where we put +\[ +\vc{r} = (x\, \vc{i} + y\, \vc{j} + z\, \vc{k}), +\] +we obtain the familiar equations for the field around a point charge +%% -----File: 192.png---Folio 178------- +in uniform motion with the velocity $u=V$ in the $X$\DPchg{-}{~}direction +\begin{gather*} +\vc{e} = Q\, \frac{\left(1 - \smfrac{u^2}{c^2}\right)\vc{r}}{s^3}, +\Tag{232} \\ +\vc{h} = \frac{1}{c}\, [\vc{u} × \vc{e}]\DPtypo{.^*}{^*.} +\Tag{233} +\end{gather*} + +\Subsubsection{166}{The Energy of a Moving Electromagnetic System.} Our +transformation equations will permit us to obtain a very important +expression for the energy of an isolated electromagnetic system in +terms of the velocity of the system and the energy of the same system +as it appears to an observer who is moving along with it. + +Consider a physical system surrounded by a shell which is impermeable +to electromagnetic radiation. This system is to be thought +of as consisting of the various mechanical parts, electric charges and +electromagnetic fields which are inside of the impermeable shell. +The system is free in space, except that it may be acted on by external +electromagnetic fields, and its energy content thus be changed. + +Let us now equate the increase in the energy of the system to the +work done by the action of the external field on the electric charges +in the system. Since the force which a magnetic field exerts on a +charge is at right angles to the motion of the charge it does no work +and we need to consider only the work done by the external electric +field and may write for the increase in the energy of the system +\[ +\Delta E %[** TN: Textstyle integral in original] + = \iiiint \rho(e_xu_x + e_yu_y + e_zu_z)\, dx\, dy\, dz\, dt, +\Tag{234} +\] +where the integration is to be taken over the total volume of the +system and over any time interval in which we may be interested. + +Let us now transform this expression with the help of our transformation +equations for the electric field~(227) for electric charge~(229), +and for velocities \DPchg{(14--15--16)}{(14), (15), (16)}. Noting that our fundamental +equations for kinematic quantities give us $dx\, dy\, dz\, dt = dx'\, dy'\, dz'\, dt'$, +we obtain +\begin{align*} +\Delta E &= \kappa \iiiint + \rho'({e_x}'{u_x}' + {e_y}'{u_y}' + {e_z}'{u_z}')\, dx'\, dy'\, dz'\, dt' \\ + &\quad + + \kappa V \iiiint \rho'\left( + {e_x}' + \frac{{u_y}'}{c}\, {h_z}' - \frac{{u_z}'}{c}\, {h_y}' + \right) dx'\, dy'\, dz'\, dt'. +\end{align*} +%% -----File: 193.png---Folio 179------- + +Consider now a system which \emph{both at the beginning and end of our +time interval is free from the action of external forces}; we may then +rewrite the above equation for this special case in the form +\[ +\Delta E = \kappa \Delta E' + + \kappa V \int \Sum {F_x}'\, dt', +\] +where, in accordance with our earlier equation~(234), $\Delta E'$~is the increase +in the energy of the system as it appears to observer~$O'$ and $\Sum {F_x}'$ +is the total force acting on the system in $X$\DPchg{-}{~}direction as measured +by~$O'$. + +The restriction that the system shall be unacted on by external +forces both at the beginning and end of our time interval is necessary +because it is only under those circumstances that an integration +between two values of~$t$ can be considered as an integration between +two definite values of~$t'$, simultaneity in different parts of the system +not being the same for observers $O$~and~$O'$. + +We may now apply this equation to a specially interesting case. +Let the system be of such a nature that we can speak of it as being +at rest with respect to~$S'$, meaning thereby that all the mechanical +parts have low velocities with respect to~$S'$ and that their center of +gravity moves permanently along with~$S'$. Under these circumstances +we may evidently put $\int\Sum {F_x}'\, dt' = 0$ and may write the +above equation in the form +\begin{align*} +\Delta E &= \frac{\Delta E_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}, \\ +\intertext{or} +\frac{\partial \Delta E}{\partial E_0} + &= \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\end{align*} +where $u$~is the velocity of the system, and $E°$~is its energy as measured +by an observer moving along with it. The energy of a system which +is \emph{unacted on by external forces} is thus a function of two variables, its +energy~$E_0$ as measured by an observer moving along with the system +and its velocity~$u$. +%% -----File: 194.png---Folio 180------- + +We may now write +\[ +E = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, E_0 + \phi(u) + \text{const.}, +\] +where $\phi(u)$ represents the energy of the system which depends solely +on the velocity of the system and not on the changes in its $E_0$~values. +$\phi(u)$~will thus evidently be the kinetic energy of the mechanical masses +in the system which we have already found~(82) to have the value +$\dfrac{m_0c^2}{\sqrt{1 - \smfrac{u^2}{c^2}}} - m_0c^2$ where $m_0$~is to be taken as the total mass of the +mechanical part of our system when at rest. We may now write +\[ +E = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, (m_0c^2 + E_0) + - m_0c^2 + \text{const.} +\] +Or, assuming as before that the constant is equal to~$m_0c^2$, which will +be equivalent to making a system which has zero energy also have +zero mass, we obtain +\[ +E = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, (m_0c^2 + E_0), +\Tag{235} +\] +which is the desired expression for the energy of an isolated system +which may contain both electrical and mechanical parts. + +\Subsubsection{167}{Relation between Mass and Energy.} This expression for the +energy of a system that contains electrical parts permits us to show +that the same relation which we found between mass and energy for +mechanical systems also holds in the case of electromagnetic energy. +Consider a system containing electromagnetic energy and enclosed +by a shell which is impermeable to radiation. Let us apply a force~$\vc{F}$ +to the system in such a way as to change the velocity of the system +without changing its $E_0$~value. We can then equate the work done +per second by the force to the rate of increase of the energy of the +system. We have +\[ +\vc{F} · \vc{u} = \frac{dE}{dt}. +\] +%% -----File: 195.png---Folio 181------- +But from equation~(235) we can obtain a value for the rate of increase +of energy~$\dfrac{dE}{dt}$, giving us +\[ +\vc{F} · \vc{u} + = F_xu_x + F_yu_y + F_zu_z + = \left(m_0 + \frac{E_0}{c^2}\right) + \frac{u\, \smfrac{du}{dt}}{\left(1 - \smfrac{u^2}{c^2}\right)^{\tfrac{3}{2}}}, +\] +and solving this equation for~$\vc{F}$ we obtain +\begin{align*} +\vc{F} &= \frac{d}{dt}\left[ + \frac{\left(m_0 + \smfrac{E_0}{c^2}\right)} + {\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{u} + \right], +\Tag{236} \\ +\intertext{which for low velocities assumes the form} +\vc{F} &= \frac{d}{dt}\left[\left(m_0 + \frac{E_0}{c^2}\right) \vc{u}\right]. +\Tag{237} +\end{align*} + +Examination of these expressions shows that our system which +contains electromagnetic energy behaves like an ordinary mechanical +system with the mass $\left(m_0 + \dfrac{E_0}{c^2}\right)$ at low velocities or $\dfrac{m_0 + \smfrac{E_0}{c^2}}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ at +any desired velocity~$u$. To the energy of the system~$E_0$, part of which +is electromagnetic, we must ascribe the mass~$\dfrac{E_0}{c^2}$ just as we found in +the case of mechanical energy. We realize again that matter and +energy are but different names for the same fundamental entity, +$10^{21}$~ergs of energy having the mass $1$~gram. + + +\Subsection{The Theory of Moving Dielectrics.} + +\Paragraph{168.} The principle of relativity proves to be very useful for the +development of the theory of moving dielectrics. + +It was first shown by Maxwell that a theory of electromagnetic +phenomena in material media can be based on a set of field equations +similar in form to those for free space, provided we introduce besides +the electric and magnetic field strengths, $\vc{E}$~and~$\vc{F}$, two new field vectors, +%% -----File: 196.png---Folio 182------- +the dielectric displacement~$\vc{D}$ and the magnetic induction~$\vc{B}$, and +also the density of electric current in the medium~$\vc{i}$. These quantities +are found to be connected by the four following equations similar in +form to the four field equations for free space: +\begin{align*} +\curl \vc{H} + &= \frac{1}{c} \left(\frac{\partial\vc{D}}{\partial t} + \vc{i}\right), + \Tag{238} \\ +\curl \vc{E} + &= -\frac{1}{c}\, \frac{\partial\vc{B}}{\partial t}, \Tag{239} \\ +\divg \vc{D} &= \rho, + \Tag{240} \\ +\divg \vc{B} &= 0. + \Tag{241} +\end{align*} + +For \emph{stationary homogeneous} media, the dielectric displacement, +magnetic induction and electric current are connected with the +electric and magnetic field strengths by the following equations: +\begin{align*} +\vc{D} &= \epsilon \vc{E}, \Tag{242}\\ +\vc{B} &= \mu \vc{H}, \Tag{243}\\ +\vc{i} &= \sigma \vc{E}, \Tag{244} +\end{align*} +where $\epsilon$~is the dielectric constant, $\mu$~the magnetic permeability and $\sigma$~the +electrical conductivity of the medium in question. + +\Subsubsection{169}{Relation between Field Equations for Material Media and +Electron Theory.} It must not be supposed that the four field equations +\DPchg{(238--241)}{(238)--(241)} for electromagnetic phenomena in \emph{material media} are +in any sense contradictory to the four equations \DPchg{(221--224)}{(221)--(224)} for free +space which we took as the fundamental basis for our development of +electromagnetic theory. As a matter of fact, one of the main achievements +of modern electron theory has been to show that the electromagnetic +behavior of material media can be explained in terms of +the behavior of the individual electrons and ions which they contain, +these electrons and ions acting in accordance with the four fundamental +field equations for free space. Thus our new equations for material +media merely express from a \emph{macroscopic} point of view the statistical +result of the behavior of the individual electrons in the material in +question. $\vc{E}$~and~$\vc{H}$ in these new equations are to be looked upon as +the average values of $\vc{e}$~and~$\vc{h}$ which arise from the action of the +individual electrons in the material, the process of averaging being so +%% -----File: 197.png---Folio 183------- +carried out that the results give the values which a \emph{macroscopic} observer +would actually find for the electric and magnetic forces acting +respectively on a unit charge and a unit pole at the point in question. +These average values, $\vc{E}$~and~$\vc{H}$, will thus pay no attention to the +rapid fluctuations of $\vc{e}$~and~$\vc{h}$ which arise from the action and motion +of the individual electrons, the macroscopic observer using in fact +differentials for time,~$dt$, and space,~$dx$, which would be large from a +microscopic or molecular viewpoint. + +Since from a microscopic point of view $\vc{E}$~and~$\vc{H}$ are not really +the instantaneous values of the field strength at an actual point in +space, it has been found necessary to introduce two new vectors, +electric displacement,~$\vc{D}$, and magnetic induction,~$\vc{B}$, whose time +rate of change will determine the curl of $\vc{E}$~and~$\vc{H}$ respectively. It will +evidently be possible, however, to relate $\vc{D}$~and~$\vc{B}$ to the actual electric +and magnetic fields $\vc{e}$~and~$\vc{h}$ produced by the individual electrons, +and this relation has been one of the problems solved by modern +electron theory, and the field equations \DPchg{(238--241)}{(238)--(241)} for material media +have thus been shown to stand in complete agreement with the most +modern views as to the structure of matter and electricity. For +the purposes of the rest of our discussion we shall merely take these +equations as expressing the experimental facts in stationary or in +moving media. + +\Subsubsection{170}{Transformation Equations for Moving Media.} Since equations +\DPchg{(238 to 241)}{(238) to (241)} are assumed to give a correct description of electromagnetic +phenomena in media whether stationary or moving with +respect to our reference system~$S$, it is evident that the equations +must be unchanged in form if we refer our measurements to a new +system of coördinates~$S'$ moving past~$S$, say, with the velocity~$V$ in the +$X$\DPchg{-}{~}direction. + +As a matter of fact, equations \DPchg{(238 to 241)}{(238) to (241)} can be transformed +into an entirely similar set +\begin{align*} +\curl \vc{H'} + &= \frac{1}{c} \left(\frac{\partial\vc{D'}}{\partial t'} + \vc{i}'\DPtypo{,}{}\right)\DPtypo{}{,} \\ +\curl \vc{E'} &= -\frac{1}{c}\, \frac{\partial\vc{B'}}{\partial t'}, \\ +\divg \vc{D'} &= \rho', \\ +\divg \vc{B'} &= 0, +\end{align*} +%% -----File: 198.png---Folio 184------- +provided we substitute for $x$,~$y$,~$z$ and~$t$ the values of $x'$,~$y'$,~$z'$ and~$t'$ +given by the fundamental transformation equations for space and +time \DPchg{(9~to~12)}{(9)~to~(12)}, and substitute for the other quantities in question the +relations +{\small% +\begin{align*}%[** TN: Re-grouping] +\begin{aligned} +{E_x}' &= E_x, & +{E_y}' &= \kappa \left(E_y - \frac{V}{c} B_z\right), & +{E_z}' &= \kappa \left(E_z + \frac{V}{c} B_y\right), \\ +% +{D_x}' &= D_x, & +{D_y}' &= \kappa \left(D_y - \frac{V}{c} H_z\right), & +{D_z}' &= \kappa \left(D_z + \frac{V}{c} H_y\right), +\end{aligned} +\Tag{245}\displaybreak[0] \\[12pt] +\begin{aligned} +{H_x}' &= H_x, & +{H_y}' &= \kappa \left(H_y + \frac{V}{c} D_z\right), & +{H_z}' &= \kappa \left(H_z - \frac{V}{c} D_y\right), \\ +{B_x}' &= B_x, & +{B_y}' &= \kappa \left(B_y + \frac{V}{c} E_z\right), & +{B_z}' &= \kappa \left(B_z - \frac{V}{c} E_y\right), +\end{aligned} +\Tag{246}\displaybreak[0] \\[12pt] +\begin{gathered} +\rho' = \kappa \left(\rho - \frac{V}{c^2}\, i_x\right),\qquad +{i_x}' = \kappa(i_x - V_\rho), \qquad +{i_y}' = i_y, \qquad +{i_z}' = i_z. +\end{gathered} +\Tag{247} +\end{align*}}% + +It will be noted that for free space these equations will reduce to +the same form as our earlier transformation equations \DPchg{(227~to~229)}{(227)~to~(229)} +since we shall have the simplifications $\vc{D} = \vc{E}$, $\vc{B} = \vc{H}$ and $\vc{i} = \rho \vc{u}$. + +We may also call attention at this point to the fact that our fundamental +%% -----File: 199.png---Folio 185------- +equations for electromagnetic phenomena \DPchg{(238--241)}{(238)--(241)} in dielectric +media might have been derived from the principle of least +action, making use of an expression for kinetic potential which could +be shown equal to $H = \ds\int dV \left(\frac{\vc{E·D}}{2} - \frac{\vc{H}·\vc{B}}{2}\right)$, and it will be noticed +that our transformation equations for these quantities are such as to +preserve that necessary invariance for $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ which we found in +\Chapref{IX} to be the general requirement for any dynamical development +which agrees with the theory of relativity. + +\Paragraph{171.} We are now in a position to handle the theory of moving +media. Consider a homogeneous medium moving past a system of +coördinates $S$ in the $X$\DPchg{-}{~}direction with the velocity~$V$; our problem is +to discover relations between the various electric and magnetic +vectors in this medium. To do this, consider a new system of coördinates~$S'$ +also moving past our original system with the velocity~$V$. +Since the medium is stationary with respect to this new system~$S'$ we +may write for measurements referred to~$S'$ in accordance with equations +\DPchg{(242~to~244)}{(242)~to~(244)} the relations +\begin{align*} +\vc{D'} &= \epsilon \vc{E'},\\ +\vc{B'} &= \mu \vc{H'},\\ +\vc{i'} &= \sigma \vc{E'}, +\end{align*} +which, as we have already pointed out, are known experimentally to +be true in the case of \emph{stationary, homogeneous} media. $\epsilon$,~$\mu$ and~$\sigma$ are +evidently the values of dielectric constant, permeability and conductivity +of the material in question, which would be found by an +experimenter with respect to whom the medium is stationary. + +Making use of our transformation equations \DPchg{(245~to~247)}{(245)~to~(247)} we can +obtain by obvious substitutions the following set of relations for +measurements made with respect to the original system of coördinates~$S$: +\begin{align*} +&\begin{aligned} +D_x &= \epsilon E_x, \\ +D_y - \frac{V}{c} H_z + &= \epsilon \left(E_y - \frac{V}{c} B_z\right), \\ +%% -----File: 200.png---Folio 186------- +D_z + \frac{V}{c} H_y + &= \epsilon \left(E_z + \frac{V}{c} B_y\right), +\end{aligned} +\Tag{248} \displaybreak[0] \\[12pt] +&\begin{aligned} +B_x &= \mu H_x, \\ +B_y + \frac{V}{c} E_z + &= \mu\left(H_y + \frac{V}{c} D_z\right), \\ +B_z - \frac{V}{c}E_y + &= \mu\left(H_z - \frac{V}{c} D_y\right), +\end{aligned} +\Tag{249} \displaybreak[0] \\[12pt] +&\begin{aligned} +\kappa (i_x - V_\rho) &= \sigma E_x, \\ +i_y &= \sigma\kappa \left(E_y - \frac{V}{c} B_z\right), \\ +i_z &= \sigma\kappa \left(E_z + \frac{V}{c} B_y\right). +\end{aligned} +\Tag{250} +\end{align*} + +\Subsubsection{172}{Theory of the Wilson Experiment.} The equations which we +have just developed for moving media are, as a matter of fact, in +complete accord with the celebrated experiment of H.~A. Wilson on +moving dielectrics and indeed all other experiments that have been +performed on moving media. + +Wilson's experiment consisted in the rotation of a hollow cylinder +of dielectric, in a magnetic field which was parallel to the axis of the +cylinder. The inner and outer surfaces of the cylinder were covered +with a thin metal coating, and arrangements made with the help of +wire brushes so that electrical contact could be made from these +coatings to the pairs of quadrants of an electrometer. By reversing +the magnetic field while the apparatus was in rotation it was possible +to measure with the electrometer the charge produced by the electrical +displacement in the dielectric. We may make use of our equations +to compute the quantitative size of the effect. +\begin{figure}[hbt] + \begin{center} + \Fig{15} + \Input[3.75in]{200} + \end{center} +\end{figure} +%% -----File: 201.png---Folio 187------- + +Let \Figref{15} represent a cross-section of the rotating cylinder. +Consider a section of the dielectric~$AA$ which is moving perpendicularly +to the plane of the paper in the $X$\DPchg{-}{~}direction with the velocity~$V$. Let +the magnetic field be in the $Y$\DPchg{-}{~}direction parallel to the axis of rotation. +The problem is to calculate dielectric displacement~$D_z$ in the $Z$\DPchg{-}{~}direction. + +Referring to equations~(248) we have +\begin{align*} +D_z + \frac{V}{c} H_y &= \epsilon \left(E_z + \frac{V}{c} B_y\right), \\ +\intertext{and, substituting the value of~$B_y$ given by equations~(249),} +B_y + \frac{V}{c} E_z &= \mu \left(H_y + \frac{V}{c} D_z\right) +\end{align*} +we obtain +\[ +\left(1 - \epsilon\mu\, \frac{V^2}{c^2}\right) D_z + = \epsilon \left(1 - \frac{V^2}{c^2}\right) E_z + + \frac{V}{c}\, (\epsilon\mu - 1)\, H_y, +\] +or, neglecting terms of orders higher than~$\dfrac{V}{c}$, we have +\[ +D_z = \epsilon E_z + \frac{V}{c}\, (\epsilon\mu - 1)\, H_y. +\Tag{251} +\] + +For a substance whose permeability is practically unity such as +Wilson actually used the equation reduces to +\[ +D_z = \epsilon E_z + \frac{V}{c}\, (\epsilon - 1)\, H_y, +\] +and this was found to fit the experimental facts, since measurements +with the electrometer show the surface charge actually to have the +magnitude $D_z$~per square centimeter in accordance with our equation +$\divg D = \rho$. + +It would be a matter of great interest to repeat the Wilson experiment +with a dielectric of high permeability so that we could test the +complete equation~(251). This is of some importance since the +original Lorentz theory led to a different equation, +\[ +D_z = \epsilon E_z + \frac{V}{c}\, (\epsilon - 1)\, \mu H_y. +\] +%% -----File: 202.png---Folio 188------- + + +\Chapter{XIII}{Four-Dimensional Analysis.} +%[** TN: Running head not hyphenated in original] +\SetRunningHeads{Chapter Thirteen.}{Four-Dimensional Analysis.} + +\Paragraph{173.} In the present chapter we shall present a four-dimensional +method of expressing the results of the Einstein theory of relativity, +a method which was first introduced by Minkowski, and in the form +which we shall use, principally developed by Wilson and Lewis. The +point of view adopted\DPtypo{,}{} consists essentially in considering the properties +of an assumed four-dimensional space in which intervals of time are +thought of as plotted along an axis perpendicular to the three Cartesian +axes of ordinary space, the science of kinematics thus becoming +the geometry of this new four-dimensional space. + +The method often has very great advantages not only because it +sometimes leads to considerable simplification of the mathematical +form in which the results of the theory of relativity are expressed, +but also because the analogies between ordinary geometry and the +geometry of this imaginary space often suggest valuable modes of +attack. On the other hand, in order to carry out actual numerical +calculations and often in order to appreciate the physical significance +of the conclusions arrived at, it is necessary to retranslate the results +obtained by this four-dimensional method into the language of ordinary +kinematics. It must further be noted, moreover, that many important +results of the theory of relativity can be more easily obtained +if we do not try to employ this four-dimensional geometry. The +reader should also be on his guard against the fallacy of thinking that +extension in time is of the same nature as extension in space merely +because intervals of space and time can both be represented by +plotting along axes drawn on the same piece of paper. + +\Subsubsection{174}{Idea of a Time Axis.} In order to grasp the method let us +consider a particle constrained to move along a single axis, say~$OX$, +and let us consider a time axis~$OT$ perpendicular to~$OX$. Then the +\emph{position} of the particle at any \emph{instant} of time can be represented by a +point in the $XT$~plane, and its motion as time progresses by a line in +the plane. If, for example, the particle were stationary, its behavior +%% -----File: 203.png---Folio 189------- +in time and space could be represented by a line parallel to the time +axis~$OT$ as shown for example by the line~$ab$ in \Figref{16}. A particle +\begin{figure}[hbt] + \begin{center} + \Fig{16} + \Input[3.5in]{203} + \end{center} +\end{figure} +moving with the uniform velocity $u = \dfrac{dx}{dt}$ could be represented by a +straight line $ac$ making an angle with the time axes, and the kinematical +behavior of an accelerated particle could be represented by a +curved line. + +By conceiving of a \emph{four}-dimensional space we can extend this +method which we have just outlined to include motion parallel to +all three space axes, and in accordance with the nomenclature of +Minkowski might call such a geometrical representation of the space-time +manifold ``the world,'' and speak of the points and lines which +represent the instantaneous positions and the motions of particles as +``world-points'' and ``world-lines.'' + +\Subsubsection{175}{Non-Euclidean Character of the Space.} It will be at once +evident that the graphical method of representing kinematical events +which is shown by \Figref[Figure]{16} still leaves something to be desired. One +of the most important conclusions drawn from the theory of relativity +was the fact that it is impossible for a particle to move with a velocity +greater than that of light, and it is evident that there is nothing in +our plot to indicate that fact, since we could draw a line making any +desired angle with the time axis, up to perpendicularity, and thus +%% -----File: 204.png---Folio 190------- +represent particles moving with any velocity up to infinity, +\[ +u = \frac{\Delta x}{\Delta t} = \infty. +\] +It is also evident that there is nothing in our plot to correspond to +that invariance in the velocity of light which is a cornerstone of the +theory of relativity. Suppose, for example, the line~$OC$, in \Figref{17}, +\begin{figure}[hbt] + \begin{center} + \Fig{17} + \Input[3.75in]{204} + \end{center} +\end{figure} +represents the trajectory of a beam of light with the velocity $\dfrac{\Delta x}{\Delta t} = c$; +there is then nothing so far introduced into our method of plotting +to indicate the fact that we could not equally well make use of another +set of axes~$OX'T'$, inclined to the first and thus giving quite a different +value, $\dfrac{\Delta x'}{\Delta t'}$, to the velocity of the beam of light. + +There are a number of methods of meeting this difficulty and +obtaining the invariance for the four-dimensional expression $x^2 + y^2 ++ z^2 - c^2t^2$ (see \Chapref{IV}) which must characterize our system of +kinematics. One of these is to conceive of a four-dimensional Euclidean +%% -----File: 205.png---Folio 191------- +space with an imaginary time axis, such that instead of plotting +real instants in time along this axis we should plot the quantity +$l = ict$ where $i = \sqrt{-1}$. In this way we should obtain invariance +for the quantity $x^2 + y^2 + z^2 + l^2 = x^2 + y^2 + z^2 - c^2t^2$, since it may +be regarded as the square of the magnitude of an imaginary four-dimensional +radius vector. This method of treatment has been +especially developed by Minkowski, Laue, and Sommerfeld. Another +method of attack, which has been developed by Wilson and Lewis +and is the one which we shall adopt in this chapter, is to use a real +time axis, for plotting the real quantity~$ct$, but to make use of a non-Euclidean +four-dimensional space in which the quantity $(x^2 + y^2 + z^2 +- c^2t^2)$ is itself taken as the square of the magnitude of a radius vector. +This latter method has of course the disadvantages that come from +using a non-Euclidean space; we shall find, however, that these reduce +largely to the introduction of certain rules as to signs. The method +has the considerable advantage of retaining a real time axis which is +of some importance, if we wish to visualize the methods of attack and +to represent them graphically. + +We may now proceed to develop an analysis for this non-Euclidean +space. We shall find this to be quite a lengthy process but at its +completion we shall have a very valuable instrument for expressing +in condensed language the results of the theory of relativity. Our +method of treatment will be almost wholly analytical, and the geometrical +analogies may be regarded merely as furnishing convenient +names for useful analytical expressions. A more geometrical method +of attack will be found in the original work of Wilson and Lewis. + + +\Section[I]{Vector Analysis of the Non-Euclidean Four-Dimensional +Manifold.} + +\Paragraph{176.} Consider a four-dimensional manifold in which the position +of a point is determined by a radius vector +\[ +\vc{r} = (x_1 \vc{k}_1 + x_2 \vc{k}_2 + x_3 \vc{k}_3 + x_4 \vc{k}_4), +\] +where $\vc{k}_1$,~$\vc{k}_2$,~$\vc{k}_3$ and~$\vc{k}_4$ may be regarded as unit vectors along four +mutually perpendicular axes and $x_1$,~$x_2$,~$x_3$, and~$x_4$ as the magnitudes +of the four components of~$\vc{r}$ along these four axes. We may identify +$x_1$,~$x_2$, and~$x_3$ with the three spatial coördinates of a point $x$,~$y$ and~$z$ +%% -----File: 206.png---Folio 192------- +with reference to an ordinary set of space axes and consider~$x_4$ as a +coördinate which specifies the time (multiplied by the velocity of +light) when the occurrence in question takes place at the point~$xyz$. +We have +\[ +x_1 = x,\qquad +x_2 = y,\qquad +x_3 = z,\qquad +x_4 = ct, +\Tag{252} +\] +and from time to time we shall make these substitutions when we +wish to interpret our results in the language of ordinary kinematics. +We shall retain the symbols $x_1$,~$x_2$,~$x_3$, and~$x_4$ throughout our development, +however, for the sake of symmetry. + +\Subsubsection{177}{Space, Time and Singular Vectors.} Our space will differ in +an important way from Euclidean space since we shall consider three +classes of one-vector, space, time and singular vectors. Considering +the coördinates $x_1$,~$x_2$,~$x_3$, and~$x_4$ which determine the end of a radius +vector, \\ +\emph{Space or $\gamma$-vectors} will have components such that +\[ +({x_1}^2 + {x_2}^2 + {x_3}^2) > {x_4}^2, +\] +and we shall put for their magnitude +\[ +s = \sqrt{{x_1}^2 + {x_2}^2 + {x_3}^2 - {x_4}^2}. +\Tag{253} +\] +\emph{Time or $\delta$-vectors} will have components such that +\[ +{x_4}^2 > ({x_1}^2 + {x_2}^2 + {x_3}^2), +\] +and we shall put for their magnitude +\[ +s = \sqrt{{x_4}^2-{x_1}^2- {x_2}^2 - {x_3}^2}. +\Tag{254} +\] +\emph{Singular or $\alpha$-vectors} will have components such that +\[ +({x_1}^2 + {x_2}^2 + {x_3}^2) = {x_4}^2, +\] +and their magnitude will be zero. + +\Subsubsection{178}{Invariance of $x^2 + y^2 + z^2 - c^2t^2$.} Since we shall naturally +consider the magnitude of a vector to be independent of any particular +choice of axes we have obtained at once by our definition of magnitude +for any rotation of axes that invariance for the expression +\[ +({x_1}^2 + {x_2}^2 + {x_3}^2 - {x_4}^2) = (x^2 + y^2 + z^2 - c^2t^2), +\] +%% -----File: 207.png---Folio 193------- +which is characteristic of the Lorentz transformation, and have thus +evidently set up an imaginary space which will be suitable for plotting +kinematical events in accordance with the requirements of the theory +of the relativity of motion. + +\Subsubsection{179}{Inner Product of One-Vectors.} We shall define the inner +product of two one-vectors with the help of the following rules for the +multiplication of unit vectors along the axes +\[ +\vc{k}_1 · \vc{k}_1 = \vc{k}_2 · \vc{k}_2 = \vc{k}_3· \vc{k}_3 = 1,\qquad +\vc{k}_4 · \vc{k}_4 = -1,\qquad \vc{k}_n · \vc{k}_m = 0. +\Tag{255} +\] + +It should be noted, of course, that there is no particular significance +in picking out the product $\vc{k}_4 · \vc{k}_4$ as the one which is negative; +it would be equally possible to develop a system in which the +products $\vc{k}_1 · \vc{k}_1, \vc{k}_2 · \vc{k}_2$ and $\vc{k}_3 · \vc{k}_3$ should be negative and $\vc{k}_4 · \vc{k}_4$ positive. + +The above rules for unit vectors are sufficient to define completely +the inner product provided we include the further requirements that +this product shall obey the \emph{associative law} for a scalar factor and the +\emph{distributive} and \emph{commutative} laws, namely +\[ +\begin{aligned} +(n\vc{a}) · \vc{b} &= n(\vc{a} · \vc{b}) = (\vc{a}· \vc{b})(n), \\ +\vc{a} · \vc{(b+c)} &= \vc{a} · \vc{b} + \vc{a} · \vc{c}, \\ +\vc{a} · \vc{b} &= \vc{b} · \vc{a}. +\end{aligned} +\Tag{256} +\] + +For the inner product of a one-vector by itself we shall have, in +accordance with these rules, +\begin{multline*} +\vc{r} · \vc{r} + = (x_1 \vc{k}_1 + x_2 \vc{k}_2 + x_3 \vc{k}_3 + x_4 \vc{k}_4) + · (x_1 \vc{k}_1 + x_2 \vc{k}_2 + x_3 \vc{k}_3 + x_4 \vc{k}_4) \\ + = (x_1^2 + x_2^2 + x_3^2 - x_4^2) +\Tag{257} +\end{multline*} +and hence may use the following expressions for the magnitudes of +vectors in terms of inner product +\[ +s = \sqrt{ \vc{r} · \vc{r}} \text{ for $\gamma$-vectors},\qquad +s = \sqrt{-\vc{r} · \vc{r}} \text{ for $\delta$-vectors}. +\Tag{258} +\] + +For curved lines we shall define interval along the curve by the +equations +\[ +\begin{aligned} +\int ds &= \int\sqrt { dr · dr} \text{ for $\gamma$-curves}, \\ +\int ds &= \int\sqrt {-dr · dr} \text{ for $\delta$-curves}. +\end{aligned} +\Tag{259} +\] +%% -----File: 208.png---Folio 194------- + +Our rules further show us that we may obtain the space components +of any one vector by taking its inner product with a unit vector +along the desired axis and may obtain the time component by taking +the negative of the corresponding product. Thus +\[ +\begin{aligned} +\vc{r}·\vc{k}_1 + &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_1 = x_1,\\ +\vc{r}·\vc{k}_2 + &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_2 = x_2,\\ +\vc{r}·\vc{k}_3 + &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_3 = x_3,\\ +\vc{r}·\vc{k}_4 + &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_4 = -x_4.\\ +\end{aligned} +\Tag{260} +\] + +We see finally moreover in general that the inner product of any +pair of vectors will be numerically equal to the product of the magnitude +of either by the projection of the other upon it, the sign depending +on the nature of the vectors involved. + +\Subsubsection{180}{Non-Euclidean Angle.} We shall define the non-Euclidean +angle~$\theta$ between two vectors $\vc{r}_1$~and~$\vc{r}_2$ in terms of their magnitudes +$s_1$~and~$s_2$ by the expressions +\[ +\pm \vc{r}_1·\vc{r}_2 + = (s_1 × \text{projection}\ s_2) + = s_1s_2\cosh\theta, +\Tag{261} +\] +the sign depending on the nature of the vectors in the way indicated +in the preceding section. We note the analogy between this equation +and those familiar in Euclidean vector-analysis, the hyperbolic +\DPtypo{trigonometeric}{trigonometric} functions taking the place of the circular functions +used in the more familiar analysis. + +For the angle between unit vectors $\vc{k}$~and~$\vc{k'}$ we shall have +\[ +\cosh\theta = \pm \vc{k}·\vc{k'}, +\Tag{262} +\] +where the sign must be chosen so as to make $\cosh\theta$ positive, the +plus sign holding if both are $\gamma$-vectors and the minus sign if both are +$\delta$-vectors. + +\Subsubsection{181}{Kinematical Interpretation of Angle in Terms of Velocity.} +At this point we may temporarily interrupt the development of our +four-dimensional analysis to consider a kinematical interpretation of +non-Euclidean angles in terms of velocity. It will be evident from +our introduction that the behavior of a moving particle can be represented +in our four-dimensional space by a $\delta$-curve,\footnote + {It is to be noted that the actual trajectories of particles are all of them represented + by $\delta$-curves since as we shall see $\gamma$-curves would correspond to velocities + greater than that of light.} +each point on +%% -----File: 209.png---Folio 195------- +this curve denoting the position of the particle at a given instant of +time, and it is evident that the velocity of the particle will be determined +by the angle which this curve makes with the axes. + +Let $\vc{r}$ be the radius vector to a given point on the curve and consider +the derivative of~$\vc{r}$ with respect to the interval $s$ along the curve; +we have +\[ +\vc{w} = \frac{d\vc{r}}{ds} + = \frac{dx_1}{ds}\, \vc{k}_1 + + \frac{dx_2}{ds}\, \vc{k}_2 + + \frac{dx_3}{ds}\, \vc{k}_3 + + \frac{dx_4}{ds}\, \vc{k}_4, +\Tag{263} +\] +and this may be regarded as a unit vector tangent to the curve at the +point in question. + +If $\phi$ is the angle between the $\vc{k}_4$~axis and the tangent to the curve +at the point in question, we have by equation~(262) +\[ +\cosh\phi = - \vc{w}·\vc{k}_4 = \frac{dx_4}{ds}; +\] +making the substitutions for $x_1$,~$x_2$,~$x_3$, and~$x_4$, in terms of $x$,~$y$,~$z$ and~$t$ +we may write, however, +\[ +ds = \sqrt{\smash[b]{dx_4^2 - dx_1^2 - dx_2^2 - dx_3^2}} + = \sqrt{1 - \frac{u^2}{c^2}}\, c\, dt, \Tag{264} +\] +which gives us +\[ +\cosh\phi = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} \Tag{265} +\] +and by the principles of hyperbolic trigonometry we may write the +further relations +\begin{gather*} +\sinh\phi = \frac{\smfrac{u}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, \Tag{266} +\displaybreak[0] \\ +\tanh\phi = \frac{u}{c}. \Tag{267} +\end{gather*} + + +%[** TN: Heading set like a \Section in original] +\Subsection{Vectors of Higher Dimensions} + +\Subsubsection{182}{Outer Products.} We shall define the outer product of two +one-vectors so that it obeys the \emph{associative law} for a scalar factor, the +%% -----File: 210.png---Folio 196------- +\emph{distributive law} and the \emph{anti-commutative law}, namely, +\[ +\begin{aligned} +(n\vc{a}) × \vc{b} &= n(\vc{a} × \vc{b}) = \vc{a} × (n\vc{b}),\\ + \vc{a} × (\vc{b} + \vc{c}) &= \vc{a} × \vc{b} + \vc{a} × \vc{c}\DPchg{}{,}\quad +( \vc{a} + \vc{b}) × \vc{c} = \vc{a} × \vc{c} + \vc{b} × \vc{c}, \\ + \vc{a} × \vc{b} &= -\vc{b}× \vc{a}. +\end{aligned} +\Tag{268} +\] + +From a geometrical point of view, we shall consider the outer +product of two one-vectors to be itself a \emph{two-vector}, namely the parallelogram, +or more generally, the area which they determine. The +sign of the two-vector may be taken to indicate the direction of progression +clockwise or anti-clockwise around the periphery. In order +to accord with the requirement that the area of a parallelogram determined +by two lines becomes zero when they are rotated into the same +direction, we may complete our definition of outer product by adding +the requirement that the outer product of a vector by itself shall be +zero. +\[ +\vc{a} × \vc{a} = 0. +\Tag{269} +\] + +We may represent the outer products of unit vectors along the +chosen axes as follows: +\[ +\begin{aligned} +\vc{k}_1 × \vc{k}_1 &= \vc{k}_2 × \vc{k}_2 = \vc{k}_3 × \vc{k}_3 = \vc{k}_4 × \vc{k}_4 = 0,\\ +\vc{k}_1 × \vc{k}_2 &= -\vc{k}_2 × \vc{k}_1 = \vc{k}_{12} = -\vc{k}_{21},\\ +\vc{k}_1 × \vc{k}_3 &= -\vc{k}_3 × \vc{k}_1 = \vc{k}_{13} = -\vc{k}_{31},\quad \text{etc.},\\ +\end{aligned} +\Tag{270} +\] +where we may regard~$\vc{k}_{12}$, for example, as a unit parallelogram in the +plane~$X_1OX_2$. + +We shall continue to use small letters in Clarendon type for one-vectors +and shall use capital letters in Clarendon type for two-vectors. +The components of a two-vector along the six mutually perpendicular +planes $X_1OX_2$,~$X_1OX_3$,~etc., may be obtained by expressing the one-vectors +involved in terms of their components along the axes and +carrying out the indicated multiplication, thus: +\[ +\begin{aligned} +\vc{A} &= \vc{a} × \vc{b} + = (a_1\vc{k}_1 + a_2\vc{k}_2 + a_3\vc{k}_3 + a_4\vc{k}_4) \\ + &\quad × (b_1\vc{k}_1 + b_2\vc{k}_2 + b_3\vc{k}_3 + b_4\vc{k}_4) \\ + &= (a_1b_2 - a_2b_1)\vc{k}_{12} + + (a_1b_3 - a_3b_1)\vc{k}_{13} + + (a_1b_4 - a_4b_1)\vc{k}_{14} \\ + &\quad + (a_2b_3 - a_3b_2)\vc{k}_{23} + + (a_2b_4 - a_4b_2)\vc{k}_{24} + + (a_3b_4 - a_4b_3)\vc{k}_{34}, +\end{aligned} +\Tag{271} +\] +%% -----File: 211.png---Folio 197------- +or, calling the quantities $(a_1b_2 - a_2b_1)$,~etc., the component magnitudes +of $\vc{A}$,~$A_{12}$,~etc., we may write +\[ +\vc{A} = A_{12}\vc{k}_{12} + A_{13}\vc{k}_{13} + A_{14}\vc{k}_{14} + + A_{23}\vc{k}_{23} + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34}. +\Tag{272} +\] + +The concept of outer product may be extended to include the +idea of vectors of higher number of dimensions than two. Thus the +outer product of three one-vectors, or of a one-vector and a two-vector +will be a three-vector which may be regarded as a \emph{directed} parallelopiped +in our four-dimensional space. The outer product of four one-vectors +will lead to a four-dimensional solid which would have direction +only in a space of more than four dimensions and hence in our case +will be called a pseudo-scalar. The outer product of vectors the +sum of whose dimensions is greater than that of the space considered +will vanish. + +The results which may be obtained from different types of outer +multiplication are tabulated below, where one-vectors are denoted +by small Clarendon type, two-vectors by capital Clarendon type, +three-vectors by Tudor black capitals, and pseudo-scalars by bold face +Greek letters. +{\small% +\begin{align*} %[** TN: Re-breaking] +&\begin{aligned} +\vc{A} + &= \vc{a} × \vc{b} = -\vc{b} × \vc{a} \\ + &= (a_1b_2 - a_2b_1)\vc{k}_{12} + + (a_1b_4 - a_3b_1)\vc{k}_{13} + + (a_1b_4 - a_4b_1)\vc{k}_{14} \\ + &+ (a_2b_3 - a_3b_2)\vc{k}_{23} + + (a_2b_4 - a_4b_2)\vc{k}_{21} + + (a_3b_4 - a_4b_3)\vc{k}_{34}, +\end{aligned} \displaybreak[0] \\[12pt] +&\begin{aligned} +\Alpha + &= \vc{c} × \vc{A} \\ + &= (c_1A_{23} - c_2A_{13} + c_3A_{12})\vc{k}_{123} + + (c_1A_{24} - c_2A_{14} + c_4A_{12})\vc{k}_{124} \\ + &+ (c_1A_{34} - c_2A_{14} + c_4A_{15})\vc{k}_{134} + + (c_2A_{34} - c_3A_{24} + c_4A_{23})\vc{k}_{234} +\end{aligned} +\Tag{273} \displaybreak[0] \\[12pt] +&\begin{aligned} +\vc{\alpha} + &= \vc{d} × \Alpha = -\Alpha × \vc{d} \\ + &= (d_1\Alpha_{234} - d_2\Alpha_{134} + + d_3\Alpha_{124} - d_4\Alpha_{123})\vc{k}_{1234}, \\ +\vc{\alpha} + &= \vc{A} × \vc{B} \\ + &= (A_{12}B_{34} - A_{13}B_{24} + A_{14}B_{23} + A_{23}B_{14} + - A_{24}B_{13} + A_{34}B_{12})\vc{k}_{1234}. +\end{aligned} +\end{align*}}% + +\emph{The signs in these expressions are determined by the general rule +that the sign of any unit vector~$\vc{\bar{k}}_{nmo}$ will be reversed by each transposition +of the order of a pair of adjacent subscripts, thus}: +\[ +k_{abcd} = - k_{bacd} = k_{bcad},\qquad \text{etc.},\ \cdots. +\Tag{274} +\] +%% -----File: 212.png---Folio 198------- + +\Subsubsection{183}{Inner Product of Vectors in General.} We have previously +defined the inner product for the special case of a pair of one-vectors, +in order to bring out some of the important characteristics of our +non-Euclidean space. We may now give a general rule for the inner +product of vectors of any number of dimensions. + +The inner product of any pair of vectors follows the \emph{associative} +law for scalar factors, and follows the \emph{distributive} and \emph{commutative} +laws. + +Since we can express any vector in terms of its components, the +above rules will completely determine the inner product of any pair +of vectors provided that we also have a rule for obtaining the inner +products of the unit vectors determined by the mutually perpendicular +axes. This rule is as follows: Transpose the subscripts of the unit +vectors involved so that the common subscripts occur at the end and +in the same order and cancel these common subscripts. If both the +unit vectors still have subscripts the product is zero; if neither vector +has subscripts the product is unity, and if one of the vectors still has +subscripts that itself will be the product. The sign is to be taken +as that resulting from the transposition of the subscripts (see equation~(274)), unless the subscript~$4$ has been cancelled, when the sign +will be changed. + +For example: +\[ +\begin{aligned} +\vc{k}_{124} · \vc{k}_{34} &= \vc{k}_{12} · \vc{k}_{3} = 0, \\ +\vc{k}_{132} · \vc{k}_{123} &= -\vc{k}_{123} · \vc{k}_{123} = -1, \\ +\vc{k}_{124} · \vc{k}_{42} &= -\vc{k}_{124} · \vc{k}_{24} = \vc{k}_{1}. +\end{aligned} +\Tag{275} +\] + +It is evident from these rules that we may obtain the magnitude +of any desired component of a vector by taking the inner product of +the vector by the corresponding unit vector, it being noticed, of course, +that when the unit vector involved contains the subscript~$4$ we obtain +the negative of the desired component. For example, we may obtain +the $k_{12}$~component of a two-vector as follows: +\[ +\begin{aligned} +A_{12} + = \vc{A} · \vc{k}_{12} + = (A_{12}\vc{k}_{12} &+ A_{13}\vc{k}_{13} + A_{14}\vc{k}_{14} \\ + &+ A_{23}\vc{k}_{23} + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34}) · \vc{k}_{12}. +\end{aligned} +\Tag{276} +\] + +\Subsubsection{184}{The Complement of a Vector.} In an $n$-dimensional space +any $m$-dimensional vector will uniquely determine a new vector of +%% -----File: 213.png---Folio 199------- +dimensions $(n-m)$ which may be called the complement of the +original vector. The complement of a vector may be exactly defined +as the inner product of the original vector with the unit pseudo-scalar +$\vc{k}_{123\cdots n}$. In general, we may denote the complement of a vector +by placing an asterisk~$*$ after the symbol. As an example we may +write as the complement of a two-vector~$\vc{A}$ in our non-Euclidean four-dimensional +space: +\[ +\begin{aligned} +\vc{A}^* &= +\begin{aligned}[t] + \vc{A} · \vc{k}_{1234} + = (A_{12}\vc{k}_{12} &+ A_{13}\vc{k}_{13} + A_{14}\vc{k}_{14} \\ + &+ + A_{23}\vc{k}_{23} + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34}) · \vc{k}_{1234} +\end{aligned} \\ + &= (A_{12}\vc{k}_{34} - A_{13}\vc{k}_{24} - A_{14}\vc{k}_{23} + + A_{23}\vc{k}_{14} + A_{24}\vc{k}_{13} - A_{34}\vc{k}_{12}). +\end{aligned} +\Tag{277} +\] + +\Subsubsection{185}{The Vector Operator, $\Qop$ or Quad.} Analogous to the familiar +three-dimensional vector-operator del, +\[ +\nabla + = \vc{k}_1\, \frac{\partial}{\partial x_1} + + \vc{k}_2\, \frac{\partial}{\partial x_2} + + \vc{k}_3\, \frac{\partial}{\partial x_3}, +\Tag{278} +\] +we may define the four-dimensional vector-operator quad, +\[ +\Qop + = \vc{k}_1\, \frac{\partial}{\partial x_1} + + \vc{k}_2\, \frac{\partial}{\partial x_2} + + \vc{k}_3\, \frac{\partial}{\partial x_3} + - \vc{k}_4\, \frac{\partial}{\partial x_4}. +\Tag{279} +\] + +If we have a scalar or a vector field we may apply these operators +by regarding them formally as one-vectors and applying the rules +for inner and outer multiplication which we have already given. + +Thus if we have a scalar function~$F$ which varies continuously +from point to point we can obtain a one-vector which we may call +the four-dimensional gradient of~$F$ at the point in question by simple +multiplication; we have +\[ +\grad F= \Qop F + = \vc{k}_1\, \frac{\partial F}{\partial x_1} + + \vc{k}_2\, \frac{\partial F}{\partial x_2} + + \vc{k}_3\, \frac{\partial F}{\partial x_3} + - \vc{k}_4\, \frac{\partial F}{\partial x_4}. +\Tag{280} +\] +If we have a one-vector field, with a vector~$\vc{f}$ whose value varies +from point to point we may obtain by inner multiplication a scalar +quantity which we may call the four-dimensional divergence of~$\vc{f}$\DPtypo{ we}{. We} +have +\[ +\divg\vc{f} = \Qop · \vc{f} + = \frac{\partial f_1}{\partial x_1} + + \frac{\partial f_2}{\partial x_2} + + \frac{\partial f_3}{\partial x_3} + + \frac{\partial f_4}{\partial x_4}. +\Tag{280} +\] +Taking the outer product with quad we may obtain a two-vector, the +%% -----File: 214.png---Folio 200------- +four-dimensional curl of~$\vc{f}$, +\[ +\begin{aligned}%[** TN: Re-aligning] +\curl \vc{f} = \Qop × \vc{f} + &= \left(\frac{\partial f_2}{\partial x_1} + - \frac{\partial f_1}{\partial x_2}\right) \vc{k}_{12} + + \left(\frac{\partial f_3}{\partial x_1} + - \frac{\partial f_1}{\partial x_3}\right) \vc{k}_{13} \\ + &+ \left(\frac{\partial f_4}{\partial x_1} + + \frac{\partial f_1}{\partial x_4}\right) \vc{k}_{14} + + \left(\frac{\partial f_3}{\partial x_2} + - \frac{\partial f_2}{\partial x_3}\right) \vc{k}_{23} \\ + &+ \left(\frac{\partial f_4}{\partial x_2} + + \frac{\partial f_2}{\partial x_4}\right) \vc{k}_{24} + + \left(\frac{\partial f_4}{\partial x_3} + + \frac{\partial f_3}{\partial x_4}\right) \vc{k}_{34}. +\end{aligned} +\Tag{282} +\] +By similar methods we could apply quad to a two-vector function~$\vc{F}$ +and obtain the one-vector function $\Qop · \vc{F}$ and the three-vector function +$\Qop × \vc{F}$. + +\Paragraph{186.} Still regarding $\Qop$ as a one-vector we may obtain a number of +important expressions containing~$\Qop$ more than once; we have: +\begin{align*} +\Qop × (\Qop F) &= 0, \quad(283) & +\Qop × (\Qop × \vc{f}) &= 0,\quad (286) \\ +% +\Qop · (\Qop · \vc{F}) &= 0, \quad (284) & +\Qop × (\Qop × \vc{F}) &= 0, \quad (287) \\ +% +\Qop · (\Qop · \frakF) &= 0, \quad (285) && +\end{align*} +\begin{align*} +\Qop · (\Qop × \vc{f}) + &= \Qop (\Qop · \vc{f}) - (\Qop · \Qop)\vc{f}, +\Tag{288} \\ +\Qop · (\Qop × \vc{F}) + &= \Qop × (\Qop · \vc{F}) + (\Qop · \Qop)\vc{F}, +\Tag{289}\\ +\Qop · (\Qop × \frakF) + &= \Qop × (\Qop · \frakF) - (\Qop · \Qop)\frakF. +\Tag{290} +\end{align*} + +The operator $\Qop · \Qop$ or~$\Qop^2$ has long been known under the name +of the D'Alembertian, +\[ +\Qop^2 = \frac{\partial^2}{\partial {x_1}^2} + + \frac{\partial^2}{\partial {x_2}^2} + + \frac{\partial^2}{\partial {x_3}^2} + - \frac{\partial^2}{\partial {x_4}^2} + = \Delta^2 - \frac{\partial^2}{c^2\, \partial t^2}. +\Tag{291} +\] + +From the definition of the complement of a vector given in the +previous section it may be shown by carrying out the proper expansions +that +\[ +(\Qop × \phi)^* = \Qop · \phi^*, +\Tag{292} +\] +where $\phi$~is a vector of any number of dimensions. + +\Subsubsection{187}{Tensors.} In analogy to three-dimensional tensors we may +define a four-dimensional tensor as a quantity with sixteen components +as given in the following table: +\[ +T = \left\{ +\begin{matrix} +T_{11} & T_{12} & T_{13} &T_{14}, \\ +T_{21} & T_{22} & T_{23} &T_{24}, \\ +T_{31} & T_{32} & T_{33} &T_{34}, \\ +T_{41} & T_{42} & T_{43} &T_{44}, +\end{matrix} +\right. +\Tag{293} +\] +%% -----File: 215.png---Folio 201------- +with the additional requirement that the divergence of the tensor, +defined as follows, shall itself be a one-vector. +\[ +\settowidth{\TmpLen}{$\ds\frac{\partial T_{12}}{\partial x_2} + +\frac{\partial T_{13}}{\partial x_3} + +\frac{\partial T_{14}}{\partial x_4}\,$}% +\begin{aligned} +\divg T &= \left\{ + \frac{\partial T_{11}}{\partial x_1} + + \frac{\partial T_{12}}{\partial x_2} + + \frac{\partial T_{13}}{\partial x_3} + + \frac{\partial T_{14}}{\partial x_4}\right\}\vc{k}_1 \\ + &+ \left\{\frac{\partial T_{21}}{\partial x_1} + + \makebox[\TmpLen][l]{$\cdots$} \right\}\vc{k}_2 \\ + &+ \left\{\frac{\partial T_{31}}{\partial x_1} + + \makebox[\TmpLen][l]{$\cdots$} \right\}\vc{k}_3 \\ + &+ \left\{\frac{\partial T_{41}}{\partial x_1} + + \makebox[\TmpLen][l]{$\cdots$} \right\}\vc{k}_4 \\ +\end{aligned} +\Tag{294} +\] + +\Subsubsection{188}{The Rotation of Axes.} Before proceeding to the application +of our four-dimensional analysis to the actual problems of relativity +theory we may finally consider the changes in the components of a +vector which would be produced by a rotation of the axes. We have +already pointed out that the quantity $({x_1}^2 + {x_2}^2 + {x_3}^2 - {x_4}^2)$ is an +invariant in our space for any set of rectangular coördinates having +the same origin since it is the square of the magnitude of a radius +vector, and have noted that in this way we have obtained for the +quantity $(x^2 + y^2 + z^2 - c^2t^2)$ the desired invariance which is characteristic +of the Lorentz transformation. In fact we may look upon +the Lorentz transformation as a rotation from a given set of axes to a +new set, with a corresponding re-expression of quantities in terms of +the new components. The particular form of Lorentz transformation, +familiar in preceding chapters, in which the new set of spatial axes +has a velocity component relative to the original set, in the $X$\DPchg{-}{~}direction +alone, will be found to correspond to a rotation of the axes in which +only the directions of the $X_1$~and~$X_4$ axes are changed, the $X_2$~and~$X_3$ +axes remaining unchanged in direction. + +Let us consider a one-vector +\[ +\vc{a} + = (a_1\vc{k}_1 + a_2\vc{k}_2 + a_3\vc{k}_3 + a_4\vc{k}_4) + = ({a_1}'\vc{k_1}' + {a_2}'\vc{k_2}' + {a_3}'\vc{k_3}' + {a_4}'\vc{k_4}'), +\] +where $a_1$,~$a_2$,~$a_3$ and~$a_4$ are the component magnitudes, using a set of +axes which have $\vc{k}_1$,~$\vc{k}_2$,~$\vc{k}_3$ and~$\vc{k}_4$ as unit vectors and ${a_1}'$,~${a_2}'$,~${a_3}'$ and~${a_4}'$ +the corresponding magnitudes using another set of mutually perpendicular +axes with the unit vectors $\vc{k_1}'$,~$\vc{k_2}'$,~$\vc{k_3}'$ and~$\vc{k_4}'$. Our problem, +%% -----File: 216.png---Folio 202------- +now, is to find relations between the magnitudes $a_1$,~$a_2$,~$a_3$ and~$a_4$ and +${a_1}'$,~${a_2}'$,~${a_3}'$ and~${a_4}'$. + +We have already seen\DPtypo{}{,} \DPchg{sections (\Secnumref{179})~and~(\Secnumref{183})}{Sections \Secnumref{179}~and~\Secnumref{183}}, that we may obtain +any desired component magnitude of a vector by taking its inner +product with a unit vector in the desired direction, reversing the +sign if the subscript~$4$ is involved. We may obtain in this way an +expression for~$a_1$ in terms of ${a_1}'$,~${a_2}'$,~${a_3}'$ and~${a_4}'$. We have +\begin{align*} +a_1 = \vc{a}·\vc{k}_1 + &= ({a_1}'{\vc{k}_1}' + {a_2}'{\vc{k}_2}' + + {a_3}'{\vc{k}_3}' + {a_4}'{\vc{k}_4}') · {\vc{k}_1} \\ + &= {a_1}'{\vc{k}_1}' · \vc{k}_1 + {a_2}'{\vc{k}_2}' · \vc{k}_1 + + {a_3}'{\vc{k}_3}' · \vc{k}_1 + {a_4}'{\vc{k}_4}' · \vc{k}_1. +\Tag{295} +\end{align*} +By similar multiplications with $\vc{k_2}$,~$\vc{k_3}$ and~$\vc{k_4}$ we may obtain expressions +for $a_2$,~$a_3$ and~$-a_4$. The results can be tabulated in the convenient +form +\[ +\begin{array}{c|*{4}{l|}} + & \Neg{a_1}' & \Neg{a_2}' & \Neg{a_3}' & \Neg{a_4}' \\ +\hline +a_1 & \Neg{\vc{k}_1}' · \vc{k}_1 & \Neg{\vc{k}_2}' · \vc{k}_1 + & \Neg{\vc{k}_3}' · \vc{k}_1 & \Neg{\vc{k}_4}' · \vc{k}_1 \\ +\hline +a_2 & \Neg{\vc{k}_1}' · \vc{k}_2 & \Neg{\vc{k}_2}' · \vc{k}_2 + & \Neg{\vc{k}_3}' · \vc{k}_2 & \Neg{\vc{k}_4}' · \vc{k}_2 \\ +\hline +a_3 & \Neg{\vc{k}_1}' · \vc{k}_3 & \Neg{\vc{k}_2}' · \vc{k}_3 + & \Neg{\vc{k}_3}' · \vc{k}_3 & \Neg{\vc{k}_4}' · \vc{k}_3 \\ +\hline +a_4 & -{\vc{k}_1}' · \vc{k}_4 & -{\vc{k}_2}' · \vc{k}_4 + & -{\vc{k}_3}' · \vc{k}_4 & -{\vc{k}_4}' · \vc{k}_4 \\ +\hline +\end{array} +\Tag{296} +\] + +Since the square of the magnitude of the vector, $({a_1}^2 + {a_2}^2 + {a_3}^2 +- {a_4}^2)$, is a quantity which is to be independent of the choice of axes, +we shall have certain relations holding between the quantities ${\vc{k}_1}'· \vc{k}_1$, +${\vc{k}_1}' · \vc{k}_2$, etc. These relations, which are analogous to the familiar +%% -----File: 217.png---Folio 203------- +conditions of orthogonality in Euclidean space, can easily be shown +to be +\[ +\begin{aligned} +({\vc{k}_1}'· \vc{k}_1)^2 + ({\vc{k}_1}'· \vc{k}_2)^2 + ({\vc{k}_1}'· \vc{k}_3)^2 - ({\vc{k}_1}'· \vc{k}_4)^2 &= 1,\\ +({\vc{k}_2}'· \vc{k}_1)^2 + ({\vc{k}_2}'· \vc{k}_2)^2 + ({\vc{k}_2}'· \vc{k}_3)^2 - ({\vc{k}_2}'· \vc{k}_4)^2 &= 1,\\ +({\vc{k}_3}'· \vc{k}_1)^2 + ({\vc{k}_3}'· \vc{k}_2)^2 + ({\vc{k}_3}'· \vc{k}_3)^2 - ({\vc{k}_3}'· \vc{k}_4)^2 &= 1,\\ +({\vc{k}_4}'· \vc{k}_1)^2 + ({\vc{k}_4}'· \vc{k}_2)^2 + ({\vc{k}_4}'· \vc{k}_3)^2 - ({\vc{k}_4}'· \vc{k}_4)^2 &= - 1, +\end{aligned} +\Tag{297} +\] +and +\begin{align*}%[** TN: Re-breaking] +({\vc{k}_1}'· \vc{k}_1)({\vc{k}_2}' · \vc{k}_1) + &+ ({\vc{k}_1}' · \vc{k}_2)({\vc{k}_2}' · \vc{k}_2) \\ + &+ ({\vc{k}_1}' · \vc{k}_3)({\vc{k}_2}' · \vc{k}_3) + - ({\vc{k}_1}' · \vc{k}_4)({\vc{k}_2}' · \vc{k}_4) = 0, +\end{align*} +etc., for each of the six pairs of vertical columns in table~(296). + +Since we shall often be interested in a simple rotation in which +the directions of the $X_2$~and~$X_3$ axes are not changed, we shall be able +to simplify this table for that particular case by writing +\[ +{\vc{k}_2}' = \vc{k}_2,\qquad +{\vc{k}_3}' = \vc{k}_3, +\] +and noting the simplifications thus introduced in the products of the +unit vectors, we shall obtain +\[ +\begin{array}{*{5}{c|}} + & \Neg {a_1}' & {a_2}' & {a_3}' & \Neg {a_4}' \\ +\hline +a_1 & \Neg{\vc{k}_1}' · \vc{k}_1 & 0 & 0 & \Neg{\vc{k}_4}' · \vc{k}_1 \\ +\hline +a_2 & \Neg 0 & 1 & 0 & \Neg 0 \\ +\hline +a_3 & \Neg 0 & 0 & 1 & \Neg 0 \\ +\hline +a_4 & -{\vc{k}_1}' · \vc{k}_4 & 0 & 0 & -{\vc{k}_4}' · \vc{k}_4 \\ +\hline +\end{array} +\Tag{298} +\] +%% -----File: 218.png---Folio 204------- + +If now we call~$\phi$ the angle of rotation between the two time axes +${OX_4}'$~and~$OX_4$, we may write, in accordance with equation~(262), +\[ +-{\vc{k}_4}' · \vc{k}_4 = \cosh \phi. +\] + +Since we must preserve the orthogonal relations~(297) and may +also make use of the well-known expression of hyperbolic trigonometry +\[ +\cosh^2 \phi - \sinh^2 \phi = 1, +\] +we may now rewrite our transformation table in the form +\[ +\begin{array}{*{5}{c|}} + & {a_1}' & {a_2}' & {a_3}' & {a_4}' \\ +\hline +a_1 & \cosh\phi & 0 & 0 & \sinh \phi \\ +\hline +a_2 & 0 & 1 & 0 & 0 \\ +\hline +a_3 & 0 & 0 & 1 & 0 \\ +\hline +a_4 & \sinh \phi & 0 & 0 & \cosh \phi \\ +\hline +\end{array} +\Tag{299} +\] + +By a similar process we may obtain transformation tables for the +components of a two-vector~$\vc{A}$. Expressing~$\vc{A}$ in terms of the unit +vectors ${\vc{k}_{12}}'$,~${\vc{k}_{13}}'$, ${\vc{k}_{14}}'$,~etc., and taking successive inner products with +the unit vectors $\vc{k}_{12}$,~$\vc{k}_{13}$, $\vc{k}_{14}$,~etc., we may obtain transformation +equations which can be expressed by the \hyperref[table:300]{tabulation~(300)} shown on +the following page.\DPnote{[** TN: No need for varioref]} +%% -----File: 219.png---Folio 205------- +\begin{sidewaystable}[p] +\phantomsection\label{table:300}% +\renewcommand{\arraystretch}{3} +\[ +\begin{array}{c|*{6}{r|}} + & \multicolumn{1}{c|}{{A_{12}}'} & \multicolumn{1}{c|}{{A_{13}}'} + & \multicolumn{1}{c|}{{A_{14}}'} & \multicolumn{1}{c|}{{A_{23}}'} + & \multicolumn{1}{c|}{{A_{24}}'} & \multicolumn{1}{c|}{{A_{34}}'} \\ +\hline +A_{12} & {\vc{k}_{12}}' · \vc{k}_{12} & {\vc{k}_{13}}' · \vc{k}_{12} + & {\vc{k}_{14}}' · \vc{k}_{12} & {\vc{k}_{23}}' · \vc{k}_{12} + & {\vc{k}_{24}}' · \vc{k}_{12} & {\vc{k}_{34}}' · \vc{k}_{12} \\ +\hline +A_{13} & {\vc{k}_{12}}' · \vc{k}_{13} & {\vc{k}_{13}}' · \vc{k}_{13} + & {\vc{k}_{14}}' · \vc{k}_{13} & {\vc{k}_{23}}' · \vc{k}_{13} + & {\vc{k}_{24}}' · \vc{k}_{13} & {\vc{k}_{34}}' · \vc{k}_{13} \\ +\hline +A_{14} &-{\vc{k}_{12}}' · \vc{k}_{14} & -{\vc{k}_{13}}' · \vc{k}_{14} + & -{\vc{k}_{14}}' · \vc{k}_{14} & -{\vc{k}_{23}}' · \vc{k}_{14} + & -{\vc{k}_{24}}' · \vc{k}_{14} & -{\vc{k}_{34}}' · \vc{k}_{14} \\ +\hline +A_{23} & {\vc{k}_{12}}' · \vc{k}_{23} & {\vc{k}_{13}}' · \vc{k}_{23} + & {\vc{k}_{14}}' · \vc{k}_{23} & {\vc{k}_{23}}' · \vc{k}_{23} + & {\vc{k}_{24}}' · \vc{k}_{23} & {\vc{k}_{34}}' · \vc{k}_{23} \\ +\hline +A_{24} & -{\vc{k}_{12}}' · \vc{k}_{24} & -{\vc{k}_{13}}' · \vc{k}_{24} + & -{\vc{k}_{14}}' · \vc{k}_{24} & -{\vc{k}_{23}}' · \vc{k}_{24} + & -{\vc{k}_{24}}' · \vc{k}_{24} & -{\vc{k}_{34}}' · \vc{k}_{24} \\ +\hline +A_{34} & -{\vc{k}_{12}}' · \vc{k}_{34} & -{\vc{k}_{13}}' · \vc{k}_{34} + & -{\vc{k}_{14}}' · \vc{k}_{34} & -{\vc{k}_{23}}' · \vc{k}_{34} + & -{\vc{k}_{24}}' · \vc{k}_{34} & -{\vc{k}_{34}}' · \vc{k}_{34} \\ +\hline +\end{array} +\Tag{300} +\] +\end{sidewaystable} + +For the particular case of a rotation in which the direction of the +$X_2$~and~$X_3$ axes are not changed we shall have +\[ +{\vc{k}_2}' = \vc{k}_2,\qquad +{\vc{k}_3}' = \vc{k}_3, +\] +and very considerable simplification will be introduced. We shall +have, for example, +\begin{alignat*}{4} +&{\vc{k}_{12}}'· \vc{k}_{12} + &&= ({\vc{k}_1}' × {\vc{k}_2}') · (\vc{k}_1 × \vc{k}_2) + &&= ({\vc{k}_1}' × \vc{k}_2) · (\vc{k}_1 × \vc{k}_2) + &&= {\vc{k}_1}' · \vc{k}_1, \\ +&{\vc{k}_{13}}' · \vc{k}_{12} + &&= ({\vc{k}_1}' × {\vc{k}_3}') · (\vc{k}_1 × \vc{k}_2) + &&= ({\vc{k}_1}' × \vc{k}_3 ) · (\vc{k}_1 × \vc{k}_2) + &&= 0, \\ +&\text{etc.} +\end{alignat*} +Making these and similar substitutions and introducing, as before, +%% -----File: 220.png---Folio 206------- +the relation $-\DPtypo{{\vc{k}'}_4}{{\vc{k}_4}'} · \vc{k}_4 = \cosh \phi$ where $\phi$~is the non-Euclidean angle +between the two time axes, we may write our transformation table +in the form +\[ +\begin{array}{*{7}{c|}} + & \Neg{A_{12}}' & \Neg{A_{13}}' & {A_{14}}' & {A_{23}}' & {A_{24}}' &{A_{34}}' \\ +\hline +A_{12} & \Neg\cosh\phi & 0 & 0 & 0 & \sinh\phi & 0 \\ +\hline +A_{13} & \Neg0 & \Neg\cosh\phi & 0 & 0 & 0 & \sinh\phi \\ +\hline +A_{14} & \Neg0 & \Neg0 & 1 & 0 & 0 & 0 \\ +\hline +A_{23} & \Neg0 & \Neg0 & 0 & 1 & 0 & 0 \\ +\hline +A_{24} & -\sinh\phi & 0 & 0 & 0 & \cosh\phi & 0 \\ +\hline +A_{34} & \Neg0 & -\sinh\phi & 0 & 0 & 0 & \cosh\phi \\ +\hline +\end{array} +\Tag{301} +\] + +\Subsubsection{189}{Interpretation of the Lorentz Transformation as a Rotation +of Axes.} We may now show that the Lorentz transformation may +be looked upon as a change from a given set of axes to a rotated set. + +Since the angle~$\phi$ which occurs in our transformation tables is +that between the $\vc{k}_4$~axis and the new ${\vc{k}_4}'$~axis, we may write, in accordance +with equations (265)~and~(266), +\[ +\cosh \phi = \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}, \qquad +\sinh \phi = \frac{\smfrac{V}{c}}{\sqrt{1 - \smfrac{V^2}{c^2}}}, +\] +where $V$~is the velocity between the two sets of space axes which +correspond to the original and the rotated set of four-dimensional +axes. This will permit us to rewrite our transformation table for the +%% -----File: 221.png---Folio 207------- +components of a one-vector in the forms +\begin{gather*} +\phantomsection\label{table:302}% +\renewcommand{\arraystretch}{2} +\begin{array}{*{5}{c|}} + & {a_1}' & {a_2}' & {a_3}' & {a_4}' \\ +\hline +a_1 & \dfrac{1} {\sqrt{1 - \smfrac{V^2}{c^2}}}& 0 & 0 + & \dfrac{V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\ +\hline +a_2 & 0 & 1 & 0 & 0 \\ +\hline +a_3 & 0 & 0 & 1 & 0 \\ +\hline +a_4 & \dfrac{V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} & 0 & 0 + & \dfrac{1} {\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\ +\hline +\end{array} \\ +\Tag{302} \\ +\renewcommand{\arraystretch}{2} +\begin{array}{*{5}{c|}} + & a_1 & a_2 & a_3 & a_4 \\ +\hline +{a_1}' & \dfrac{1} {\sqrt{1 - \smfrac{V^2}{c^2}}} & 0 & 0 + & \dfrac{-V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\ +\hline +{a_2}' & 0 & 1 & 0 & 0 \\ +\hline +{a_3}' & 0 & 0 & 1 & 0 \\ +\hline +{a_4}' & \dfrac{-V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} & 0 & 0 + & \dfrac{1} {\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\ +\hline +\end{array} +\end{gather*} + +Consider now any point $P(x_1, x_2, x_3, x_4)$. The radius vector from +the origin to this point will be $\vc{r} = (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)$, or, +making use of the relations between $x_1$,~$x_2$, $x_3$,~$x_4$ and $x$,~$y$, $z$,~$t$ given +by equations~(252), we may write +\[ +\vc{r} = (x\vc{k}_1 + y\vc{k}_2 + z\vc{k}_3 + ct\vc{k}_4). +\] +Applying our transformation table to the components of this one-vector, +we obtain the familiar equations for the Lorentz transformation +\begin{align*} +x' &= \frac{x - Vt}{\sqrt{1 - \smfrac{V^2}{c^2}}}, \\ +%% -----File: 222.png---Folio 208------- +y' &= y, \\ +z' &= z, \\ +t' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\left(t - \frac{V}{c^2}\, x\right). +\end{align*} + +We thus see that the Lorentz transformation is to be interpreted +in our four-dimensional analysis as a rotation of axes. + +\Subsubsection{190}{Graphical Representation.} Although we have purposely restricted +ourselves in the foregoing treatment to methods of attack +which are almost purely analytical rather than geometrical in nature, +the importance of a graphical representation of our four-dimensional +manifold should not be neglected. The difficulty of representing all +four axes on a single piece of two-dimensional paper is not essentially +different from that encountered in the graphical representation of the +facts of ordinary three-dimensional solid geometry, and these difficulties +can often be solved by considering only one pair of axes at a +time, say $OX_1$~and~$OX_4$, and plotting the occurrences in the $X_1OX_4$ +plane. The fact that the geometry of this plane is a non-Euclidean +one presents a more serious complication since the figures that we +draw on our sheet of paper will obviously be Euclidean in nature, +but this difficulty also can be met if we make certain conventions as +to the significance of the lines we draw, conventions which are fundamentally +not so very unlike the conventions by which we interpret as +solid, a figure drawn in ordinary perspective. + +Consider for example the diagram shown in \Figref{18}, where we +have drawn a pair of perpendicular axes, $OX_1$,~and~$OX_4$ and the +two unit hyperbolæ given by the equations +\[ +\begin{aligned} +{x_1}^2 - {x_4}^2 &= 1, \\ +{x_1}^2 - {x_4}^2 &= -1, +\end{aligned} +\Tag{303} +\] +together with their asymptotes, $OA$~and~$OB$, given by the equation +\[ +{x_1}^2 - {x_4}^2 = 0. +\Tag{304} +\] +This purely Euclidean figure permits, as a matter of fact, a fairly +satisfactory representation of the non-Euclidean properties of the +manifold with which we have been dealing. +%% -----File: 223.png---Folio 209------- + +$OX_1$~and~$OX_4$ may be considered as perpendicular axes in the +non-Euclidean $X_1OX_4$~plane. Radius vectors lying in the quadrant~$AOB$\DPtypo{,}{} +will have a greater component along the~$X_4$ than along the $X_1$~axis +and hence will be $\delta$-vectors with the magnitude $s = \sqrt{{x_4}^2 - {x_1}^2}$, +where $x_1$~and~$x_4$ are the coördinates of the terminal of the vector. +\begin{figure}[hbt] + \begin{center} + \Fig{18} + \Input[4in]{223} + \end{center} +\end{figure} +$\gamma$-radius-vectors will lie in the quadrant~$BOC$ and will have the magnitude +$s = \sqrt{{x_1}^2 - {x_4}^2}$. Radius vectors lying along the asymptotes +$OA$~and~$OB$ will have zero magnitudes ($s = \sqrt{{x_1}^2 - {x_4}^2} = 0$) and +hence will be singular vectors. + +Since the two hyperbolæ have the equations ${x_1}^2 - {x_4}^2 = 1$ and +${x_1}^2 - {x_4}^2 = -1$, rays such as $Oa$,~${Oa}'$, $Ob$,~etc., starting from the +origin and terminating on the hyperbolæ, will all have unit magnitude. +Hence we may consider the hyperbolæ as representing unit pseudo-circles +in our non-Euclidean plane and consider the rays as representing +the radii of these pseudo-circles. + +A non-Euclidean rotation of axes will then be represented by +changing from the axes $OX_1$~and~$OX_4$ to ${OX_1}'$~and~${OX_4}'$, and taking +${Oa}'$~and~${Ob}'$ as unit distances along the axes instead of $Oa$~and~$Ob$. +%% -----File: 224.png---Folio 210------- + +It is easy to show, as a matter of fact, that such a change of axes +and units does correspond to the Lorentz transformation. Let $x_1$~and~$x_4$ +be the coördinates of any point with respect to the original +axes $OX_1$~and~$OX_4$, and ${x_1}''$~and~${x_4}''$ the coördinates of the same point +referred to the oblique axes ${OX_1}'$~and~${OX_4}'$, no change having yet +been made in the actual lengths of the units of measurement. Then, +by familiar equations of analytical geometry, we shall have +\[ +\begin{aligned} +x_1 &= {x_1}'' \cos\theta + {x_4}'' \sin\theta, \\ +x_4 &= {x_1}'' \sin\theta + {x_4}'' \cos\theta, +\end{aligned} +\Tag{305} +\] +where $\theta$ is the angle~$X_1O{X_1}'$. + +We have, moreover, from the properties of the hyperbola, +\[ +\frac{{Oa}'}{Oa} = \frac{{Ob}'}{Ob} + = \frac{1}{\sqrt{\cos^2\theta - \sin^2\theta}}, +\] +and hence if we represent by ${x_1}'$~and~${x_4}'$ the coördinates of the point +with respect to the oblique axes and use $O{a}'$~and~$O{b}'$ as unit distances +instead of $Oa$~and~$Ob$, we shall obtain +\begin{align*} +x_1 &= {x_1}'\, \frac{\cos\theta}{\sqrt{\cos^2\theta - \sin^2\theta}} + + {x_4}'\, \frac{\sin\theta}{\sqrt{\cos^2\theta - \sin^2\theta}}, \\ +x_4 &= {x_1}'\, \frac{\sin\theta}{\sqrt{\cos^2\theta - \sin^2\theta}} + + {x_4}'\, \frac{\cos\theta}{\sqrt{\cos^2\theta - \sin^2\theta}}. +\end{align*} + +It is evident, however, that we may write +\[ +\frac{\sin\theta}{\cos\theta} = \tan\theta = \frac{dx_1}{ dx_4} = \frac{V}{c}, +\] +where $V$ may be regarded as the relative velocity of our two sets of +space axes. Introducing this into the above equations and also +writing $x_1 = x$, $x_4 = ct$, ${x_1}' = x'$, ${x_4}' = ct'$, we may obtain the familiar +equations +\begin{align*} +x &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, (x' + Vt'), \\ +t &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}} \left(t' + \frac{V}{c^2}\, x'\right). +\end{align*} +%% -----File: 225.png---Folio 211------- +We thus see that our diagrammatic representation of non-Euclidean +rotation in the ${X_1}OX_4$~plane does as a matter of fact correspond to +the Lorentz transformation. + +Diagrams of this kind can now be used to study various kinematical +events. $\delta$-curves can be drawn in the quadrant~$AOB$ to represent +the space-time trajectories of particles, their form can be investigated +using different sets of rotated axes, and the equations for +the transformation of velocities and accelerations thus studied. +$\gamma$-lines perpendicular to the particular time axis used can be drawn to +correspond to the instantaneous positions of actual lines in ordinary +space and studies made of the Lorentz shortening. Singular vectors +along the asymptote~$OB$ can be used to represent the trajectory of a +ray of light and it can be shown that our rotation of axes is so devised +as to leave unaltered, the angle between such singular vectors and the +$OX_4$~axis, corresponding to the fact that the velocity of light must +appear the same to all observers. Further development of the possibilities +of graphical representation of the properties of our non-Euclidean +space may be left to the reader. + + +\Section[II]{Applications of the Four-Dimensional Analysis.} + +\Paragraph{191.} We may now apply our four-dimensional methods to a +number of problems in the fields of kinematics, mechanics and electromagnetics. +Our general plan will be to express the laws of the particular +field in question in four-dimensional language, making use of +four-dimensional vector quantities of a kinematical, mechanical, or +electromagnetic nature. Since the components of these vectors +along the three spatial axes and the temporal axis will be closely +related to the ordinary quantities familiar in kinematical, mechanical, +and electrical discussions, there will always be an easy transition from +our four-dimensional language to that ordinarily used in such discussions, +and necessarily used when actual numerical computations +are to be made. We shall find, however, that our four-dimensional +language introduces an extraordinary brevity into the statement of a +number of important laws of physics. + + +%[** TN: Heading set like a \Section in original] +\Subsection{Kinematics.} + +\Subsubsection{192}{Extended Position.} The position of a particle and the particular +instant at which it occupies that position can both be indicated +%% -----File: 226.png---Folio 212------- +by a point in our four-dimensional space. We can call this +the extended position of the particle and determine it by stating the +value of a four-dimensional radius vector +\[ +\vc{r} = (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4). +\Tag{306} +\] + +\Subsubsection{193}{Extended Velocity.} Since the velocity of a real particle can +never exceed that of light, its changing position in space and time +will be represented by a $\delta$-curve. + +The equation for a unit vector tangent to this $\delta$-curve will be +\[ +\vc{w} = \frac{d\vc{r}}{ds} + = \left(\frac{dx_1}{ds}\, \vc{k}_1 + \frac{dx_2}{ds}\, \vc{k}_2 + + \frac{dx_3}{ds}\, \vc{k}_3 + \frac{dx_4}{ds}\, \vc{k}_4\right), +\Tag{307} +\] +where $ds$~indicates interval along the $\delta$-curve; and this important +vector~$\vc{w}$ may be called the extended velocity of the particle. + +Remembering that for a $\delta$-curve +\[ +ds = \sqrt{d{x_4}^2 - d{x_1}^2 - d{x_2}^2 - d{x_3}^2} + = c\, dt \sqrt{1 - \frac{u^2}{c^2}}, +\Tag{308} +\] +we may rewrite our expression for extended velocity in the form +\[ +\vc{w} = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} + \left\{\frac{\vc{u}}{c} + \vc{k}_4\right\}, +\Tag{309} +\] +where $\vc{u}$ is evidently the ordinary three-dimensional velocity of the +particle. + +Since $\vc{w}$ is a four-dimensional vector in our imaginary space, we +may use our tables for transforming the components of~$\vc{w}$ from one +set of axes to another. We shall find that we may thus obtain transformation +equations for velocity identical with those already familiar +in \Chapref{IV}. + +The four components of $\vc{w}$ are +\[ +\frac{\smfrac{u_x}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{k}_1, \qquad +\frac{\smfrac{u_y}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{k}_2, \qquad +\frac{\smfrac{u_z}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{k}_3, \qquad +\frac{\vc{k}_4}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +and with the help of \hyperref[table:302]{table~(302)} we may easily obtain, by making +simple algebraic substitutions, the following familiar transformation +%% -----File: 227.png---Folio 213------- +equations: +\begin{gather*}%[** TN: Re-breaking] + {u_x}' = \frac{u_x - V}{1 - \smfrac{u_xV}{c^2}},\qquad + {u_y}' = \frac{u_y\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}},\qquad + {u_z}' = \frac{u_z\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}},\\ +\frac{1}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}} + = \frac{1 - \smfrac{u_xV}{c^2}} + {\sqrt{1 - \smfrac{u^2}{c^2}}\, \sqrt{1 - \smfrac{V^2}{c^2}}}. +\end{gather*} + +This is a good example of the ease with which we can derive our +familiar transformation equations with the help of the four-dimensional +method. + +\Subsubsection{194}{Extended Acceleration.} We may define the extended acceleration +of a particle as the rate of curvature of the $\delta$-line which determines +its four-dimensional position. We have +\[ +c = \frac{d^2\vc{r}}{ds^2} = \frac{d\vc{w}}{ds} + = \frac{d}{ds}\left[ + \frac{\smfrac{\vc{u}}{c} + \vc{k}_4} + {\sqrt{1 - \smfrac{u^2}{c^2}}}\right]. +\Tag{310} +\] +Or, introducing as before the relation $ds = c\, dt \sqrt{1 - \dfrac{u^2}{c^2}}$, we may write +\begin{multline*} +c = \frac{1}{c^2} \Biggl\{ + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \frac{d\vc{u}}{dt} + + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, \frac{u}{c^2}\, + \frac{du}{dt}\, \vc{u} \\ + + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, \frac{u}{c}\, + \frac{du}{dt}\, \vc{k}_4\Biggr\}, +\Tag{311} +\end{multline*} +%% -----File: 228.png---Folio 214------- +where $\vc{u}$ is evidently the ordinary three-dimensional velocity, and $\dfrac{d\vc{u}}{dt}$ +the three-dimensional acceleration; and we might now use our transformation +table to determine the transformation equations for acceleration +which we originally obtained in \Chapref{IV}. + +\Subsubsection{195}{The Velocity of Light.} As an interesting illustration of the +application to kinematics of our four-dimensional methods, we may +point out that the trajectory of a ray of light will be represented by a +singular line. Since the magnitude of all singular vectors is zero by +definition, we have for any singular line +\[ +{dx_1}^2 + {dx_2}^2 + {dx_3}^2 = {dx_4}^2, +\] +or, since the magnitude will be independent of any particular choice +of axes, we may also write +\[ +{{dx_1}'}^2 + {{dx_2}'}^2 + {{dx_3}'}^2 = {{dx_4}'}^2. +\] +Transforming the first of these equations we may write +\[ +\frac{{dx_1}^2 + {dx_2}^2 + {dx_3}^2 }{{dx_4}^2} + = \frac{dx^2 + dy^2 + dz^2 }{c^2\, dt^2} = 1 +\] +or +\[ +\frac{dl}{dt} = c. +\] +Similarly we could obtain from the second equation +\[ +\frac{dl'}{dt'} = c. +\] +We thus see that a singular line does as a matter of fact correspond +to the four-dimensional trajectory of a ray of light having the velocity~$c$, +and that our four-dimensional analysis corresponds to the requirements +of the second postulate of relativity that a ray of light shall +have the same velocity for all reference systems. + + +%[** TN: Heading set like a \Section in original] +\Subsection{The Dynamics of a Particle.} + +\Subsubsection{196}{Extended Momentum.} We may define the extended momentum +of a material particle as equal to the product~$m_0\vc{w}$ of its mass~$m_0$, +measured when at rest, and its extended velocity~$\vc{w}$. In accordance +%% -----File: 229.png---Folio 215------- +with equation~(309) for extended velocity, we may write then, for +the extended momentum, +\[ +m_0\vc{w} = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} + \left(\frac{\vc{u}}{c} + \vc{k}_4\right). +\Tag{312} +\] +Or, if in accordance with our considerations of \Chapref{VI} we put +for the mass of the particle at the velocity~$u$ +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +we may write +\[ +m_0\vc{w} = m\, \frac{\vc{u}}{c} + m\vc{k}_4. +\Tag{313} +\] +We note that the space component of this vector is ordinary momentum +and the time component has the magnitude of mass, and by +applying our \hyperref[table:302]{transformation table~(302)} we can derive very simply +the transformation equations for mass and momentum already +obtained in \Chapref{VI}. + +\Subsubsection{197}{The Conservation Laws.} We may now express the laws for +the dynamics of a system of particles in a very simple form by stating +the principle that the extended momentum of a system of particles is a +quantity which remains constant in all interactions of the particles, +we have then +\[ +\Sum m_0\vc{w} + = \Sum\left(\frac{m\vc{u}}{c} + m\vc{k}_4 \right) + = \text{ a constant}, +\Tag{314} +\] +where the summation $\Sum$ extends over all the particles of the system. + +It is evident that this one principle really includes the three +principles of the conservation of momentum, mass, and energy. +This is true because in order for the vector~$\Sum m_0\vc{w}$ to be a constant +quantity, its components along each of the four axes must be constant, +and as will be seen from the above equation this necessitates +the constancy of the momentum~$\Sum m\vc{u}$, of the total mass~$\Sum m$, and of +the total energy~$\Sum \dfrac{m}{c^2}$. +%% -----File: 230.png---Folio 216------- + + +%[** TN: Heading set like a \Section in original] +\Subsection{The Dynamics of an Elastic Body.} + +Our four-dimensional methods may also be used to present the +results of our theory of elasticity in a very compact form. + +\Subsubsection{198}{The Tensor of Extended Stress.} In order to do this we shall +first need to define an expression which may be called the four-dimensional +stress in the elastic medium. For this purpose we may take the +symmetrical tensor~$T_m$ defined by the following table: +\[ +T_m = \left\{ +\begin{matrix} +p_{xx} & p_{xy} & p_{xz} & cg_x, \\ +p_{yx} & p_{yy} & p_{yz} & cg_y, \\ +p_{zx} & p_{zy} & p_{zz} & cg_z, \\ +\dfrac{s_x}{c} & \dfrac{s_y}{c} & \dfrac{s_z}{c} & w, +\end{matrix} +\right. +\Tag{315} +\] +where the spatial components of~$T_m$ are equal to the components of +the symmetrical tensor~$\vc{p}$ which we have already defined in \Chapref{X} +and the time components are related to the density of momentum~$\vc{g}$, +density of energy flow~$\vc{s}$ and energy density~$w$, as shown in the tabulation. + +From the symmetry of this tensor we may infer at once the simple +relation between density of momentum and density of energy flow: +\[ +\vc{g} = \frac{\vc{s}}{c^2}, +\Tag{316} +\] +with which we have already become familiar in \Secref{132}. + +\Subsubsection{199}{The Equation of Motion.} We may, moreover, express the +equation of motion for an elastic medium unacted on by external +forces in the very simple form +\[ +\divg T_m = 0. +\Tag{317} +\] + +It will be seen from our definition of the divergence of a four-dimensional +tensor, \Secref{187}, that this one equation is in reality +equivalent to the two equations +\begin{align*} +\divg\vc{p} + \frac{\partial\vc{g}}{\partial t} &= 0 +\Tag{318} \\ +\intertext{and} +\divg\vc{s} + \frac{\partial w}{\partial t} &= 0. +\end{align*} +%% -----File: 231.png---Folio 217------- +The first of these equations is identical with~(184) of Chapter~X, %[** TN: Not a useful cross-reference] +which we found to be the equation for the motion of an elastic medium +in the absence of external forces, and the second of these equations +expresses the principle of the conservation of energy. + +The elegance and simplicity of this four-dimensional method of +expressing the results of our laborious calculations in \Chapref{X} cannot +fail to be appreciated. + + +%[** TN: Heading set like a \Section in original] +\Subsection{Electromagnetics.} + +We also find it possible to express the laws of the electromagnetic +field very simply in our four-dimensional language. + +\Subsubsection{200}{Extended Current.} We may first define the extended current, +a simple but important one-vector, whose value at any point will depend +on the density and velocity of charge at that point. We shall +take as the equation of definition +\[ +\vc{q} = \rho_0\vc{w} + = \rho \left\{\frac{\vc{u}}{c} + \vc{k}_4\right\}, +\Tag{319} +\] +where +\[ +\rho = \frac{\rho_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} +\] +is the density of charge at the point in question. + +\Subsubsection{201}{The Electromagnetic Vector $\vc{M}$.} We may further define a +two-vector~$\vc{M}$ which will be directly related to the familiar vectors +strength of electric field~$\vc{e}$ and strength of magnetic field~$\vc{h}$ by the +equation of definition +\begin{align*} +\vc{M} &= (h_1\vc{k}_{23} + h_2\vc{k}_{31} + h_3\vc{k}_{12} + - e_1\vc{k}_{14} - e_2\vc{k}_{24} - e_3\vc{k}_{34}) \\ +%[** TN: Hack to get equation number vertically centered] +\intertext{or\hfill(320)} +\vc{M^*} &= (e_1\vc{k}_{23} + e_2\vc{k}_{31} + e_3\vc{k}_{12} + + h_1\vc{k}_{14} + h_2\vc{k}_{24} + h_3\vc{k}_{34}), +\end{align*} +where $e_1$,~$e_2$,~$e_3$, and $h_1$,~$h_2$,~$h_3$ are the components of $\vc{e}$~and~$\vc{h}$. + +\Subsubsection{202}{The Field Equations.} We may now state the laws of the +electromagnetic field in the extremely simple form +\begin{align*} +\Qop · \vc{M} &= \vc{q}, \Tag{321} \\ +\Qop × \vc{M} &= 0. \Tag{322} +\end{align*} +%% -----File: 232.png---Folio 218------- + +These two simple equations are, as a matter of fact, completely +equivalent to the four field equations which we made fundamental +for our treatment of electromagnetic theory in \Chapref{XII}. Indeed +if we treat~$\Qop$ formally as a one-vector +\[ +\left(\vc{k}_1\, \frac{\partial}{\partial x_1} + + \vc{k}_2\, \frac{\partial}{\partial x_2} + + \vc{k}_3\, \frac{\partial}{\partial x_3} + - \vc{k}_4\, \frac{\partial}{\partial x_4}\right) +\] +and apply it to the electromagnetic vector~$\vc{M}$ expressed in the extended +form given in the equation of definition~(320) we shall obtain from~(321) +the two equations +\begin{align*} +\curl \vc{h} - \frac{1}{c}\, \frac{\partial\vc{e}}{\partial t} + &= \rho\, \frac{\vc{u}}{c}, \\ +\divg\vc{e} &= \rho, \\ +\intertext{and from (322)} +\divg \vc{h} &= 0,\\ +\curl \vc{e} + \frac{1}{c}\, \frac{\partial\vc{h}}{\partial t} &= 0, +\end{align*} +where we have made the substitution $x_4 = ct$. These are of course +the familiar field equations for the Maxwell-Lorentz theory of electromagnetism. + +\Subsubsection{203}{The Conservation of Electricity.} We may also obtain very +easily an equation for the conservation of electric charge. In accordance +with equation~(284) we may write as a necessary mathematical +identity +\[ +\Qop · (\Qop · \vc{M}) = 0. +\Tag{323} +\] +Noting that $\Qop · \vc{M} = \vc{q}$, this may be expanded to give us the equation +of continuity. +\[ +\divg \rho\vc{u} + \frac{\partial\rho}{\partial t} = 0. +\Tag{324} +\] + +\Subsubsection{204}{The Product $\vc{M}·\vc{q}$.} We have thus shown the form taken by +the four field equations when they are expressed in four dimensional +language. Let us now consider with the help of our four-dimensional +methods what can be said about the forces which determine the +motion of electricity under the action of the electromagnetic field. + +Consider the inner product of the electromagnetic vector and +%% -----File: 233.png---Folio 219------- +the extended current: +\begin{multline*} +\vc{M} · \vc{q} + = (h_1\vc{k}_{23} + h_2\vc{k}_{31} + h_3\vc{k}_{12} + - e_1\vc{k}_{14} - e_2\vc{k}_{24} - e_3\vc{k}_{34}) + · \rho \left\{\frac{\vc{u}}{c} + \vc{k}_4\right\} \\ + = \rho \left\{\vc{e} + \frac{[\vc{u} × \vc{h}]^*}{c}\right\} + + \rho\, \frac{\vc{e} · \vc{h}}{c}\vc{k}_4. +\Tag{325} +\end{multline*} +We see that the space component of this vector is equal to the expression +which we have already found in \Chapref{XII} as the force +acting on the charge contained in unit volume, and the time component +is proportional to the work done by this force on the moving +charge; hence we may write the equation +\[ +\vc{M} · \vc{q} = \left\{\vc{f} + \frac{\vc{f} · \vc{u}}{c}\, \vc{k}_4\right\}, +\Tag{326} +\] +an expression which contains the same information as that given by +the so-called fifth fundamental equation of electromagnetic theory, +$\vc{f}$~being the force exerted by the electromagnetic field per unit volume +of charged material. + +\Subsubsection{205}{The Extended Tensor of Electromagnetic Stress.} We may +now show the possibility of defining a four-dimensional tensor~$T_e$, such +that the important quantity $\vc{M} · \vc{q}$ shall be equal to~$-\divg T_e$. This +will be valuable since we shall then be able to express the equation +of motion for a combined mechanical and electrical system in a very +simple and beautiful form. + +Consider the symmetrical tensor +\[ +T_e = +\left\{ +\begin{matrix} +T_{11} & T_{12} & T_{13} & T_{14}, \\ +T_{21} & T_{22} & T_{23} & T_{24}, \\ +T_{31} & T_{32} & T_{33} & T_{34}, \\ +T_{41} & T_{42} & T_{43} & T_{44}, +\end{matrix} +\right. +\Tag{327} +\] +defined by the expression +\[ +\begin{aligned} +T_{jk} &= \tfrac{1}{2} + \{M_{j1}M_{k1} + M_{j2}M_{k2} + M_{j3}M_{k3} - M_{j4}M_{k4} \\ + &\qquad + + {M_{j1}}^*{M_{k1}}^* + {M_{j2}}^*{M_{k2}}^* + + {M_{j3}}^*{M_{k3}}^* - {M_{j4}}^*{M_{k4}}^*\}, +\end{aligned} +\Tag{328} +\] +where $j$, $k = 1$, $2$, $3$, $4$. +%% -----File: 234.png---Folio 220------- + +It can then readily be shown by expansion that +\[ +-\divg T_e = \vc{M} · (\Qop · \vc{M}) + \vc{M}^* · (\Qop · \vc{M}^*). +\] +But, in accordance with equations (321),~(326),~(292) and~(322), this +is equivalent to +\[ +-\divg T_e = \vc{M} · \vc{q} +%[** TN: Keeping () in numerator, cf. (326) above] + = \left\{\vc{f} + \frac{(\vc{f} · \vc{u})}{c}\, \vc{k}_4\right\}. +\Tag{329} +\] + +Since in free space the value of the force~$\vc{f}$ is zero, we may write +for free space the equation +\[ +\divg T_e = 0. +\Tag{330} +\] + +This one equation is equivalent, as a matter of fact, to two important +and well-known equations of electromagnetic theory. If we +develop the components $T_{11}$,~$T_{12}$,~etc., of our tensor in accordance +with equations (328)~and~(320) we find that we can write +\[ +T_e = +\left\{ +\renewcommand{\arraystretch}{2} +\begin{matrix} +\psi_{xx} & \psi_{xy} & \psi_{xz} & \dfrac{S_x}{c}, \\ +\psi_{yx} & \psi_{yy} & \psi_{yz} & \dfrac{S_y}{c}, \\ +\psi_{zx} & \psi_{zxy} & \psi_{zz} & \dfrac{S_z}{c}, \\ +\dfrac{s_x}{c}& \dfrac{s_x}{c} & \dfrac{s_x}{c} & w, +\end{matrix} +\right. +\Tag{331} +\] +where we shall have +\[ +\begin{aligned} +\psi_{xx} + &= -\tfrac{1}{2}({e_x}^2 - {e_y}^2 - {e_z}^2 + {h_x}^2 - {h_y}^2 - {h_z}^2), \\ +\psi_{xy} + &= -(e_xh_y + h_xh_y), \\ +\text{etc.}& \\ +s_x &= c(e_yh_z - e_zh_y), \\ +\text{etc.}& \\ +w &= \tfrac{1}{2}(e^2 + h^2), +\end{aligned} +\Tag{332} +\] +$\psi$ thus being equivalent to the well-known Maxwell three-dimensional +stress tensor, $s_x$,~$s_y$,~etc., being the components of the Poynting vector +$c\, [\vc{e} × \vc{h}]^*$, and $w$~being the familiar expression for density of electromagnetic +%% -----File: 235.png---Folio 221------- +energy $\dfrac{e^2 + h^2}{s}$. We thus see that equation~(330) is equivalent +to the two equations +\begin{align*} +\divg \psi + \frac{1}{c^2}\, \frac{\partial s}{\partial t} = 0, \\ +\divg \vc{s} + \frac{\partial w}{\partial t} = 0. +\end{align*} +The first of these is the so-called equation of electromagnetic momentum, +and the second, Poynting's equation for the flow of electromagnetic +energy. + +\Subsubsection{206}{Combined Electrical and Mechanical Systems.} For a point +not in free space where mechanical and electrical systems are both +involved, taking into account our previous considerations, we may +now write the equation of motion for a combined electrical and +mechanical system in the very simple form +\[ +\divg T_m + \divg T_e = 0. +\] +And we may point out in closing that we may reasonably expect all +forces to be of such a nature that our most general equation of motion +for any continuous system can be written in the form +\[ +\divg T_1 + \divg T_2 + \cdots = 0. +\] +%% -----File: 236.png---Folio 222------- + + +\Appendix{I}{Symbols for Quantities.} + +\AppSection{Scalar Quantities}{Scalar Quantities. \(Indicated by Italic type.\)} + +\begin{longtable}{rl} +$c$& speed of light.\\ +$e$& electric charge.\\ +$E$& energy.\\ +$H$& kinetic potential.\\ +$K$& kinetic energy.\\ +$l$, $m$, $n$& direction cosines.\\ +$L$& Lagrangian function.\\ +$p$& pressure.\\ +$Q$& quantity of electricity.\\ +$S$& entropy.\\ +$t$& time.\\ +$T$& temperature, function $\ds\Sum m_0c^2 \left(1-\sqrt{1-\frac{u^2}{c^2}}\;\right)$.\\ +$U$& potential energy.\\ +$v$& volume.\\ +$V$& relative speed of coördinate systems, volume.\\ +$w$& energy density.\\ +$W$& work.\\ +$\epsilon$&dielectric constant.\\ +$\kappa$ &$\dfrac{1}{\sqrt{1-\smfrac{V^2}{c^2}}}$.\\ +$\mu$ &index of refraction, magnetic permeability.\\ +$\nu $ &frequency.\\ +$\rho$ &density of charge.\\ +$\sigma$ &electrical conductivity.\\ +$\phi$ &non-Euclidean angle between time axes.\\ +$\phi_1\phi_2\phi_3 \cdots $& generalized coördinates.\\ +$\psi$ &scalar potential.\\ +$\psi_1\psi_2\psi_3\cdots$ & generalized momenta. +\end{longtable} +%% -----File: 237.png---Folio 223------- + + +\AppSection{Vector Quantities}{Vector Quantities. \(Indicated by Clarendon type.\)} + +\begin{longtable}{r l} +$\vc{B}$& magnetic induction.\\ +$\vc{c}$& extended acceleration.\\ +$\vc{D}$& dielectric displacement.\\ +$\vc{e}$& electric field strength in free space.\\ +$\vc{E}$& electric field strength in a medium.\\ +$\vc{f}$& force per unit volume.\\ +$\vc{F}$& force acting on a particle.\\ +$\vc{g}$& density of momentum.\\ +$\vc{h}$& magnetic field strength in free space.\\ +$\vc{H}$& magnetic field strength in a medium.\\ +$\vc{i}$& density of electric current.\\ +$\vc{M}$& angular momentum, electromagnetic vector.\\ +$\vc{p}$& symmetrical elastic stress tensor.\\ +$\vc{q}$& extended current.\\ +$\vc{r}$& radius vector\DPtypo{}{.}\\ +$\vc{s}$& density of energy flow.\\ +$\vc{t}$& unsymmetrical elastic stress tensor.\\ +$\vc{u}$& velocity.\\ +$\vc{w}$& extended velocity.\\ +$\vc{\phi}$& vector potential. +\end{longtable} +%% -----File: 238.png---Folio 224------- + + +\Appendix{II}{Vector Notation.} + +\AppSection{Three Dimensional Space}{Three Dimensional Space.} + +%[** TN: No periods after items in this section.] +Unit Vectors, $\vc{i}\ \vc{j}\ \vc{k}$ + +Radius Vector, $\vc{r} = x\vc{i} + y\vc{j} + z\vc{k}$ + +Velocity, +\begin{align*} +\vc{u} = \frac{d\vc{r}}{dt} + &= \dot{x}\vc{i} + \dot{y}\vc{j} + \dot{z}\vc{k} \\ + &= u_x\vc{i} + u_y\vc{j} + u_z\vc{k} \\ +\intertext{\indent Acceleration,} +\dot{\vc{u}} = \frac{d^2\vc{r}}{dt^2} + &= \ddot{x}\vc{i} + \ddot{y}\vc{j} + \ddot{z}\vc{k} \\ + &= \dot{u}_x\vc{i} + \dot{u}_y\vc{j} + \dot{u}_z\vc{k} +\end{align*} + +Inner Product, +\[ +\vc{a}·\vc{b} = a_xb_x + a_yb_y + a_zb_z +\] + +Outer Product, +\[ +\vc{a} × \vc{b} + = (a_xb_y - a_yb_x)\vc{ij} + + (a_yb_z - a_zb_y)\vc{jk} + + (a_zb_x - a_xb_z)\vc{ki} +\] + +Complement of Outer Product, +\[ +[\vc{a} × \vc{b}]^* + = (a_yb_z - a_zb_y)\vc{i} + + (a_zb_x - a_xb_z)\vc{j} + + (a_xb_y - a_yb_x)\vc{k} +\] + +The Vector Operator Del or~$\nabla$, +\[ +\nabla + = \vc{i}\, \frac{\partial}{\partial x} + + \vc{j}\, \frac{\partial}{\partial y} + + \vc{k}\, \frac{\partial}{\partial z} +\] +\begin{align*} +\grad A &= \nabla A + = \vc{i}\, \frac{\partial A}{\partial x} + + \vc{j}\, \frac{\partial A}{\partial y} + + \vc{k}\, \frac{\partial A}{\partial z} \\ +\divg\vc{a} &= \nabla · \vc{a} + = \frac{\partial a_x}{\partial x} + + \frac{\partial a_y}{\partial y} + + \frac{\partial a_z}{\partial z} \\ +\curl\vc{a} &= [\nabla × \vc{a}]^* \\ + &= \left(\frac{\partial a_z}{\partial y} + - \frac{\partial a_y}{\partial z}\right) \vc{i} + + \left(\frac{\partial a_x}{\partial z} + - \frac{\partial a_z}{\partial x}\right) \vc{j} + + \left(\frac{\partial a_y}{\partial x} + - \frac{\partial a_x}{\partial y}\right) \vc{k} +\end{align*} +%% -----File: 239.png---Folio 225------- + +\AppSection{Non-Euclidean Four Dimensional Space.}{Non-Euclidean Four Dimensional Space.} + +Unit Vectors, $\vc{k}_1$ $\vc{k}_2$ $\vc{k}_3$ $\vc{k}_4$ + +Radius Vector, +\begin{align*} +\vc{r} &= x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4 \\ + &= x\vc{i} + y\vc{j} + z\vc{k} + ct\vc{k}_4 +\end{align*} + +One Vector, +\[ +\vc{a} = a_1\vc{k}_1 + a_2\vc{k}_2 + a_3\vc{k}_3 + a_4\vc{k}_4 +\] + +Two Vector, +\[ +\vc{A} = A_{12}\vc{k}_{12} + A_{13}\vc{k}_{13} + + A_{14}\vc{k}_{14} + A_{23}\vc{k}_{23} + + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34} +\] + +Three Vector, +\[ +\Alpha = \frakA_{123}\vc{k}_{123} + \frakA_{124}\vc{k}_{124} + + \frakA_{134}\vc{k}_{134} + \frakA_{234}\vc{k}_{234} +\] + +Pseudo Scalar, +\[ +\vc{\alpha} = \alpha\vc{k}_{1234} +\] + +Transposition of Subscripts, +\[ +\vc{k}_{abc\cdots} = -\vc{k}_{bac\cdots} = \vc{k}_{bca\cdots} +\] + +Inner Product of One Vectors, + +(\textit{See \Secref{183}}). + +Outer Product of One Vectors, +\[ +\vc{k}_{ab\cdots} × \vc{k}_{nm\cdots} = \vc{k}_{ab\cdots nm\cdots} +\] + +Complement of a Vector, +\[ +\vc{\phi}^* = \phi·\vc{k}_{1234} +\] + +The Vector Operator Quad or~$\Qop$, +\[ +\Qop = \vc{k}_1\frac{\partial}{\partial x_1} + + \vc{k}_2\frac{\partial}{\partial x_2} + + \vc{k}_3\frac{\partial}{\partial x_3} + + \vc{k}_4\frac{\partial}{\partial x_4} +\] + +\cleardoublepage +\backmatter + +%%%% LICENSE %%%% +\pagenumbering{Alph} +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} +\pdfbookmark[0]{Project Gutenberg License}{License} +\fancyhf{} +\fancyhead[C]{\CtrHeading{Project Gutenberg License}} + +\begin{PGtext} +End of the Project Gutenberg EBook of The Theory of the Relativity of Motion, by +Richard Chace Tolman + +*** END OF THIS PROJECT GUTENBERG EBOOK THE THEORY OF THE RELATIVITY *** + +***** This file should be named 32857-pdf.pdf or 32857-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/2/8/5/32857/ + +Produced by Andrew D. 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{\makebox(0,0){\hbox{\color{rgb_000000}$\times$}}} +\pgftext[at={\pgfpoint{-0.055348in}{1.5in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$S$}}} +\pgftext[at={\pgfpoint{1.55535in}{1.44465in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$A$}}} +\pgftext[at={\pgfpoint{1.71517in}{3.04151in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{3.10267in}{1.5in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{1.52767in}{0in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$O$}}} +\pgftext[at={\pgfpoint{1.5in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~1.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/032.xp b/32857-t/images/sources/032.xp new file mode 100644 index 0000000..8d8faa1 --- /dev/null +++ b/32857-t/images/sources/032.xp @@ -0,0 +1,37 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double dX(0.05), sc(2.5); + +int main() +{ + picture(P(-1,-1), P(1,1), "3 x 3in"); + + begin(); + degrees(); + bold(); + mirror(P(0,0), -45); + + mirror(P(0,1), 90); + mirror(P(1,0), 0); + + dashed(); + dash_size(6); + line(P(-1,0), P(1,0)); + line(P(0,-1), P(0,1)); + marker(P(-1,0), PLUS); + marker(P(-1,0), TIMES); + + label(P(-1,0), P(-4,0), "$S$", l); + label(P(0,0), P(4,-4), "$A$", br); + label(P(sc*dX,1), P(2,3), "$B$", r); + label(P(1 +dX,0), P(2,0), "$C$", r); + label(P(0,-1), P( 2,0), "$O$", tr); + + font_face("sc"); + label(P(0,-1), P(0,-12), "Fig.~1.", b); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/036.eepic b/32857-t/images/sources/036.eepic new file mode 100644 index 0000000..889df34 --- /dev/null +++ b/32857-t/images/sources/036.eepic @@ -0,0 +1,66 @@ +%% Generated from 036.xp on Sun May 30 14:08:44 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-5,5] x [-1,1] +%% Actual size: 5 x 1in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (5in,1in); +\pgfsetlinewidth{0.8pt} +\draw (5in,0.75in)--(4.75in,0.75in)--(4.5in,0.75in); +\pgfsetlinewidth{0.4pt} +\draw (5in,0.75in)--(4.5in,0.75in); +\pgfsetfillcolor{rgb_000000} +\draw [fill](4.52076in,0.75in)--(4.58302in,0.777674in)-- + (4.5in,0.75in)--(4.58302in,0.722326in)--(4.52076in,0.75in)--cycle; +\draw (5in,0.75in)--(5.01038in,0.734433in)--(5.02076in,0.718867in); +\draw (5in,0.75in)--(5.01038in,0.765567in)--(5.02076in,0.781133in); +\draw (4.97924in,0.75in)--(4.98962in,0.734433in)--(5in,0.718867in); +\draw (4.97924in,0.75in)--(4.98962in,0.765567in)--(5in,0.781133in); +\draw (4.95849in,0.75in)--(4.96887in,0.734433in)--(4.97924in,0.718867in); +\draw (4.95849in,0.75in)--(4.96887in,0.765567in)--(4.97924in,0.781133in); +\draw (4.93773in,0.75in)--(4.94811in,0.734433in)--(4.95849in,0.718867in); +\draw (4.93773in,0.75in)--(4.94811in,0.765567in)--(4.95849in,0.781133in); +\draw 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(4.58302in,0.25in)--(4.57264in,0.265567in)--(4.56227in,0.281133in); +\draw (4.58302in,0.25in)--(4.57264in,0.234433in)--(4.56227in,0.218867in); +\pgfsetlinewidth{0.8pt} +\draw (1.5in,0.25in)--(2.9375in,0.25in)--(4.375in,0.25in); +\draw (1.5in,0.75in)--(2.9375in,0.75in)--(4.375in,0.75in); +\pgftext[at={\pgfpoint{0in}{0.5in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$+$}}} +\pgftext[at={\pgfpoint{0in}{0.5in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\times$}}} +\pgftext[at={\pgfpoint{0in}{0.416978in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$S$}}} +\pgftext[at={\pgfpoint{2in}{0.75in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{4in}{0.75in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{3in}{0.75in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{2in}{0.805348in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$a$}}} +\pgftext[at={\pgfpoint{4in}{0.805348in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$a'$}}} +\pgftext[at={\pgfpoint{3in}{0.805348in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$A$}}} +\pgftext[at={\pgfpoint{2in}{0.25in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{4in}{0.25in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{3in}{0.25in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{2in}{0.194652in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$b$}}} +\pgftext[at={\pgfpoint{4in}{0.194652in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$b'$}}} +\pgftext[at={\pgfpoint{3in}{0.194652in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{2.5in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~2.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/036.xp b/32857-t/images/sources/036.xp new file mode 100644 index 0000000..981767f --- /dev/null +++ b/32857-t/images/sources/036.xp @@ -0,0 +1,49 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double x1(-1), x2(3), x3(1), Y1(-0.5), Y2(0.5); + +int main() +{ + picture(P(-5,-1), P(5,1), "5 x 1in"); + + begin(); + degrees(); + arrow_init(); + + Arrow(P(xmax(), Y2), P(4, Y2)); + Arrow(P(4, Y1), P(xmax(), Y1)); + + bold(); + + line(P(-2, Y1), P(3.75, Y1)); + line(P(-2, Y2), P(3.75, Y2)); + + marker(P(xmin(),0), PLUS); + marker(P(xmin(),0), TIMES); + + label(P(xmin(),0), P(0, -6), "$S$", b); + + h_axis_tick(P(x1,Y2)); + h_axis_tick(P(x2,Y2)); + h_axis_tick(P(x3,Y2)); + + label(P(x1,Y2), P(0,4), "$a$", t); + label(P(x2,Y2), P(0,4), "$a'$", t); + label(P(x3,Y2), P(0,4), "$A$", t); + + h_axis_tick(P(x1,Y1)); + h_axis_tick(P(x2,Y1)); + h_axis_tick(P(x3,Y1)); + + label(P(x1,Y1), P(0,-4), "$b$", b); + label(P(x2,Y1), P(0,-4), "$b'$", b); + label(P(x3,Y1), P(0,-4), "$B$", b); + + font_face("sc"); + label(P(0,ymin()), P(0,-12), "Fig.~2.", b); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/038.eepic b/32857-t/images/sources/038.eepic new file mode 100644 index 0000000..1c5370f --- /dev/null +++ b/32857-t/images/sources/038.eepic @@ -0,0 +1,129 @@ +%% Generated from 038.xp on Sun May 30 14:11:46 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,1] x [0,1] +%% Actual size: 1.75 x 1.75in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (1.75in,1.75in); +\pgfsetlinewidth{0.8pt} +\draw (0.9625in,0.875in)--(1.35625in,0.875in)--(1.75in,0.875in); +\pgfsetlinewidth{0.4pt} +\draw (0.9625in,0.875in)--(1.75in,0.875in); +\pgfsetfillcolor{rgb_000000} +\draw [fill](1.72924in,0.875in)--(1.66698in,0.847326in)-- + (1.75in,0.875in)--(1.66698in,0.902674in)--(1.72924in,0.875in)--cycle; +\draw (0.9625in,0.875in)--(0.952122in,0.890567in)--(0.941744in,0.906133in); +\draw 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+dX_mir,0), P(2,0), "$B$", r); + + font_size("footnotesize"); + label(P(0.75, 0.5), P(0,-6), "\\textit{Direction of Earth's Motion}", b); + + font_size(); + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), P(0,-12), "Fig.~3.", b); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/039.eepic b/32857-t/images/sources/039.eepic new file mode 100644 index 0000000..1807649 --- /dev/null +++ b/32857-t/images/sources/039.eepic @@ -0,0 +1,395 @@ +%% Generated from 039.xp on Sun May 30 14:11:48 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-0.5,3.5] x [-1.5,1.5] +%% Actual size: 4 x 3in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\xdefinecolor{rgb_ffffff}{rgb}{1,1,1}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4in,3in); +\pgfsetlinewidth{0.8pt} +\draw (0.55in,0.5in)--(0.549931in,0.502617in)-- + (0.549726in,0.505226in)--(0.549384in,0.507822in)-- + 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(0.549384in,2.99218in)--(0.549726in,2.99477in)-- + (0.549931in,2.99738in)--(0.55in,3in)--cycle; +\draw (0.5in,3in)--(1.1in,3in); +\draw [fill](1.07924in,3in)--(1.01698in,2.97233in)--(1.1in,3in)-- + (1.01698in,3.02767in)--(1.07924in,3in)--cycle; +\pgftext[at={\pgfpoint{0.403141in}{3.09686in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}$A$}}} +\pgftext[at={\pgfpoint{0.75in}{3.05535in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}$u$}}} +\pgftext[at={\pgfpoint{2in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~4.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/039.xp b/32857-t/images/sources/039.xp new file mode 100644 index 0000000..dac260d --- /dev/null +++ b/32857-t/images/sources/039.xp @@ -0,0 +1,49 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double rad(0.05), Rad(0.5); + +void orbit(const P& loc) +{ + circle(loc, rad); + aarrow(loc + P(rad + 0.5*dX_mir,0), loc + P(3,0)); + + dashed(); + ellipse(loc, Rad*E_1, Rad*E_2, 0, 360, 30); + solid(); + masklabel(loc + P(1.5,0), "$l$"); + label(loc + P(3,0), P(4,0), "$O$", r); + label(loc + P(3,0), P(0,-4), "\\textit{Observer}", b); +} + +void direct(const P&loc, const P& dir, const std::string& msg) +{ + circle(loc + Rad*J(dir), rad); + arrow(loc + Rad*J(dir), loc + Rad*J(dir) + 1.2*Rad*dir); + label(loc + Rad*J(dir), -7*(dir-J(dir)), msg, c); + label(loc + Rad*J(dir) + 0.5*Rad*dir, 4*J(dir), "$u$",c); +} + +int main() +{ + picture(P(-0.5,-1.5), P(3.5,1.5), "4 x 3in"); + + begin(); + degrees(); + arrow_init(); + + bold(); + + orbit(P(0,-1)); + orbit(P(0,1)); + + direct(P(0,-1), -E_1, "$B$"); + direct(P(0, 1), E_1, "$A$"); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), P(0,-12), "Fig.~4.", b); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/042.eepic b/32857-t/images/sources/042.eepic new file mode 100644 index 0000000..696234d --- /dev/null +++ b/32857-t/images/sources/042.eepic @@ -0,0 +1,76 @@ +%% Generated from 042.xp on Sun May 30 14:08:46 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,2] x [0,1] +%% Actual size: 4 x 2in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4in,2in); +\pgfsetlinewidth{0.8pt} +\draw (3.55in,0.9in)--(3.725in,0.9in)--(3.9in,0.9in); +\pgfsetlinewidth{0.4pt} +\draw (3.55in,0.9in)--(3.9in,0.9in); +\pgfsetfillcolor{rgb_000000} +\draw [fill](3.87924in,0.9in)--(3.81698in,0.872326in)-- + (3.9in,0.9in)--(3.81698in,0.927674in)--(3.87924in,0.9in)--cycle; +\draw (3.55in,0.9in)--(3.53962in,0.915567in)--(3.52924in,0.931133in); +\draw (3.55in,0.9in)--(3.53962in,0.884433in)--(3.52924in,0.868867in); +\draw (3.57076in,0.9in)--(3.56038in,0.915567in)--(3.55in,0.931133in); +\draw (3.57076in,0.9in)--(3.56038in,0.884433in)--(3.55in,0.868867in); +\draw (3.59151in,0.9in)--(3.58113in,0.915567in)--(3.57076in,0.931133in); 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{\makebox(0,0)[l]{\hbox{\color{rgb_000000}$a$}}} +\pgftext[at={\pgfpoint{2.24465in}{1.7in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$b$}}} +\pgftext[at={\pgfpoint{3.35535in}{1.7in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$b$}}} +\pgftext[at={\pgfpoint{0.944652in}{0.6in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$A$}}} +\filldraw[color=rgb_000000] (1in,0.6in) circle(0.013837in); +\pgftext[at={\pgfpoint{2.74465in}{0.6in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$B$}}} +\filldraw[color=rgb_000000] (2.8in,0.6in) circle(0.013837in); +\pgftext[at={\pgfpoint{1.3in}{1.1in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}$S$}}} +\pgftext[at={\pgfpoint{3.1in}{1.1in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}$S'$}}} +\pgftext[at={\pgfpoint{2in}{1in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\includegraphics[width=4in]{042_nolabels.eps}}}} +\pgftext[at={\pgfpoint{2in}{0.166044in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}\textsc{Fig.~5.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/042.xp b/32857-t/images/sources/042.xp new file mode 100644 index 0000000..b9a286b --- /dev/null +++ b/32857-t/images/sources/042.xp @@ -0,0 +1,50 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double Y1(0.85); + +int main() +{ + picture(P(0,0), P(2,1), "4 x 2in"); + + begin(); + arrow_init(); + + P pta1(0.25, Y1), pta2(0.75, Y1), ptA(0.5, 0.3); + P ptb1(1.15, Y1), ptb2(1.65, Y1), ptB(1.4, 0.3); + + Arrow(P(1.775, 0.45), P(1.95, 0.45)); + + bold(); + line(pta1, pta2); + line(ptb1, ptb2); + + dashed(); + dash_size(6); + line(0.5*(pta1 + pta2), ptA); + solid(); + + label(0.5*(pta1 + pta2), P(0,2), "$m$", t); + label(P(1.85, 0.45), P(0,-4), "$V$", b); + + label(pta1, P(-4,0), "$a$", l); + label(pta2, P( 4,0), "$a$", r); + + label(ptb1, P(-4,0), "$b$", l); + label(ptb2, P( 4,0), "$b$", r); + + ddot(ptA, P(-4,0), "$A$", l); + ddot(ptB, P(-4,0), "$B$", l); + + label(P(0.65, 0.55), "$S$"); + label(P(1.55, 0.55), "$S'$"); + + label(P(1,0.5), "\\includegraphics[width=4in]{042_nolabels.eps}"); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), P(0,12), "Fig.~5.", t); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/043.eepic b/32857-t/images/sources/043.eepic new file mode 100644 index 0000000..8562c3e --- /dev/null +++ b/32857-t/images/sources/043.eepic @@ -0,0 +1,140 @@ +%% Generated from 043.xp on Sun May 30 14:08:48 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-1,1] x [0,1] +%% Actual size: 3.5 x 1.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3.5in,1.5in); +\pgfsetlinewidth{0.8pt} +\draw (-0.4375in,1.5in)--(0.4375in,1.5in)--(1.3125in,1.5in); +\draw (0.4375in,1.5in)--(0.4375in,1.4625in); +\draw (0.4375in,1.3875in)--(0.4375in,1.35in); +\draw (0.4375in,1.35in)--(0.4375in,1.3125in); +\draw 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+\draw (3.24297in,1.15625in)--(3.25937in,1.125in); +\draw (3.25937in,1.125in)--(3.27578in,1.09375in); +\draw (3.30859in,1.03125in)--(3.325in,1in); +\draw (3.325in,1in)--(3.34141in,0.96875in); +\draw (3.37422in,0.90625in)--(3.39062in,0.875in); +\draw (3.39062in,0.875in)--(3.40703in,0.84375in); +\draw (3.43984in,0.78125in)--(3.45625in,0.75in); +\draw (3.45625in,0.75in)--(3.47266in,0.71875in); +\draw (3.50547in,0.65625in)--(3.52187in,0.625in); +\draw (3.52187in,0.625in)--(3.53828in,0.59375in); +\draw (3.57109in,0.53125in)--(3.5875in,0.5in); +\draw (3.5875in,0.5in)--(3.60391in,0.46875in); +\draw (3.63672in,0.40625in)--(3.65313in,0.375in); +\draw (3.65313in,0.375in)--(3.66953in,0.34375in); +\draw (3.70234in,0.28125in)--(3.71875in,0.25in); +\draw (3.71875in,0.25in)--(3.73516in,0.21875in); +\draw (3.76797in,0.15625in)--(3.78437in,0.125in); +\draw (3.78437in,0.125in)--(3.80078in,0.09375in); +\draw (3.83359in,0.03125in)--(3.85in,0in); +\pgftext[at={\pgfpoint{2.04465in}{0in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{4.08035in}{0in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$B'$}}} +\pgftext[at={\pgfpoint{1.75in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~6.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/043.xp b/32857-t/images/sources/043.xp new file mode 100644 index 0000000..9dbae82 --- /dev/null +++ b/32857-t/images/sources/043.xp @@ -0,0 +1,48 @@ +/* -*-ePiX-*- */ +#include "epix.h" +using namespace ePiX; + +double my_dX(0.5); + +void tee(const P& loc, const std::string& m1, const std::string& m2, + const std::string& m3, const P& off, epix_label_posn A, bool f=false) +{ + P tmp(my_dX,0); + line(loc - tmp, loc + tmp); + dashed(); + line(loc, loc - P(0,1)); + + label(loc, P(0,4), m2, t); + label(loc - tmp, P(-2,0), m1, l); + label(loc + tmp, P( 2,0), m1, r); + + label(loc - P(0, 1), off, m3, A); + + if (f) + { + line(loc - P(0,1) - 1.1*tmp, loc - P(0,1) + 1.1*tmp); + line(loc, loc - P(0,1) - 0.9*tmp); + line(loc, loc - P(0,1) + 0.9*tmp); + } + solid(); +} + +int main() +{ + picture(P(-1,0), P(1,1), "3.5 x 1.5in"); + + begin(); + + bold(); + dash_size(6); + tee(P(-0.75, 1), "$a$", "$m$", "$A$", P(-4,0), l); + tee(P( 0.75, 1), "$b$", "$n$", "$p$", P(0,-4), b, true); + + label(P(0.75) - 1.1*P(my_dX), P(-4,0), "$B$", l); + label(P(0.75) + 1.1*P(my_dX), P(4,0), "$B'$", r); + + font_face("sc"); + label(P(0,0.5*(xmin() + xmax()), ymin()), P(0,-12), "Fig.~6.", b); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/045.eepic b/32857-t/images/sources/045.eepic new file mode 100644 index 0000000..08808cb --- /dev/null +++ b/32857-t/images/sources/045.eepic @@ -0,0 +1,45 @@ +%% Generated from 045.xp on Sun May 30 14:08:49 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,2] x [0,1] +%% Actual size: 4 x 2in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4in,2in); +\pgfsetlinewidth{0.8pt} +\draw (3.1in,0.625in)--(3.5in,0.625in)--(3.9in,0.625in); +\pgfsetlinewidth{0.4pt} +\draw (3.1in,0.625in)--(3.9in,0.625in); +\pgfsetfillcolor{rgb_000000} +\draw [fill](3.87924in,0.625in)--(3.81698in,0.597326in)-- + (3.9in,0.625in)--(3.81698in,0.652674in)--(3.87924in,0.625in)--cycle; +\draw (3.1in,0.625in)--(3.08962in,0.640567in)--(3.07924in,0.656133in); +\draw (3.1in,0.625in)--(3.08962in,0.609433in)--(3.07924in,0.593867in); +\draw (3.12076in,0.625in)--(3.11038in,0.640567in)--(3.1in,0.656133in); +\draw (3.12076in,0.625in)--(3.11038in,0.609433in)--(3.1in,0.593867in); +\draw (3.14151in,0.625in)--(3.13113in,0.640567in)--(3.12076in,0.656133in); +\draw (3.14151in,0.625in)--(3.13113in,0.609433in)--(3.12076in,0.593867in); +\draw (3.16227in,0.625in)--(3.15189in,0.640567in)--(3.14151in,0.656133in); +\draw (3.16227in,0.625in)--(3.15189in,0.609433in)--(3.14151in,0.593867in); +\draw (3.18302in,0.625in)--(3.17264in,0.640567in)--(3.16227in,0.656133in); +\draw (3.18302in,0.625in)--(3.17264in,0.609433in)--(3.16227in,0.593867in); +\pgfsetlinewidth{0.8pt} +\draw (0.5in,1.65in)--(1.6in,1.65in)--(2.7in,1.65in); +\draw (0.5in,0.625in)--(1.6in,0.625in)--(2.7in,0.625in); +\pgftext[at={\pgfpoint{0.8in}{1.70535in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$A$}}} +\pgftext[at={\pgfpoint{2.3in}{1.70535in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$m$}}} +\pgftext[at={\pgfpoint{0.8in}{0.680348in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{2.3in}{0.680348in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$n$}}} +\pgftext[at={\pgfpoint{0.8in}{1.65in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{2.3in}{1.65in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{0.8in}{0.625in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{2.3in}{0.625in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{3.5in}{0.569652in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$V$}}} +\pgftext[at={\pgfpoint{2in}{1in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\includegraphics[width=4in]{045_nolabels.eps}}}} +\pgftext[at={\pgfpoint{2in}{0in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\textsc{Fig.~7.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/045.xp b/32857-t/images/sources/045.xp new file mode 100644 index 0000000..6ca2615 --- /dev/null +++ b/32857-t/images/sources/045.xp @@ -0,0 +1,44 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double Y1(0.825), Y2(0.3125); + +int main() +{ + picture(P(0,0), P(2,1), "4 x 2in"); + + begin(); + arrow_init(); + + P pta1(0.25, Y1), pta2(1.35, Y1), ptA(0.4, Y1), ptm(1.15, Y1); + P ptb1(0.25, Y2), ptb2(1.35, Y2), ptB(0.4, Y2), ptn(1.15, Y2); + + Arrow(P(1.55, Y2), P(1.95, Y2)); + + bold(); + line(pta1, pta2); + line(ptb1, ptb2); + + label(ptA, P(0,4), "$A$", t); + label(ptm, P(0,4), "$m$", t); + + label(ptB, P(0,4), "$B$", t); + label(ptn, P(0,4), "$n$", t); + + h_axis_tick(ptA); + h_axis_tick(ptm); + + h_axis_tick(ptB); + h_axis_tick(ptn); + + label(P(1.75,Y2), P(0,-4), "$V$", b); + + label(P(1,0.5), "\\includegraphics[width=4in]{045_nolabels.eps}"); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), "Fig.~7."); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/046.eepic b/32857-t/images/sources/046.eepic new file mode 100644 index 0000000..51f084f --- /dev/null +++ b/32857-t/images/sources/046.eepic @@ -0,0 +1,25 @@ +%% Generated from 046.xp on Sun May 30 14:08:51 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,1] x [-0.25,0.25] +%% Actual size: 3 x 0.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3in,0.5in); +\pgfsetlinewidth{0.8pt} +\draw (0in,0.25in)--(1.5in,0.25in)--(3in,0.25in); +\pgftext[at={\pgfpoint{0.15in}{0.305348in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{0.9in}{0.305348in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$B'$}}} +\pgftext[at={\pgfpoint{2.4in}{0.305348in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$n$}}} +\pgftext[at={\pgfpoint{3in}{0.305348in}}] {\makebox(0,0)[b]{\hbox{\color{rgb_000000}$n'$}}} +\pgftext[at={\pgfpoint{0.15in}{0.25in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{0.9in}{0.25in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{2.4in}{0.25in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{3in}{0.25in}}] {\makebox(0,0){\hbox{\color{rgb_000000}$\rule{0.5pt}{6pt}$}}} +\pgftext[at={\pgfpoint{1.5in}{0in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\textsc{Fig.~8.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/046.xp b/32857-t/images/sources/046.xp new file mode 100644 index 0000000..d1e4e84 --- /dev/null +++ b/32857-t/images/sources/046.xp @@ -0,0 +1,36 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double Y1(0.825), Y2(0.3125); + +int main() +{ + picture(P(0,-0.25), P(1,0.25), "3 x 0.5in"); + + begin(); + arrow_init(); + + P ptB1(0.05,0), ptB2(0.3,0), ptn1(0.8,0), ptn2(1,0); + + bold(); + line(P(xmin()), P(xmax())); + + label(ptB1, P(0,4), "$B$", t); + label(ptB2, P(0,4), "$B'$", t); + + label(ptn1, P(0,4), "$n$", t); + label(ptn2, P(0,4), "$n'$", t); + + h_axis_tick(ptB1); + h_axis_tick(ptB2); + + h_axis_tick(ptn1); + h_axis_tick(ptn2); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), "Fig.~8."); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/052.eepic b/32857-t/images/sources/052.eepic new file mode 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+\draw (2.385in,0.913256in)--(2.36943in,0.902878in)--(2.35387in,0.8925in); +\draw (2.385in,0.913256in)--(2.40057in,0.902878in)--(2.41613in,0.8925in); +\draw (2.385in,0.934011in)--(2.36943in,0.923633in)--(2.35387in,0.913256in); +\draw (2.385in,0.934011in)--(2.40057in,0.923633in)--(2.41613in,0.913256in); +\draw (2.385in,0.954767in)--(2.36943in,0.944389in)--(2.35387in,0.934011in); +\draw (2.385in,0.954767in)--(2.40057in,0.944389in)--(2.41613in,0.934011in); +\draw (2.385in,0.975522in)--(2.36943in,0.965144in)--(2.35387in,0.954767in); +\draw (2.385in,0.975522in)--(2.40057in,0.965144in)--(2.41613in,0.954767in); +\pgftext[at={\pgfpoint{1.35in}{2.1in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}$S$}}} +\pgftext[at={\pgfpoint{0.6in}{1.05in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}$S'$}}} +\pgftext[at={\pgfpoint{2.25in}{1.5in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\includegraphics[width=4.5in]{052_nolabels.eps}}}} +\pgftext[at={\pgfpoint{2.25in}{0in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\textsc{Fig.~9.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/052.xp b/32857-t/images/sources/052.xp new file mode 100644 index 0000000..c183ec0 --- /dev/null +++ b/32857-t/images/sources/052.xp @@ -0,0 +1,46 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double rad(0.06), scale(1.5); + +void ball(const P& loc, const P& dir, const std::string& msg) +{ + P ctr(loc - 5.5*rad*dir), DX(scale*rad*J(dir)); + circle(ctr, rad); + dashed(); + dash_size(6); + line(loc, loc - 8*rad*dir); + solid(); + label(ctr + DX, 2*J(dir), msg, c); + Arrow(ctr - DX - 1.25*rad*dir, ctr - DX + 2.25*rad*dir); +} + + +int main() +{ + picture(P(0,0), P(3,2), "4.5 x 3in"); + + begin(); + + arrow_inset(0.75); + arrow_ratio(3); + arrow_width(4); + + Arrow(P(2.5, 0.7), P(2.9, 0.7)); + label(P(2.7, 0.7), P(0,-4), "$V$", b); + + ball(P(1.9, 1.1), -E_2, "$A$"); + ball(P(1.5, 1), E_2, "$B$"); + + label(P(0.9, 1.4), "$S$"); + label(P(0.4, 0.7), "$S'$"); + + label(P(1.5,1), "\\includegraphics[width=4.5in]{052_nolabels.eps}"); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), "Fig.~9."); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/056.eepic b/32857-t/images/sources/056.eepic new file mode 100644 index 0000000..ef411fb --- /dev/null +++ b/32857-t/images/sources/056.eepic @@ -0,0 +1,47 @@ +%% Generated from 056.xp on Sun May 30 14:08:54 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,5] x [0,2] +%% Actual size: 5 x 2in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (5in,2in); +\pgfsetlinewidth{0.8pt} +\draw (0.75in,1in)--(1.5in,1in)--(2.25in,1in); +\draw (0.75in,1in)--(0.75in,1.5in)--(0.75in,2in); +\draw (0.75in,1in)--(0.375in,0.625in)--(0in,0.25in); +\pgftext[at={\pgfpoint{2.30535in}{1in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$X$}}} +\pgftext[at={\pgfpoint{0.805348in}{2in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$Y$}}} +\pgftext[at={\pgfpoint{0.055348in}{0.25in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$Z$}}} +\pgftext[at={\pgfpoint{0.694652in}{1in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$O$}}} +\draw (3.5in,1in)--(4.25in,1in)--(5in,1in); +\draw (3.5in,1in)--(3.5in,1.5in)--(3.5in,2in); +\draw (3.5in,1in)--(3.125in,0.625in)--(2.75in,0.25in); +\pgftext[at={\pgfpoint{5.05535in}{1in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$X'$}}} +\pgftext[at={\pgfpoint{3.55535in}{2in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$Y'$}}} +\pgftext[at={\pgfpoint{2.80535in}{0.25in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$Z'$}}} +\pgftext[at={\pgfpoint{3.44465in}{1in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$O'$}}} +\draw (4in,0.67in)--(4.4in,0.67in)--(4.8in,0.67in); +\pgfsetlinewidth{0.4pt} +\draw (4in,0.67in)--(4.8in,0.67in); +\pgfsetfillcolor{rgb_000000} +\draw [fill](4.77924in,0.67in)--(4.71698in,0.642326in)-- + (4.8in,0.67in)--(4.71698in,0.697674in)--(4.77924in,0.67in)--cycle; +\draw (4in,0.67in)--(3.98962in,0.685567in)--(3.97924in,0.701133in); +\draw (4in,0.67in)--(3.98962in,0.654433in)--(3.97924in,0.638867in); +\draw (4.02076in,0.67in)--(4.01038in,0.685567in)--(4in,0.701133in); +\draw (4.02076in,0.67in)--(4.01038in,0.654433in)--(4in,0.638867in); +\draw (4.04151in,0.67in)--(4.03113in,0.685567in)--(4.02076in,0.701133in); +\draw (4.04151in,0.67in)--(4.03113in,0.654433in)--(4.02076in,0.638867in); +\draw (4.06227in,0.67in)--(4.05189in,0.685567in)--(4.04151in,0.701133in); +\draw (4.06227in,0.67in)--(4.05189in,0.654433in)--(4.04151in,0.638867in); +\draw (4.08302in,0.67in)--(4.07264in,0.685567in)--(4.06227in,0.701133in); +\draw (4.08302in,0.67in)--(4.07264in,0.654433in)--(4.06227in,0.638867in); +\pgftext[at={\pgfpoint{4.4in}{0.614652in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$V$}}} +\pgftext[at={\pgfpoint{2.5in}{0in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\textsc{Fig.~10.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/056.xp b/32857-t/images/sources/056.xp new file mode 100644 index 0000000..251aeb3 --- /dev/null +++ b/32857-t/images/sources/056.xp @@ -0,0 +1,47 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +void axes(const P& loc, bool pr = false) +{ + P pX(loc + P(1.5,0)), pY(loc + P(0,1)), pZ(loc - P(0.75,0.75)); + bold(); + line(loc, pX); + line(loc, pY); + line(loc, pZ); + + if (!pr) + { + label(pX, P(4,0), "$X$", r); + label(pY, P(4,0), "$Y$", br); + label(pZ, P(4,0), "$Z$", r); + label(loc, P(-4,0), "$O$", l); + } + else + { + label(pX, P(4,0), "$X'$", r); + label(pY, P(4,0), "$Y'$", br); + label(pZ, P(4,0), "$Z'$", r); + label(loc, P(-4,0), "$O'$", l); + + Arrow(loc + P(0.5, -0.33), loc + P(1.3, -0.33)); + label(loc + P(0.9, -0.33), P(0,-4), "$V$", b); + } +} + +int main() +{ + picture(P(0,0), P(5,2), "5 x 2in"); + + begin(); + arrow_init(); + + axes(P(0.75, 1)); + axes(P(3.5, 1), true); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), "Fig.~10."); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/078.eepic b/32857-t/images/sources/078.eepic new file mode 100644 index 0000000..8b4e59e --- /dev/null +++ b/32857-t/images/sources/078.eepic @@ -0,0 +1,139 @@ +%% Generated from 078.xp on Sun May 30 14:08:56 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-3,3] x [-1,1] +%% Actual size: 3 x 1in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (3in,1in); +\pgfsetlinewidth{0.8pt} +\draw (1.175in,0.75in)--(1.17493in,0.752617in)-- + (1.17473in,0.755226in)--(1.17438in,0.757822in)-- + (1.17391in,0.760396in)--(1.1733in,0.762941in)-- + (1.17255in,0.765451in)--(1.17168in,0.767918in)-- + 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{\makebox(0,0)[t]{\hbox{\color{rgb_000000}$-u$}}} +\pgftext[at={\pgfpoint{1.5in}{0in}}] {\makebox(0,0)[c]{\hbox{\color{rgb_000000}\textsc{Fig.~11.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/078.xp b/32857-t/images/sources/078.xp new file mode 100644 index 0000000..fe91ee6 --- /dev/null +++ b/32857-t/images/sources/078.xp @@ -0,0 +1,41 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double rad(0.1); +void particle(const P& loc, double sgn) +{ + P utip(loc - P(3*sgn*rad,0)), vtip(loc - P(0,sgn*0.75)); + bold(); + circle(loc, rad); + arrow(loc - P(3*sgn, 0), utip); + arrow(loc - P(0, 2*sgn*rad), vtip); + + if (sgn < 0) + { + label(vtip, P(4,2*sgn), "$-v$", r); + label(loc - P(1.5*sgn,0), P(0,-2), "$-u$", b); + } + else + { + label(vtip, P(4,2*sgn), "$+v$", r); + label(loc - P(1.5*sgn,0), P(0, 2), "$+u$", t); + } +} + +int main() +{ + picture(P(-3,-1), P(3,1), "3 x 1in"); + + begin(); + arrow_init(); + + particle(P(-0.75, 0.5), 1); + particle(P( 0.75, -0.5), -1); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), "Fig.~11."); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/088.eepic b/32857-t/images/sources/088.eepic new file mode 100644 index 0000000..9b288e7 --- /dev/null +++ b/32857-t/images/sources/088.eepic @@ -0,0 +1,171 @@ +%% Generated from 088.xp on Sun May 30 14:08:57 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,1] x [0,1] +%% Actual size: 2.5 x 2.5in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (2.5in,2.5in); +\pgfsetlinewidth{0.8pt} +\draw (0in,0in)--(1.25in,0in)--(2.5in,0in); +\draw (0in,0in)--(0in,1.25in)--(0in,2.5in); +\draw (0.6875in,0.625in)--(0.687414in,0.628271in)-- + (0.687158in,0.631533in)--(0.686731in,0.634777in)-- + (0.686134in,0.637994in)--(0.68537in,0.641176in)-- + 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{\makebox(0,0)[r]{\hbox{\color{rgb_000000}$u_y$}}} +\pgftext[at={\pgfpoint{1.27767in}{1.47233in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$u$}}} +\pgftext[at={\pgfpoint{1.25in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~12.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/088.xp b/32857-t/images/sources/088.xp new file mode 100644 index 0000000..9f098c5 --- /dev/null +++ b/32857-t/images/sources/088.xp @@ -0,0 +1,44 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double rad(0.025); + +P loc(0.25, 0.25), ux(0.5,0), uy(0,0.7), u(ux+uy); + +int main() +{ + picture(P(0,0), P(1,1), "2.5 x 2.5in"); + + begin(); + arrow_init(); + + bold(); + line(P(0,0), P(1,0)); + line(P(0,0), P(0,1)); + + circle(loc, rad); + arrow(loc, loc + ux); + arrow(loc, loc + uy); + arrow(loc, loc + u); + + dashed(); + dash_size(6); + line(loc + ux, loc + u); + line(loc + uy, loc + u); + + label(P(0,0), P(-2,-2), "$O$", bl); + label(P(1,0), P( 0,-2), "$X$", bl); + label(P(0,1), P(-2, 0), "$Y$", bl); + + label(loc, P(-8,0), "$m$", l); + label(loc + 0.5*ux, P(0,-2), "$u_x$", b); + label(loc + 0.5*uy, P(-2,0), "$u_y$", l); + label(loc + 0.5*u, P( 2,-2), "$u$", br); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), P(0,-12), "Fig.~12.", b); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/094.eepic b/32857-t/images/sources/094.eepic new file mode 100644 index 0000000..dab47e0 --- /dev/null +++ b/32857-t/images/sources/094.eepic @@ -0,0 +1,176 @@ +%% Generated from 094.xp on Sun May 30 14:08:59 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,1] x [0,1.5] +%% Actual size: 2 x 3in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (2in,3in); +\pgfsetlinewidth{0.8pt} +\draw (0.1155in,0in)--(1.05775in,0in)--(2in,0in); +\draw 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(0.198in,1.6155in)--(0.198in,2.05775in)--(0.198in,2.5in); +\pgfsetlinewidth{0.4pt} +\draw (0.198in,1.6155in)--(0.198in,2.5in); +\draw [fill](0.198in,2.47924in)--(0.225674in,2.41698in)-- + (0.198in,2.5in)--(0.170326in,2.41698in)--(0.198in,2.47924in)--cycle; +\draw (0.198in,1.6155in)--(0.182433in,1.60512in)--(0.166867in,1.59474in); +\draw (0.198in,1.6155in)--(0.213567in,1.60512in)--(0.229133in,1.59474in); +\draw (0.198in,1.63626in)--(0.182433in,1.62588in)--(0.166867in,1.6155in); +\draw (0.198in,1.63626in)--(0.213567in,1.62588in)--(0.229133in,1.6155in); +\draw (0.198in,1.65701in)--(0.182433in,1.64663in)--(0.166867in,1.63626in); +\draw (0.198in,1.65701in)--(0.213567in,1.64663in)--(0.229133in,1.63626in); +\draw (0.198in,1.67777in)--(0.182433in,1.66739in)--(0.166867in,1.65701in); +\draw (0.198in,1.67777in)--(0.213567in,1.66739in)--(0.229133in,1.65701in); +\draw (0.198in,1.69852in)--(0.182433in,1.68814in)--(0.166867in,1.67777in); +\draw (0.198in,1.69852in)--(0.213567in,1.68814in)--(0.229133in,1.67777in); +\pgftext[at={\pgfpoint{-0.107511in}{0in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$e$}}} +\pgftext[at={\pgfpoint{-0.107511in}{1.5in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$e_1$}}} +\pgftext[at={\pgfpoint{2in}{-0.027674in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$X$}}} +\pgftext[at={\pgfpoint{-0.027674in}{3in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$Y$}}} +\pgftext[at={\pgfpoint{0.341174in}{1.41698in}}] {\makebox(0,0)[tl]{\hbox{\color{rgb_000000}$U_x = V$}}} +\pgftext[at={\pgfpoint{0.239511in}{2in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$U_y}}} +\pgftext[at={\pgfpoint{1in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~13.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/094.xp b/32857-t/images/sources/094.xp new file mode 100644 index 0000000..2a35a9b --- /dev/null +++ b/32857-t/images/sources/094.xp @@ -0,0 +1,42 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double rad(0.033), scale(1.75), dr(scale*rad); + +P loc(0.25, 0.25), ux(0.5,0), uy(0,0.7), u(ux+uy); + +int main() +{ + picture(P(0,0), P(1,1.5), "2 x 3in"); + + begin(); + arrow_init(); + + bold(); + line(P(dr,0), P(1,0)); + line(P(0,dr), P(0,0.75 - dr)); + line(P(0,0.75 + dr), P(0, 1.5)); + + circle(P(0,0), rad); + circle(P(0,0.75), rad); + + P loc(3*rad, 0.75); + Arrow(loc + P(dr,0), loc + P(0.4,0)); + Arrow(loc + P(0,dr), loc + P(0,0.5)); + + label(P(-rad,0), P(-3, 0), "$e$", l); + label(P(-rad,0.75), P(-3, 0), "$e_1$", l); + + label(P(xmax(),0), P( 0,-2), "$X$", bl); + label(P(0,ymax()), P(-2, 0), "$Y$", bl); + + label(loc + P(dr,0), P(2,-6), "$U_x = V$", br); + label(loc + P(0, 0.25), P(3,0), "$U_y", r); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), P(0,-12), "Fig.~13.", b); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/166.eepic b/32857-t/images/sources/166.eepic new file mode 100644 index 0000000..27f9184 --- /dev/null +++ b/32857-t/images/sources/166.eepic @@ -0,0 +1,49 @@ +%% Generated from 166.xp on Wed Jun 16 20:29:02 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,1] x [0,1] +%% Actual size: 2.25 x 2.25in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\xdefinecolor{rgb_ffffff}{rgb}{1,1,1}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (2.25in,2.25in); +\pgfsetlinewidth{0.8pt} +\draw (0in,2.25in)--(1.125in,2.25in)--(2.25in,2.25in); +\draw (0in,2.25in)--(0in,1.125in)--(0in,0in); +\draw (0.3375in,1.9125in)--(1.29375in,1.9125in)--(2.25in,1.9125in); +\draw (0.3375in,1.9125in)--(0.3375in,0.95625in)--(0.3375in,0in); +\draw (0in,0in)--(0.84375in,0in); +\pgfsetfillcolor{rgb_000000} +\draw [fill](0.822994in,0in)--(0.760728in,-0.027674in)-- + (0.84375in,0in)--(0.760728in,0.027674in)--(0.822994in,0in)--cycle; +\draw (2.25in,2.25in)--(2.25in,1.40625in); +\draw [fill](2.25in,1.42701in)--(2.22233in,1.48927in)-- + (2.25in,1.40625in)--(2.27767in,1.48927in)--(2.25in,1.42701in)--cycle; +\draw (-0.16875in,1.125in)--(-0.16875in,0in); +\draw [fill](-0.16875in,0.0207555in)--(-0.196424in,0.083022in)-- + (-0.16875in,0in)--(-0.141076in,0.083022in)--(-0.16875in,0.0207555in)--cycle; +\draw (-0.16875in,1.125in)--(-0.16875in,2.25in); +\draw [fill](-0.16875in,2.22924in)--(-0.141076in,2.16698in)-- + (-0.16875in,2.25in)--(-0.196424in,2.16698in)--(-0.16875in,2.22924in)--cycle; +\draw (1.125in,2.41875in)--(0in,2.41875in); +\draw [fill](0.0207555in,2.41875in)--(0.083022in,2.44642in)-- + (0in,2.41875in)--(0.083022in,2.39108in)--(0.0207555in,2.41875in)--cycle; +\draw (1.125in,2.41875in)--(2.25in,2.41875in); +\draw [fill](2.22924in,2.41875in)--(2.16698in,2.39108in)-- + (2.25in,2.41875in)--(2.16698in,2.44642in)--(2.22924in,2.41875in)--cycle; +\pgftext[at={\pgfpoint{-0.16875in}{1.125in}}] {\makebox(0,0)[c]{\rotatebox{90}{\colorbox{rgb_ffffff}{\hbox{\color{rgb_000000}$l_1$}}}}} +\pgftext[at={\pgfpoint{1.125in}{2.41875in}}] {\makebox(0,0)[c]{\colorbox{rgb_ffffff}{\hbox{\color{rgb_000000}$l_2$}}}} +\filldraw[color=rgb_ffffff] (0.16875in,2.08125in) circle(0.0207555in); +\draw[color=rgb_000000] (0.16875in,2.08125in) circle(0.0207555in); +\pgftext[at={\pgfpoint{0.556875in}{-0.055348in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$F_1$}}} +\pgftext[at={\pgfpoint{2.30535in}{1.69312in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$F_2$}}} +\pgftext[at={\pgfpoint{-0.027674in}{2.27767in}}] {\makebox(0,0)[br]{\hbox{\color{rgb_000000}$B$}}} +\pgftext[at={\pgfpoint{2.27767in}{2.27767in}}] {\makebox(0,0)[bl]{\hbox{\color{rgb_000000}$C$}}} +\pgftext[at={\pgfpoint{-0.027674in}{-0.027674in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$A$}}} +\pgftext[at={\pgfpoint{1.125in}{-0.249066in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~14.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/166.xp b/32857-t/images/sources/166.xp new file mode 100644 index 0000000..c053f92 --- /dev/null +++ b/32857-t/images/sources/166.xp @@ -0,0 +1,50 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double wd(0.15), dL(0.075), len(2.5); + +P loc(0.25, 0.25), ux(0.5,0), uy(0,0.7), u(ux+uy); + +int main() +{ + picture(P(0,0), P(1,1), "2.25 x 2.25in"); + + begin(); + arrow_init(); + degrees(); + + bold(); + line(P(0,1), P(1,1)); + line(P(0,1), P(0,0)); + + line(P(wd,1-wd), P(1,1-wd)); + line(P(wd,1-wd), P(wd,0)); + + arrow(P(0,0), P(len*wd,0)); + arrow(P(1,1), P(1,1 - len*wd)); + + aarrow(P(-dL,0), P(-dL, 1)); + aarrow(P(0,1 + dL), P(1, 1 + dL)); + + label_angle(90); + masklabel(P(-dL, 0.5), "$l_1$"); + label_angle(0); + + masklabel(P(0.5, 1+dL), "$l_2$"); + + circ(P(0,1) + 0.5*wd*P(1,-1)); + + label(P(0.66*len*wd,0), P(0,-4), "$F_1$", b); + label(P(1, 1 - 0.66*len*wd), P(4,0), "$F_2$", r); + + label(P(0,1), P(-2,2), "$B$", tl); + label(P(1,1), P( 2,2), "$C$", tr); + label(P(0,0), P(-2,-2), "$A$", bl); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), P(0,-18), "Fig.~14.", b); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/200.eepic b/32857-t/images/sources/200.eepic new file mode 100644 index 0000000..0c55d67 --- /dev/null +++ b/32857-t/images/sources/200.eepic @@ -0,0 +1,170 @@ +%% Generated from 200.xp on Sun May 30 14:09:03 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,7] x [0,2] +%% Actual size: 4.2 x 1.2in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\xdefinecolor{rgb_cccccc}{rgb}{0.8,0.8,0.8}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4.2in,1.2in); +\pgfsetlinewidth{0.8pt} +\draw (0.6in,0.6in)--(0.9in,0.6in)--(1.2in,0.6in); +\draw (0.6in,0.6in)--(0.6in,0.9in)--(0.6in,1.2in); +\draw (0.6in,0.6in)--(0.387868in,0.387868in)--(0.175736in,0.175736in); 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b/32857-t/images/sources/200.xp @@ -0,0 +1,80 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +double wd(0.6), Lft(3), Rght(6.5); + +double tmin(10), tmax(180 - tmin), dt(5); +P ctr(2.75,1), v1(0, 0.3), v2(-0.25,0); + +void axes(const P& loc) +{ + line(loc, loc + P(1,0)); + line(loc, loc + P(0,1)); + line(loc, loc + cis(-135)); + + label(loc + P(1,0), P(0,-2), "$Y$", bl); + label(loc + P(0,1), P(2,-2), "$Z$", r); + label(loc + cis(-135), P(4,0), "$X$", r); +} + +P my_loc(double t) +{ + return ctr + Cos(t)*v1 + Sin(t)*v2; +} + +void my_Arrow(){ + + ellipse(ctr, v1, v2, tmin, tmax); + arrow(my_loc(tmax - 5*dt), my_loc(tmax)); + + double pip(pt_to_screen(1.5)); + plain(); + + for (int i=0; i<5; ++i) + { + double t(tmin + 1.5*i*dt); + P tail(my_loc(t)), head(my_loc(t + dt)); + P dL(head - tail); + dL *= pip/norm(dL); + + line(tail, tail - sc_arr*dL + 1.5*J(dL)); + line(tail, tail - sc_arr*dL - 1.5*J(dL)); + } +} + +int main() +{ + picture(P(0,0), P(7,2), "4.2 x 1.2in"); + + begin(); + degrees(); + arrow_init(); + + bold(); + axes(P(1, 1)); + + line(P(2.5,1), P(xmax(),1)); + + my_Arrow(); + + fill(Black(0.2)); + rect(P(Lft, 2 - wd), P(Rght, 2)); + rect(P(Lft, 0), P(Rght, wd)); + nofill(); + + dashed(); + dash_size(6); + + line(P(Lft,0), P(Lft,2)); + line(P(Rght,0), P(Rght,2)); + + label(P(Rght, 2 - 0.5*wd), P(2,0), "$A$", r); + label(P(Lft, 2 - 0.5*wd), P(-2,0), "$A$", l); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), P(0,-12), "Fig.~15.", b); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/203.eepic b/32857-t/images/sources/203.eepic new file mode 100644 index 0000000..a9317eb --- /dev/null +++ b/32857-t/images/sources/203.eepic @@ -0,0 +1,242 @@ +%% Generated from 203.xp on Sun May 30 14:09:05 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,4] x [0,3.5] +%% Actual size: 3.6 x 3.15in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% 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+\pgftext[at={\pgfpoint{2.745in}{1.54465in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}$\Delta x$}}} +\pgftext[at={\pgfpoint{3.20535in}{2.05in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$\Delta t$}}} +\pgftext[at={\pgfpoint{1.8in}{-0.249066in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~16.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/203.xp b/32857-t/images/sources/203.xp new file mode 100644 index 0000000..e2abee0 --- /dev/null +++ b/32857-t/images/sources/203.xp @@ -0,0 +1,45 @@ +/* -*-ePiX-*- */ +#include "epix.h" +//#include "tolman.h" +using namespace ePiX; + +double xa(1), dt(1), dx(0.9), t0(3.5), m(dt/dx); + +int main() +{ + picture(P(0,0), P(4,3.5), "3.6 x 3.15in"); + + begin(); + // degrees(); + // arrow_init(); + + bold(); + + line(P(0,0), P(xmax(),0)); + line(P(0,0), P(0,ymax())); + + dashed(); + line(P(xa,0), P(xa, ymax())); + line(P(xa,0), P(xmax(), m*(xmax() - xa))); + + dash_size(6); + line(P(t0,0), P(t0, m*(t0 - xa))); + line(P(t0 - dx,0), P(t0 - dx, m*(t0 - dx - xa))); + line(P(t0, m*(t0 - dx - xa)), P(t0 - dx, m*(t0 - dx - xa))); + + label(P(0,0), P(-2,-2), "$O$", bl); + label(P(xmax(),0), P(0,-2), "$X$", bl); + label(P(0,ymax()), P(-2,0), "$T$", bl); + + label(P(xa,0), P(0,-4), "$a$", b); + label(P(xa,ymax()), P(4,0), "$b$", r); + label(P(xmax(), m*(xmax() - xa)), P(2,-2), "$c$", br); + + label(P(t0 - 0.5*dx, m*(t0 - dx - xa)), P(0,-4), "$\\Delta x$", b); + label(P(t0, m*(t0 - dx - xa) + 0.5*dt), P(4,0), "$\\Delta t$", r); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), P(0,-18), "Fig.~16.", b); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/204.eepic b/32857-t/images/sources/204.eepic new file mode 100644 index 0000000..eb04902 --- /dev/null +++ b/32857-t/images/sources/204.eepic @@ -0,0 +1,560 @@ +%% Generated from 204.xp on Sun May 30 14:09:06 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [0,4] x [-1,4] +%% Actual size: 3.6 x 4.5in +%% Figure offset: left by 0in, down by 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+\pgftext[at={\pgfpoint{-0.027674in}{0.872326in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}$O$}}} +\pgftext[at={\pgfpoint{3.62767in}{0.9in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}$X$}}} +\pgftext[at={\pgfpoint{-0.027674in}{4.5in}}] {\makebox(0,0)[tr]{\hbox{\color{rgb_000000}$T$}}} +\pgftext[at={\pgfpoint{1.8in}{-0.166044in}}] {\makebox(0,0)[t]{\hbox{\color{rgb_000000}\textsc{Fig.~17.}}}} +\end{tikzpicture} diff --git a/32857-t/images/sources/204.xp b/32857-t/images/sources/204.xp new file mode 100644 index 0000000..b5b5353 --- /dev/null +++ b/32857-t/images/sources/204.xp @@ -0,0 +1,66 @@ +/* -*-ePiX-*- */ +#include "epix.h" +//#include "tolman.h" +using namespace ePiX; + +int main() +{ + picture(P(0,-1), P(4,4), "3.6 x 4.5in"); + + begin(); + degrees(); + // arrow_init(); + + double t1(0.9), t2(2), t3(2.25), t4(3.25), th(-15); + + P O(0,0), ptc(4.5,4.5); + P p1a(t1,t1), p2a(t2,t2), p3a(t2,t1); + + P b1(polar(t3, th)), b2(polar(t4, th)); + Segment L1(O, ptc), L2(b1, b1 + cis(th + 90)), L3(b2, b2 + cis(th + 90)); + P p1b(L1*L2), p2b(L1*L3), p3b(p1b + polar(t4 - t3, th)); + + bold(); + + line(O, P(xmax(),0)); + line(O, P(0,ymax())); + + dashed(); + line(O, polar(xmax(), th)); + line(O, polar(ymax(), th + 90)); + + label(polar(xmax(), th), polar(4, th), "$X'$", br); + label(polar(ymax(), th + 90), P(-2,-2), "$T'$", bl); + + dash_size(6); + line(P(t1,0), p1a); + line(P(t2,0), p2a); + line(p1a, p3a); + + label(0.5*(p1a + p3a), P(0,-4), "$\\Delta x$", b); + label(0.5*(p2a + p3a), P(4, 0), "$\\Delta t$", r); + + line(polar(t3, th), p1b); + line(polar(t4, th), p2b); + line(p1b, p3b); + + line_style("- ---------- -"); + line(O, 1.1*p2b); + + label_angle(th); + + label(1.1*p2b, P(2,-2), "$c$", br); + + label(0.5*(p1b + p3b), P(0,-4), "$\\Delta x'$", b); + label(0.5*(p2b + p3b), P(4, 0), "$\\Delta t'$", r); + + label_angle(0); + label(P(0,0), P(-2,-2), "$O$", l); + label(P(xmax(),0), P(2,0), "$X$", r); + label(P(0,ymax()), P(-2,0), "$T$", bl); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), P(0,-12), "Fig.~17.", b); + tikz_format(); + end(); +} diff --git a/32857-t/images/sources/223.eepic b/32857-t/images/sources/223.eepic new file mode 100644 index 0000000..d786647 --- /dev/null +++ b/32857-t/images/sources/223.eepic @@ -0,0 +1,223 @@ +%% Generated from 223.xp on Sun May 30 14:09:09 EDT 2010 by +%% ePiX-1.2.4 +%% +%% Cartesian bounding box: [-4,4] x [-4,4] +%% Actual size: 4 x 4in +%% Figure offset: left by 0in, down by 0in +%% +%% usepackages tikz +%% +\xdefinecolor{rgb_000000}{rgb}{0,0,0}% +\xdefinecolor{rgb_ffffff}{rgb}{1,1,1}% +\begin{tikzpicture} +\pgfsetlinewidth{0.4pt} +\useasboundingbox (0in,0in) rectangle (4in,4in); +\pgfsetlinewidth{0.8pt} +\draw (3.125in,0.992853in)--(3.02917in,1.10045in)-- + (2.90533in,1.24527in)--(2.8019in,1.37307in)-- + (2.71654in,1.48674in)--(2.64734in,1.58884in)-- + (2.59273in,1.68167in)--(2.55149in,1.76733in)-- + (2.52267in,1.84774in)--(2.50564in,1.92472in)--(2.5in,2in)-- + 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b/32857-t/images/sources/223.xp @@ -0,0 +1,102 @@ +/* -*-ePiX-*- */ +#include "epix.h" +#include "tolman.h" +using namespace ePiX; + +P f1p(double t) +{ + return P(cosh(t), sinh(t)); +} + +P f1m(double t) +{ + return P(-cosh(t), sinh(t)); +} + +P f2p(double t) +{ + return P(sinh(t), cosh(t)); +} + +P f2m(double t) +{ + return P(sinh(t), -cosh(t)); +} + +double MAX(2.25), sc(0.95), Rad(3); + +int main() +{ + picture(P(-4,-4), P(4,4), "4 x 4in"); + + begin(); + arrow_init(); + degrees(); + bold(); + P pB(sc*MAX, sc*MAX), pD(-sc*MAX, -sc*MAX), O(0,0); + P pA(-sc*MAX, sc*MAX), pC(sc*MAX, -sc*MAX); + + double th(20), t0(atanh(Tan(th))); + + P pa1(f1p(0)), pa2(f1p(t0)), pb1(f2p(0)), pb2(f2p(t0)); + + clip_box(P(-MAX,-MAX,-1), P(MAX,MAX,1)); + plot(f1p, -3, 3, 40); + plot(f1m, -3, 3, 40); + + plot(f2p, -3, 3, 40); + plot(f2m, -3, 3, 40); + clip_box(); + + line(P(xmin(), 0), P(xmax(), 0)); + line(P(0, ymin()), P(0, ymax())); + + line(pA, pC); + line(pB, pD); + + label(pA, P(-2, 2), "$A$", tl); + label(pC, P( 2,-2), "$C$", br); + + label(pB, P( 2, 2), "$B$", tr); + label(pD, P(-2,-2), "$D$", bl); + + label(O, P(-12,2), "$O$", tl); + + label(pa1, P(2,2), "$a$", tr); + label(pa2, P(4,4), "$a'$", tr); + + label(pb1, P(2,2), "$b$", tr); + label(pb2, P(4,4), "$b'$", tr); + + label(P(xmax(),0), P(2,0), "$X_1$", r); + label(P(0,ymax()), P(2,0), "$X_4$", br); + + label(polar(4, th), P(2,0), "$X_1'$", r); + label(polar(4, 90-th), P(2,0), "$X_4'$", r); + + arc_arrow(O, Rad, 0.5*th, 0); + arc_arrow(O, Rad, 0.5*th, th); + + arc_arrow(O, Rad, 90 - 0.5*th, 90); + arc_arrow(O, Rad, 90 - 0.5*th, 90 - th); + + masklabel(polar(Rad, 0.5*th), "$\\theta$"); + masklabel(polar(Rad, 90 - 0.5*th), "$\\theta$"); + + dashed(); + line(polar(-4,th), polar(4, th)); + line(polar(-4,90-th), polar(4, 90-th)); + + dash_size(6); + P tmp1(polar(2.75,90-th)), tmp2(polar(4, 90-th)), tmp3(tmp2.x1(), tmp1.x2()); + line(tmp1, tmp3); + line(tmp2, tmp3); + + label(0.5*(tmp1+tmp3), P(0,-4), "$dx_1$", b); + label(0.5*(tmp2+tmp3), P(4, 0), "$dx_4$", r); + + font_face("sc"); + label(P(0.5*(xmin() + xmax()),ymin()), P(0,-12), "Fig.~18.", b); + tikz_format(); + end(); +} diff --git a/32857-t/old/32857-t.tex b/32857-t/old/32857-t.tex new file mode 100644 index 0000000..baedce9 --- /dev/null +++ b/32857-t/old/32857-t.tex @@ -0,0 +1,11906 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% The Project Gutenberg EBook of The Theory of the Relativity of Motion, by +% Richard Chace Tolman % +% % +% This eBook is for the use of anyone anywhere at no cost and with % +% almost no restrictions whatsoever. You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: The Theory of the Relativity of Motion % +% % +% Author: Richard Chace Tolman % +% % +% Release Date: June 17, 2010 [EBook #32857] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK THE THEORY OF THE RELATIVITY *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{32857} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% fontenc: For boldface small-caps. Required. %% +%% %% +%% calc: Infix arithmetic. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. Required. %% +%% %% +%% alltt: Fixed-width font environment. Required. %% +%% array: Enhanced tabular features. Required. %% +%% longtable: Tables spanning multiple pages. Required. %% +%% %% +%% indentfirst: Optional. %% +%% textcase: \MakeUppercase et al. ignore math. Required. %% +%% bm: Bold math. Optional. %% +%% %% +%% footmisc: Extended footnote capabilities. Required. %% +%% %% +%% fancyhdr: Enhanced running headers and footers. Required. %% +%% %% +%% graphicx: Standard interface for graphics inclusion. Required. %% +%% wrapfig: Illustrations surrounded by text. Required. %% +%% rotating: Need to rotate a large table. Required. %% +%% %% +%% geometry: Enhanced page layout package. Required. %% +%% hyperref: Hypertext embellishments for pdf output. 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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: The Theory of the Relativity of Motion + +Author: Richard Chace Tolman + +Release Date: June 17, 2010 [EBook #32857] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE THEORY OF THE RELATIVITY *** +\end{PGtext} +\end{minipage} +\end{center} + + +%%%% Credits and transcriber's note %%%% +\clearpage +\thispagestyle{empty} + +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Berj Zamanian, Joshua +Hutchinson and the Online Distributed Proofreading Team +at http://www.pgdp.net (This file was produced from images +from the Cornell University Library: Historical Mathematics +Monographs collection.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter +\normalsize + + +%% -----File: 001.png---------- +\iffalse +Production Note + +Cornell University Library produced +this volume to replace the +irreparably deteriorated original. +It was scanned using Xerox software +and equipment at 600 dots +per inch resolution and compressed +prior to storage using +CCITT Group 4 compression. The +digital data were used to create +Cornell's replacement volume on +paper that meets the ANSI Standard +Z39.48-1984. The production +of this volume was supported in +part by the Commission on Preservation +and Access and the Xerox +Corporation. Digital file copyright +by Cornell University +Library 1992. +\fi +%% -----File: 002.png---------- +%[Blank Page] +%% -----File: 003.png---------- +\iffalse %[** TN: Cornell University Ex Libris page] + +%[Illustration: Cornell insignia] + +Cornell University Library +Ithaca, New York + +BOUGHT WITH THE INCOME OF THE +SAGE ENDOWMENT FUND +THE GIFT OF +HENRY W. SAGE + +1891 +\fi +%% -----File: 004.png---------- +%[Blank Page] +%% -----File: 005.png---Folio i------- +%% Title page +\begin{center} +\setlength{\TmpLen}{24pt}% +\LARGE\textbf{THE THEORY OF \\[\TmpLen] +THE RELATIVITY OF MOTION} \\[3\TmpLen] +\footnotesize BY \\[0.5\TmpLen] +\large RICHARD C. TOLMAN +\vfill + +\footnotesize UNIVERSITY OF CALIFORNIA PRESS \\ +BERKELEY \\[8pt] +1917 +\end{center} +%% -----File: 006.png---Folio ii------- +%% Verso +\clearpage +\null\vfill +\begin{center} +\scshape\tiny Press of \\ +The New Era Printing Company \\ +Lancaster, Pa +\end{center} +\vfill +%% -----File: 007.png---Folio iii------- +\clearpage +\null\vfill +\begin{center} +\footnotesize TO + +\large H. E. +\end{center} +\vfill +%% -----File: 008.png---Folio iv------- +%[Blank Page] +%% -----File: 009.png---Folio v------- +\cleardoublepage +%\pagestyle{fancy}**** +\phantomsection\pdfbookmark[0]{Table of Contents}{Contents} +\small +\tableofcontents +\normalsize + +\iffalse +%%%% Table of contents auto-generated; Scanned ToC commented out. %%%% +%[** TN: Heading below is printed by \contentsname] +THE THEORY OF THE RELATIVITY OF MOTION. +BY +RICHARD C. TOLMAN, PH.D. +TABLE OF CONTENTS. + +\textsc{Preface} 1 +\textsc{Chapter} I. Historical Development of Ideas as to the Nature of Space and +Time 5 +\textit{Part I}. The Space and Time of Galileo and Newton 5 +Newtonian Time 6 +Newtonian Space 7 +The Galileo Transformation Equations 9 +\textit{Part II}. The Space and Time of the Ether Theory 10 +Rise of the Ether Theory 10 +Idea of a Stationary Ether 12 +Ether in the Neighborhood of Moving Bodies 12 +Ether Entrained in Dielectrics 13 +The Lorentz Theory of a Stationary Ether 13 +\textit{Part III}. Rise of the Einstein Theory of Relativity 17 +The Michelson-Morley Experiment 17 +The Postulates of Einstein 18 +\textsc{Chapter} II. The Two Postulates of the Einstein Theory of Relativity 20 +The First Postulate of Relativity 20 +The Second Postulate of the Einstein Theory of Relativity 21 +Suggested Alternative to the Postulate of the Independence of the +Velocity of Light and the Velocity of the Source 23 +Evidence against Emission Theories of Light 24 +Different Forms of Emission Theory 25 +Further Postulates of the Theory of Relativity 27 +\textsc{Chapter} III. Some Elementary Deductions 28 +Measurements of Time in a Moving System 28 +Measurements of Length in a Moving System 30 +The Setting of Clocks in a Moving System 33 +The Composition of Velocities 35 +The Mass of a Moving Body 37 +The Relation between Mass and Energy 39 +\textsc{Chapter} IV. The Einstein Transformation Equations for Space and Time 42 +The Lorentz Transformation 42 +Deduction of the Fundamental Transformation Equations 43 +The Three Conditions to be Fulfilled 44 +The Transformation Equations 45 +Further Transformation Equations 47 +Transformation Equations for Velocity 47 +Transformation Equations for the Function $\dfrac{1}{\sqrt{1-\frac{u^2}{c^2}}}$ 47 +%% -----File: 010.png---Folio vi------- +Transformation Equations for Acceleration 48 +Chapter V. Kinematical Applications 49 +The Kinematical Shape of a Rigid Body 49 +The Kinematical Rate of a Clock 50 +The Idea of Simultaneity 51 +The Composition of Velocities 52 +The Case of Parallel Velocities 52 +Composition of Velocities in General 53 +Velocities Greater than that of Light 54 +Applications to Optical Problems 56 +The Doppler Effect 57 +The Aberration of Light 59 +Velocity of Light in Moving Media 60 +Group Velocity 61 +Chapter VI. The Dynamics of a Particle 62 +The Laws of Motion 62 +Difference between Newtonian and Relativity Mechanics 62 +The Mass of a Moving Particle 63 +Transverse Collision 63 +Mass the Same in all Directions 66 +Longitudinal Collision 67 +Collision of any Type 68 +Transformation Equations for Mass 72 +The Force Acting on a Moving Particle 73 +Transformation Equations for Force 73 +The Relation between Force and Acceleration 74 +Transverse and Longitudinal Acceleration 76 +The Force Exerted by a Moving Charge 77 +The Field around a Moving Charge 79 +Application to a Specific Problem 80 +Work 81 +Kinetic Energy 81 +Potential Energy 82 +The Relation between Mass and Energy 83 +Application to a Specific Problem 85 +Chapter VII. The Dynamics of a System of Particles 88 +On the Nature of a System of Particles 88 +The Conservation of Momentum 89 +The Equation of Angular Momentum 90 +The Function $T$ 92 +The Modified Lagrangian Function 93 +The Principle of Least Action 93 +Lagrange's Equations 95 +Equations of Motion in the Hamiltonian Form 96 +Value of the Function $T'$ 97 +The Principle of the Conservation of Energy 99 +On the Location of Energy in Space 100 +%% -----File: 011.png---Folio vii------- +\textsc{Chapter} VIII. The Chaotic Motion of a System of Particles 102 +The Equations of Motion 102 +Representation in Generalized Space 103 +Liouville's Theorem 103 +A System of Particles 104 +Probability of a Given Statistical State 105 +Equilibrium Relations 106 +The Energy as a Function of the Momentum 108 +The Distribution Law 109 +Polar Coördinates 110 +The Law of Equipartition 110 +Criterion for Equality of Temperature 112 +Pressure Exerted by a System of Particles 113 +The Relativity Expression for Temperature 114 +The Partition of Energy 117 +Partition of Energy for Zero Mass 117 +Approximate Partition for Particles of any Mass 118 +\textsc{Chapter} IX. The Principle of Relativity and the Principle of Least Action. 121 +The Principle of Least Action 121 +The Equations of Motion in the Lagrangian Form 122 +Introduction of the Principle of Relativity 124 +Relation between $\int W'dt'$ and $\int Wdt$ 124 +Relation between $H'$ and $H$ 127 +\textsc{Chapter} X. The Dynamics of Elastic Bodies 130 +On the Impossibility of Absolutely Rigid Bodies 130 +\textit{Part I}. Stress and Strain 130 +Definition of Strain 130 +Definition of Stress 132 +Transformation Equations for Strain 133 +Variation in the Strain 134 +\textit{Part II}. Introduction of the Principle of Least Action 137 +The Kinetic Potential for an Elastic Body 137 +Lagrange's Equations 138 +Transformation Equations for Stress 139 +Value of $E°$ 139 +The Equations of Motion in the Lagrangian Form 140 +Density of Momentum 142 +Density of Energy 142 +Summary of Results from the Principle of Least Action 142 +\textit{Part III}. Some Mathematical Relations 143 +The Unsymmetrical Stress Tensor $\mathrm{t}$ 143 +The Symmetrical Tensor $\mathrm{p}$ 145 +Relation between div $\mathrm{t}$ and $\mathrm{t}_n$ 146 +The Equations of Motion in the Eulerian Form 147 +\textit{Part IV}. Applications of the Results 148 +Relation between Energy and Momentum 148 +The Conservation of Momentum 149 +%% -----File: 012.png---Folio viii------- +The Conservation of Angular Momentum 150 +Relation between Angular Momentum and the Unsymmetrical +Stress Tensor 151 +The Right-Angled Lever 152 +Isolated Systems in a Steady State 154 +The Dynamics of a Particle 154 +Conclusion 154 +\textsc{Chapter} XI. The Dynamics of a Thermodynamic System 156 +The Generalized Coördinates and Forces 156 +Transformation Equation for Volume 156 +Transformation Equation for Entropy 157 +Introduction of the Principle of Least Action. The Kinetic +Potential 157 +The Lagrangian Equations 158 +Transformation Equation for Pressure 159 +Transformation Equation for Temperature 159 +The Equations of Motion for Quasistationary Adiabatic Acceleration +160 +The Energy of a Moving Thermodynamic System 161 +The Momentum of a Moving Thermodynamic System 161 +The Dynamics of a Hohlraum 162 +\textsc{Chapter} XII. Electromagnetic Theory 164 +The Form of the Kinetic Potential 164 +The Principle of Least Action 165 +The Partial Integrations 165 +Derivation of the Fundamental Equations of Electromagnetic +Theory 166 +The Transformation Equations for $\mathrm{e}$, $\mathrm{h}$ and $\rho$ 168 +The Invariance of Electric Charge 170 +The Relativity of Magnetic and Electric Fields 171 +Nature of Electromotive Force 172 +Derivation of the Fifth Fundamental Equation 172 +Difference between the Ether and the Relativity Theories of Electromagnetics +173 +Applications to Electromagnetic Theory 176 +The Electric and Magnetic Fields around a Moving Charge 176 +The Energy of a Moving Electromagnetic System 178 +Relation between Mass and Energy 180 +The Theory of Moving Dielectrics 181 +Relation between Field Equations for Material Media and +Electron Theory 182 +Transformation Equations for Moving Media 183 +Theory of the Wilson Experiment 186 +\textsc{Chapter} XIII. Four-Dimensional Analysis 188 +Idea of a Time Axis 188 +Non-Euclidean Character of the Space 189 +%% -----File: 013.png---Folio ix------- +Part I. Vector Analysis of the Non-Euclidean Four-Dimensional Manifold +191 +Space, Time and Singular Vectors 192 +Invariance of $x^2 + y^2 + z^2 - c^2t^2$ 192 +Inner Product of One-Vectors 193 +Non-Euclidean Angle 194 +Kinematical Interpretation of Angle in Terms of Velocity 194 +Vectors of Higher Dimensions 195 +Outer Products 195 +Inner Product of Vectors in General 198 +The Complement of a Vector 198 +The Vector Operator, $\Diamond$ or Quad 199 +Tensors 200 +The Rotation of Axes 201 +Interpretation of the Lorentz Transformation as a Rotation of +Axes 206 +Graphical Representation 208 +Part II. Applications of the Four-Dimensional Analysis 211 +Kinematics 211 +Extended Position 211 +Extended Velocity 212 +Extended Acceleration 213 +The Velocity of Light 214 +The Dynamics of a Particle 214 +Extended Momentum 214 +The Conservation Laws 215 +The Dynamics of an Elastic Body 216 +The Tensor of Extended Stress 216 +The Equation of Motion 216 +Electromagnetics 217 +Extended Current 218 +The Electromagnetic Vector $\vc{M}$ 217 +The Field Equations 217 +The Conservation of Electricity 218 +The Product $\vc{M} · \vc{q}$ 218 +The Extended Tensor of Electromagnetic Stress 219 +Combined Electrical and Mechanical Systems 221 +Appendix I. Symbols for Quantities 222 +Scalar Quantities 222 +Vector Quantities 223 +Appendix II. Vector Notation 224 +Three Dimensional Space 224 +Non-Euclidean Four Dimensional Space 225 +\fi +%%%% End of commented table of contents %%%% +%% +%% -----File: 014.png---Folio x------- +%[Blank Page] +%% -----File: 015.png---Folio 1------- +\mainmatter +\phantomsection\pdfbookmark[-1]{Main Matter}{Main Matter} + +\Preface + +Thirty or forty years ago, in the field of physical science, there +was a widespread feeling that the days of adventurous discovery had +passed forever, and the conservative physicist was only too happy to +devote his life to the measurement to the sixth decimal place of +quantities whose significance for physical theory was already an old +story. The passage of time, however, has completely upset such +bourgeois ideas as to the state of physical science, through the discovery +of some most extraordinary experimental facts and the development +of very fundamental theories for their explanation. + +On the experimental side, the intervening years have seen the +discovery of radioactivity, the exhaustive study of the conduction of +electricity through gases, the accompanying discoveries of cathode, +canal and X-rays, the isolation of the electron, the study of the +distribution of energy in the hohlraum, and the final failure of all +attempts to detect the earth's motion through the supposititious +ether. During this same time, the theoretical physicist has been +working hand in hand with the experimenter endeavoring to correlate +the facts already discovered and to point the way to further research. +The theoretical achievements, which have been found particularly +helpful in performing these functions of explanation and prediction, +have been the development of the modern theory of electrons, the +application of thermodynamic and statistical reasoning to the phenomena +of radiation, and the development of Einstein's brilliant +theory of the relativity of motion. + +It has been the endeavor of the following book to present an +introduction to this theory of relativity, which in the decade since +the publication of Einstein's first paper in 1905 (\textit{Annalen der Physik}) +has become a necessary part of the theoretical equipment of every +physicist. Even if we regard the Einstein theory of relativity merely +as a convenient tool for the prediction of electromagnetic and optical +phenomena, its importance to the physicist is very great, not only +because its introduction greatly simplifies the deduction of many +%% -----File: 016.png---Folio 2------- +theorems which were already familiar in the older theories based on a +stationary ether, but also because it leads simply and directly to correct +conclusions in the case of such experiments as those of Michelson +and Morley, Trouton and Noble, and Kaufman and Bucherer, which +can be made to agree with the idea of a stationary ether only by the +introduction of complicated and \textit{ad~hoc} assumptions. Regarded from +a more philosophical point of view, an acceptance of the Einstein +theory of relativity shows us the advisability of completely remodelling +some of our most fundamental ideas. In particular we shall now +do well to change our concepts of space and time in such a way as +to give up the old idea of their complete independence, a notion +which we have received as the inheritance of a long ancestral experience +with bodies moving with slow velocities, but which no longer proves +pragmatic when we deal with velocities approaching that of light. + +The method of treatment adopted in the following chapters is +to a considerable extent original, partly appearing here for the first +time and partly already published elsewhere.\footnote + {\textit{Philosophical Magazine}, vol.~18, p.~510 (1909); + \textit{Physical Review}, vol.~31, p.~26 (1910); + \textit{Phil.\ Mag.}, vol.~21, p.~296 (1911); + \textit{ibid}., vol.~22, p.~458 (1911); + \textit{ibid}., vol.~23, p.~375 (1912); + \textit{Phys.\ Rev.}, vol.~35, p.~136 (1912); + \textit{Phil.\ Mag.}, vol.~25, p.~150 (1913); + \textit{ibid}., vol.~28, p.~572 (1914); + \textit{ibid}., vol.~28, p.~583 (1914).} +\Chapref{III} follows +a method which was first developed by Lewis and Tolman,\footnote + {\textit{Phil.\ Mag.}, vol.~18, p.~510 (1909).} +and the +\Chapnumref[XIII]{last chapter} a method developed by Wilson and Lewis.\footnote + {\textit{Proceedings of the American Academy of Arts and Sciences}, + vol.~48, p.~389 (1912).} +The writer +must also express his special obligations to the works of Einstein, +Planck, Poincaré, Laue, Ishiwara and Laub. + +It is hoped that the mode of presentation is one that will be found +well adapted not only to introduce the study of relativity theory to +those previously unfamiliar with the subject but also to provide the +necessary methodological equipment for those who wish to pursue +the theory into its more complicated applications. + +After presenting, in the \Chapnumref[I]{first chapter}, a brief outline of the historical +development of ideas as to the nature of the space and time of science, +we consider, in \Chapref{II}, the two main postulates upon which the +theory of relativity rests and discuss the direct experimental evidence +for their truth. The \Chapnumref[III]{third chapter} then presents an elementary and +%% -----File: 017.png---Folio 3------- +non-mathematical deduction of a number of the most important +consequences of the postulates of relativity, and it is hoped that this +chapter will prove especially valuable to readers without unusual +mathematical equipment, since they will there be able to obtain a +real grasp of such important new ideas as the change of mass with +velocity, the non-additivity of velocities, and the relation of mass +and energy, without encountering any mathematics beyond the +elements of analysis and geometry. + +In \Chapref{IV} we commence the more analytical treatment of +the theory of relativity by obtaining from the two postulates of +relativity Einstein's transformation equations for space and time as +well as transformation equations for velocities, accelerations, and +for an important function of the velocity. \Chapref{V} presents +various kinematical applications of the theory of relativity following +quite closely Einstein's original method of development. In particular +we may call attention to the ease with which we may handle +the optics of moving media by the methods of the theory of relativity +as compared with the difficulty of treatment on the basis of the ether +theory. + +In Chapters \Chapnumref{VI},~\Chapnumref{VII} and~\Chapnumref{VIII} we develop and apply a theory of +the dynamics of a particle which is based on the Einstein transformation +equations for space and time, Newton's three laws of motion, +and the principle of the conservation of mass. + +We then examine, in \Chapref{IX}, the relation between the theory +of relativity and the principle of least action, and find it possible to +introduce the requirements of relativity theory at the very start into +this basic principle for physical science. We point out that we +might indeed have used this adapted form of the principle of least +action, for developing the dynamics of a particle, and then proceed +in Chapters \Chapnumref{X},~\Chapnumref{XI} and~\Chapnumref{XII} to develop the dynamics of an elastic +body, the dynamics of a thermodynamic system, and the dynamics +of an electromagnetic system, all on the basis of our adapted form +of the principle of least action. + +Finally, in \Chapref{XIII}, we consider a four-dimensional method +of expressing and treating the results of relativity theory. This +chapter contains, in Part~I, an epitome of some of the more important +methods in four-dimensional vector analysis and it is hoped that it +%% -----File: 018.png---Folio 4------- +can also be used in connection with the earlier parts of the book as a +convenient reference for those who are not familiar with ordinary +three-dimensional vector analysis. + +In the present book, the writer has confined his considerations to +cases in which there is a \emph{uniform} relative velocity between systems of +coördinates. In the future it may be possible greatly to extend the +applications of the theory of relativity by considering accelerated +systems of coördinates, and in this connection Einstein's latest work +on the relation between gravity and acceleration is of great interest. +It does not seem wise, however, at the present time to include such +considerations in a book which intends to present a survey of accepted +theory. + +The author will feel amply repaid for the work involved in the +preparation of the book if, through his efforts, some of the younger +American physicists can be helped to obtain a real knowledge of the +important work of Einstein. He is also glad to have this opportunity +to add his testimony to the growing conviction that the conceptual +space and time of science are not God-given and unalterable, but are +rather in the nature of human constructs devised for use in the description +and correlation of scientific phenomena, and that these +spatial and temporal concepts should be altered whenever the discovery +of new facts makes such a change pragmatic. + +The writer wishes to express his indebtedness to Mr.~William~H. +Williams for assisting in the preparation of Chapter~I\@. %[** TN: Not a useful cross-reference] +%% -----File: 019.png---Folio 5------- + + +\Chapter{I}{Historical Development of Ideas as to the Nature of +Space and Time.} +\SetRunningHeads{Chapter One.}{Historical Development.} + +\Paragraph{1.} Since the year 1905, which marked the publication of Einstein's +momentous article on the theory of relativity, the development of +scientific thought has led to a complete revolution in accepted ideas +as to the nature of space and time, and this revolution has in turn +profoundly modified those dependent sciences, in particular mechanics +and electromagnetics, which make use of these two fundamental +concepts in their considerations. + +In the following pages it will be our endeavor to present a description +of these new notions as to the nature of space and time,\footnote + {Throughout this work by ``space'' and ``time'' we shall mean the \emph{conceptual} + space and time of science.} +and to give a partial account of the consequent modifications which +have been introduced into various fields of science. Before proceeding +to this task, however, we may well review those older ideas +as to space and time which until now appeared quite sufficient for +the correlation of scientific phenomena. We shall first consider the +space and time of Galileo and Newton which were employed in the +development of the classical mechanics, and then the space and time +of the ether theory of light. + + +\Section[I]{The Space and Time of Galileo and Newton.} + +\Paragraph{2.} The publication in 1687 of Newton's \textit{Principia} laid down so +satisfactory a foundation for further dynamical considerations, that +it seemed as though the ideas of Galileo and Newton as to the nature +of space and time, which were there employed, would certainly remain +forever suitable for the interpretation of natural phenomena. And +indeed upon this basis has been built the whole structure of classical +mechanics which, until our recent familiarity with very high velocities, +has been found completely satisfactory for an extremely large number +of very diverse dynamical considerations. +%% -----File: 020.png---Folio 6------- + +An examination of the fundamental laws of mechanics will show +how the concepts of space and time entered into the Newtonian +system of mechanics. Newton's laws of motion, from which the +whole of the classical mechanics could be derived, can best be stated +with the help of the equation +\[ +\vc{F} = \frac{d}{dt} (m\vc{u}). +\Tag{1} +\] +This equation defines the force~$\vc{F}$ acting on a particle as equal to the +rate of change in its momentum (\ie, the product of its mass~$m$ and +its velocity~$\vc{u}$), and the whole of Newton's laws of motion may be +summed up in the statement that in the case of two interacting particles +the forces which they mutually exert on each other are equal in +magnitude and opposite in direction. + +Since in Newtonian mechanics the mass of a particle is assumed +constant, equation~(1) may be more conveniently written +\[ +\vc{F} + = m \frac{d\vc{u}}{dt} + = m \frac{d}{dt} \left( \frac{d\vc{r}}{dt} \right), +\] +or +\[ +\begin{aligned} + F_x &= m \frac{d}{dt} \left( \frac{dx}{dt} \right),\\ + F_y &= m \frac{d}{dt} \left( \frac{dy}{dt} \right),\\ + F_z &= m \frac{d}{dt} \left( \frac{dz}{dt} \right), +\end{aligned} +\Tag{2} +\] +and this definition of force, together with the above-stated principle +of the equality of action and reaction, forms the starting-point for +the whole of classical mechanics. + +The necessary dependence of this mechanics upon the concepts +of space and time becomes quite evident on an examination of this +fundamental equation~(2), in which the expression for the force acting +on a particle is seen to contain both the variables $x$,~$y$, and~$z$, which +specify the position of the particle in \emph{space}, and the variable~$t$, which +specifies the \emph{time}. + +\Subsubsection{3}{Newtonian Time.} To attempt a definite statement as to the +%% -----File: 021.png---Folio 7------- +meaning of so fundamental and underlying a notion as that of time +is a task from which even philosophy may shrink. In a general +way, conceptual time may be thought of as a \emph{one-dimensional}, \emph{unidirectional}, +\emph{one-valued} continuum. This continuum is a sort of framework +in which the instants at which actual occurrences take place +find an ordered position. Distances from point to point in the +continuum, that is intervals of time, are measured by the periods of +certain continually recurring cyclic processes such as the daily rotation +of the earth. A unidirectional nature is imposed upon the time +continuum among other things by an acceptance of the second law +of thermodynamics, which requires that actual progression in time +shall be accompanied by an increase in the entropy of the material +world, and this same law requires that the continuum shall be one-valued +since it excludes the possibility that time ever returns upon +itself, either to commence a new cycle or to intersect its former path +even at a single point. + +In addition to these characteristics of the time continuum, which +have been in no way modified by the theory of relativity, the \emph{Newtonian +mechanics always assumed a complete independence of time and +the three-dimensional space continuum} which exists along with it. +In dynamical equations time entered as an \emph{entirely independent} variable +in no way connected with the variables whose specification +determines position in space. In the following pages, however, we +shall find that the theory of relativity requires a very definite interrelation +between time and space, and in the Einstein transformation +equations we shall see the exact way in which measurements of time +depend upon the choice of a set of variables for measuring position +in space. + +\Subsubsection{4}{Newtonian Space.} An exact description of the concept of +space is perhaps just as difficult as a description of the concept of time. +In a general way we think of space as a \emph{three-dimensional}, \emph{homogeneous}, +\emph{isotropic} continuum, and these ideas are common to the +conceptual spaces of Newton, Einstein, and the ether theory of light. +The space of Newton, however, differs on the one hand from that of +Einstein because of a tacit assumption of the complete independence +of space and time measurements; and differs on the other hand from +that of the ether theory of light by the fact that ``free'' space was +%% -----File: 022.png---Folio 8------- +assumed completely empty instead of filled with an all-pervading +quasi-material medium---the ether. A more definite idea of the particularly +important characteristics of the Newtonian concept of space +may be obtained by considering somewhat in detail the actual methods +of space measurement. + +Positions in space are in general measured with respect to some +arbitrarily fixed system of reference which must be threefold in +character corresponding to the three dimensions of space. In particular +we may make use of a set of Cartesian axes and determine, +for example, the position of a particle by specifying its three Cartesian +coördinates $x$,~$y$ and~$z$. + +In Newtonian mechanics the particular set of axes chosen for +specifying position in space has in general been determined in the +first instance by considerations of convenience. For example, it is +found by experience that, if we take as a reference system lines drawn +upon the surface of the earth, the equations of motion based on Newton's +laws give us a simple description of nearly all dynamical phenomena +which are merely terrestrial. When, however, we try to +interpret with these same axes the motion of the heavenly bodies, we +meet difficulties, and the problem is simplified, so far as planetary +motions are concerned, by taking a new reference system determined +by the sun and the fixed stars. But this system, in its turn, becomes +somewhat unsatisfactory when we take account of the observed +motions of the stars themselves, and it is finally convenient to take a +reference system relative to which the sun is moving with a velocity +of twelve miles per second in the direction of the constellation Hercules. +This system of axes is so chosen that the great majority of stars have +on the average no motion with respect to it, and the actual motion +of any particular star with respect to these coördinates is called the +peculiar motion of the star. + +Suppose, now, we have a number of such systems of axes in uniform +relative motion; we are confronted by the problem of finding +some method of transposing the description of a given kinematical +occurrence from the variables of one of these sets of axes to those of +another. For example, if we have chosen a system of axes~$S$ and +have found an equation in $x$,~$y$,~$z$, and~$t$ which accurately describes the +motion of a given point, what substitutions for the quantities involved +%% -----File: 023.png---Folio 9------- +can be made so that the new equation thereby obtained will again +correctly describe the same phenomena when we measure the displacements +of the point relative to a new system of reference~$S'$ +which is in uniform motion with respect to~$S$? The assumption of +Galileo and Newton that ``free'' space is entirely empty, and the +further tacit assumption of the complete independence of space and +time, led them to propose a very simple solution of the problem, and +the transformation equations which they used are generally called +the Galileo Transformation Equations to distinguish them from the +Einstein Transformation Equations which we shall later consider. + +\Subsubsection{5}{The Galileo Transformation Equations.} Consider two systems +of right-angled coördinates, $S$~and~$S'$, which are in relative motion in +the $X$~direction with the velocity~$V$; for convenience let the $X$~axes, +$OX$~and~$O'X'$, of the two systems coincide in direction, and for further +simplification let us take as our zero point for time measurements the +instant when the two origins $O$~and~$O'$ coincide. Consider now a +point which at the time~$t$ has the coördinates $x$,~$y$ and~$z$ measured in +system~$S$. Then, according to the space and time considerations of +Galileo and Newton, the coördinates of the point with reference to +system~$S'$ are given by the following transformation equations: +\begin{align*} +x' &= x-Vt, \Tag{3}\displaybreak[0] \\ +y' &= y, \Tag{4}\displaybreak[0] \\ +z' &= z, \Tag{5}\displaybreak[0] \\ +t' &= t. \Tag{6} +\end{align*} +These equations are fundamental for Newtonian mechanics, and may +appear to the casual observer to be self-evident and bound up with +necessary ideas as to the nature of space and time. Nevertheless, +the truth of the first and the last of these equations is absolutely +dependent on the unsupported assumption of the complete independence +of space and time measurements, and since in the Einstein +theory we shall find a very definite relation between space and time +measurements we shall be led to quite a different set of transformation +equations. Relations (3),~(4),~(5) and~(6) will be found, however, to +be the limiting form which the correct transformation equations assume +when the velocity between the systems~$V$ becomes small compared +%% -----File: 024.png---Folio 10------- +with that of light. Since until very recent times the human +race in its entire past history has been familiar only with velocities +that are small compared with that of light, it need not cause surprise +that the above equations, which are true merely at the limit, should +appear so self-evident. + +\Paragraph{6.} Before leaving the discussion of the space and time system of +Newton and Galileo we must call attention to an important characteristic +which it has in common with the system of Einstein but +which is not a feature of that assumed by the ether theory. If we +have two systems of axes such as those we have just been considering, +we may with equal right consider either one of them at rest and the +other moving past it. All we can say is that the two systems are in +relative motion; it is meaningless to speak of either one as in any +sense ``\textit{absolutely}'' at rest. The equation $x' = x - Vt$ which we +use in transforming the description of a kinematical event from the +variables of system $S$ to those of system $S'$ is perfectly symmetrical +with the equation $x = x' + Vt'$ which we should use for a transformation +in the reverse direction. Of all possible systems no particular +set of axes holds a unique position among the others. We +shall later find that this important principle of the relativity of motion +is permanently incorporated into our system of physical science as +the \textit{first postulate of relativity}. This principle, common both to the +space of Newton and to that of Einstein, is not characteristic of the +space assumed by the classical theory of light. The space of this +theory was supposed to be filled with a stationary medium, the +luminiferous ether, and a system of axes stationary with respect to +this ether would hold a unique position among the other systems +and be the one peculiarly adapted for use as the ultimate system of +reference for the measurement of motions. + +We may now briefly sketch the rise of the ether theory of light and +point out the permanent contribution which it has made to physical +science, a contribution which is now codified as the second postulate +of relativity. + + +\Section[II]{The Space and Time of the Ether Theory.} + +\Subsubsection{7}{Rise of the Ether Theory.} Twelve years before the appearance +of the \textit{Principia}, Römer, a Danish astronomer, observed that an +%% -----File: 025.png---Folio 11------- +eclipse of one of the satellites of Jupiter occurred some ten minutes +later than the time predicted for the event from the known period +of the satellite and the time of the preceding eclipse. He explained +this delay by the hypothesis that it took light twenty-two minutes +to travel across the earth's orbit. Previous to Römer's discovery, +light was generally supposed to travel with infinite velocity. Indeed +Galileo had endeavored to find the speed of light by direct experiments +over distances of a few miles and had failed to detect any lapse of +time between the emission of a light flash from a source and its observation +by a distant observer. Römer's hypothesis has been repeatedly +verified and the speed of light measured by different methods +with considerable exactness. The mean of the later determinations +is $2.9986 × 10^{10}$ cm.~per second. + +\Paragraph{8.} At the time of Römer's discovery there was much discussion +as to the nature of light. Newton's theory that it consisted of particles +or corpuscles thrown out by a luminous body was attacked by +Hooke and later by Huygens, who advanced the view that it was +something in the nature of wave motions in a supposed space-filling +medium or ether. By this theory Huygens was able to explain +reflection and refraction and the phenomena of color, but assuming +\emph{longitudinal} vibrations he was unable to account for polarization. +Diffraction had not yet been observed and Newton contested the +Hooke-Huygens theory chiefly on the grounds that it was contradicted +by the fact of rectilinear propagation and the formation of +shadows. The scientific prestige of Newton was so great that the +emission or corpuscular theory continued to hold its ground for a +hundred and fifty years. Even the masterly researches of Thomas +Young at the beginning of the nineteenth century were unable to +dislodge the old theory, and it was not until the French physicist, +Fresnel, about 1815, was independently led to an undulatory theory +and added to Young's arguments the weight of his more searching +mathematical analysis, that the balance began to turn. From this +time on the wave theory grew in power and for a period of eighty +years was not seriously questioned. This theory has for its essential +postulate the existence of an all-pervading medium, the ether, in +which wave disturbances can be set up and propagated. And the +physical properties of this medium became an enticing field of inquiry +and speculation. +%% -----File: 026.png---Folio 12------- + +\Subsubsection{9}{Idea of a Stationary Ether.} Of all the various properties with +which the physicist found it necessary to endow the ether, for us the +most important is the fact that it must apparently remain stationary, +unaffected by the motion of matter through it. This conclusion was +finally reached through several lines of investigation. We may first +consider whether the ether would be dragged along by the motion of +nearby masses of matter, and, second, whether the ether enclosed in a +moving medium such as water or glass would partake in the latter's +motion. + +\Subsubsection{10}{Ether in the Neighborhood of Moving Bodies.} About the +year 1725 the astronomer Bradley, in his efforts to measure the +parallax of certain fixed stars, discovered that the apparent position +of a star continually changes in such a way as to trace annually a +small ellipse in the sky, the apparent position always lying in the +plane determined by the line from the earth to the center of the +ellipse and by the direction of the earth's motion. On the corpuscular +theory of light this admits of ready explanation as Bradley himself +discovered, since we should expect the earth's motion to produce an +apparent change in the direction of the oncoming light, in just the +same way that the motion of a railway train makes the falling drops +of rain take a slanting path across the window pane. If $\DPtypo{\vc{c}}{c}$~be the +velocity of a light particle and $\DPtypo{\vc{v}}{v}$~the earth's velocity, the apparent or +relative velocity would be $\DPtypo{\vc{c - v}}{c - v}$ and the tangent of the angle of +aberration would be~$\dfrac{v}{c}$. + +Upon the wave theory, it is obvious that we should \emph{also} expect a +similar aberration of light, provided only that the ether shall be +quite stationary and unaffected by the motion of the earth through it, +and this is one of the important reasons that most ether theories have +assumed a \emph{stationary ether unaffected by the motion of neighboring +matter}.\footnote + {The most notable exception is the theory of Stokes, which did assume that + the ether moved along with the earth and then tried to account for aberration with + the help of a velocity potential, but this led to difficulties, as was shown by Lorentz.} + +In more recent years further experimental evidence for assuming +that the ether is not dragged along by the neighboring motion of +large masses of matter was found by Sir Oliver Lodge. His final +experiments were performed with a large rotating spheroid of iron +%% -----File: 027.png---Folio 13------- +with a narrow groove around its equator, which was made the path +for two rays of light, one travelling in the direction of rotation and +the other in the opposite direction. Since by interference methods +no difference could be detected in the velocities of the two rays, here +also the conclusion was reached that \emph{the ether was not appreciably +dragged along by the rotating metal}. + +\Subsubsection{11}{Ether Entrained in Dielectrics.} With regard to the action of +a moving medium on the ether which might be entrained within it, +experimental evidence and theoretical consideration here too finally +led to the supposition that the ether itself must remain perfectly +stationary. The earlier view first expressed by Fresnel, in a letter +written to Arago in 1818, was that the entrained ether did receive a +fraction of the total velocity of the moving medium. Fresnel gave +to this fraction the value~$\dfrac{\mu^2-1}{\mu^2}$, where $\mu$~is the index of refraction of +the substance forming the medium. On this supposition, Fresnel +was able to account for the fact that Arago's experiments upon the +reflection and refraction of stellar rays show no influence whatever +of the earth's motion, and for the fact that Airy found the same angle +of aberration with a telescope filled with water as with air. Moreover, +the later work of Fizeau and the accurate determinations of +Michelson and Morley on the velocity of light in a moving stream +of water did show that the speed was changed by an amount corresponding +to Fresnel's fraction. The fuller theoretical investigations +of Lorentz, however, did not lead scientists to look upon this increased +velocity of light in a moving medium as an evidence that the ether +is pulled along by the stream of water, and we may now briefly sketch +the developments which culminated in the Lorentz theory of a completely +stationary ether. + +\Subsubsection{12}{The Lorentz Theory of a Stationary Ether.} The considerations +of Lorentz as to the velocity of light in moving media became +possible only after it was evident that optics itself is a branch of the +wider science of electromagnetics, and it became possible to treat +transparent media as a special case of dielectrics in general. In 1873, +in his \textit{Treatise on Electricity and Magnetism}, Maxwell first advanced +the theory that electromagnetic phenomena also have their seat in +the luminiferous ether and further that light itself is merely an electromagnetic +%% -----File: 028.png---Folio 14------- +disturbance in that medium, and Maxwell's theory was +confirmed by the actual discovery of electromagnetic waves in 1888 +by Hertz. + +The attack upon the problem of the relative motion of matter and +ether was now renewed with great vigor both theoretically and experimentally +from the electromagnetic side. Maxwell in his treatise had +confined himself to phenomena in stationary media. Hertz, however, +extended Maxwell's considerations to moving matter on the assumption +that the entrained ether is carried bodily along by it. It is evident, +however, that in the field of optical theory such an assumption +could not be expected to account for the Fizeau experiment, which +had already been explained on the assumption that the ether receives +only a fraction of the velocity of the moving medium; while in the +field of electromagnetic theory it was found that Hertz's assumptions +would lead us to expect \emph{no} production of a magnetic field in the +neighborhood of a rotating electric condenser providing the plates of +the condenser and the dielectric move together with the same speed +and this was decisively disproved by the experiment of Eichenwald. +The conclusions of the Hertz theory were also out of agreement with +the important experiments of H.~A.~Wilson on moving dielectrics. +It remained for Lorentz to develop a general theory for moving +dielectrics which was consistent with the facts. + +The theory of Lorentz developed from that of Maxwell by the +addition of the idea of the \emph{electron}, as the atom of electricity, and his +treatment is often called the ``electron theory.'' This atomistic +conception of electricity was foreshadowed by Faraday's discovery +of the quantitative relations between the amount of electricity associated +with chemical reactions in electrolytes and the weight of +substance involved, a relation which indicates that the atoms act as +carriers of electricity and that the quantity of electricity carried by a +single particle, whatever its nature, is always some small multiple of a +definite quantum of electricity, the electron. Since Faraday's time, +the study of the phenomena accompanying the conduction of electricity +through gases, the study of radioactivity, and finally indeed +the isolation and exact measurement of these atoms of electrical +charge, have led us to a very definite knowledge of many of the +properties of the electron. +%% -----File: 029.png---Folio 15------- + +While the experimental physicists were at work obtaining this +more or less first-hand acquaintance with the electron, the theoretical +physicists and in particular Lorentz were increasingly successful in +explaining the electrical and optical properties of matter in general +on the basis of the behavior of the electrons which it contains, the +properties of conductors being accounted for by the presence of movable +electrons, either free as in the case of metals or combined with +atoms to form ions as in electrolytes, while the electrical and optical +properties of dielectrics were ascribed to the presence of electrons +more or less bound by quasi-elastic forces to positions of equilibrium. +This Lorentz electron theory of matter has been developed in great +mathematical detail by Lorentz and has been substantiated by numerous +quantitative experiments. Perhaps the greatest significance +of the Lorentz theory is that such properties of matter as electrical +conductivity, magnetic permeability and dielectric inductivity, which +occupied the position of rather accidental experimental constants in +Maxwell's original theory, are now explainable as the statistical result +of the behavior of the individual electrons. + +With regard now to our original question as to the behavior of +\emph{moving} optical and dielectric media, the Lorentz theory was found +capable of accounting quantitatively for all known phenomena, including +Airy's experiment on aberration, Arago's experiments on the +reflection and refraction of stellar rays, Fresnel's coefficient for the +velocity of light in moving media, and the electromagnetic experiments +upon moving dielectrics made by Röntgen, Eichenwald, H.~A.~Wilson, +and others. For us the particular significance of the Lorentz +method of explaining these phenomena is that he does \emph{not} assume, as +did Fresnel, that the ether is partially dragged along by moving +matter. His investigations show rather that the ether must remain +perfectly stationary, and that such phenomena as the changed velocity +of light in moving media are to be accounted for by the modifying +influence which the electrons in the moving matter have upon the +propagation of electromagnetic disturbances, rather than by a dragging +along of the ether itself. + +Although it would not be proper in this place to present the +mathematical details of Lorentz's treatment of moving media, we +may obtain a clearer idea of what is meant in the Lorentz theory by a +%% -----File: 030.png---Folio 16------- +stationary ether if we look for a moment at the five fundamental +equations upon which the theory rests. These familiar equations, of +which the first four are merely Maxwell's four field equations, modified +by the introduction of the idea of the electron, may be written +\begin{align*} +\curl \vc{h} + &= \frac{1}{c}\, \frac{\partial \vc{e}}{\partial t} + + \rho\, \frac{\vc{u}}{c},\\ +\curl \vc{e} + &= -\frac{1}{c}\, \frac{\partial \vc{h}}{\partial t},\\ +\divg \vc{e} &= \rho,\\ +\divg \vc{h} &= 0,\\ +\vc{f} &= \rho\left\{ + \vc{e} + \left[\frac{\vc{u}}{c} × \vc{h}\right]^*\right\} +\end{align*} +in which the letters have their usual significance. (See \Chapref{XII}\@.) +Now the whole of the Lorentz theory, including of course his treatment +of moving media, is derivable from these five equations, and +the fact that the idea of a stationary ether does lie at the basis of +his theory is most clearly shown by the first and last of these equations, +which contain the velocity~$\vc{u}$ with which the charge in question +is moving, and \emph{for Lorentz this velocity is to be measured with respect +to the assumed stationary ether}. + +We have devoted this space to the Lorentz theory, since his work +marks the culmination of the ether theory of light and electromagnetism, +and for us the particularly significant fact is that by this +line of attack science was \emph{inevitably led to the idea of an absolutely +immovable and stationary ether}. + +\Paragraph{13.} We have thus briefly traced the development of the ether +theory of light and electromagnetism. We have seen that the space +continuum assumed by this theory is not empty as was the space of +Newton and Galileo but is assumed filled with a stationary medium, +the ether, and in conclusion should further point out that the \emph{time +continuum} assumed by the ether theory was apparently the same as +that of Newton and Galileo, and in particular that the \emph{old ideas as to +the absolute independence of space and time were all retained}. +%% -----File: 031.png---Folio 17------- + + +\Section[III]{Rise of the Einstein Theory of Relativity.} + +\Subsubsection{14}{The Michelson-Morley Experiment.} In spite of all the brilliant +achievements of the theory of a stationary ether, we must now +call attention to an experiment, performed at the very time when +the success of the ether theory seemed most complete, whose result +was in direct contradiction to its predictions. This is the celebrated +Michelson-Morley experiment, and to the masterful interpretation of +its consequences at the hands of Einstein we owe the whole theory of +relativity, a theory which will nevermore permit us to assume that +space and time are independent. + +If the theory of a stationary ether were true we should find, contrary +to the expectations of Newton, that systems of coördinates in +relative motion are not symmetrical, a system of axes fixed relatively +to the ether would hold a unique position among all other systems +moving relative to it and would be peculiarly adapted for the measurement +of displacements and velocities. Bodies at rest with respect +to this system of axes fixed in the ether would be spoken of as ``absolutely'' +at rest and bodies in motion through the ether would be +said to have ``absolute'' motion. From the point of view of the +ether theory one of the most important physical problems would be +to determine the velocity of various bodies, for example that of the +earth, through the ether. + +Now the Michelson-Morley experiment was devised for the very +purpose of determining the relative motion of the earth and the ether. +The experiment consists essentially in a comparison of the velocities +of light parallel and perpendicular to the earth's motion in its orbit. +A ray of light from the source~$S$ falls on the half silvered mirror~$A$, +where it is divided into two rays, one of which travels to the mirror~$B$ +and the other to the mirror~$C$, where they are totally reflected. The +rays are recombined and produce a set of interference fringes at~$\DPtypo{0}{O}$. +(See \Figref{1}.) + +We may now think of the apparatus as set so that one of the +divided paths is parallel to the earth's motion and the other perpendicular +to it. On the basis of the stationary ether theory, the +velocity of the light with reference to the apparatus would evidently +be different over the two paths, and hence on rotating the apparatus +%% -----File: 032.png---Folio 18------- +through an angle of ninety degrees we should expect a shift in the +position of the fringes. Knowing the magnitude of the earth's +velocity in its orbit and the dimensions of the apparatus, it is quite +possible to calculate the magnitude of the expected shift, a quantity +\begin{figure}[hbt] + \begin{center} + \Fig{1} + \Input[3in]{032} + \end{center} +\end{figure} +entirely susceptible of experimental determination. Nevertheless the +most careful experiments made at different times of day and at +different seasons of the year entirely failed to show any such shift +at all. + +This result is in direct contradiction to the theory of a stationary +ether and could be reconciled with that theory only by very arbitrary +assumptions. Instead of making such assumptions, the Einstein +theory of relativity finds it preferable to return in part to the older +ideas of Newton and Galileo. + +\Subsubsection{15}{The Postulates of Einstein.} In fact, in accordance with the +results of this work of Michelson-Morley and other confirmatory +experiments, the Einstein theory takes as its \emph{first postulate} the idea +familiar to Newton of the relativity of all motion. It states that +there is nothing out in space in the nature of an ether or of a fixed +set of coördinates with regard to which motion can be measured, +that there is no such thing as absolute motion, and that all we can +speak of is the relative motion of one body with respect to another. +%% -----File: 033.png---Folio 19------- + +Although we thus see that the Einstein theory of relativity has +returned in part to the ideas of Newton and Galileo as to the nature +of space, it is not to be supposed that the ether theory of light and +electromagnetism has made no lasting contribution to physical science. +Quite on the contrary, not only must the ideas as to the periodic and +polarizable nature of the light disturbance, which were first appreciated +and understood with the help of the ether theory, always +remain incorporated in every optical theory, but in particular the +Einstein theory of relativity takes as the basis for its \emph{second postulate} +a principle that has long been familiar to the ether theory, namely +that the velocity of light is independent of the velocity of the source. +We shall see in following chapters that it is the combination of this +principle with the first postulate of relativity that leads to the whole +theory of relativity and to our new ideas as to the nature and interrelation +of space and time. +%% -----File: 034.png---Folio 20------- + + + +\Chapter{II}{The Two Postulates of the Einstein Theory of +Relativity.} +\SetRunningHeads{Chapter Two.}{The Two Postulates.} + +\Paragraph{16.} There are two general methods of evaluating the theoretical +development of any branch of science. One of these methods is to +test by direct experiment the fundamental postulates upon which +the theory rests. If these postulates are found to agree with the facts, +we may feel justified in assuming that the whole theoretical structure +is a valid one, providing false logic or unsuspected and incorrect +assumptions have not later crept in to vitiate the conclusions. The +other method of testing a theory is to develop its interlacing chain of +propositions and theorems and examine the results both for their +internal coherence and for their objective validity. If we find that +the conclusions drawn from the theory are neither self-contradictory +nor contradictory of each other, and furthermore that they agree +with the facts of the external world, we may again feel that our theory +has achieved a measure of success. In the present chapter we shall +present the two main postulates of the theory of relativity, and indicate +the direct experimental evidence in favor of their truth. In following +chapters we shall develop the consequences of these postulates, show +that the system of consequences stands the test of internal coherence, +and wherever possible compare the predictions of the theory with +experimental facts. + + +\Subsection{The First Postulate of Relativity.} + +\Paragraph{17.} The first postulate of relativity as originally stated by Newton +was that it is impossible to measure or detect absolute translatory +motion through space. No objections have ever been made to this +statement of the postulate in its original form. In the development +of the theory of relativity, the postulate has been modified to include +the impossibility of detecting translatory motion through any medium +or ether which might be assumed to pervade space. + +In support of the principle is the general fact that no effects due +to the motion of the earth or other body through the supposed ether +%% -----File: 035.png---Folio 21------- +have ever been observed. Of the many unsuccessful attempts to +detect the earth's motion through the ether we may call attention to +the experiments on the refraction of light made by Arago, Respighi, +Hoek, Ketteler and Mascart, the interference experiments of Ketteler +and Mascart, the work of Klinkerfuess and Haga on the position of +the absorption bands of sodium, the experiment of Nordmeyer on the +intensity of radiation, the experiments of Fizeau, Brace and Strasser +on the rotation of the plane of polarized light by transmission through +glass plates, the experiments of Mascart and of Rayleigh on the +rotation of the plane of polarized light in naturally active substances, +the electromagnetic experiments of Röntgen, Des Coudres, J.~Koenigsberger, +Trouton, Trouton and Noble, and Trouton and Rankine, and +finally the Michelson and Morley experiment, with the further work +of Morley and Miller. For details as to the nature of these experiments +the reader may refer to the original articles or to an excellent +discussion by Laub of the experimental basis of the theory of relativity.\footnote + {\textit{Jahrbuch der Radioaktivität}, vol.~7, p.~405 (1910).} + +In none of the above investigations was it possible to detect any +effect attributable to the earth's motion through the ether. Nevertheless +a number of these experiments \emph{are} in accord with the final +form given to the ether theory by Lorentz, especially since his work +satisfactorily accounts for the Fresnel coefficient for the changed +velocity of light in moving media. Others of the experiments mentioned, +however, could be made to accord with the Lorentz theory +only by very arbitrary assumptions, in particular those of Michelson +and Morley, Mascart and Rayleigh, and Trouton and Noble. For +the purposes of our discussion we shall accept the principle of the +relativity of motion as an experimental fact. + + +\Subsection{The Second Postulate of the Einstein Theory of Relativity.} + +\Paragraph{18.} The second postulate of relativity states that \emph{the velocity of +light in free space appears the same to all observers regardless of the +relative motion of the source of light and the observer}. This postulate +may be obtained by combining the first postulate of relativity with a +principle which has long been familiar to the ether theory of light. +This principle states that the velocity of light is unaffected by a +motion of the emitting source, in other words, that the velocity with +%% -----File: 036.png---Folio 22------- +which light travels past any observer is not increased by a motion +of the source of light towards the observer. The first postulate of +relativity adds the idea that a motion of the source of light towards +the observer is identical with a motion of the observer towards the +source. The second postulate of relativity is seen to be merely a +combination of these two principles, since it states that the velocity +of light in free space appears the same to all observers regardless \emph{both} +of the motion of the source of light and of the observer. + +\Paragraph{19.} It should be pointed out that the two principles whose combination +thus leads to the second postulate of Einstein have come +from very different sources. The first postulate of relativity practically +denies the existence of any stationary ether through which +the earth, for instance, might be moving. On the other hand, the +principle that the velocity of light is unaffected by a motion of the +source was originally derived from the idea that light is transmitted +by a stationary medium which does not partake in the motion of the +source. This combination of two principles, which from a historical +point of view seem somewhat contradictory in nature, has given to +the second postulate of relativity a very extraordinary content. +Indeed it should be particularly emphasized that the remarkable +conclusions as to the nature of space and time forced upon science +by the theory of relativity are the special product of the second +postulate of relativity. + +A simple example of the conclusions which can be drawn from +this postulate will make its extraordinary nature evident. +\begin{figure}[hbt] + \begin{center} + \Fig{2} + \Input{036} + \end{center} +\end{figure} + +$S$~is a source of light and $A$~and~$B$ two moving systems. $A$~is +moving \emph{towards} the source~$S$, and $B$~\emph{away} from it. Observers on the +systems mark off equal distances $aa'$~and~$bb'$ along the path of the light +and determine the time taken for light to pass from $a$~to~$a'$ and $b$~to~$b'$ +respectively. Contrary to what seem the simple conclusions of +common sense, the second postulate requires that the time taken +%% -----File: 037.png---Folio 23------- +for the light to pass from $a$~to~$a'$ shall measure the same as the time +for the light to go from $b$~to~$b'$. Hence if the second postulate of +relativity is correct it is not surprising that science is forced in general +to new ideas as to the nature of space and time, ideas which are in +direct opposition to the requirements of so-called common sense. + + +\Subsection{Suggested Alternative to the Postulate of the Independence of the +Velocity of Light and the Velocity of the Source.} + +\Paragraph{20.} Because of the extraordinary conclusions derived by combining +the principle of the relativity of motion with the postulate +that the velocity of light is independent of the velocity of its source, +a number of attempts have been made to develop so-called \emph{emission} +theories of relativity based on the principle of the relativity of motion +and the further postulate that the velocity of light and the velocity +of its source are additive. + +Before examining the available evidence for deciding between the +rival principles as to the velocity of light, we may point out that +this proposed postulate, of the additivity of the velocity of source +and light, would as a matter of fact lead to a very simple kind of +relativity theory without requiring any changes in our notions of +space and time. For if light or other electromagnetic disturbance +which is being emitted from a source did partake in the motion of +that source in such a way that the velocity of the source is added to +the velocity of emission, it is evident that a system consisting of the +source and its surrounding disturbances would act as a whole and +suffer no \emph{permanent} change in configuration if the velocity of the +source were changed. This result would of course be in direct agreement +with the idea of the relativity of motion which merely requires +that the physical properties of a system shall be independent of its +velocity through space. + +As a particular example of the simplicity of emission theories we +may show, for instance, how easily they would account for the negative +\begin{wrapfigure}{l}{2in}%[** TN: Width-dependent line break] + \Fig{3} + \Input[2in]{038} +\end{wrapfigure} +result of the Michelson-Morley experiment. If~$O$, \Figref{3}, is a +source of light and $A$~and~$B$ are mirrors placed a meter away from~$O$, the +Michelson-Morley experiment shows that the time taken for light to +travel to~$A$ and back is the same as for the light to travel to~$B$ and +back, in spite of the fact that the whole apparatus is moving through +space in the direction $O - B$, due to the earth's motion around the sun. +%% -----File: 038.png---Folio 24------- +The basic assumption of emission theories, however, would require +exactly this result, since it says that light travels out from~$O$ with a +constant velocity in all directions with +respect to~$O$, and not with respect to +some ether through which $O$~is supposed +to be moving. + +The problem now before us is to +decide between the two rival principles +as to the velocity of light, and we shall +find that the bulk of the evidence is all +in favor of the principle which has led +to the Einstein theory of relativity with +its complete revolution in our ideas as to space and time, and against +the principle which has led to the superficially simple emission theories +of relativity. + +\Subsubsection{21}{Evidence Against Emission Theories of Light.} All emission +theories agree in assuming that light from a moving source has a +velocity equal to the vector sum of the velocity of light from a stationary +source and the velocity of the source itself at the instant of +emission. And without first considering the special assumptions +which distinguish one emission theory from another we may first +present certain astronomical evidence which apparently stands in +contradiction to this basic assumption of all forms of emission +theory. This evidence was pointed out by Comstock\footnote + {\textit{Phys.\ Rev}., vol.~30, p.~291 (1910).} +and later by +de Sitter.\footnote + {\textit{Phys.\ Zeitschr}., vol.~14, pp.~429, 1267 (1913).} + +Consider the rotation of a binary star as it would appear to an +observer situated at a considerable distance from the star and in its +plane of rotation. (See \Figref{4}.) If an emission theory of light +be true, the velocity of light from the star in position~$A$ will be $c + u$, +where $u$~is the velocity of the star in its orbit, while in the position~$B$ +the velocity will be $c - u$. Hence the star will be observed to arrive +in position~$A$, $\dfrac{l}{c+u}$~seconds after the event has actually occurred, and +in position~$B$, $\dfrac{l}{c-u}$~seconds after the event has occurred. This will +%% -----File: 039.png---Folio 25------- +make the period of half rotation from $A$~to~$B$ appear to be +\[ +\Delta t - \frac{l}{c+u} + \frac{l}{c-u} = \Delta t + \frac{2ul}{c^2}, +\] +where $\Delta t$~is the actual time of a half rotation in the orbit, which for +\begin{figure}[hbt] + \begin{center} + \Fig{4} + \Input[3.25in]{039} + \end{center} +\end{figure} +simplicity may be taken as circular. On the other hand, the period +of the next half rotation from $B$ back to~$A$ would appear to be +\[ +\Delta t - \frac{2ul}{c^2}. +\] + +Now in the case of most spectroscopic binaries the quantity~$\dfrac{2ul}{c^2}$ +is not only of the same order of magnitude as~$\Delta t$ but oftentimes probably +even larger. Hence, if an emission theory of light were true, +we could hardly expect without correcting for the variable velocity +of light to find that these orbits obey Kepler's laws, as is actually +the case. This is certainly very strong evidence against any form +of emission theory. It may not be out of place, however, to state +briefly the different forms of emission theory which have been tried. + +\Subsubsection{22}{Different Forms of Emission Theory.} As we have seen, emission +theories all agree in assuming that light from a moving source +%% -----File: 040.png---Folio 26------- +has a velocity equal to the vector sum of the velocity of light from a +stationary source and the velocity of the source itself at the instant +of emission. Emission theories differ, however, in their assumptions +as to the velocity of light after its reflection from a mirror. The three +assumptions which up to this time have been particularly considered +are (1)~that the excited portion of the reflecting mirror acts as a new +source of light and that the reflected light has the same velocity~$c$ +with respect to the mirror as has original light with respect to its source; +(2)~that light reflected from a mirror acquires a component of velocity +equal to the velocity of the mirror image of the original source, and +hence has the velocity~$c$ with respect to this mirror image; and (3)~that +light retains throughout its whole path the component of velocity +which it obtained from its original moving source, and hence after +reflection spreads out with velocity~$c$ in a spherical form around a +center which moves with the same speed as the original source. + +Of these possible assumptions as to the velocity of reflected light, +the first seems to be the most natural and was early considered by the +author but shown to be incompatible, not only with an experiment +which he performed on the velocity of light from the two limbs of +the sun,\footnote + {\textit{Phys.\ Rev}., vol.~31, p.~26 (1910).} +but also with measurements of the Stark effect in canal +rays.\footnote + {\textit{Phys.\ Rev}., vol.~35, p.~136 (1912).} +The second assumption as to the velocity of light was made +by Stewart,\footnote + {\textit{Phys.\ Rev}., vol.~32, p.~418 (1911).} +but has also been shown\footnotemark[2] %[** TN: Repeated footnote here, below] +to be incompatible with +measurements of the Stark effect in canal rays. Making use of the +third assumption as to the velocity of reflected light, a somewhat +complete emission theory has been developed by Ritz,\footnote + {\textit{Ann.\ de chim.\ et phys}., vol.~13, p.~145 (1908); + \textit{Arch.\ de Génève} vol.~26, p.~232 + (1908); \textit{Scientia}, vol.\ 5 (1909).} +and unfortunately +optical experiments for deciding between the Einstein +and Ritz relativity theories have never been performed, although +such experiments are entirely possible of performance.\footnotemark[2] Against the +Ritz theory, however, we have of course the general astronomical +evidence of Comstock and de Sitter which we have already described +above. + +For the present, the observations described above, comprise the +whole of the direct experimental evidence against emission theories +%% -----File: 041.png---Folio 27------- +of light and in favor of the principle which has led to the second +postulate of the Einstein theory. One of the consequences of the +Einstein theory, however, has been the deduction of an expression +for the mass of a moving body which has been closely verified by the +Kaufmann-Bucherer experiment. Now it is very interesting to note, +that starting with what has thus become an \emph{experimental} expression +for the mass of a moving body, it is possible to work backwards to a +derivation of the second postulate of relativity. For the details of +the proof we must refer the reader to the original article.\footnote + {\textit{Phys.\ Rev}., vol.\ 31, p.\ 26 (1910).} + + +\Subsection{Further Postulates of the Theory of Relativity.} + +\Paragraph{23.} In the development of the theory of relativity to which we +shall now proceed we shall of course make use of many postulates. +The two which we have just considered, however, are the only ones, +so far as we are aware, which are essentially different from those +common to the usual theoretical developments of physical science. +In particular in our further work we shall assume without examination +all such general principles as the homogeneity and isotropism of the +space continuum, and the unidirectional, one-valued, one-dimensional +character of the time continuum. In our treatment of the dynamics +of a particle we shall also assume Newton's laws of motion, and the +principle of the conservation of mass, although we shall find, of course, +that the Einstein ideas as to the connection between space and time +will lead us to a non-Newtonian mechanics. We shall also make +extensive use of the principle of least action, which we shall find a +powerful principle in all the fields of dynamics. +%% -----File: 042.png---Folio 28------- + + +\Chapter{III}{Some Elementary Deductions.} +\SetRunningHeads{Chapter Three.}{Some Elementary Deductions.} + +\Paragraph{24.} In order gradually to familiarize the reader with the consequences +of the theory of relativity we shall now develop by very +elementary methods a few of the more important relations. In this +preliminary consideration we shall lay no stress on mathematical +elegance or logical exactness. It is believed, however, that the +chapter will present a substantially correct account of some of the +more important conclusions of the theory of relativity, in a form +which can be understood even by readers without mathematical +equipment. + + +\Subsection{Measurements of Time in a Moving System.} + +\Paragraph{25.} We may first derive from the postulates of relativity a relation +connecting measurements of time intervals as made by observers in +systems moving with different velocities. Consider a system~$S$ +(\Figref[Fig.]{5}) provided with a plane mirror~$a\, a$, and an observer~$A$, who +\begin{figure}[hbt] + \begin{center} + \Fig{5} + \Input[3.25in]{042} + \end{center} +\end{figure} +has a clock so that he can determine the time taken for a beam of +light to travel up to the mirror and back along the path~$A\, m\, A$. +Consider also another similar system~$S'$, provided with a mirror~$b\, b$, +and an observer~$B$, who also has a clock for measuring the time it +takes for light to go up to his mirror and back. System~$S'$ is moving +past~$S$ with the velocity~$V$, the direction of motion being parallel +to the mirrors $a\, a$~and~$b\, b$, the two systems being arranged, moreover, +%% -----File: 043.png---Folio 29------- +so that when they pass one another the two mirrors $a\,a$~and~$b\,b$ +will coincide, and the two observers $A$~and~$B$ will also come into +coincidence. + +$A$,~considering his system at rest and the other in motion, measures +the time taken for a beam of light to pass to his mirror and return, +over the path~$A\, m\, A$, and compares the time interval thus obtained +with that necessary for the performance of a similar experiment +by~$B$, in which the light has to pass over a longer path such as~$B\, n\, B'$, +shown in \Figref{6}, where $B\, B'$~is the distance through which the +\begin{figure}[hbt] + \begin{center} + \Fig{6} + \Input{043} + \end{center} +\end{figure} +observer~$B$ has moved during the time taken for the passage of the +light up to the mirror and back. + +Since, in accordance with the second postulate of relativity, the +velocity of light is independent of the velocity of its source, it is +evident that the ratio of these two time intervals will be proportional +to the ratio of the two paths $A\, m\, A$~and~$B\, n\, B'$, and this can easily +be calculated in terms of the velocity of light~$c$ and the velocity~$V$ +of the system~$S'$. + +From \Figref{6} we have +\[ +(A\, m)^2 = (p\, n)^2 = (B\, n)^2 - (B\, p)^2. +\] +Dividing by $(B\, n)^2$, +\[ +\frac{(A\, m)^2}{(B\, n)^2} = 1 - \frac{(B\, p)^2}{(B\, n)^2}. +\] +But the distance $B\, p$ is to $B\, n$ as $V$ is to~$c$, giving us +\[ +\frac{A\, m }{ B\, n} = \sqrt{1 - \frac{V^2}{c^2}}, +\] +%% -----File: 044.png---Folio 30------- +and hence $A$~will find, either by calculation or by direct measurement +if he has arranged clocks at $B$~and~$B'$, that it takes a longer time for +the performance of $B$'s~experiment than for the performance of his +own in the ratio $1: \sqrt{ 1 - \dfrac{V^2}{c^2}}$. + +It is evident from the first postulate of relativity, however, that +$B$~himself must find exactly the same length of time for the light to +pass up to his mirror and come back as did~$A$ in his experiment, +because the two systems are, as a matter of fact, entirely symmetrical +and we could with equal right consider $B$'s~system to be the one at +rest and $A$'s~the one in motion. + +\emph{We thus find that two observers, $A$~and~$B$, who are in relative motion +will not in general agree in their measurements of the time interval necessary +for a given event to take place}, the event in this particular case, +for example, having been the performance of $B$'s~experiment; indeed, +making use of the ratio obtained in a preceding paragraph, we may +go further and make the quantitative statement that measurements of +\emph{time intervals made with a moving clock must be multiplied by the quantity +$\dfrac{1}{\sqrt{ 1 - \smfrac{V^2}{c^2}}}$ in order to agree with measurements made with a stationary +system of clocks}. + +It is sometimes more convenient to state this principle in the +form: A stationary observer using a set of stationary clocks will +obtain a greater measurement in the ratio $1: \sqrt{ 1 - \dfrac{V^2}{c^2}}$ for a given +time interval than an observer who uses a clock moving with the +velocity~$V$. + + +\Subsection{Measurements of Length in a Moving System.} + +\Paragraph{26.} We may now extend our considerations, to obtain a relation +between measurements of \emph{length} made in stationary and moving +systems. + +As to measurements of length \emph{perpendicular} to the line of motion +of the two systems $S$~and~$S'$, a little consideration will make it at once +evident that both $A$~and~$B$ must obtain identical results. This is +true because the possibility is always present of making a direct comparison +%% -----File: 045.png---Folio 31------- +of the meter sticks which $A$~and~$B$ use for such measurements +by holding them perpendicular to the line of motion. When the +relative motion of the two systems brings such meter sticks into +juxtaposition, it is evident from the first postulate of relativity that +$A$'s~meter and $B$'s~meter must coincide in length. Any difference in +length could be due only to the different velocity of the two systems +through space, and such an occurrence is ruled out by our first postulate. +\emph{Hence measurements made with a moving meter stick held perpendicular +to its line of motion will agree with those made with stationary meter +sticks.} + +\Paragraph{27.} With regard to measurements of length \emph{parallel} to the line of +motion of the systems, the affair is much more complicated. Any +direct comparison of the lengths of meter sticks in the two systems +would be very difficult to carry out; the consideration, however, of a +simple experiment on the velocity of light parallel to the motion of +the systems will lead to the desired relation. + +Let us again consider two systems $S$~and~$S'$ (\Figref[fig.]{7}), $S'$~moving +past~$S$ with the velocity~$V$. +\begin{figure}[hbt] + \begin{center} + \Fig{7} + \Input[3.5in]{045} + \end{center} +\end{figure} + +$A$ and $B$ are observers on these systems provided with clocks and +meter sticks. The two observers lay off, each on his own system, +paths for measuring the velocity of light. $A$~lays off a distance of +one meter,~$A\, m$, so that he can measure the time for light to travel +to the mirror~$m$ and return, and $B$, using a meter stick which has +the same length as~$A$'s when they are both at rest, lays off the distance~$B\, n$. + +Each observer measures the length of time it takes for light to +travel to his mirror and return, and will evidently have to find the +same length of time, since the postulates of relativity require that the +velocity of light shall be the same for all observers. +%% -----File: 046.png---Folio 32------- + +Now the observer~$A$, taking himself as at rest, finds that $B$'s~light +travels over a path~$B\, n'\, B'$ (\Figref[fig.]{8}), where $n\, n'$~is the distance +\begin{figure}[hbt] + \begin{center} + \Fig{8} + \Input[3in]{046} + \end{center} +\end{figure} +through which the mirror~$n$ moves while the light is travelling up to +it, and $B\, B'$~is the distance through which the source travels before +the light gets back. It is easy to calculate the length of this path. + +We have +\[ +\frac{n\,n'}{B\,n'} = \frac{V}{c} +\] +and +\[ +\frac{B\,B'}{B\,n'\,B'} = \frac{V}{c}. +\] +Also, from the figure, +\begin{align*} +B\,n' &= B\,n + n\,n',\\ +B\,n'\,B' &= B\,n\,B + 2\,n\,n'- B\, B'. +\end{align*} +Combining, we obtain +\[ +\frac{B\,n'\,B'}{B\,n\,B} = \frac{1}{1 - \smfrac{V^2}{c^2}}. +\] + +Thus $A$ finds that the path traversed by $B$'s~light, instead of being +exactly two meters as was his own, will be longer in the ratio of +$1:\left(1 - \dfrac{V^2}{c^2}\right)$. For this reason $A$~is rather surprised that $B$~does +not report a longer time interval for the passage of the light than he +himself found. He remembers, however, that he has already found +that measurements of time made with a moving clock must be multiplied +by the quantity $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$ in order to agree with his own, and +sees that this will account for part of the discrepancy between the +expected and observed results. To account for the remaining discrepancy +the further conclusion is now obtained \emph{that measurements of +%% -----File: 047.png---Folio 33------- +length made with a moving meter stick, parallel to its motion, must be +multiplied by the quantity $\sqrt{1 - \dfrac{V^2}{c^2}}$ in order to agree with those made +in a stationary system}. + +In accordance with this principle, a stationary observer will +obtain a smaller measurement for the length of a moving body than +will an observer moving along with the object. This has been called +the Lorentz shortening, the shortening occurring in the ratio +\[ +\sqrt{1 - \frac{V^2}{c^2}}:1 +\] +in the line of motion. + + +\Subsection{The Setting of Clocks in a Moving System.} + +\Paragraph{28.} It will be noticed that in our considerations up to this point +we have considered cases where only a \emph{single} moving clock was needed +in performing the desired experiment, and this was done purposely, +since we shall find, not only that a given time interval measures +shorter on a moving clock than on a system of stationary clocks, +but that a system of moving clocks which have been set in synchronism +by an observer moving along with them will not be set in synchronism +for a stationary observer. + +Consider again two systems $S$~and~$S'$ in relative motion with the +velocity~$V$. An observer~$A$ on system~$S$ places two carefully compared +clocks, unit distance apart, in the line of motion, and has the +time on each clock read when a given point on the other system +passes it. An observer~$B$ on system~$S'$ performs a similar experiment. +The time interval obtained in the two sets of readings must be the +same, since the first postulate of relativity obviously requires that the +relative velocity of the two systems $V$~shall have the same value for +both observers. + +The observer~$A$, however, taking himself as at rest, and familiar +with the change in the measurements of length and time in the moving +system which have already been deduced, expects that the velocity +as measured by~$B$ will be greater than the value that he himself +obtains in the ratio $\dfrac{1}{1 - \smfrac{V^2}{c^2}}$, since any particular one of $B$'s~clocks +%% -----File: 048.png---Folio 34------- +gives a shorter value for a given time interval than his own, while +$B$'s~measurements of the length of a moving object are greater than +his own, each by the factor $\sqrt{1 - \dfrac{V^2}{c^2}}$. In order to explain the actual +result of $B$'s~experiment he now has to conclude that the clocks which +for $B$ are set synchronously are not set in synchronism for himself. + +From what has preceded it is easy to see that in the moving system, +from the point of view of the stationary observer, clocks must be set +further and further ahead as we proceed towards the rear of the +system, since otherwise $B$~would not obtain a great enough difference +in the readings of the clocks as they come opposite the given point +on the other system. Indeed, if two clocks are situated in the moving +system,~$S'$, one in front of the other by the distance $l'$, as measured +by~$B$, then for $A$ it will appear as though $B$~had set his rear clock ahead +by the amount~$\dfrac{l'V}{c^2}$. + +\Paragraph{29.} We have now obtained all the information which we shall +need in this chapter as to measurements of time and length in systems +moving with different velocities. We may point out, however, before +proceeding to the application of these considerations, that our choice +of $A$'s system as the one which we should call stationary was of course +entirely arbitrary and immaterial. We can at any time equally well +take $B$'s~system as the one to which we shall ultimately refer all our +measurements, and indeed all that we shall mean when we call one of +our systems stationary is that for reasons of convenience we have +picked out that particular system as the one with reference to which +we particularly wish to make our measurements. We may also +point out that of course $B$~has to subject $A$'s~measurements of time +and length to just the same multiplications by the factor $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$ +as did~$A$ in order to make them agree with his own. + +These conclusions as to measurements of space and time are of course +very startling when first encountered. The mere fact, however, that +they appear strange to so-called ``common sense'' need cause us +no difficulty, since the older ideas of space and time were obtained +from an ancestral experience which never included experiments with +%% -----File: 049.png---Folio 35------- +high relative velocities, and it is only when the ratio $\dfrac{V^2}{c^2}$ becomes +appreciable that we obtain unexpected results. To those scientists +who do not wish to give up their ``common sense'' ideas of space +and time we can merely say that if they accept the two postulates +of relativity then they will also have to accept the consequences +which can be deduced therefrom. The remarkable nature of these +consequences merely indicates the very imperfect nature of our older +conceptions of space and time. + + +\Subsection{The Composition of Velocities.} + +\Paragraph{30.} Our conclusions as to the setting of clocks make it possible +to obtain an important expression for the composition of velocities. +Suppose we have a system~$S$, which we shall take as stationary, and +on the system an observer~$A$. Moving past~$S$ with the velocity~$V$ +is another system~$S'$ with an observer~$B$, and finally moving past~$S'$ +in the same direction is a body whose velocity is~$u'$ as measured by +observer~$B$. What will be the velocity~$u$ of this body as measured +by~$A$? + +Our older ideas led us to believe in the simple additivity of velocities +and we should have calculated~$u$ in accordance with the simple +expression +\[ +u = V + u'. +\] +We must now allow, however, for the fact that $u'$~is measured with +clocks which to~$A$ appear to be set in a peculiar fashion and running +at a different rate from his own, and with meter sticks which give +longer measurements than those used in the stationary system. + +The determination of~$u'$ by observer~$B$ would be obtained by +measuring the time interval necessary for the body in question to +move a given distance~$l'$ along the system~$S'$. If $t'$~is the difference +in the respective clock readings when the body reaches the ends of +the line~$l'$, we have +\[ +u' = \frac{l'}{t'}. +\] +We have already seen, however, that the two clocks are for~$A$ set $\dfrac{l'V}{c^2}$~units +apart and hence for clocks set together the time interval would +%% -----File: 050.png---Folio 36------- +have measured $t' + \dfrac{l'V}{c^2}$. Furthermore these moving clocks give +time measurements which are shorter in the ratio $\sqrt{1 - \dfrac{V^2}{c^2}}:1$ than +those obtained by~$A$, so that for~$A$ the time interval for the body to +move from one end of~$l'$ to the other would measure +\[ +\frac{t' + \smfrac{l'V}{c^2}}{\sqrt{1 - \smfrac{V^2}{c^2}}}; +\] +furthermore, owing to the difference in measurements of length, this +line~$l'$ has for~$A$ the length $l'\sqrt{1 - \dfrac{V^2}{c^2}}$. Hence $A$~finds that the +body is moving past~$S'$ with the velocity, +\[ +\frac{\ l'\sqrt{1 - \smfrac{V^2}{c^2}}\ } + {\smfrac{t' + \smfrac{l'V}{c^2}}{\sqrt{1 - \smfrac{V^2}{c^2}}}} + = \frac{\smfrac{l'}{t'} \left(1 - \smfrac{V^2}{c^2}\right)} + {1 + \smfrac{l'}{t'}\, \smfrac{V}{c^2}} + = \frac{u' \left(1 - \smfrac{V^2}{c^2}\right)} + {1+ \smfrac{u'V}{c^2}}. +\] +This makes the total velocity of the body past~$S$ equal to the sum +\[ +u = V + \frac{u' \left(1 - \smfrac{V^2}{c^2}\right)}{1 + \smfrac{u'V}{c^2}}, +\] +or +\[ +u = \frac{V + u'}{1 + \smfrac{u'V}{c^2}}. +\] + +This new expression for the composition of velocities is extremely +important. When the velocities $u'$~and~$V$ are small compared with +the velocity of light~$c$, we observe that the formula reduces to the simple +additivity principle which we know by common experience to be true +%% -----File: 051.png---Folio 37------- +for all ordinary velocities. Until very recently the human race has +had practically no experience with high velocities and we now see +that for velocities in the neighborhood of that of light, the simple +additivity principle is nowhere near true. + +In particular it should be noticed that by the composition of +velocities which are themselves less than that of light we can never +obtain any velocity greater than that of light. As an extreme case, +suppose for example that the system~$S'$ were moving past $S$~itself +with the velocity of light (\ie, $V = c$) and that in the system~$S'$ a +particle should itself be given the velocity of light in the same direction +(\ie, $u' = c$); we find on substitution that the particle still has +only the velocity of light with respect to~$S$. We have +\[ +u = \frac{c + c}{1 + \smfrac{c^2}{c^2}} = \frac{2c}{2} = c. +\] + +By the consideration of such conclusions as these the reader will +appreciate more and more the necessity of abandoning his older +naïve ideas of space and time which are the inheritance of a long +human experience with physical systems in which only slow velocities +were encountered. + + +\Subsection{The Mass of a Moving Body.} + +\Paragraph{31.} We may now obtain an important relation for the mass of a +moving body. Consider again two similar systems, $S$~at rest and $S'$~moving +past with the velocity~$V$. The observer~$A$ on system~$S$ has a +sphere made from some rigid elastic material, having a mass of $m$~grams, +and the observer~$B$ on system~$S'$ is also provided with a similar +sphere. The two spheres are made so that they are exactly alike +when both are at rest; thus $B$'s~sphere, since it is at rest with respect +to him, looks to him just the same as the other sphere does to~$A$. +As the two systems pass each other (\Figref[fig.]{9}) each of these clever experimenters +rolls his sphere towards the other system with a velocity of +$u$~cm.~per second, so that they will just collide and rebound in a line +perpendicular to the direction of motion. Now, from the first postulate +of relativity, system~$S'$ appears to~$B$ just the same as system $S$~appears +to~$A$, and $B$'s~ball appears to him to go through the same +evolutions that $A$~finds for his ball. $A$~finds that his ball on collision +%% -----File: 052.png---Folio 38------- +undergoes the algebraic change of velocity~$2u$, $B$~finds the same change +in velocity~$2u$ for his ball. $B$~reports this fact to~$A$, and $A$~knowing +that $B$'s~measurements of length agree with his own in this transverse +\begin{figure}[hbt] + \begin{center} + \Fig{9} + \Input{052} + \end{center} +\end{figure} +direction, but that his clock gives time intervals that are shorter than +his own in the ratio $\sqrt{1 - \dfrac{V^2}{c^2}}:1$, calculates that the change in velocity +of $B$'s~ball must be~$2u\sqrt{1 - \dfrac{V^2}{c^2}}$. + +From the principle of the conservation of momentum, however, +$A$~knows that the change in momentum of $B$'s~ball must be the same +as that of his own and hence can write the equation +\[ +m_au = m_bu\sqrt{1 - \frac{V^2}{c^2}}, +\] +where $m_a$~is the mass of $A$'s~ball and $m_b$~is the mass of $B$'s~ball. Solving +we have +\[ +m_b = \frac{m_a}{\sqrt{1 - \smfrac{V^2}{c^2}}}. +\] + +In other words, $B$'s~ball, which had the same mass~$m_a$ as~$A$'s when +%% -----File: 053.png---Folio 39------- +both were at rest, is found to have the larger mass $\dfrac{m_a}{\sqrt{1 - \smfrac{v^2}{c^2}}}$ when +placed in a system that is moving with the velocity~$V$.\footnote + {In carrying out this experiment the transverse velocities of the balls should + be made negligibly small in comparison with the relative velocity of the systems~$V$.} + +The theory of relativity thus leads to the general expression +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{v^2}{c^2}}} +\] +for the mass of a body moving with the velocity~$u$ and having the +mass~$m_0$ when at rest. + +Since we have very few velocities comparable with that of light +it is obvious that the quantity $\sqrt{1 - \dfrac{v^2}{c^2}}$ seldom differs much from +unity, which makes the experimental verification of this expression +difficult. In the case of electrons, however, which are shot off from +radioactive substances, or indeed in the case of cathode rays produced +with high potentials, we do have particles moving with velocities +comparable to that of light, and the experimental work of Kaufmann, +Bucherer, Hupka and others in this field provides one of the most +striking triumphs of the theory of relativity. + + +\Subsection{The Relation Between Mass and Energy.} + +\Paragraph{32.} The theory of relativity has led to very important conclusions +as to the nature of mass and energy. In fact, we shall see that matter +and energy are apparently different names for the same fundamental +entity. + +When we set a body in motion it is evident from the previous +section that we increase both its mass as well as its energy. Now +we can show that there is a definite ratio between the amount of +energy that we give to the body and the amount of mass that we +give to it. + +If the force~$f$ acts on a particle which is free to move, its increase in +kinetic energy is evidently +\[ +\Delta E = \int f\, dl. +\] +But the force acting\DPtypo{, is}{ is,} by definition, equal to the rate of increase in +%% -----File: 054.png---Folio 40------- +the momentum of the particle +\[ +f = +\frac{d}{dt}(mu). +\] +Substituting we have +\[ +\Delta E + = \int \frac{d(mu)}{dt}\, dl + = \int \frac{dl}{dt}\, d(mu) + = \int u\, d(mu). +\] +We have, however, from the previous section, +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +which, solved for~$u$, gives us +\[ +u = c \sqrt{1 - \frac{{m_0}^2}{m^2}}. +\] +Substituting this value of~$u$ in our equation for~$\Delta E$ we obtain, after +simplification, +\[ +\Delta E = \int c^2\, dm = c^2\, \Delta m. +\] + +This says that the increase of the kinetic energy of the particle, +in ergs, is equal to the increase in mass, in grams, multiplied by the +square of the velocity of light. If now we bring the particle to rest +it will give up both its kinetic energy and its excess mass. Accepting +the principles of the conservation of mass and energy, we know, however, +that neither this energy nor the mass has been destroyed; they +have merely been passed on to other bodies. There is, moreover, +every reason to believe that this mass and energy, which were associated +together when the body was in motion and left the body when +it was brought to rest, still remain always associated together. For +example, if the body should be brought to rest by setting another +body into motion, it is of course a necessary consequence of our considerations +that the kinetic energy and the excess mass both pass +on together to the new body which is set in motion. A similar conclusion +would be true if the body is brought to rest by frictional forces, +since the heat produced by the friction means an increase in the kinetic +energies of ultimate particles. +%% -----File: 055.png---Folio 41------- + +In general we shall find it pragmatic to consider that matter and +energy are merely different names for the same fundamental entity. +One gram of matter is equal to $10^{21}$~ergs of energy. +\[ +c^2 = (2.9986 × 10^{10})^2 = \text{approx.\ }10^{21}. +\] + +This apparently extraordinary conclusion is in reality one which +produces the greatest simplification in science. Not to mention +numerous special applications where this principle is useful, we may +call attention to the fact that the great laws of the conservation of +mass and of energy have now become identical. We may also point +out that those opposing camps of philosophic materialists who defend +matter on the one hand or energy on the other as the fundamental +entity of the universe may now forever cease their unimportant bickerings. +%% -----File: 056.png---Folio 42------- + + +\Chapter{IV}{The Einstein Transformation Equations for Space +and Time.} +\SetRunningHeads{Chapter Four.}{Transformation Equations for Space and Time.} + +\Subsection{The Lorentz Transformation.} + +\Paragraph{33.} We may now proceed to a systematic study of the consequences +of the theory of relativity. + +The fundamental problem that first arises in considering +spatial and temporal measurements is that of transforming the +description of a given kinematical occurrence from the variables of +one system of coördinates to those of another system which is in +motion relative to the first. + +Consider two systems of right-angled Cartesian coördinates $S$~and~$S'$ +(\Figref[fig.]{10}) in relative motion in the $X$~direction with the velocity~$V$. +\begin{figure}[hbt] + \begin{center} + \Fig{10} + \Input{056} + \end{center} +\end{figure} +The \emph{position} of any given point in space can be determined by specifying +its coördinates $x$,~$y$, and~$z$ with respect to system~$S$ or its coördinates +$x'$,~$y'$ and~$z'$ with respect to system~$S'$. Furthermore, for the +purpose of determining the \emph{time} at which any event takes place, we +may think of each system of coördinates as provided with a whole +series of clocks placed at convenient intervals throughout the system, +the clocks of each series being set and regulated\footnote + {We may think of the clocks as being set in any of the ways that are usual + in practice. Perhaps the simplest is to consider the clocks as mechanisms which + have been found to ``keep time'' when they are all together where they can be + examined by one individual observer. The assumption can then be made, in accordance + with our ideas of the homogeneity of space, that they will continue to + ``keep time'' after they have been distributed throughout the system.} +by observers in the +%% -----File: 057.png---Folio 43------- +corresponding system. The time at which the event in question +takes place may be denoted by~$t$ if determined by the clocks belonging +to system~$S$ and by~$t'$ if determined by the clocks of system~$S'$. + +For convenience the two systems $S$~and~$S'$ are chosen so that the +axes $OX$~and~$O'X'$ lie in the same line, and for further simplification +we choose, as our starting-point for time measurements, $t$~and~$t'$ both +equal to zero when the two origins come into coincidence. + +The specific problem now before us is as follows: If a given kinematical +occurrence has been observed and described in terms of the +variables $x'$,~$y'$,~$z'$ and~$t'$, what substitutions must we make for the +values of these variables in order to obtain a correct description of the +\emph{same} kinematical event in terms of the variables $x$,~$y$,~$z$ and~$t$? In +other words, we want to obtain a set of transformation equations +from the variables of system~$S'$ to those of system~$S$. The equations +which we shall present were first obtained by Lorentz, and the process +of changing from one set of variables to the other has generally been +called the Lorentz transformation. The significance of these equations +from the point of view of the theory of relativity was first appreciated +by Einstein. + + +\Subsection{Deduction of the Fundamental Transformation Equations.} + +\Paragraph{34.} It is evident that these transformation equations are going +to depend on the relative velocity $V$ of the two systems, so that we +may write for them the expressions +\begin{align*} +x' &= F_1(V, x, y, z, t), \displaybreak[0] \\ +y' &= F_2(V, x, y, z, t), \displaybreak[0] \\ +z' &= F_3(V, x, y, z, t), \displaybreak[0] \\ +t' &= F_4(V, x, y, z, t), +\end{align*} +where $F_1$,~$F_2$,~etc., are the unknown functions whose form we wish +to determine. + +It is possible at the outset, however, greatly to simplify these +relations. If we accept the idea of the homogeneity of space it is +evident that any other line parallel to~$OXX'$ might just as well have +been chosen as our line of $X$-axes, and hence our two equations for +$x'$~and~$t'$ must be independent of $y$~and~$z$. Moreover, as to the equations +%% -----File: 058.png---Folio 44------- +for $y'$~and~$z'$ it is at once evident that the only possible solutions +are $y' = y$ and $z' = z$. This is obvious because a meter stick held +in the system~$S'$ perpendicular to the line of relative motion,~$OX'$, +of the system can be directly compared with meter sticks held similarly +in system~$S$, and in accordance with the first postulate of relativity +they must agree in length in order that the systems may be entirely +symmetrical. We may now rewrite our transformation equations +in the simplified form +\begin{align*} +x' &= F_1(V, t, x), \\ +y' &= y, \\ +z' &= z, \\ +t' &= F_2(V, t, x), +\end{align*} +and have only two functions, $F_1$~and~$F_2$, whose form has to be determined. + +To complete the solution of the problem we may make use of three +further conditions which must govern the transformation equations. + +\Subsubsection{35}{Three Conditions to be Fulfilled.} In the first place, when the +velocity~$V$ between the systems is small, it is evident that the transformation +equations must reduce to the form that they had in Newtonian +mechanics, since we know both from measurements and from +everyday experience that the Newtonian concepts of space and time +are correct as long as we deal with slow velocities. Hence the limiting +form of the equations as $V$~approaches zero will be (cf.~\Chapref{I}, +equations \DPchg{3--4--5--6}{(3),~(4), (5),~(6)}) +\begin{align*} +x' &= x - Vt,\\ +y' &= y, \\ +z' &= z, \\ +t' &= t. +\end{align*} + +\Paragraph{36.} A second condition is imposed upon the form of the functions +$F_1$~and~$F_2$ by the first postulate of relativity, which requires that the +two systems $S$~and~$S'$ shall be entirely symmetrical. Hence the +transformation equations for changing from the variables of system~$S$ +to those of system~$S'$ must be of exactly the same form as those used +in the reverse transformation, containing, however, $-V$~wherever +$+V$~occurs in the latter equations. Expressing this requirement in +%% -----File: 059.png---Folio 45------- +mathematical form, we may write as true equations +\begin{align*} +x &= F_1(-V, t', x'), \\ +t &= F_2(-V, t', x'), +\end{align*} +where $F_1$~and~$F_2$ must have the same form as above. + +\Paragraph{37.} A final condition is imposed upon the form of $F_1$~and~$F_2$ by +the second postulate of relativity, which states that the velocity of a +beam of light appears the same to all observers regardless of the +motion of the source of light or of the observer. Hence our transformation +equations must be of such a form that a given beam of +light has the same velocity,~$c$, when measured in the variables of either +system. Let us suppose, for example, that at the instant $t = t' = 0$, +when the two origins come into coincidence, a light impulse is started +from the common point occupied by $O$~and~$O'$. Then, measured in +the coördinates of either system, the optical disturbance which is +generated must spread out from the origin in a spherical form with +the velocity~$c$. Hence, using the variables of system~$S$, the coördinates +of any point on the surface of the disturbance will be given by the +expression +\[ +x^2 + y^2 + z^2 = c^2t^2, +\Tag{7} +\] +while using the variables of system~$S'$ we should have the similar +expression +\[ +x'^2 + y'^2 + z'^2 = c^2t'^2. +\Tag{8} +\] +Thus we have a particular kinematical occurrence, the spreading out +of a light disturbance, whose description is known in the variables +of either system, and our transformation equations must be of such +a form that their substitution will change equation (8) to equation (7). +In other words, the expression $x^2 + y^2 + z^2 - c^2t^2$ is to be an invariant +for the Lorentz transformation. + +\Subsubsection{38}{The Transformation Equations.} The three sets of conditions +which, as we have seen in the last three paragraphs, are imposed upon +the form of $F_1$~and~$F_2$ are sufficient to determine the solution of the +problem. The natural method of solution is obviously that of trial, +%% -----File: 060.png---Folio 46------- +and we may suggest the solution: +\begin{align*} +x' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, (x - Vt) + = \kappa(x - Vt), \Tag{9}\\ +y' &= y, \Tag{10}\\ +z' &= z, \Tag{11}\\ +t' &= \frac{1}{\sqrt{1- \smfrac{V^2}{c^2}}} \left(t - \frac{V}{c^2}\, x\right) + = \kappa \left(t - \frac{V}{c^2}\, x\right), \Tag{12} +\end{align*} +where we have placed~$\kappa$ to represent the important and continually +recurring quantity $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$. + +It will be found as a matter of fact by examination that these\rule{0em}{1.8em} +solutions do fit all three requirements which we have stated. Thus, +when $V$~becomes small compared with the velocity of light,~$c$, the +equations do reduce to those of Galileo and Newton. Secondly, if +the equations are solved for the unprimed quantities in terms of the +primed, the resulting expressions have an unchanged form except for +the introduction of~$-V$ in place of~$+V$, thus fulfilling the requirements +of symmetry imposed by the first postulate of relativity. And +finally, if we substitute the expressions for $x'$,~$y'$,~$z'$ and~$t'$ in the polynomial +$x'^2 + y'^2 + z'^2 = c^2t'^2$, we shall obtain the expression $x^2 + y^2 ++ z^2 - c^2t^2$ and have thus secured the invariance of $x^2 + y^2 + z^2 - c^2t^2$ +which is required by the second postulate of relativity. + +We may further point out that the whole series of possible Lorentz +transformations form a group such that the result of two successive +transformations could itself be represented by a single transformation +provided we picked out suitable magnitudes and directions for the +velocities between the various systems. + +It is also to be noted that the transformation becomes imaginary +for cases where $V > c$, and we shall find that this agrees with ideas +obtained in other ways as to the speed of light being an upper limit +for the magnitude of all velocities. +%% -----File: 061.png---Folio 47------- + + +\Subsection{Further Transformation Equations.} + +\Paragraph{39.} Before making any applications of our equations we shall find +it desirable to obtain by simple substitutions and differentiations a +series of further transformation equations which will be of great value +in our future work. + +By the simple differentiation of equation~(12) we can obtain +\[ +\frac{dt'}{dt} = \kappa\left(1 - \frac{\dot{x}V}{c^2}\right), +\Tag{13} +\] +where we have put~$\dot{x}$ for~$\dfrac{dx}{dt}$. + +\Subsubsection{40}{Transformation Equations for Velocity.} By differentiation of +the equations for $x'$,~$y'$ and~$z'$, nos.\ (9),~(10) and~(11), and substitution +of the value just found for~$\dfrac{dt'}{dt}$ we may obtain the following transformation +equations for velocity: +\begin{alignat*}{3} +\dot{x}' &= \frac{\dot{x} - V}{1 - \smfrac{\dot{x}V}{c^2}} + &&\qquad\text{or}\qquad& + u'_x &= \frac{u_x - V}{1 - \smfrac{u_xV}{c^2}}, \Tag{14} \\ +% +\dot{y}' &= \frac{\dot{y}\kappa^{-1}}{1 - \smfrac{\dot{y}V}{c^2}} &&& + u'_y &= \frac{u_y\kappa^{-1}}{1 - \smfrac{u_xV}{c^2}}, \Tag{15} \\ +% +\dot{z}' &= \frac{\dot{z}\kappa^{-1}}{1 - \smfrac{\dot{z}V}{c^2}} &&& + u'_z &= \frac{u_z\kappa^{-1}}{1 - \smfrac{u_xV}{c^2}}, \Tag{16} +\end{alignat*} +where the placing of a dot has the familiar significance of differentiation +with respect to time, $\dfrac{dx}{dt}$~being represented by~$\dot{x}$ and $\dfrac{dx'}{dt'}$ by~$\dot{x}'$. + +The significance of these equations for the transformation of +velocities is as follows: If for an observer in system~$S$ a point appears +to be moving with the uniform velocity $(\dot{x}, \dot{y}, \dot{z})$ its velocity $(\dot{x}', \dot{y}', \dot{z}')$, +as measured by an observer in system~$S'$, is given by these expressions +(14),~(15) and~(16). + +\Subsubsection{41}{Transformation Equations for the Function $\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}$.} These +%% -----File: 062.png---Folio 48------- +three transformation equations for the velocity components of a point\DPtypo{,}{} +permit us to obtain a further transformation equation for an important +function of the velocity which we shall find continually recurring in +our later work. This is the function $\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, where we have indicated +the total velocity of the point by~$u$, according to the expression +$u^2 = \dot{x}^2 + \dot{y}^2 + \dot{z}^2$. By the substitution of equations (14),~(15) and~(16) +we obtain the transformation equation +\[ +\frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = \frac{\left(1 - \smfrac{u_xV}{c^2}\right) \kappa} + {\sqrt{1 - \smfrac{u^2}{c^2}}}. +\Tag{17} +\] + +\Subsubsection{42}{Transformation Equations for Acceleration.} By further differentiating +equations (14),~(15) and~(16) and simplifying, we easily +obtain three new equations for transforming measurements of \emph{acceleration} +from system $S'$~to~$S$, viz.: +\begin{align*} +\ddot{x}' &= \left(1 - + \frac{\dot{x}V}{c^2}\right)^{-3}\kappa^{-3}\ddot{x}, +\Tag{18} \\ +% +\ddot{y}' &= \left(1 - \frac{\dot{x}V}{c^2}\right)^{-2} \kappa^{-2}\ddot{y} + + \dot{y}\, \frac{V}{c^2} + \left(1 - \frac{\dot{x}V}{c^2}\right)^{-3} \kappa^{-2}\ddot{x}, +\Tag{19} \\ +% +\ddot{z}' &= \left(1 - \frac{\dot{x}V}{c^2}\right)^{-2} \kappa^{-2}\ddot{z} + + \dot{z} \frac{V}{c^2} + \left(1 - \frac{\dot{x}V}{c^2}\right)^{-3} \kappa^{-2}\ddot{x}, +\Tag{20} +\intertext{or} +{\dot{u}_x}' &= \left(1 - \frac{u_xV}{c^2}\right)^{-3} \kappa^{-3}\ddot{u}_x, +\Tag{18} \\ +% +{\dot{u}_y}' &= \left(1 - \frac{u_xV}{c^2}\right)^{-2} \kappa^{-2}\ddot{u}_y + + u_y\, \frac{V}{c^2} + \left(1 - \frac{u_xV}{c^2}\right)^{-3} \kappa^{-2}\dot{u}_x, +\Tag{19} \\ +% +{\dot{u}_z}' &= \left(1 - \frac{u_xV}{c^2}\right)^{-2} \kappa^{-2}\ddot{u}_z + + u_z\, \frac{V}{c^2} + \left(1 - \frac{u_xV}{c^2}\right)^{-3} \kappa^{-2}\dot{u}_x. +\Tag{20} +\end{align*} +%% -----File: 063.png---Folio 49------- + + +\Chapter{V}{Kinematical Applications.} +\SetRunningHeads{Chapter Five.}{Kinematical Applications.} + +\Paragraph{43.} The various transformation equations for spatial and temporal +measurements which were derived in the \Chapnumref[IV]{previous chapter} may now be +used for the treatment of a number of kinematical problems. In +particular it will be shown in the latter part of the chapter that a +number of optical problems can be handled with extraordinary facility +by the methods now at our disposal. + + +\Subsection{The Kinematical Shape of a Rigid Body.} + +\Paragraph{44.} We may first point out that the conclusions of relativity theory +lead us to quite new ideas as to what is meant by the shape of a rigid +body. We shall find that the shape of a rigid body will depend entirely +upon the relative motion of the body and the observer who is making +measurements on it. + +Consider a rigid body which is at rest with respect to system~$S'$. +Let $x_1'$,~$y_1'$,~$z_1'$ and $x_2'$,~$y_2'$,~$z_2'$ be the coördinates in system~$S'$ of two +points in the body. The coördinates of the same points as measured +in system~$S$ can be found from transformation equations (9),~(10) +and~(11), and by subtraction we can obtain the following expressions +\begin{gather*} +(x_2 - x_1) = \sqrt{1 - \frac{V^2}{c^2}}\, ({x_2}' - {x_1}'), +\Tag{21} \\ +(y_2 - y_1) = (y_2' - y_1'), +\Tag{22} \\ +(z_2 - \DPtypo{y_2}{z_1}) = (z_2' - z_1'), +\Tag{23} +\end{gather*} +connecting the distances between the pair of points as viewed in the +two systems. In making this subtraction terms containing~$t$ have +been cancelled out since we are interested in the \emph{simultaneous} positions +of the points. In accordance with these equations we may distinguish +then between the \emph{geometrical shape} of a body, which is the shape that +it has when measured on a system of axes which are at rest relative +to it, and its \emph{kinematical shape}, which is given by the coördinates which +%% -----File: 064.png---Folio 50------- +express the \emph{simultaneous} positions of its various points when it is in +motion with respect to the axes of reference. We see that the kinematical +shape of a rigid body differs from its geometrical shape by a +shortening of all its dimensions in the line of motion in the ratio +$\sqrt{1 - \dfrac{V^2}{c^2}}:1$; thus a sphere, for example, becomes a Heaviside ellipsoid. + +In order to avoid incorrectness of speech we must be very careful +not to give the idea that the kinematical shape of a body is in +any sense either more or less real than its geometrical shape. We +must merely learn to realize that the shape of a body is entirely dependent +on the particular set of coördinates chosen for the making +of space measurements. + + +\Subsection{The Kinematical Rate of a Clock.} + +\Paragraph{45.} Just as we have seen that the shape of a body depends upon +our choice of a system of coördinates, so we shall find that the rate of +a given clock depends upon the relative motion of the clock and its +observer. Consider a clock or any mechanism which is performing +a periodic action. Let the clock be at rest with respect to system~$S'$ +and let a given period commence at~${t_1}'$ and end at~${t_2}'$, the length of +the interval thus being $\Delta t' = {t_2}' - {t_1}'$. + +From transformation equation~(12) we may obtain +\begin{align*} +t_1' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}} + \left(t_1 - \frac{V}{c^2}\, x_1\right), \\ +t_2' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}} + \left(t_2 - \frac{V}{c^2}\, x_2\right), +\end{align*} +and by subtraction, since $x_2 - x_1$ is obviously equal to~$Vt$, we have +\begin{align*}%[** TN: Not aligned in orig.] +t_2 - t_1 &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, ({t_2}' - {t_1}'), \\ +\Delta t &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, \Delta t'. +\end{align*} +%% -----File: 065.png---Folio 51------- +Thus an observer who is moving past a clock finds a longer period for +the clock in the ratio $1 : \sqrt{1 - \dfrac{V^2}{c^2}}$ than an observer who is stationary +with respect to it. Suppose, for example, we have a particle which +is turning alternately red and blue. For an observer who is moving +past the particle the periods for which it remains a given color measure +longer in the ratio $1 : \sqrt{1 -\dfrac{V^2}{c^2}}$ than they do to an observer who is +stationary with respect to the particle. + +\Paragraph{46.} A possible opportunity for testing this interesting conclusion +of the theory of relativity is presented by the phenomena of canal +rays. We may regard the atoms which are moving in these rays as +little clocks, the frequency of the light which they emit corresponding +to the period of the clock. If now we should make spectroscopic +observations on canal rays of high velocity, the frequency of the +emitted light ought to be less than that of light from stationary atoms +of the same kind if our considerations are correct. It would of course +be necessary to view the canal rays at right angles to their direction +of motion, to prevent a confusion of the expected shift in the spectrum +with that produced by the ordinary Doppler effect (see \Secref{54}). + + +\Subsection{The Idea of Simultaneity.} + +\Paragraph{47.} We may now also point out that the idea of the \emph{absolute} simultaneity +of two events must henceforth be given up. Suppose, for +example, an observer in the system~$S$ is interested in two events +which take place simultaneously at the time~$t$. Suppose one of these +events occurs at a point having the $X$~coördinate~$x_1$ and the other +at a point having the coördinate~$x_2$; then by transformation equation~(12) +it is evident that to an observer in system~$S'$, which is moving +relative to~$S$ with the velocity~$V$, the two events would take place +respectively at the times +\begin{align*} +{t_1}' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}} + \left(t - \frac{V}{c^2}\, x_1\right) \\ +\intertext{and} +{t_2}' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}} + \left(t - \frac{V}{c^2}\, x_2\right) +\end{align*} +%% -----File: 066.png---Folio 52------- +or the difference in time between the occurrence of the events would +appear to this other observer to be +\[ +{t_2}' - {t_1}' + = \frac{V}{c^2\, \sqrt{1 - \smfrac{V^2}{c^2}}}\, (x_1 - x_2). +\Tag{25} +\] + + +\Subsection{The Composition of Velocities.} + +\Subsubsection{48}{The Case of Parallel Velocities.} We may now present one of +the most important characteristics of Einstein's space and time, +which can be best appreciated by considering transformation equation~(14), +or more simply its analogue for the transformation in the reverse +direction +\[ +u_x = \frac{{u_x}' + V}{1 + \smfrac{{u_x}'V}{c^2}}. +\Tag{26} +\] + +Consider now the significance of the above equation. If we +have a particle which is moving in the $X$~direction with the velocity~$u_x'$ +as measured in system~$S'$, its velocity~$u_x$ with respect to system~$S$ +is to be obtained by adding the relative velocity of the two systems~$V$ +\emph{and dividing the sum of the two velocities by} $1 + \dfrac{{u_x}'V}{c^2}$. Thus we see +that we must completely throw overboard our old naïve ideas of the +direct additivity of velocities. Of course, in the case of very slow +velocities, when $u_x'$~and~$V$ are both small compared with the velocity +of light, the quantity~$\dfrac{{u_x}'V}{c^2}$ is very nearly zero and the direct addition +of velocities is a close approximation to the truth. In the case of +velocities, however, which are in the neighborhood of the speed of +light, the direct addition of velocities would lead to extremely erroneous +results. + +\Paragraph{49.} In particular it should be noticed that by the composition of +velocities which are themselves less than that of light we can never +obtain any velocity greater than that of light. Suppose, for example, +that the system~$S'$ were moving past~$S$ with the velocity of light +(\ie, $V = c$), and that in the system~$S'$ a particle should itself be +given the velocity of light in the $X$~direction (\ie, $u_x' = c$); we find +on substitution that the particle still has only the velocity of light +%% -----File: 067.png---Folio 53------- +with respect to~$S$. We have +\[ +u_x = \frac{c + c}{1 + \smfrac{c^2}{c^2}} = \frac{2c}{2} = c. +\] + +If the relative velocity between the systems should be one half +the velocity of light,~$\dfrac{c}{2}$, and an experimenter on~$S'$ should shoot off a +particle in the $X$~direction with half the velocity of light, the total +velocity with respect to~$S$ would be +\[ +u_x = \frac{\frac{1}{2}c + \frac{1}{2}c}{1 + \smfrac{\frac{1}{4}c^2}{c^2}} + = \frac{4}{5}\, c. +\] + +\Subsubsection{50}{Composition of Velocities in General.} In the case of particles +which have components of velocity in other than the $X$~direction it +is obvious that our transformation equations will here also provide +methods of calculation to supersede the simple addition of velocities. +If we place +\begin{align*} +u^2 &= {u_x}^2 + {u_y}^2 + {u_z}^2 ,\\ +{u'}^2 &= {{u_x}'}^2 + {{u_y}'}^2 + {{u_z}'}^2 , +\end{align*} +we may obtain by the substitution of equations (14),~(15) and~(16) +\[ +u = \frac{\left({u'}^2 + V^2 + 2u'V \cos\alpha + - \smfrac{{u'}^2V^2 \sin^2\alpha}{c^2}\right)^{1/2}} + {1 + \smfrac{u'V \cos\alpha}{c^2}}, +\Tag{27} +\] +where $\alpha$ is the angle in the system~$S'$ between the $X'$~axis and the +velocity of the particle~$u'$. For the particular case that $V$~and~$u'$ +are in the same direction, the equation obviously reduces to the +simpler form +\[ +u = \frac{u' + V}{1 + \smfrac{u'V}{c^2}}, +\] +which we have already considered. + +\Paragraph{51.} We may also call attention at this point to an interesting characteristic +of the equations for the transformation of velocities. It will +%% -----File: 068.png---Folio 54------- +be noted from an examination of these equations that if to any observer +a particle appears to have a constant velocity, \ie, to be +unacted on by any force, it will also appear to have a \emph{uniform} although +of course different velocity to any observer who is himself in uniform +motion with respect to the first. An examination, however, of the +transformation equations for acceleration (18),~(19),~(20) will show +that here a different state of affairs is true, since it will be seen that a +point which has \emph{uniform acceleration} $(\ddot{x}, \ddot{y}, \ddot{z})$ with respect to an observer +in system~$S$ will not in general have a uniform acceleration in +another system~$S'$, since the acceleration in system~$S'$ depends not +only on the constant acceleration but also on the velocity in system~$S$, +which is necessarily varying. + + +\Subsection{Velocities Greater than that of Light.} + +\Paragraph{52.} In the preceding section we have called attention to the fact +that the mere composition of velocities which are not themselves +greater than that of light will never lead to a speed that is greater +than that of light. The question naturally arises whether velocities +which are greater than that of light could ever possibly be obtained +in any way. + +This problem can be attacked in an extremely interesting manner. +Consider two points $A$~and~$B$ on the $X$~axis of the system~$S$, and +suppose that some impulse originates at~$A$, travels to~$B$ with the +velocity~$u$ and at~$B$ produces some observable phenomenon, the starting +of the impulse at~$A$ and the resulting phenomenon at~$B$ thus +being connected by the relation of \emph{cause and effect}. + +The time elapsing between the cause and its effect as measured +in the units of system~$S$ will evidently be +\[ +\Delta t = t_B - t_A = \frac{x_B - x_A}{u}, +\Tag{28} +\] +where $x_A$~and~$x_B$ are the coördinates of the two points $A$~and~$B$. + +Now in another system~$S'$, which has the velocity~$V$ with respect +to~$S$, the time elapsing between cause and effect would evidently be +\[ +\Delta t' = {t'}_B - {t'}_A + = \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\left(t_B - \frac{V}{c^2}\, x_B\right) + - \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\left(t_A - \frac{V}{c^2}\, x_A\right), +\] +%% -----File: 069.png---Folio 55------- +where we have substituted for $t'_B$~and~$t'_A$ in accordance with equation~(12). +Simplifying and introducing equation~(28) we obtain +\[ +\Delta t' + = \frac{1 - \smfrac{uV}{ c^2}} + {\sqrt{1 - \smfrac{V^2}{c^2}}}\, \Delta t. +\Tag{29} +\] +Let us suppose now that there are no limits to the possible magnitude +of the velocities $u$~and~$V$, and in particular that the causal impulse +can travel from~$A$ to~$B$ with a velocity~$u$ greater than that of +light. It is evident that we could then take a velocity~$u$ great enough +so that $\dfrac{uV}{c^2}$~would be greater than unity and $\Delta t'$~would become negative. +In other words, for an observer in system~$S'$ the effect which +occurs at~$B$ would \emph{precede} in time its cause which originates at~$A$. +Such a condition of affairs might not be a logical impossibility; nevertheless +its extraordinary nature might incline us to believe that no +causal impulse can travel with a velocity greater than that of light. + +We may point out in passing, however, that in the case of kinematic +occurrences in which there is no causal connection there is no +reason for supposing that the velocity must be less than that of light. +Consider, for example, a set of blocks arranged side by side in a long +row. For each block there could be an \emph{independent} time mechanism +like an alarm clock which would go off at just the right instant so +that the blocks would fall down one after another along the line. +The velocity with which the phenomenon would travel along the +line of blocks could be arranged to have any value. In fact, the +blocks could evidently all be fixed to fall just at the same instant, +which would correspond to an infinite velocity. It is to be noticed +here, however, that there is no causal connection between the falling +of one block and that of the next, and no transfer of energy. + +%[** TN: ToC entry reads "Applications to Optical Problems"] +\Subsection{Application of the Principles of Kinematics to Certain Optical Problems.} + +\Paragraph{53.} Let us now apply our kinematical considerations to some +problems in the field of optics. We may consider a beam of light +as a periodic electromagnetic disturbance which is propagated through +a vacuum with the velocity~$c$. At any point in the path of a beam of +%% -----File: 070.png---Folio 56------- +light the intensity of the electric and magnetic fields will be undergoing +periodic changes in magnitude. Since the intensities of both the +electric and the magnetic fields vary together, the statement of a +single vector is sufficient to determine the instantaneous condition +at any point in the path of a beam of light. It is customary to call +this vector (which might be either the strength of the electric or of +the magnetic field) the light vector. + +For the case of a simple plane wave (\ie, a beam of monochromatic +light from a distant source) the light vector at any point in the path +of the light may be put proportional to +\[ +\sin\omega \left(t - \frac{lx + my + nz}{c}\right), +\Tag{30} +\] +where $x$,~$y$ and~$z$ are the coördinates of the point under observation, +$t$~is the time, $l$,~$m$ and~$n$ are the cosines of the angles $\alpha$,~$\beta$ and~$\gamma$ which +determine the direction of the beam of light with reference to our +system, and $\omega$~is a constant which determines the period of the light. + +If now this same beam of light were examined by an observer in +system~$S'$ which is moving past the original system in the $X$~direction +with the velocity~$V$, we could write the light vector proportional to +\[ +\sin\omega' \left(t' - \frac{l'x' + m'y' + n'z'}{c}\right), +\Tag{31} +\] +It is not difficult to show that the transformation equations which +we have already developed must lead to the following relations between +the measurements in the two systems\footnote + {Methods for deriving the relation between the accented and unaccented + quantities will be obvious to the reader. For example, consider the relation between + $\omega$~and~$\omega'$. At the origin of coördinates $x = y = z = 0$ in system~$S$, we shall have + in accordance with expression~(30) the light vector proportional to $\sin \omega t$, and hence + similarly at the point~$O'$, which is the origin of coördinates in system~$S'$, we shall + have the light vector proportional to $\sin \omega' t'$. But the point~$O'$ as observed from + system~$S$ moves with the velocity~$V$ along the $X$\DPchg{-}{~}axis and at any instant has the + position $x = Vt$; hence substituting in expression~(30) we have the light vector at + the point~$O'$ as measured in system~$S$ proportional to + \[ + \sin\omega t \left(1 - l\, \frac{V}{c}\right), + \Tag{36} + \] + while as measured in system~$S'$ the intensity is proportional to + \[ + \sin\omega' t'. + \Tag{37} + \] + We have already obtained, however, a transformation equation for~$t'$, namely, + \[ + t' = \kappa \left(t - \frac{V}{c^2}\, x\right), + \] + and further may place $x = Vt$. Making these substitutions and comparing expressions + (36)~and~(37) we see that we must have the relation + \[ + \omega' = \omega \kappa \left(1 - l\, \frac{V}{c}\right). + \] + Methods of obtaining the relation between the cosines $l$,~$m$ and~$n$ and the corresponding + cosines $l'$,~$m'$, and~$n'$ as measured in system~$S'$ may be left to the reader.} +%% -----File: 071.png---Folio 57------- +\begin{align*}%[* TN: Aligning; centered in original] +\omega' &= \omega\kappa \left(1 - l\, \smfrac{V}{c}\right), \Tag{32} \\ +l' &= \frac{l - \smfrac{V}{c}}{1 - l\, \smfrac{V}{c}}, \Tag{33} \\ +m' &= \frac{m}{\kappa\left(1 - l\smfrac{V}{c}\right)}, \Tag{34} \\ +n' &= \frac{n}{\kappa\left(1 - l\smfrac{V}{c}\right)}. \Tag{35} +\end{align*} + +With the help of these equations we may now treat some important +optical problems. + +\Subsubsection{54}{The Doppler Effect.} At the origin of coördinates, $x = y = z += 0$, in system~$S$ we shall evidently have from expression~(30) the +light vector proportional to $\sin \omega t$. That means that the vector +becomes zero whenever $\omega t = 2N \pi$, where $N$~is any integer; in other +words, the period of the light is $p = \dfrac{2\pi}{\omega}$ or the frequency +\[ +\nu = \frac{\omega}{2\pi}. +\] +Similarly the frequency of the light as measured by an observer in +system~$S'$ would be +\[ +\nu' = \frac{\omega'}{2\pi}. +\] +%% -----File: 072.png---Folio 58------- +Combining these two equations and substituting the equation connecting +$\omega$~and~$\omega'$ we have +\[ +\nu = \frac{\nu'}{\kappa \left(1 - l\smfrac{V}{c}\right)}. +\] +This is the relation between the frequencies of a given beam of light +as it appears to observers who are in relative motion. + +If we consider a source of light at rest with respect to system~$S'$ +and at a considerable distance from the observer in system~$S$, we +may substitute for~$\nu'$ the frequency of the source itself,~$\nu_0$, and for~$l$ +we may write~$\cos\phi$, where $\phi$~is the angle between the line connecting +source and observer and the direction of motion of the source, leading +to the expression +\[ +\nu = \frac{\nu_0}{\kappa \left(1 - \cos\phi\, \smfrac{V}{c}\right)}. +\Tag{38} +\] + +This is the most general equation for the \emph{Doppler effect}. When +the source of light is moving directly in the line connecting source +and observer, we have $\cos\phi = 1$, and the equation reduces to +\[ +\nu = \frac{\nu_0}{\kappa \left(1 - \smfrac{V}{c}\right)}, +\Tag{39} +\] +which except for second order terms is identical with the older expressions +for the Doppler effect, and hence agrees with experimental +determinations. + +We must also observe, however, that even when the source of +light moves at right angles to the line connecting source and observer +there still remains a second-order effect on the observed frequency, +in contradiction to the predictions of older theories. We have in this +case $\cos\phi = 0$, +\[ +\nu = \nu_0\, \sqrt{1 - \frac{V^2}{c^2}}. +\Tag{40} +\] +This is the change in frequency which we have already considered +when we discussed the rate of a moving clock. The possibilities of +%% -----File: 073.png---Folio 59------- +direct experimental verification should not be overlooked (see \Secref[section]{46}). + +\Subsubsection{55}{The Aberration of Light.} Returning now to our transformation +equations, we see that equation~(33) provides an expression for +calculating the \emph{aberration of light}. Let us consider that the source +of light is stationary with respect to system~$S$, and let there be an +observer situated at the origin of \DPchg{coordinates}{coördinates} of system~$S'$ and thus +moving past the source with the velocity~$V$ in the $X$~direction. Let $\phi$~be +the angle between the $X$\DPchg{-}{~}axis and the line connecting source of +light and observer and let $\phi'$~be the same angle as it appears to the +moving observer; then we can obviously substitute in equation~(33), +$\cos\phi = l$, $\cos\phi' = l'$, giving us +\[ +\cos\phi' = \frac{\cos\phi - \smfrac{V}{c}}{1 - \cos\phi\, \smfrac{V}{c}}. +\Tag{41} +\] +This is a general equation for the aberration of light. + +For the particular case that the direction of the beam of light is +perpendicular to the motion of the observer we have $\cos\phi = 0$ +\[ +\cos\phi' = - \frac{V}{c}, +\Tag{42} +\] +which, except for second-order differences, is identical with the familiar +expression which makes the tangent of the angle of aberration numerically +equal to~$V/c$.\DPnote{** Slant fractions start here} The experimental verification of the formula +by astronomical measurements is familiar. + +\Subsubsection{56}{Velocity of Light in Moving Media.} It is also possible to treat +very simply by kinematic methods the problem of the velocity of +light in moving media. We shall confine ourselves to the particular +case of a beam of light in a medium which is itself moving parallel +to the light. + +Let the medium be moving with the velocity~$V$ in the $X$~direction, +and let us consider the system of coördinates~$S'$ as stationary with +respect to the medium. Now since the medium appears to be stationary +with respect to observers in~$S'$ it is evident that the velocity +of the light with respect to~$S'$ will be~$c/\mu$, where $\mu$~is index of refraction +%% -----File: 074.png---Folio 60------- +for the medium. If now we use our equation~(26) for the addition of +velocities we shall obtain for the velocity of light, as measured by +observers in~$S$, +\[ +u = \frac{\smfrac{c}{\mu} + V}{1 + \smfrac{V\, \smfrac{c}{\mu}}{c^2}}. +\Tag{43} +\] +Carrying out the division and neglecting terms of higher order we +obtain +\[ +u = \frac{c}{\mu} + \left(\frac{\mu^2 - 1}{\mu^2}\right) V. +\Tag{44} +\] +The equation thus obtained is identical with that of Fresnel, the +quantity $\left(\dfrac{\mu^2 - 1}{\mu^2}\right)$ being the well-known Fresnel coefficient. The +empirical verification of this equation by the experiments of Fizeau +and of Michelson and Morley is too well known to need further +mention. + +For the case of a dispersive medium we should obviously have to +substitute in equation~(44) the value of~$\mu$ corresponding to the particular +frequency,~$\nu'$, which the light has in system~$S'$. It should be +noticed in this connection that the frequencies $\nu'$~and~$\nu$ which the +light has respectively in system~$S$ and system~$S'$, although nearly +enough the same for the practical use of equation~(44), are in reality +connected by an expression which can easily be shown (see \Secref[section]{54}) +to have the form +\[ +\nu' = \kappa \left(1 - \frac{V}{c}\right)\nu. +\Tag{45} +\] + +\Subsubsection{57}{Group Velocity.} In an entirely similar way we may treat the +problem of group velocity and obtain the equation +\[ +G = \frac{G' + V}{1 + \smfrac{G'V}{c^2}}, +\Tag{46} +\] +where $G'$ is the group velocity as it appears to an observer who is +%% -----File: 075.png---Folio 61------- +stationary with respect to the medium. $G'$~is, of course, an experimental +quantity, connected with frequency and the properties of the +medium, in a way to be determined by experiments on the stationary +medium. + +In conclusion we wish to call particular attention to the extraordinary +simplicity of this method of handling the optics of moving +media as compared with those that had to be employed before the +introduction of the principle of relativity. +%% -----File: 076.png---Folio 62------- + + +\Chapter{VI}{The Dynamics of a Particle.} +\SetRunningHeads{Chapter Six.}{Dynamics of a Particle.} + +\Paragraph{58.} In this chapter and the two following, we shall present a +system of ``relativity mechanics'' based on Newton's three laws of +motion, the Einstein transformation equations for space and time, +and the principle of the conservation of mass. + + +\Subsection{The Laws of Motion.} + +Newton's laws of motion may be stated in the following form: + +I\@. Every particle continues in its state of rest or of uniform motion +in a straight line, unless it is acted upon by an external force. + +II\@. The rate of change of the momentum of the particle is equal +to the force acting and is in the same direction. + +III\@. For the action of every force there is an equal force acting +in the opposite direction. + +Of these laws the first two merely serve to define the concept of +force, and their content may be expressed in mathematical form by +the following equation of definition +\[ +\vc{F} + = \frac{d}{dt}(m\vc{u}) + = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u}, +\Tag{47} +\] +where $\vc{F}$ is the force acting on a particle of mass~$m$ which has the +velocity~$\vc{u}$, and hence the momentum~$m\vc{u}$. + +Quite different in its nature from the first two laws, which merely +give us a definition of force, the third law states a very definite physical +postulate, since it requires for every change in the momentum of a +body an equal and opposite change in the momentum of some other +body. The truth of this postulate will of course be tested by comparing +with experiment the results of the theory of mechanics which +we base upon its assumption. + + +\Subsection{Difference between Newtonian and Relativity Mechanics.} + +\Paragraph{59.} Before proceeding we may point out the particular difference +between the older Newtonian mechanics, which were based on the +laws of motion and the \emph{Galilean} transformation equations for space +%% -----File: 077.png---Folio 63------- +and time, and our new system of relativity mechanics based on +those same laws of motion and the \emph{Einstein} transformation equations. + +In the older mechanics there was no reason for supposing that the +mass of a body varied in any way with its velocity, and hence force +could be defined interchangeably as the rate of change of momentum +or as mass times acceleration, since the two were identical. In relativity +mechanics, however, we shall be forced to conclude that the +mass of a body increases in a perfectly definite way with its velocity, +and hence in our new mechanics we must define force as equal to the +total rate of change of momentum +\[ +\frac{d(m\vc{u})}{dt} + = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u} +\] +instead of merely as mass times acceleration $m\, \dfrac{d\vc{u}}{dt}$. If we should try +to define force in ``relativity mechanics'' as merely equal to mass +times acceleration, we should find that the application of Newton's +third law of motion would then lead to very peculiar results, which +would make the mass of a body different in different directions and +force us to give up the idea of the conservation of mass. + + +\Subsection{The Mass of a Moving Particle.} + +\Paragraph{60.} In \Secref{31} we have already obtained in an elementary way +an expression for the mass of a moving particle, by considering a +collision between elastic particles and calculating how the resulting +changes in velocity would appear to different observers who are +themselves in relative motion. Since we now have at our command +general formulæ for the transformation of velocities, we are now in +a position to handle this problem much more generally, and \emph{in particular +to show that the expression obtained for the mass of a moving particle +is entirely independent of the consideration of any particular type of +collision}. + +\Subsubsection{61}{Transverse Collision.} Let us first treat the case of a so-called +``transverse'' collision. Consider a system of coördinates and two +\begin{wrapfigure}{l}{3in}%[** TN: Width-dependent break] + \Fig{11} + \Input[3in]{078} +\end{wrapfigure} +exactly similar elastic particles, each having the mass~$m_0$ when at +rest, one moving in the $X$~direction with the velocity~$+u$ and the +other with the velocity~$-u$. (See \Figref{11}.) Besides the large +components of velocity $+u$~and~$-u$ which they have in the $X$~direction +%% -----File: 078.png---Folio 64------- +let them also have small components of velocity in the $Y$~direction, +$+v$~and~$-v$. The experiment is so arranged that the particles +will just undergo a glancing collision as they pass each other and +rebound with components +of velocity in the $Y$~direction +of the same magnitude,~$v$, +which they originally had, +but in the reverse direction. +(It is evident from the symmetry of the arrangement that the experiment +would actually occur as we have stated.) + +We shall now be interested in the way this experiment would appear +to an observer who is in motion in the $X$ direction with the velocity~$V$ +relative to our original system of coördinates. + +From equation~(14) for the transformation of velocities, it can +be seen that this \emph{new observer} would find for the $X$~component velocities +of the two particles the values +\[ +u_1 = \frac{ u - V}{1 - \smfrac{uV}{c^2}} \qquad\text{and}\qquad +u_2 = \frac{-u - V}{1 + \smfrac{uV}{c^2}} +\Tag{48} +\] +and from equation~(15) for the $Y$~component velocities would find the +values +\[ +v_1 = \pm \frac{v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}} +\qquad\text{and}\qquad +v_2 = \mp \frac{v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}}, +\Tag{49} +\] +the signs depending on whether the velocities are measured before or +after the collision. + +Now from Newton's third law of motion (\ie, the principle of +the equality of action and reaction) it is evident that on collision +the two particles must undergo the same numerical change in momentum. + +For the experiment that we have chosen the only change in momentum +is in the $Y$~direction, and the observer whose measurements +we are considering finds that one particle undergoes the total change +%% -----File: 079.png---Folio 65------- +in velocity +\begin{align*} +2v_1 &= \frac{2v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}} \\ +\intertext{and that the other particle undergoes the change in velocity} +2v_2 &= \frac{2v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 + \smfrac{uV}{c^2}}. +\end{align*} + +Since these changes in the velocities of the particles are not equal, +it is evident that their masses must also be unequal if the principle +of the equality of action and reaction is true for all observers, as we +have assumed. This difference in the mass of the particles, each of +which has the mass~$m_0$ when at rest, arises from the fact that the mass +of a particle is a function of its velocity and for the observer in question +the two particles are not moving with the same velocity. + +Using the symbols $m_1$~and~$m_2$ for the masses of the particles, we +may now write as a mathematical expression of the requirements of +the third law of motion +\[ +\frac{2m_1v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{uV}{c^2}} = +\frac{2m_2v\, \sqrt{1 - \smfrac{V^2}{c^2}}}{1 + \smfrac{uV}{c^2}}. +\] + +Simplifying, we obtain by direct algebraic transformation +%[** TN: Setting innermost denominator fractions textstyle for clarity] +\[ +\frac{m_1}{m_2} + = \frac{1 - \smfrac{uV}{c^2}}{1 + \smfrac{uV}{c^2}} + = \frac{\sqrt{ + 1 - \smfrac{\Biggl(\smfrac{-u - V}{1 + \tfrac{uV}{c^2}}\Biggr)^2}{c^2}}} + {\sqrt{ + 1 - \smfrac{\Biggl(\smfrac{ u - V}{1 - \tfrac{uV}{c^2}}\Biggr)^2}{c^2}}}, +\] +%% -----File: 080.png---Folio 66------- +which on the substitution of equations~(48) gives us +\[ +\frac{m_1}{m_2} + = \frac{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}} + {\sqrt{1 - \smfrac{{u_1}^2}{c^2}}}. +\Tag{50} +\] + +This equation thus shows that the mass of a particle moving with +the velocity~$u$\footnote + {For simplicity of calculation we consider the case where the components of + velocity in the $Y$~direction are small enough to be negligible in their effect on the + mass of the particles compared with the large components of velocity $u_1$~and~$u_2$ in + the $X$~direction.} +is inversely proportional to $\sqrt{1 - \dfrac{u^2}{c^2}}$, and, denoting +the mass of the particle at rest by~$m_0$, we may write as a \emph{general expression +for the mass of a moving particle} +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}. +\Tag{51} +\] + +\Subsubsection{62}{Mass the Same in All Directions.} The method of derivation +that we have just used to obtain this expression for the mass of a +moving particle is based on the consideration of a so-called ``transverse +collision,'' and in fact the expression obtained has often been +spoken of as that for the \emph{transverse mass} of a moving particle, while +a different expression, $\dfrac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}$, has been used for the so-called +\emph{longitudinal mass} of the particle. These expressions $\dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ and +$\dfrac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}$ are, as a matter of fact, the values of the electric force +necessary to give a charged particle unit acceleration respectively +at right angles and in the same direction as its original velocity, and +hence such expressions would be proper for the mass of a moving particle +if we should define force as mass times acceleration. As already +%% -----File: 081.png---Folio 67------- +stated, however, it has seemed preferable to retain, for force, Newton's +original definition which makes it equal to the rate of change of +momentum, and we shall presently see that this more suitable definition +is in perfect accord with the idea that the mass of a particle is +the same in all directions. + +Aside from the unnecessary complexity which would be introduced, +the particular reason making it unfortunate to have different +expressions for mass in different directions is that under such conditions +it would be impossible to retain or interpret the principle of +the conservation of mass. And we shall now proceed to show that +by introducing the principle of the conservation of mass, the consideration +of a ``longitudinal collision'' will also lead to exactly the +same expression, $\dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, for the mass of a moving particle as we +have already obtained from the consideration of a transverse collision. + +\Subsubsection{63}{Longitudinal Collision.} Consider a system of coördinates and +two elastic particles moving in the $X$~direction with the velocities +$+u$~and~$-u$ so that a ``longitudinal'' (\ie, head-on) collision will +occur. Let the particles be exactly alike, each of them having the +mass~$m_0$ when at rest. On collision the particles will evidently come +to rest, and then under the action of the elastic forces developed start +up and move back over their original paths with the respective velocities +$-u$~and~$+u$ of the same magnitude as before. + +Let us now consider how this collision would appear to an observer +who is moving past the original system of coördinates with the velocity~$V$ +in the $X$~direction. Let $u_1$~and~$u_2$ be the velocities of the particles +as they appear to this new observer before the collision has taken +place. Then, from our formula for the transformation of velocities~(14), +it is evident that we shall have +\[ +u_1 = \frac{ u - V}{1 - \smfrac{uV}{c^2}}\qquad\text{and}\qquad +u_2 = \frac{-u - V}{1 + \smfrac{uV}{c^2}}. +\Tag{52} +\] + +Since these velocities $u_1$~and~$u_2$ are not of the same magnitude, +the two particles which have the same mass when at rest do not have +the same mass for this observer. Let us call the masses before collision +$m_1$~and~$m_2$. +%% -----File: 082.png---Folio 68------- + +Now during the collision the velocities of the particles will all the +time be changing, but from the principle of the conservation of mass +the sum of the two masses must all the time be equal to $m_1 + m_2$. +When in the course of the collision the particles have come to relative +rest, they will be moving past our observer with the velocity~$-V$, +and their momentum will be $-(m_1 + m_2)V$. But, from the principle +of the equality of action and reaction, it is evident that this momentum +must be equal to the original momentum before collision occurred. +This gives us the equation $-(m_1 + m_2)V = m_1 u_1 + m_2 u_2$. Substituting +our values~(52) for $u_1$~and~$u_2$ we have +\[ +\frac{m_1}{\left(1 - \smfrac{uV}{c^2}\right)} = +\frac{m_2}{\left(1 + \smfrac{uV}{c^2}\right)}, +\] +and by direct algebraic transformation, as in the previous proof, +this can be shown to be identical with +\[ +\frac{m_1}{m_2} + = \frac{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}} + {\sqrt{1 - \smfrac{{u_1}^2}{c^2}}}, +\] +leading to the same expression that we obtained before for the mass +of a moving particle, viz.: +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}. +\] + +\Subsubsection{64}{Collision of Any Type.} We have derived this formula for the +mass of a moving particle first from the consideration of a transverse +and then of a longitudinal collision between particles which are elastic +and have the same mass when at rest. It seems to be desirable to +show, however, that the consideration of any type of collision between +particles of any mass leads to the same formula for the mass of a +moving particle. + +For the mass~$m$ of a particle moving with the velocity~$u$ let us +write the equation $m = m_0 F(u^2)$, where $F(\:)$~is the function whose +form we wish to determine. The mass is written as a function of +%% -----File: 083.png---Folio 69------- +the square of the velocity, since from the homogeneity of space the +mass will be independent of the direction of the velocity, and the +mass is made proportional to the mass at rest, since a moving body +may evidently be thought of as divided into parts without change in +mass. It may be further remarked that the form of the function~$F(\:)$ +must be such that its value approaches unity as the variable +approaches zero. + +Let us now consider two particles having respectively the masses +$m_0$~and~$n_0$ when at rest, moving with the velocities $u$~and~$w$ before +collision, and with the velocities $U$~and~$W$ after a collision has taken +place. + +From the principle of the conservation of mass we have +\begin{multline*} +m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2) + +n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2) \\ += m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2) ++ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2), +\Tag{53} +\end{multline*} +and from the principle of the equality of action and reaction (\ie, +Newton's third law of motion) +\begin{multline*} +m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2)u_x + +n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2)w_x \\ += m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2)U_x ++ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2)W_x, +\Tag{54} +\end{multline*} +\begin{multline*} +m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2)u_y + +n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2)w_y \\ += m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2)U_y ++ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2)W_y, +\Tag{55} +\end{multline*} +\begin{multline*} +m_0 F({u_x}^2 + {u_y}^2 + {u_z}^2)u_z + +n_0 F({w_x}^2 + {w_y}^2 + {w_z}^2)w_z \\ += m_0 F({U_x}^2 + {U_y}^2 + {U_z}^2)U_z ++ n_0 F({W_x}^2 + {W_y}^2 + {W_z}^2)W_z. +\Tag{56} +\end{multline*} + +These velocities, $u_x$,~$u_y$,~$u_x$, $w_x$,~$w_y$,~$w_z$, $U_x$,~etc., are measured, of +course, with respect to some definite system of ``space-time'' coördinates. +An observer moving past this system of coördinates with the +velocity~$V$ in the $X$~direction would find for the corresponding component +velocities the values +\[ +\frac{u_x - V}{1 - \smfrac{u_xV}{c^2}},\quad +\frac{\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}}\, u_y,\quad +\frac{\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}}\, u_z,\quad +\frac{w_x - V}{1 - \smfrac{w_xV}{c^2}},\quad\text{etc.}, +\] +as given by our transformation equations for velocity \DPchg{(14, 15, 16)}{(14),~(15),~(16)}. +%% -----File: 084.png---Folio 70------- + +Since the law of the conservation of mass and Newton's third +law of motion must also hold for the measurements of the new observer, +we may write the following new relations corresponding to +equations \DPchg{53~to~56}{(53)~to~(56)}: + +{\footnotesize% +\[ +\begin{aligned} +m_0 F&\left\{ + \left(\frac{u_x - V}{1 - \smfrac{u_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_y}{1 - \smfrac{u_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_z}{1 - \smfrac{u_x V}{c^2}}\right)^2 +\right\} \\ ++ n_0F&\left\{ +\left(\frac{w_x - V}{1 - \smfrac{w_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_y}{1 - \smfrac{w_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_z}{1 - \smfrac{w_x V}{c^2}}\right)^2 +\right\} \\ += m_0F&\left\{ +\left(\frac{U_x - V}{1 - \smfrac{U_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_y}{1 - \smfrac{U_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_z}{1 - \smfrac{U_x V}{c^2}}\right)^2 +\right\} \\ ++ n_0F&\left\{ +\left(\frac{W_x - V}{1 - \smfrac{W_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_y}{1 - \smfrac{W_x V}{c^2}}\right)^2 ++ \left(\frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_z}{1 - \smfrac{W_x V}{c^2}}\right)^2 +\right\}, +\end{aligned} +\Tag{53\textit{a}} +\]}% +\[ +\begin{aligned} +&m_0F\{u_x\cdots\}\, \frac{u_x - V}{1 - \smfrac{u_xV}{c^2}} + +n_0F\{w_x\cdots\}\, \frac{w_x - V}{1 - \smfrac{w_xV}{c^2}} \\ +&\qquad= +m_0F\{U_x\cdots\}\, \frac{U_x - V}{1 - \smfrac{U_xV}{c^2}} + +n_0F\{W_x\cdots\}\, \frac{W_x - V}{1 - \smfrac{W_xV}{c^2}}, +\end{aligned} +\Tag{54\textit{a}} +\] +{\small% +\[ +\begin{aligned} +&m_0F\{u_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_y}{1 - \smfrac{u_xV}{c^2}} + +n_0F\{w_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_y}{1 - \smfrac{w_xV}{c^2}} \\ +&\qquad= +m_0F\{U_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_y}{1 - \smfrac{U_xV}{c^2}} + +n_0F\{W_x\cdots\}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_y}{1 - \smfrac{W_xV}{c^2}}, +\end{aligned} +\Tag{55\textit{a}} +\]}% +%% -----File: 085.png---Folio 71------- +\[ +\begin{aligned} +&m_0F{u_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, u_x}{1 - \smfrac{u_xV}{c^2}} + +n_0F{w_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, w_x}{1 - \smfrac{w_xV}{c^2}} \\ +&\qquad= +m_0F{U_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, U_x}{1 - \smfrac{U_xV}{c^2}} + +n_0F{W_x\cdots}\, \frac{\sqrt{1 - \smfrac{V^2}{c^2}}\, W_x}{1 - \smfrac{W_xV}{c^2}}. +\end{aligned} +\Tag{56\textit{a}} +\] + +It is evident that these equations \DPchg{(53\textit{a}--56\textit{a})}{(53\textit{a})--(56\textit{a})} must be true no +matter what the velocity between the new observer and the original +system of coördinates, that is, true for all values of~$V$. The velocities +$u_x$,~$u_y$,~$u_z$, $w_x$,~etc., are, however, perfectly definite quantities, measured +with reference to a definite system of coördinates and entirely independent +of~$V$. If these equations are to be true for perfectly definite +values of $u_x$,~$u_y$,~$u_z$, $w_x$,~etc., and for all values of~$V$, it is evident that +the function~$F(\:\,)$ must be of such a form that the equations are +identities in~$V$. As a matter of fact, it is found by trial that $V$~can +be cancelled from all the equations if we make $F(\:\,)$ of the form +$\dfrac{1}{\sqrt{1 - \smfrac{(\:)}{c^2}}}$; and we see that the expected relation is a solution of the +equations, although perhaps not necessarily a unique solution. + +Before proceeding to use our formula for the mass of a moving +particle for the further development of our system of mechanics, +we may call attention in passing to the fact that the experiments of +Kaufmann, Bucherer, and Hupka have in reality shown that the mass +of the electron increases with its velocity according to the formula +which we have just obtained. We shall consider the dynamics of the +electron more in detail in the chapter devoted to \Chapnumref[XII]{electromagnetic +theory}. We wish to point out now, however, that in this derivation +we have made no reference to any electrical charge which might be +carried by the particle whose mass is to be determined. Hence we +may reject the possibility of explaining the Kaufmann experiment +by assuming that the charge of the electron decreases with its velocity, +since the increase in mass is alone sufficient to account for the results +of the measurement. +%% -----File: 086.png---Folio 72------- + + +\Subsection{Transformation Equations for Mass.} + +\Paragraph{65.} Since the velocity of a particle depends on the particular +system of coördinates chosen for the measurement, it is evident that +the mass of the particle will also depend on our reference system of +coördinates. For the further development of our system of dynamics, +we shall find it desirable to obtain transformation equations for mass +similar to those already obtained for velocity, acceleration, etc. + +We have +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +where the velocity~$u$ is measured with respect to some definite system +of coördinates,~$S$. Similarly with respect to a system of coördinates~$S'$ +which is moving relatively to~$S$ with the velocity~$V$ in the $X$~direction +we shall have +\[ +m' = \frac{m_0}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}. +\] +We have already obtained, however, a transformation equation~(17) +for the function of the velocity occurring in these equations and on +substitution we obtain the desired transformation equation +\[ +m' = \left(1 - \frac{u_x V}{c^2}\right) \kappa m, +\Tag{57} +\] +where $\kappa$ has the customary significance +\[ +\kappa = \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}. +\] + +By differentiation of~(57) with respect to the time and simplification, +we obtain the following transformation equation for the +\emph{rate at which the mass of a particle is changing} owing to change in +velocity +\[ +\dot{m}' = \dot{m} - \frac{mV}{c^2} + \left(1 - \frac{u_xV}{c^2}\right)^{-1} \frac{du_x}{dt}. +\Tag{58} +\] +%% -----File: 087.png---Folio 73------- + +%[** TN: ToC entry reads "The Force Acting on a Moving Particle"] +\Subsection{Equation for the Force Acting on a Moving Particle.} + +\Paragraph{66.} We are now in a position to return to our development of the +dynamics of a particle. In the first place, the equation which we +have now obtained for the mass of a moving particle will permit +us to rewrite the original equation by which we defined force, in a +number of ways which will be useful for future reference. + +We have our equation of definition~(47) +\[ +\vc{F} = \frac{d}{dt}(m\vc{u}) + = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u}, +\] +which, on substitution of the expression for~$m$, gives us +\[ +\vc{F} + = \frac{d}{dt} \Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{u}\Biggr] + = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt} + + \frac{d}{dt} \Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr] \vc{u} +\Tag{59} +\] +or, carrying out the indicated differentiation, +\[ +\vc{F} + = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt} + + \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, + \frac{u}{c^2}\, \frac{du}{dt}\, \vc{u}. +\Tag{60} +\] + + +\Subsection{Transformation Equations for Force.} + +\Paragraph{67.} We are also in position to obtain transformation equations for +force. We have +\[ +\vc{F} = \frac{d}{dt}(m\vc{u}) = m\vc{\dot{u}} + \dot{m}\vc{u} +\] +or +\begin{align*} +F_x &= m\dot{u}_x + \dot{m}u_x, \\ +F_y &= m\dot{u}_y + \dot{m}u_y, \\ +F_z &= m\dot{u}_z + \dot{m}u_z. +\end{align*} +We have transformation equations, however, for all the quantities +on the right-hand side of these equations. For the velocities we +have equations (14),~(15) and~(16), for the accelerations (18),~(19) +and~(20), for mass, equation~(57) and for rate of change of mass, +equation~(58). Substituting above we obtain as our \emph{transformation +%% -----File: 088.png---Folio 74------- +equations for force} +\begin{align*} +F_x' &= \frac{F_x - \dot{m}V}{1 - \smfrac{u_xV}{c^2}} + = F_x - \frac{u_y V}{c^2 - u_x V}\, F_y + - \frac{u_z V}{c^2 - u_x V}\, F_z, \Tag{61} \\ +F_y' &= \frac{\kappa^{-1}}{1 - \smfrac{u_x V}{c^2}}\, F_y, \Tag{62}\\ +F_z' &= \frac{\kappa^{-1}}{1 - \smfrac{u_x V}{c^2}}\, F_z. \Tag{63} +\end{align*} + +We may now consider a few applications of the principles governing +the dynamics of a particle. + + +\Subsection{The Relation between Force and Acceleration.} + +\Paragraph{68.} If we examine our equation~(59) for the force acting on a +particle +\[ +F = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt} + + \frac{d}{dt} \Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr] \vc{u}, +\Tag{59} +\] +we see that the force is equal to the sum of two vectors, one of which +is in the direction of the acceleration $\dfrac{d\vc{u}}{dt}$ and the other in the direction +of +\begin{wrapfigure}[17]{l}{2.5in}%[** TN: Width-dependent break] + \Fig{12} + \Input[2.5in]{088} +\end{wrapfigure} +the existing velocity~$\vc{u}$, so that \emph{in general force and the acceleration +it produces are not in the same direction}. +We shall find it interesting +to see, however, that if the force +which does produce acceleration in +a given direction be resolved perpendicular +and parallel to the acceleration, +the two components will +be connected by a definite relation. + +Consider a particle (\Figref[fig.]{12}) in +plane space moving with the velocity +\[ +\vc{u} = {u_x}\vc{i} + {u_y}\vc{j}. +\] +%% -----File: 089.png---Folio 75------- +Let it be accelerated in the $X$~direction by the action of the component +forces $F_x$~and~$F_y$. + +From our general equation~(59) for the force acting on a particle +we have for these component forces +\begin{align*} +F_x &= \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{du_x}{dt} + + \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]u_x, +\Tag{64} \\ +F_y &= \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{du_y}{dt} + + \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]u_y. +\Tag{65} +\end{align*} + +Introducing the condition that all the acceleration is to be in the $Y$~direction, +which makes $\dfrac{du_x}{dt} = 0$, and further noting that $u^2 = u_x^2 + u_y^2$, +by the division of equation~(64) by~(65), we obtain +\begin{align*} +\frac{F_x}{F_y} &= \frac{u_x u_y}{c^2 - {u_x}^2}, \\ +F_x &= \frac{u_x u_y}{c^2 - {u_x}^2}\, F_y. +\Tag{66} +\end{align*} + +\emph{Hence, in order to accelerate a particle in a given direction, we may +apply any force~$F_y$ in the desired direction, but must at the same time +apply at right angles another force~$F_x$ whose magnitude is given by +equation~\upshape{(66)}.} + +Although at first sight this state of affairs might seem rather +unexpected, a simple qualitative consideration will show the necessity +of a component of force perpendicular to the desired acceleration. +Refer again to \Figref{12}; since the particle is being accelerated in the $Y$~direction, +its total velocity and hence its mass are increasing. This +increasing mass is accompanied by increasing momentum in the $X$~direction +even when the velocity in that direction remains constant. +The component force~$F_x$ is necessary for the production of this increase +in $X$-momentum. + +In a later paragraph we shall show an application of equation~(66) +in electrical theory. +%% -----File: 090.png---Folio 76------- + + +\Subsection{Transverse and Longitudinal Acceleration.} + +\Paragraph{69.} An examination of equation~(66) shows that there are two +special cases in which the component force~$F_x$ disappears and the +force and acceleration are in the same direction. $F_x$~will disappear +when either $u_x$~or~$u_y$ is equal to zero, so that force and acceleration +will be in the same direction when the force acts exactly at right +angles to the line of motion of the particle, or in the direction of the +motion (or of course also when $u_x$~and~$u_y$ are both equal to zero and +the particle is at rest). It is instructive to obtain simplified expressions +for force for these two cases of transverse and longitudinal +acceleration. + +Let us again examine our equation~(60) for the force acting on a +particle +\[ +\vc{F} + = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt} + + \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, + \frac{u}{c^2}\, \frac{du}{dt} \vc{u}. +\Tag{60}%[** TN: [sic] Repeated equation] +\] + +For the case of a \emph{transverse acceleration} there is no component of +force in the direction of the velocity~$\vc{u}$ and the second term of the +equation is equal to zero, giving us +\[ +\vc{F} = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt}. +\Tag{67} +\] + +For the case of \emph{longitudinal acceleration}, the velocity~$\vc{u}$ and the +acceleration~$\dfrac{d\vc{u}}{dt}$ are in the same direction, so that we may rewrite the +second term of~(60), giving us +\[ +\vc{F} + = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{d\vc{u}}{dt} + + \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, + \frac{u^2}{c^2}\, \frac{d\vc{u}}{dt}, +\] +and on simplification this becomes +\[ +\vc{F} + = \frac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, \frac{d\vc{u}}{dt}. +\Tag{68} +\] +%% -----File: 091.png---Folio 77------- +An examination of this expression shows the reason why $\dfrac{m_0}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}$ +is sometimes spoken of as the expression for the \emph{longitudinal mass} of a +particle. + + +\Subsection{The Force Exerted by a Moving Charge.} + +\Paragraph{70.} In a \Chapnumref[XII]{later chapter} we shall present a consistent development +of the fundamentals of electromagnetic theory based on the Einstein +transformation equations for space and time and the four field equations. +At this point, however, it may not be amiss to point out that +the principles of mechanics themselves may sometimes be employed +to obtain a simple and direct solution of electrical problems. + +Suppose, for example, we wish to calculate the force with which a +\emph{point charge in uniform motion} acts on any other point charge. We +can solve this problem by considering a system of coördinates which +move with the same velocity as the charge itself. An observer +making use of the new system of coördinates could evidently calculate +the force exerted by the charge in question by Coulomb's familiar +inverse square law for static charges, and the magnitude of the force +as measured in the original system of coördinates can then be determined +from our transformation equations for force. Let us proceed +to the specific solution of the problem. + +Consider a system of coördinates~$S$, and a charge~$e$ in uniform +motion along the $X$~axis with the velocity~$V$. We desire to know +the force acting at the time~$t$ on any other charge~$e_1$ which has any +desired coördinates $x$,~$y$, and~$z$ and any desired velocity $u_x$,~$u_y$ and~$u_z$. + +Assume a system of coördinates,~$S'$, moving with the same velocity +as the charge~$e$ which is taken coincident with the origin. To an +observer moving with the system~$S'$, the charge~$e$ appears to be +always at rest and surrounded by a pure electrostatic field. Hence +in system~$S'$ the force with which $e$~acts on~$e_1$ will be, in accordance +with Coulomb's law\footnote + {It should be noted that in its original form Coulomb's law merely stated + that the force between two stationary charges was proportional to the product of + the charges and inversely to the distance between them. In the present derivation + we have extended this law to apply to the instantaneous force exerted by a stationary + charge upon any other charge. + + The fact that a charge of electricity appears the same to observers in all systems + is obviously also necessary for the setting up of equations (69),~(70),~(71). That + such is the case, however, is an evident consequence of the atomic nature of electricity. + The charge~$e$ would appear of the same magnitude to observers both in + system~$S$ and system~$S'$, since they would both count the same number of electrons + on the charge. (See \Secref{157}.)} +\[ +\vc{F'} = \frac{e e_1 \vc{r'}}{{r'}^3} +\] +%% -----File: 092.png---Folio 78------- +or +\begin{align*} +F_x' &= \frac{ee_1x'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \Tag{69} \\ +F_y' &= \frac{ee_1x'}{({y'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \Tag{70} \\ +F_z' &= \frac{ee_1x'}{({z'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \Tag{71} +\end{align*} +where $x'$,~$y'$, and~$z'$ are the coördinates of the charge~$e_1$ at the time~$t'$. +For simplicity let us consider the force at the time $t' = 0$; then from +transformation equations (9),~(10), (11),~(12) we shall have +\[ +x' = \kappa^{-1} x,\qquad y' = y, \qquad z'= z. +\] +Substituting in (69),~(70),~(71) and also using our transformation +equations for force (61),~(62),~(63), we obtain the following equations +for the force acting on~$e_1$, as it appears to an observer in system~$S$: +\begin{align*}%[** TN: Aligning on "="s] +F_x &= \frac{ee_1x'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}} + \left(x + \frac{V}{c^2}\, \kappa^2(yu_y + zu_z)\right), +\Tag{72} \\ +F_y &= \frac{ee_1\left(1 - \smfrac{u_xV}{c^2}\right) \kappa y} + {({\kappa^{-2}x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, +\Tag{73} \\ +F_z &= \frac{ee_1\left(1 - \smfrac{u_xV}{c^2}\right) \kappa z} + {({\kappa^{-2}x'}^2 + {y'}^2 + {z'}^2)^{3/2}}. +\Tag{74} +\end{align*} + +These equations give the force acting on~$e_1$ at the time~$t$. From +transformation equation~(12) we have $t = \dfrac{V}{c^2}\, x$, since $t' = 0$. At this +time the charge~$e$, which is moving with the uniform velocity~$V$ along +%% -----File: 093.png---Folio 79------- +the $X$~axis, will evidently have the position +\[ +x_e = \frac{V^2}{c^2}\, x,\qquad +y_e = 0, \qquad +z_e = 0. +\] + +For convenience we may now refer our results to a system of +coördinates whose origin coincides with the position of the charge~$e$ +at the instant under consideration. If $X$,~$Y$ and~$Z$ are the coördinates +of~$e_1$ with respect to this new system, we shall evidently have +the relations +\begin{gather*} +X = x - \frac{V^2 }{c^2}\, x = \kappa^{-2} x,\qquad Y = y,\qquad Z = z,\\ +U_x = u_x, \qquad U_y = u_y, \qquad U_z = u_z. +\end{gather*} +Substituting into (72),~(73),~(74) we obtain +\begin{align*}%[** TN: Aligning on "="s] +F_x &= \frac{ee_1}{s^3} \left(1 - \frac{V^2}{c^2}\right) + \left(X + \frac{V}{c^2}\, (YU_y + ZU_z)\right), +\Tag{75} \\ +F_y &= \frac{ee_1}{s^3} \left(1 - \frac{V^2}{c^2}\right) + \left(1 - \frac{U_xV}{c^2})\right) Y, +\Tag{76} \\ +F_z &= \frac{ee_1}{s^3} \left(1 - \frac{V^2}{c^2}\right) + \left(1 - \frac{U_xV}{c^2})\right) Z, +\Tag{77} +\end{align*} +where for simplicity we have placed +\[ +s = \sqrt{X^2 + \left(1 - \frac{V^2}{c^2}\right)(Y^2 + Z^2)}. +\] + +These are the same equations which would be obtained by substituting +the well-known formulæ for the strength of the electric and +magnetic field around a moving point charge into the fifth fundamental +equation of the Maxwell-Lorentz theory, $\vc{f} = \rho \left(\vc{e} + \dfrac{1}{c}\, [\vc{u} × \vc{h}]^*\right)$. +They are really obtained in this way more easily, however, and are +seen to come directly from Coulomb's law. + +%[** TN: Unnumbered, but has a ToC entry] +\Subsubsection{}{The Field around a Moving Charge.} Evidently we may also use +these considerations to obtain an expression for the electric field +produced by a moving charge~$e$, if we consider the particular case +that the charge~$e_1$ is stationary (\ie, $U_x = U_y = U_z = 0$) and equal +%% -----File: 094.png---Folio 80------- +to unity. Making these substitutions in (75),~(76),~(77) we obtain +the well-known expression for the electrical field in the neighborhood +of a moving point charge +\[ +\vc{F} = e = \frac{\vc{e}}{s^3} \left(1 - \frac{V^2}{c^2}\right)\vc{r}, +\Tag{78} +\] +where +\[ +\vc{r} = X\vc{i} + Y\vc{j} + Z\vc{k}. +\] + +\Subsubsection{71}{Application to a Specific Problem.} Equations (75), (76), (77) +can also be applied in the solution of a +rather interesting specific problem. + +Consider a charge~$e$ constrained to +move in the $X$~direction with the velocity~$V$ +and at the instant under consideration +let it coincide with the origin +of a system of stationary coördinates +$YeX$ (\Figref[fig.]{13}). Suppose now a second +charge~$e_1$, situated at the point $X = 0$, +$Y = Y$ and moving in the $X$~direction +with the same velocity~$V$ as the charge~$e$, +and also having a component velocity +in the $Y$~direction~$U_y$. Let us +%[** TN: Move down past page break; width-dependent line break] +\begin{wrapfigure}{l}{2.25in} + \Fig{13} + \Input[2.25in]{094} +\end{wrapfigure} +predict +the nature of its motion under the influence +of the charge~$e$, it being otherwise +unconstrained. + +From the simple qualitative considerations placed at our disposal +by the theory of relativity, it seems evident that the charge~$e_1$ ought +merely to increase its component of velocity in the $Y$~direction and +retain unchanged its component in the $X$~direction, since from the +point of view of an observer moving along with~$e$ the phenomenon is +merely one of ordinary \emph{electrostatic} repulsion. + +Let us see whether our equations for the force exerted by a moving +charge actually lead to this result. By making the obvious substitutions +in equations (75)~and~(76) we obtain for the component +forces on~$e_1$ +\begin{align*}%[** TN: Aligning on "="s] +F_x &= \frac{e e_1}{s^3}\left(1 - \frac{V^2}{c^2}\right) + \frac{V}{c^2}\, Y U_y, +\Tag{79} \\ +F_x &= \frac{e e_1}{s^3}\left(1 - \frac{V^2}{c^2}\right)^2 Y. +\Tag{80} +\end{align*} +%% -----File: 095.png---Folio 81------- + +Now under the action of the component force~$F_x$ we might at +first sight expect the charge~$e_1$ to obtain an acceleration in the $X$~direction, +in contradiction to the simple qualitative prediction that +we have just made on the basis of the theory of relativity. We +remember, however, that equation~(66) prescribes a definite ratio +between the component forces $F_x$~and~$F_y$ if the acceleration is to be +in the $Y$~direction, and dividing~(79) by~(80) we actually obtain the +necessary relation +\[ +\frac{F_x}{F_y} = \frac{V U_y}{c^2 - V^2}. +\] + +Other applications of the new principles of dynamics to electrical, +magnetic and gravitational problems will be evident to the reader. + + +\Subsection{Work.} + +\Paragraph{72.} Before proceeding with the further development of our theory +of dynamics we shall find it desirable to define the quantities work, +kinetic, and potential energy. + +We have already obtained an expression for the force acting on a +particle and shall define the work done on the particle as the integral +of the force times the distance through which the particle is displaced. +Thus +\[ +W = \int \vc{F} · d\vc{r}, +\Tag{81} +\] +where $\vc{r}$ is the radius vector determining the position of the particle. + + +\Subsection{Kinetic Energy.} + +\Paragraph{73.} When a particle is brought from a state of rest to the velocity~$\vc{u}$ +by the action of an unbalanced force~$\vc{F}$, we shall define its kinetic +energy as numerically equal to the work done in producing the velocity. +Thus +\[ +K = W = \int \vc{F} · d\vc{r}. +\] + +Since, however, the kinetic energy of a particle turns out to be +entirely independent of the particular choice of forces used in producing +the final velocity, it is much more useful to have an expression +for kinetic energy in terms of the mass and velocity of the particle. + +We have +\[ +K = \int \vc{F} · d\vc{r} + = \int \vc{F} · \frac{d\vc{r}}{dt}\, dt + = \int \vc{F} · \vc{u}\, dt. +\] +%% -----File: 096.png---Folio 82------- +Substituting the value of~$\vc{F}$ given by the equation of definition~(47) +we obtain +\begin{align*} +K &= \int m\, \frac{d\vc{u}}{dt} · \vc{u}\, dt + + \int \frac{dm}{dt}\, \vc{u} · \vc{u}dt \\ + &= \int m\, \vc{u} · d\vc{u} + \int \vc{u} · \vc{u}\, dm \\ + &= \int mu\, du + \int u^2\, dm. +\end{align*} +Introducing the expression~(51) for the mass of a moving particle +$m = \dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, we obtain +\[ +K = \int m_0\, \frac{u}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, du + + \int\frac{m_0}{c^2}\, \frac{u^3}{\left(1 - \smfrac{u^2}{c^2}\right)^{3/2}}\, du +\] +and on integrating and evaluating the constant of integration by +placing the kinetic energy equal to zero when the velocity is zero, +we easily obtain the desired expression for the kinetic energy of a +particle: +\begin{align*} +K &= m_0 c^2 \Biggl[\frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} - 1\Biggr], +\Tag{82} \\ + &= c^2(m - m_0). +\Tag{83} +\end{align*} + +It should be noticed, as was stated above, that the kinetic energy +of a particle \emph{does} depend merely on its mass and final velocity and is +entirely independent of the particular choice of forces which happened +to be used in producing the state of motion. + +It will also be noticed, on expansion into a series, that our expression~(82) +for the kinetic energy of a particle approaches at low +velocities the form familiar in the older Newtonian mechanics, +\[ +K = \tfrac{1}{2} m_0 u^2. +\] + + +\Subsection{Potential Energy.} + +\Paragraph{74.} When a moving particle is brought to rest by the action of a +%% -----File: 097.png---Folio 83------- +\emph{conservative}\footnote + {A conservative force is one such that any work done by displacing a system + against it would be completely regained if the motion of the system should be reversed. + + Since we believe that the forces which act on the ultimate particles and constituents + of matter are in reality all of them conservative, we shall accept the general + principle of the conservation of energy just as in Newtonian mechanics. (For a + logical deduction of the principle of the conservation of energy in a system of particles, + see the next chapter, \Secref[section]{89}.)} %[** TN: Not a useful chapter cross-ref] +force we say that its kinetic energy has been transformed +into potential energy. The increase in the potential energy +of the particle is equal to the kinetic energy which has been destroyed +and hence equal to the work done by the particle against the force, +giving us the equation +\[ +\Delta U = -W = -\int \vc{F} · d\vc{r}. +\Tag{84} +\] + + +\Subsection{The Relation between Mass and Energy.} + +\Paragraph{75.} We may now consider a very important relation between the +mass and energy of a particle, which was first pointed out in our +chapter on ``\Chapnumref[III]{Some Elementary Deductions}.'' + +When an isolated particle is set in motion, both its mass and +energy are increased. For the increase in mass we may write +\[ +\Delta m = m - m_0, +\] +and for the increase in energy we have the expression for kinetic energy +given in equation~(83), giving us +\[ +\Delta E = c^2(m-m_0), +\] +or, combining with the previous equation, +\[ +\Delta E = c^2 \Delta m. +\Tag{85} +\] + +Thus the increase in the kinetic energy of a particle always bears +the same definite ratio (the square of the velocity of light) to its +increase in mass. Furthermore, when a moving particle is brought +to rest and thus loses both its kinetic energy and its extra (``kinetic'') +mass, there seems to be every reason for believing that this mass +and energy which are associated together when the particle is in +motion and leave the particle when it is brought to rest will still +remain always associated together. For example, if the particle is +brought to rest by collision with another particle, it is an evident +%% -----File: 098.png---Folio 84------- +consequence of our considerations that the energy and the mass +corresponding to it do remain associated together since they are both +passed on to the new particle. On the other hand, if the particle +is brought to rest by the action of a conservative force, say for example +that exerted by an elastic spring, the kinetic energy which leaves the +particle will be transformed into the potential energy of the stretched +spring, and since the mass which has undoubtedly left the particle +must still be in existence, we shall believe that this mass is now associated +with the potential energy of the stretched spring. + +\Paragraph{76.} Such considerations have led us to believe that matter and +energy may be best regarded as different names for the same fundamental +entity: \emph{matter}, the name which has been applied when we +have been interested in the property of mass or inertia possessed +by the entity, and \emph{energy}, the name applied when we have been +interested in the part taken by the entity in the production of motion +and other changes in the physical universe. We shall find these +ideas as to the relations between matter, energy and mass very fruitful +in the simplification of physical reasoning, not only because it +identifies the two laws of the conservation of mass and the conservation +of energy, but also for its frequent application in the solution +of specific problems. + +\Paragraph{77.} We must call attention to the great difference in size between +the two units, the gram and the erg, both of which are used for the +measurement of the one fundamental entity, call it matter or energy +as we please. Equation~(85) gives us the relation +\[ +E = c^2 m, +\Tag{86} +\] +where $E$~is expressed in ergs and $m$~in grams; hence, taking the velocity +of light as $3 × 10^{10}$~centimeters per second, we shall have +\[ +1\text{ gram} = 9 × 10^{20}\text{ ergs}. +\Tag{87} +\] +The enormous number of ergs necessary for increasing the mass of +a system to the amount of a single gram makes it evident that experimental +proofs of the relation between mass and energy will be hard to +find, and outside of the experimental work on electrons at high velocities, +already mentioned in \Secref{64} and the well-known relations +%% -----File: 099.png---Folio 85------- +between the energy and momentum of a beam of light, such evidence +has not yet been forthcoming. + +As to the possibility of obtaining further direct experimental +evidence of the relation between mass and energy, we certainly cannot +look towards thermal experiments with any degree of confidence, +since even on cooling a body down to the absolute zero of temperature +it loses but an inappreciable fraction of its mass at ordinary temperatures.\footnote + {It should be noticed that our theory points to the presence of enormous + stores of interatomic energy which are still left in substances cooled to the absolute + zero.} +In the case of some radioactive processes, however, we may +find a transfer of energy large enough to bring about measurable +differences in mass. And making use of this point of view we might +account for the lack of exact relations between the atomic weights of +the successive products of radioactive decomposition.\footnote + {See, for example, Comstock, \textit{Philosophical Magazine}, vol.~15, p.~1 (1908).} + +\Subsubsection{78}{Application to a Specific Problem.} We may show an interesting +application of our ideas as to the relation between mass and +energy, in the treatment of a specific problem. Consider, just as in +\Secref{63}, two elastic particles both of which have the mass~$m_0$ at +rest, one moving in the $X$~direction with the velocity~$+u$ and the +other with the velocity~$-u$, in such a way that a head-on collision +between the particles will occur and they will rebound over their +original paths with the respective velocities $-u$~and~$+u$ of the +same magnitude as before. + +Let us now consider how this collision would appear to an observer +who is moving past the original system of coördinates with the velocity~$V$ +in the $X$~direction. To this new observer the particles will be +moving before the collision with the respective velocities +\[ +u_1 = \frac{ u - V}{1 - \smfrac{uV}{c^2}}\qquad\text{and}\qquad +u_2 = \frac{-u - V}{1 - \smfrac{uV}{c^2}}, +\Tag{88} +\] +as given by equation~(14) for the transformation of velocities. Furthermore, +when in the course of the collision the particles have come +to relative rest they will obviously be moving past our observer with +the velocity~$-V$. +%% -----File: 100.png---Folio 86------- + +Let us see what the masses of the particles will be both before and +during the collision. Before the collision, the mass of the first particle +will be +\[ +\frac{m_0}{\sqrt{1 - \smfrac{{u_1}^2}{c^2}}} = +\frac{m_0}{\sqrt{1 - \smfrac{\Biggl[\smfrac{ u - V}{1 - \tfrac{uV}{c^2}}\Biggr]^2}{c^2}}} = +\frac{m_0 \left(1 - \smfrac{uV}{c^2}\right)} + {\sqrt{\left(1 - \smfrac{V^2}{c^2}\right)\left(1 - \smfrac{u^2}{c^2}\right)}} +\] +and the mass of the second particle will be +\[ +\frac{m_0}{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}} = +\frac{m_0}{\sqrt{1 - \smfrac{\Biggl[\smfrac{-u - V}{1 + \tfrac{uV}{c^2}}\Biggr]^2}{c^2}}} = +\frac{m_0 \left(1 + \smfrac{uV}{c^2}\right)} + {\sqrt{\left(1 - \smfrac{V^2}{c^2}\right)\left(1 - \smfrac{u^2}{c^2}\right)}}. +\] +Adding these two expressions, we obtain for the sum of the masses of +the two particles before collision, +\[ +\frac{2m_0}{\sqrt{\left(1 - \smfrac{V^2}{c^2}\right) + \left(1 - \smfrac{u^2}{c^2}\right)}}. +\] + +Now during the collision, when the two particles have come to +relative rest, they will evidently both be moving past our observer +with the velocity~$-V$ and hence the sum of their masses at the +instant of relative rest would appear to be +\[ +\frac{2m_0}{\sqrt{1 - \smfrac{V^2}{c^2}}}, +\] +a quantity which is smaller than that which we have just found for +the sum of the two masses before the collision occurred. This apparent +discrepancy between the total mass of the system before and during +the collision, is removed, however, if we realize that when the particles +%% -----File: 101.png---Folio 87------- +have come to relative rest an amount of potential energy of +elastic deformation has been produced, which is just sufficient to restore +them to their original velocities, and the mass corresponding to +this potential energy will evidently be just sufficient to make the +total mass of the system the same as before collision. + +In the following chapter on the dynamics of a system of particles +we shall make further use of our ideas as to the mass corresponding +to potential energy. +%% -----File: 102.png---Folio 88------- + + +\Chapter{VII}{The Dynamics of a System of Particles.} +\SetRunningHeads{Chapter Seven.}{Dynamics of a System of Particles.} + +\Paragraph{79.} In the \Chapnumref[VI]{preceding chapter} we discussed the laws of motion +of a particle. With the help of those laws we shall now derive some +useful general dynamical principles which describe the motions of a +system of particles, and in the \Chapnumref[VIII]{following chapter} shall consider an +application of some of these principles to the kinetic theory of gases. + +The general dynamical principles which we shall present in this +chapter will be similar \emph{in form} to principles which are already familiar +in the classical Newtonian mechanics. Thus we shall deduce principles +corresponding to the principles of the conservation of momentum, +of the conservation of moment of momentum, of least action and of +\textit{vis~viva}, as well as the equations of motion in the Lagrangian and +Hamiltonian (canonical) forms. For cases where the velocities of all +the particles involved are slow compared with that of light, we shall +find, moreover, that our principles become identical in content, as +well as in form, with the corresponding principles of the classical +mechanics. Where high velocities are involved, however, our new +principles will differ from those of Newtonian mechanics. In particular +we shall find among other differences that in the case of high +velocities it will no longer be possible to define the Lagrangian function +as the difference between the kinetic and potential energies of the +system, nor to define the generalized momenta used in the Hamiltonian +equations as the partial differential of the kinetic energy with +respect to the generalized velocity. + + +\Subsection{On the Nature of a System of Particles.} + +\Paragraph{80.} Our purpose in this chapter is to treat dynamical systems +consisting of a finite number of particles, each obeying the equation +of motion which we have already written in the forms, +\begin{gather*} +\vc{F} + = \frac{d}{dt}(m\vc{u}) + = m\, \frac{d\vc{u}}{dt} + \frac{dm}{dt}\, \vc{u}, +\Tag{47} \displaybreak[0] \\ +\vc{F} + = \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c}}}\,\vc{u}\Biggr] + = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c}}}\, \frac{d\vc{u}}{dt} + + \frac{d}{dt}\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c}}}\Biggr] \vc{u}. +\Tag{59} +\end{gather*} +%% -----File: 103.png---Folio 89------- + +It is not to be supposed, however, that the total mass of such a +system can be taken as located solely in these particles. It is evident +rather, since potential energy has mass, that there will in general be +mass distributed more or less continuously throughout the space in +the neighborhood of the particles. Indeed we have shown at the +end of the preceding chapter\DPnote{** TN: Not a useful cross-ref} (\Secref{78}) that unless we take account +of the mass corresponding to potential energy we can not maintain +the principle of the conservation of mass, and we should also find it +impossible to retain the principle of the conservation of momentum +unless we included the momentum corresponding to potential energy. + +For a continuous distribution of mass we may write for the force +acting at any point on the material in a small volume,~$\delta V$, +\[ +\vc{f}\, \delta V = \frac{d}{dt} (\vc{g}\, \delta V), +\Tag{47\textit{A}} +\] +where $\vc{f}$ is the force per unit volume and $\vc{g}$~is the density of momentum. +This equation is of course merely an equation of definition for the +intensity of force at a point. We shall assume, however, that Newton's +third law, that is, the principle of the equality of action and +reaction, holds for forces of this type as well as for those acting on +particles. In later chapters we shall investigate the way in which $\vc{g}$~depends +on velocity, state of strain, etc., but for the purposes of this +chapter we shall not need any further information as to the nature +of the distributed momentum. + +Let us proceed to the solution of our specific problems. + + +\Subsection{The Conservation of Momentum.} + +\Paragraph{81.} We may first show from Newton's third law of motion that +the momentum of an isolated system of particles remains constant. + +Considering a system of particles of masses $m_1$,~$m_2$, $m_3$,~etc., we +may write in accordance with equation~\DPtypo{47}{(47)}, +\[ +\begin{aligned} +\vc{F}_1 + \vc{I}_1 &= \frac{d}{dt} (m_1 \vc{u}_1), \\ +\vc{F}_2 + \vc{I}_2 &= \frac{d}{dt} (m_2 \vc{u}_2), \\ +\text{etc.,}\quad & +\end{aligned} +\Tag{89} +\] +%% -----File: 104.png---Folio 90------- +where $\vc{F}_1$,~$\vc{F}_2$,~etc., are the external forces impressed on the individual +particles from outside the system and $\vc{I}_1$,~$\vc{I}_2$,~etc., are the internal +forces arising from mutual reactions within the interior of the system. +Considering the distributed mass in the system, we may also write, +in accordance with~(47\textit{A}) the further equation +\[ +(\vc{f} + \vc{i})\, \delta V = \frac{d}{dt}(\vc{g}\, \delta V), +\Tag{90} +\] +where $\vc{f}$~and~$\vc{i}$ are respectively the external and internal forces acting +\emph{per unit volume} of the distributed mass. Integrating throughout the +whole volume of the system~$V$ we have +\[ +\int (\vc{f} + \vc{i})\, dV = \frac{d\vc{G}}{dt}, +\Tag{91} +\] +where $\vc{G}$ is the total distributed momentum in the system. Adding +this to our previous equations~(89) for the forces acting on the individual +particles, we have +\[ +%[** TN: \textstyle \sum in original] +\Sum \vc{F}_1 + \Sum \vc{I}_1 + \int \vc{f}\, dV + \int \vc{i}\, dV + = \frac{d}{dt} \Sum m_1 u_1 + \frac{d\vc{G}}{dt}. +\] + +But from Newton's third law of motion (\ie, the principle of the +equality of action and reaction) it is evident that the sum of the +internal forces, $\Sum \vc{I}_1 + \int \vc{i}\, dV$, which arise from mutual reactions within +the system must be equal to zero, which leads to the desired equation +of momentum +\[ +\Sum \vc{F}_1 + \int \vc{f}\, dv = \frac{d}{dt}(\Sum m_1 u_1 + \vc{G}). +\Tag{92} +\] + +In words this equation states that at any given instant the vector +sum of the external forces acting on the system is equal to the rate +at which the total momentum of the system is changing. + +For the particular case of an isolated system there are no external +forces and our equation becomes a statement of the principle of the +\emph{conservation of momentum}. + + +\Subsection{The Equation of Angular Momentum.} + +\Paragraph{82.} We may next obtain an equation for the moment of momentum +of a system about a point. +%% -----File: 105.png---Folio 91------- +Consider a particle of mass~$m_1$ and velocity~$u_1$. Let $\vc{r}_1$~be the +radius vector from any given point of reference to the particle. Then +for the moment of momentum of the particle about the point we may +write +\[ +\vc{M}_1 = \vc{r}_1 × m_1\vc{u}_1, +\] +and summing up for all the particles of the system we may write +\[ +\Sum \vc{M}_1 = \Sum (\vc{r}_1 × m_1\vc{u}_1). +\Tag{93} +\] +Similarly, for the moment of momentum of the \emph{distributed mass} we +may write +\[ +\vc{M}_{\text{dist.}} = \int (\vc{r} × \vc{g})\, dV, +\Tag{94} +\] +where $\vc{r}$ is the radius vector from our chosen point of reference to a +point in space where the density of momentum is~$\vc{g}$ and the integration +is to be taken throughout the whole volume,~$V$, of the system. + +Adding these two equations (93)~and~(94), we obtain for the total +amount of momentum of the system about our chosen point +\[ +\vc{M} = \Sum(\vc{r}_1 × m_1\vc{u}_1) + \int (\vc{r} × \vc{g})\, dV; +\] +and differentiating with respect to the time we have, for the rate of +change of the moment of momentum, +\begin{multline*} +\frac{d\vc{M}}{dt} + = \Sum \left\{\vc{r}_1 × \frac{d}{dt}(m_1\vc{u}_1)\right\} + + \Sum \left(\frac{d\vc{r}_1}{dt} × m_1\vc{u}_1\right) \\ + + \int \left(\vc{r} × \frac{d\vc{g}}{dt} \right) dV + + \int \left(\frac{d\vc{r}}{dt} × \vc{g} \right) dV; +\end{multline*} +or, making the substitutions given by equations (89)~and~(90), and +writing $\dfrac{d\vc{r}_1}{dt} = \vc{u}_1$, etc.\DPtypo{}{,} we have +\begin{multline*} +\frac{d\vc{M}}{dt} + = \Sum (\vc{r}_1 × \vc{F}_1) + \Sum (\vc{r}_1 × \vc{I}_1) + + \Sum (\vc{u}_1 × m_1\vc{u}_1) \\ + + \int (\vc{r} × \vc{f})\, dV + \int (\vc{r} × \vc{i})\, dV + + \int (\vc{u} × \vc{g})\, dV. +\end{multline*} +To simplify this equation we may note that the third term is equal to +zero because it contains the outer product of a vector by itself. Furthermore, +if we accept the principle of the equality of action and +%% -----File: 106.png---Folio 92------- +reaction, together with the further requirement that forces are not +only equal and opposite but that their points of application be in the +same straight line, we may put the moment of all the internal forces +equal to zero and thus eliminate the second and fifth terms. We +obtain as the equation of angular momentum +\[ +\frac{d\vc{M}}{dt} = \Sum(\vc{r}_1 × \vc{F}_1) + + \int (\vc{r} × \vc{f})\, dV + \int (\vc{u} × \vc{g})\, dV. +\Tag{95} +\] + +We may call attention to the inclusion in this equation of the +interesting term $\int(\vc{u} × \vc{g})\, dV$. If density of momentum and velocity +should always be in the same direction this term would vanish, since +the outer product of a vector by itself is equal to zero. In our consideration +of the ``Dynamics of Elastic Bodies,'' however, we shall +find bodies with a component of momentum at right angles to their +direction of motion and hence must include this term in a general +treatment. For a completely isolated system it can be shown, however, +that this term vanishes along with the external forces and we +then have the principle of the \emph{conservation of moment of momentum.} + + +\Subsection{The Function $T$.} + +\Paragraph{83.} We may now proceed to the definition of a function which +will be needed in our treatment of the principle of least action. + +One of the most valuable properties of the Newtonian expression, +$\frac{1}{2}m_0u^2$, for kinetic energy was the fact that its derivative with respect +to velocity is evidently the Newtonian expression for momentum,~$m_0u$. +It is not true, however, that the derivative of our new expression +for kinetic energy (see \Secref{73}), $m_0c^2 \Biggl[\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} - 1\Biggr]$, with respect +to velocity is equal to momentum, and for that reason in our non-Newtonian +mechanics we shall find it desirable to define a new function,~$T$, +by the equation, +\[ +T = m_0c^2\left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right). +\Tag{96} +\] + +For slow velocities (\ie, small values of~$u$) this reduces to the +Newtonian expression for kinetic energy and at all velocities we have +%% -----File: 107.png---Folio 93------- +the relation, +\[ +\frac{dT}{du} + = -m_0 c^2\, \frac{d}{du} \sqrt{1 - \frac{u^2}{c^2}} + = \frac{m_0u}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = mu, +\Tag{97} +\] +showing that the differential of~$T$ with respect to velocity is momentum. + +For a system of particles we shall define~$T$ as the summation of +the values for the individual particles: +\[ +T = \Sum m_0 c^2 \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right). +\Tag{98} +\] + + +\Subsection{The Modified Lagrangian Function.} + +\Paragraph{84.} In the older mechanics the Lagrangian function for a system +of particles was defined as the difference between the kinetic and +potential energies of the system. The value of the definition rested, +however, on the fact that the differential of the kinetic energy with +respect to velocity was equal to momentum, so that we shall now +find it advisable to define the Lagrangian function with the help of +our new function~$T$ in accordance with the equation +\[ +L = T - U. +\Tag{99} +\] + + +\Subsection{The Principle of Least Action.} + +\Paragraph{85.} We are now in a position to derive a principle corresponding +to that of least action in the older mechanics. Consider the path +by which our dynamical system actually moves from state~(1) to +state~(2). The motion of any particle in the system of mass $m$ will +be governed by the equation +\[ +\vc{F} = \frac{d}{dt} (m\vc{u}). +\Tag{100} +\] + +Let us now compare the actual path by which the system moves +from state~(1) to state~(2) with a slightly displaced path in which the +laws of motion are not obeyed, and let the displacement of the particle +at the instant in question be~$\delta \vc{r}$. + +Let us take the inner product of both sides of equation~(100) with~$\delta \vc{r}$; +%% -----File: 108.png---Folio 94------- +we have +\begin{gather*} +\begin{aligned} +\vc{F} ·\delta\vc{r} + &= \frac{d}{dt}(m \vc{u}) · \delta \vc{r} \\ + &= \frac{d}{dt}(m\vc{u} · \delta\vc{r}) + - m\vc{u} · \frac{d\, \delta\vc{r}}{dt} \\ + &= \frac{d}{dt}(m\vc{u} · \delta\vc{r}) - m\vc{u} · \delta\vc{u}) +\end{aligned} \\ +(m\vc{u} · \delta\vc{u} + \vc{F} · \delta\vc{r})\, dt + = d(m\vc{u} · \delta\vc{r}). +\end{gather*} + +Summing up for all the particles of the system and integrating +between the limits $t_1$~and~$t_2$, we have +\[ +\int_{t_1}^{t_2} \left(\Sum m\vc{u}· \delta\vc{u} + \Sum \vc{F} · \delta\vc{r}\right) dt + = \left[\Sum m\vc{u} · \delta\vc{r} \right]_{t_1}^{t_2}. +\] +Since $t_1$~and~$t_2$ are the times when the actual and displaced motions +coincide, we have at these times $\delta\vc{r} = 0$; furthermore we also have +$\vc{u} · \delta\vc{u} = u\, \delta u$, so that we may write +\[ +\int_{t_1}^{t_2}\left(\Sum mu\, \delta u + \vc{F} · \delta\vc{r}\right) dt = 0. +\] +With the help of equation~(97), however, we see that $\Sum mu\, \delta u = \delta T$, +giving us +\[ +\int_{t_1}^{t_2} (\delta T + \vc{F} · \delta r)\, dt = 0. +\Tag{101} +\] +\emph{If the forces~$F$ are conservative}, we may write $\vc{F} · \delta r = -\delta U$, where +$\delta U$~is the difference between the potential energies of the displaced +and the actual configurations. This gives us +\[ +\delta \int_{t_1}^{t_2} (T - U)\, dt = 0 +\] +or +\[ +\delta \int_{t_1}^{t_2} L\, dt = 0, +\Tag{102} +\] +which is the modified principle of least action. The principle evidently +requires that for the actual path by which the system goes +%% -----File: 109.png---Folio 95------- +from state~(1) to state~(2), the quantity $\ds\int_{t_1}^{t_2} L\, dt$ shall be a minimum (or +maximum). + + +\Subsection{Lagrange's Equations.} + +\Paragraph{86.} We may now derive the Lagrangian equations of motion from +the above principle of least action. Let us suppose that the position +of each particle of the system under consideration is completely determined +by $n$~\emph{independent} generalized coördinates $\phi_{1}$,~$\phi_{2}$, $\phi_{3} \cdots \phi_{n}$ and +hence that $L$~is some function of $\phi_{1}$,~$\phi_{2}$, $\phi_{3} \cdots \phi_{n}$, $\dot{\phi}_{1}$,~$\dot{\phi}_{2}$, $\dot{\phi}_{3} \cdots \dot{\phi}_{n}$, +where for simplicity we have put $\dot{\phi}_{1} = \dfrac{d\phi_1}{dt}$, $\dot{\phi}_{2} = \dfrac{d\phi_2}{dt}$,~etc. + +%%%% Use of "1" as a subscript in the original starts here %%%% +From equation~(102) we have +\[ +\int_{t_1}^{t_2} (\delta L)\, dt = \int_{t_1}^{t_2} \left( + \Sum_1^n \frac{\partial L}{\partial\phi_{\1}}\, \delta\phi_{\1} + + \Sum_1^n \frac{\partial L}{\partial\dot{\phi}_{\1}}\, \delta\dot{\phi}_{\1} + \right)dt = 0. +\Tag{103} +\] +But +\[ +\delta\dot{\phi}_{\1} = \frac{d}{dt}(\delta\phi_{\1})\DPchg{}{,} +\] +which gives us +\begin{align*} +\int_{t_1}^{t_2} \frac{\partial L}{\partial\dot{\phi}_{\1}}\, + \delta\dot{\phi}_{\1}\, dt + &= \int_{t_1}^{t_2} \frac{\partial L}{\partial\dot{\phi}_{\1}}\, + \frac{d}{dt}(\delta\phi_{\1})\, dt \\ + &= \left[\frac{\partial L}{\partial\dot{\phi}_{\1}}\, \delta\phi_{\1}\right]_{t_1}^{t_2} + - \int_{t_1}^{t_2} \delta\phi_{\1}\, + \frac{d}{dt} \left(\frac{\partial L}{\partial\dot{\phi}_{\1}}\right) dt +\end{align*} +or, since at times $t_{1}$~and~$t_{2}$, $\delta \phi_{\1}$~is zero, the first term in this expression +disappears and on substituting in equation~(103) we obtain +\[ +\int_{t_1}^{t2} \left[\Sum_{1}^{n} \delta \phi_{\1} + \left\{ \frac{\partial L}{\partial\phi_{\1}} + - \frac{d}{dt} \left(\frac{\partial L}{\partial\dot{\phi}_{\1}}\right) + \right\} \right] dt = 0. +\] +Since, however, the limits $t_{1}$~and~$t_{2}$ are entirely at our disposal we must +have at every instant +\[ +\Sum_{1}^{n} \delta \phi_{\1} + \left\{ \frac{\partial L}{\partial\phi_{\1}} + - \frac{d}{dt} \left(\frac{\partial L}{\partial\dot{\phi}_{\1}}\right) + \right\} = 0. +\] +Finally, moreover, since the $\phi$'s~are independent parameters, we can +assign perfectly arbitrary values to $\delta\phi_{1}$,~$\delta\phi_{2}$,~etc., and hence must have +%% -----File: 110.png---Folio 96------- +the series of equations +\[ +\begin{aligned} +&\frac{d}{dt} \left( \frac{\partial L}{\partial\dot{\phi}_1} \right) + - \frac{\partial L}{\partial\phi_1} = 0, \\ +&\frac{d}{dt} \left( \frac{\partial L}{\partial\dot{\phi}_2} \right) + - \frac{\partial L}{\partial\phi_2} = 0, \\ +&\text{etc.} +\end{aligned} +\Tag{104} +\] +These correspond to Lagrange's equations in the older mechanics, +differing only in the definition of~$L$. + + +\Subsection{Equations of Motion in the Hamiltonian Form.} + +\Paragraph{87.} We shall also find it desirable to obtain equations of motion +in the Hamiltonian or canonical form. + +Let us define the \emph{generalized momentum}~$\psi_{\1}$ corresponding to the +coördinate~$\phi_{\1}$ by the equation, +\[ +\psi_{\1} = \frac{\partial T}{\partial\dot{\phi}_{\1}}. +\Tag{105} +\] + +It should be noted that the generalized momentum is not as in +ordinary mechanics the derivative of the kinetic energy with respect +to the generalized velocity but approaches that value at low velocities. + +Consider now a function~$T'$ defined by the equation +\[ +T' = \psi_1\dot{\phi}_1 + \psi_{2}\dot{\phi}_2 + \cdots - T. +\Tag{106} +\] +Differentiating we have +\begin{align*} +dT' &= \psi_1\, d\dot{\phi}_1 + \psi_2\, d\dot{\phi}_2 + \cdots \\ + &\quad+ \dot{\phi}_1\, d\psi_{1} + \dot{\phi}_2\, d\psi_{2} + \cdots \\ + &\quad- \frac{\partial T}{\partial\phi_1}\, d\phi_1 + - \frac{\partial T}{\partial\phi_2}\, d\phi_2 - \cdots \\ + &\quad- \frac{\partial T}{\partial\dot{\phi}_1}\, d\dot{\phi}_1 + - \frac{\partial T}{\partial\dot{\phi}_2}\, d\dot{\phi}_2 - \cdots, +\end{align*} +and this, by the introduction of~(105), becomes +\[ +dT' = \dot{\phi}_1\, d\psi_1 + \dot{\phi}_2\, d\psi_{2} + \cdots + - \frac{\partial T}{\partial\phi_1}\, d\phi_1 + - \frac{\partial T}{\partial\phi_2}\, d\phi_2 - \cdots. +\Tag{107} +\] +%% -----File: 111.png---Folio 97------- +Examining this equation we have +\begin{align*} +\frac{\partial T'}{\partial\phi_{\1}} + &= - \frac{\partial T}{\partial\phi_{\1}}, +\Tag{108} \\ +\frac{\partial T'}{\partial\psi_{\1}} + & = \dot{\phi}_{\1}. +\Tag{109} +\end{align*} +In Lagrange's equations we have +\[ +\frac{d}{dt}\left\{ \frac{\partial}{\partial\dot{\phi}_{\1}}(T - U)\right\} + - \frac{\partial}{\partial\phi_{\1}}(T - U) = 0. +\] +But since $U$ is independent of~$\psi_{\1}$ we may write +\[ +\frac{\partial(T - U)}{\partial\dot{\phi}_{\1}} + = \frac{\partial T}{\partial\dot{\phi}_{\1}} = \psi_{\1}, +\] +and furthermore by~(108), +\[ +\frac{\partial T}{\partial\phi_{\1}} = -\frac{\partial T'}{\partial\phi_{\1}}. +\] +Substituting these two expressions in Lagrange's equations we obtain +\[ +\frac{d\psi_{\1}}{dt} = -\frac{\partial(T' + U)}{\partial\phi_{\1}} +\] +or, writing $T' + U = E$, we have +\[ +\frac{d\psi_{\1}}{dt} = -\frac{\partial E}{\partial\phi_{\1}} +\Tag{110} +\] +and since $U$~is independent of~$\psi_{\1}$ we may rewrite equation~(109) in +the form +\[ +\frac{d\phi_{\1}}{dt} = \frac{\partial E}{\partial\psi_{\1}}. +\Tag{111} +\] + +The set of equations corresponding to (110)~and~(111) for all the +coördinates $\phi_{1}$,~$\phi_{2}$, $\phi_{3}, \cdots \phi_{n}$ and the momenta $\psi_{1}$,~$\psi_{2}$, $\psi_{3}, \cdots \psi_{n}$ are +the desired equations of motion in the canonical form. + +\Subsubsection{88}{Value of the Function $T'$.} We have given the symbol~$E$ to +the quantity $T' + U$, since $T'$~actually turns out to be identical with +%% -----File: 112.png---Folio 98------- +the expression by which we defined kinetic energy, thus making +$E = T' + U$ the sum of the kinetic and potential energies of the +system. + +To show that $T'$~is equal to~$K$, the kinetic energy, we have by the +equation of definition~(106) +\begin{align*} +T' &= \phi_1\psi_1 + \phi_2\psi_2 + \cdots - T, \\ + &= \phi_1\, \frac{\partial T}{\partial\dot{\phi}_1} + + \phi_2\, \frac{\partial T}{\partial\dot{\phi}_2} + \cdots - T. +\end{align*} +But $T$ by definition, equation~(98), is +\begin{align*} +T &= \Sum c^2m_0 \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right), \\ +\intertext{which gives us} +\frac{\partial T}{\partial\dot{\phi}_{\1}} + &= \Sum m_0 \left(1 - \frac{u^2}{c^2}\right)^{-1/2} + u\, \frac{\partial u}{\partial\dot{\phi}_{\1}} \\ + &= \Sum mu\, \frac{\partial u}{\partial\dot{\phi}_{\1}} +\end{align*} +and substituting we obtain +\[ +\begin{aligned} +T' &= \dot{\phi}_1 \Sum mu\, \frac{\partial u}{\partial\dot{\phi}_1} + + \dot{\phi}_2 \Sum mu\, \frac{\partial u}{\partial\dot{\phi}_2} + + \cdots - T \\ + &= \Sum mu \left\{ + \dot{\phi}_1\, \frac{\partial u}{\partial\dot{\phi}_1} + + \dot{\phi}_2\, \frac{\partial u}{\partial\dot{\phi}_2} + + \cdots \right\} - T. +\end{aligned} +\Tag{112} +\] +We can show, however, that the term in parenthesis is equal to~$u$. +If the \DPchg{coordinates}{coördinates} $x$,~$y$,~$z$ determine the position of the particle in +question, we have, +\begin{align*} +x &= f(\phi_1\phi_2\phi_3 \cdots \phi_n), \\ +\dot{x} = \frac{dx}{dt} + &= \dot{\phi}_1\, \frac{\partial f(\:)}{\partial\phi_1} + + \dot{\phi}_2\, \frac{\partial f(\:)}{\partial\phi_2} + + \dot{\phi}_3\, \frac{\partial f(\:)}{\partial\phi_3} + \cdots +\end{align*} +and differentiating with respect to the~$\dot{\phi}$'s, we obtain, +\[ +\frac{\partial\dot{x}}{\partial\dot{\phi}_1} + = \frac{\partial f(\:)}{\partial\phi_1} + = \frac{\partial x}{\partial\phi_1}, \quad +\frac{\partial\dot{x}}{\partial\dot{\phi}_2} + = \frac{\partial x}{\partial\phi_2}, \quad +\frac{\partial\dot{x}}{\partial\dot{\phi}_3} + = \frac{\partial x}{\partial\phi_3}, \quad \text{etc.}\DPtypo{,}{} +\] +%% -----File: 113.png---Folio 99------- +Similarly +\begin{alignat*}{3} +\frac{\partial\dot{y}}{\partial\dot{\phi}_1} + &= \frac{\partial y}{\partial\phi_1}, +&\qquad +\frac{\partial\dot{y}}{\partial\dot{\phi}_2} + &= \frac{\partial y}{\partial\phi_2}, &\qquad \text{etc.}, \\ +\frac{\partial\dot{z}}{\partial\dot{\phi}_1} + &= \frac{\partial z}{\partial\phi_1}, +&\qquad +\frac{\partial\dot{z}}{\partial\dot{\phi}_2} + &= \frac{\partial z}{\partial\phi_2}, &\qquad \text{etc.}, +\end{alignat*} +Let us write now +\begin{align*} +u &= \sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}, \\ +\frac{\partial u}{\partial\dot{\phi}_{\1}} + &= \frac{1}{\sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}} + \left(\dot{x}\, \frac{\partial\dot{x}}{\partial\dot{\phi}_{\1}} + + \dot{y}\, \frac{\partial\dot{y}}{\partial\dot{\phi}_{\1}} + + \dot{z}\, \frac{\partial\dot{z}}{\partial\dot{\phi}_{\1}}\right), +\end{align*} +or making the substitutions for $\dfrac{\partial\dot{x}}{\partial\dot{\phi}_{\1}}$, $\dfrac{\partial\dot{y}}{\partial\dot{\phi}_{\1}}$, etc., given above, we have, +\[ +\frac{\partial u}{\partial\dot{\phi}_{\1}} + = \frac{1}{u} + \left(\dot{x}\, \frac{\partial x}{\partial\phi_{\1}} + + \dot{y}\, \frac{\partial y}{\partial\phi_{\1}} + + \dot{z}\, \frac{\partial z}{\partial\phi_{\1}}\right). +\] +%%%% Use of "1" as a subscript in the original ends here %%%% +Substituting now in~(112) we shall obtain, +{\footnotesize% +\begin{align*} +T'& = \Sum mu +\begin{aligned}[t] +\Biggl\{\frac{\dot{x}}{u} + \left(\phi_1\, \frac{\partial x}{\partial\phi_1} + + \phi_2\, \frac{\partial x}{\partial\phi_2} + \cdots \right) + &+ \frac{\dot{y}}{u} + \left(\phi_1\, \frac{\partial y}{\partial\phi_1} + + \phi_2\, \frac{\partial y}{\partial\phi_2} + \cdots \right) \\ + &+ \frac{\dot{z}}{u} + \left(\phi_1\, \frac{\partial z}{\partial\phi_1} + + \phi_2\, \frac{\partial z}{\partial\phi_2} + \cdots \right) + \Biggr\} - T +\end{aligned} \\ + &= \Sum mu^2 - T +\end{align*}}% +or, introducing the value of~$T$ given by equation~(98), we have +\begin{align*} +T' &= \Sum \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} + \left\{u^2 - c^2 \sqrt{1 - \frac{u^2}{c^2}} + + c^2 \left(1 - \frac{u^2}{c^2}\right)\right\} \\ + &= \Sum c^2(m - m_0), +\end{align*} +which is the expression~(83) for kinetic energy. + +Hence we see that the Hamiltonian function $E = T' + U$ is the +sum of the kinetic and potential energies of the system as in Newtonian +mechanics. + + +\Subsection{The Principle of the Conservation of Energy.} + +\Paragraph{89.} We may now make use of our equations of motion in the +canonical form to show that the total energy of a system of interacting +%% -----File: 114.png---Folio 100------- +particles remains constant. If such were not the case it is obvious +that our definitions of potential and kinetic energy would not be +very useful. + +Since $E = T' + U$ is a function of $\phi_1$,~$\phi_2$, $\phi_3, \cdots$ $\psi_1$,~$\psi_2$, $\psi_3, \cdots$, we +may write +\begin{align*} +\frac{dE}{dt} + &= \frac{\partial E}{\partial\phi_1}\, \dot{\phi}_1 + + \frac{\partial E}{\partial\phi_2}\, \dot{\phi}_2 + \cdots \\ + &\quad + + \frac{\partial E}{\partial\psi_1}\, \dot{\psi}_1 + + \frac{\partial E}{\partial\psi_2}\, \dot{\psi}_2 + \cdots. +\end{align*} +Substituting the values of $\dfrac{\partial E}{\partial\phi_1}$, $\dfrac{\partial E}{\partial\psi_1}$, etc., given by the canonical +equations of motion (110)~and~(111), we have +\begin{align*} +\frac{dE}{dt} + &= -\dot{\psi}_1\dot{\phi}_1 - \dot{\psi}_2\dot{\phi}_2 - \cdots \\ + &\quad + + \dot{\psi}_1\dot{\phi}_1 + \dot{\psi}_2\dot{\phi}_2 + \cdots \\ + &= 0, +\end{align*} +which gives us the desired proof that just as in the older Newtonian +mechanics the total energy of an isolated system of particles is a +conservative quantity. + + +\Subsection{On the Location of Energy in Space.} + +\Paragraph{90.} This proof of the conservation of energy in a system of interacting +particles justifies us in the belief that the concept of energy +will not fail to retain in the newer mechanics the position of great +importance which it gradually acquired in the older systems of physical +theory. Indeed, our newer considerations have augmented the +important rôle of energy by adding to its properties the attribute of +mass or inertia, and thus leading to the further belief that matter +and energy are in reality different names for the same fundamental +entity. + +The importance of this entity, energy, makes it very interesting +to consider the possibility of ascribing a definite location in space to +any given quantity of energy. In the older mechanics we had a +hazy notion that the kinetic energy of a moving body was probably +located in some way in the moving body itself, and possibly a vague +%% -----File: 115.png---Folio 101------- +idea that the potential energy of a raised weight might be located in +the space between the weight and the earth. Our discovery of the +relation between mass and energy has made it possible, however, to +give a much more definite, although not a complete, answer to inquiries +of this kind. + +In our discussions of the dynamics of a particle (Chapter~VI, %[** TN: Not a useful cross-reference] +\Secref{61}) we saw that an acceptance of Newton's principle of the +equality of action and reaction forced us to ascribe an increased mass +to a moving particle over that which it has at rest. This increase in +the mass of the moving particle is necessarily located either in the +particle itself or distributed in the surrounding space in such a way +that its center of mass always coincides with the position of the +particle, and since the kinetic energy of the particle is the energy +corresponding to this increased mass we may say that \emph{the kinetic energy +of a moving particle is so distributed in space that its center of mass +always coincides with the position of the particle}. + +If now we consider the transformation of kinetic energy into +potential energy we can also draw somewhat definite conclusions as to +the location of potential energy. By the principle of the conservation +of mass we shall be able to say that the mass of any potential +energy formed is just equal to the ``kinetic'' mass which has disappeared, +and by the principle of the conservation of momentum we +can say that the velocity of this potential energy is just that necessary +to keep the total momentum of the system constant. Such considerations +will often permit us to reach a good idea as to the location +of potential energy. + +Consider, for example, a pair of similar attracting particles which +are moving apart from each other with the velocities $+u$~and~$-u$ +and are gradually coming to rest under the action of their mutual +attraction, their kinetic energy thus being gradually changed into +potential energy. Since the total momentum of the system must +always remain zero, we may think of the potential energy which is +formed as left stationary in the space between the two particles. +%% -----File: 116.png---Folio 102------- + + +\Chapter{VIII}{The Chaotic Motion of a System of Particles.} +\SetRunningHeads{Chapter Eight.}{Chaotic Motion of a System of Particles.} + +The discussions of the \Chapnumref[VII]{previous chapter} have placed at our disposal +generalized equations of motion for a system of particles similar in +form to those familiar in the classical mechanics, and differing only +in the definition of the Lagrangian function. With the help of these +equations it is possible to carry out investigations parallel to those +already developed in the classical mechanics, and in the present +chapter we shall discuss the chaotic motion of a system of particles. +This problem has received much attention in the classical mechanics +because of the close relations between the theoretical behavior of +such an ideal system of particles and the actual behavior of a monatomic +gas. We shall find no more difficulty in handling the problem +than was experienced in the older mechanics, and our results will of +course reduce to those of Newtonian mechanics in the case of slow +velocities. Thus we shall find a distribution law for momenta which +reduces to that of Maxwell for slow velocities, and an equipartition +law for the average value of a function which at low velocities becomes +identical with the kinetic energy of the particles. + +\Subsubsection{91}{The Equations of Motion.} It has been shown that the Hamiltonian +equations of motion +\[ +\begin{aligned} +&\frac{\partial E}{\partial\phi_1} = -\frac{d\psi_1}{dt} = -\dot{\psi}_1, \\ +&\frac{\partial E}{\partial\psi_1} = \frac{d\phi_1}{dt} = \dot{\phi}_1, \\ +&\text{etc.}, +\end{aligned} +\Tag{113} +\] +will hold in relativity mechanics provided we define the generalized +momenta $\psi_1$,~$\psi_2$,~etc., \emph{not} as the differential of the kinetic energy +with respect to the generalized velocities $\dot{\phi}_1$,~$\dot{\phi}_2$,~etc., but as the differential +with respect to $\dot{\phi}_1$,~$\dot{\phi}_2$,~etc., of a function +\[ +T = \Sum m_0c^2 \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right), +\] +%% -----File: 117.png---Folio 103------- +where $m_0$~is the mass of a particle having the velocity~$u$ and the summation~$\Sum$ +extends over all the particles of the system. + +\Subsubsection{92}{Representation in Generalized Space.} Consider now a system +defined by the $n$~generalized coördinates $\phi_1$,~$\phi_2$, $\phi_3, \cdots, \phi_n$, and the +corresponding momenta $\psi_1$,~$\psi_2$, $\psi_3, \cdots, \psi_n$. Employing the methods +so successfully used by Jeans,\footnote + {Jeans, \textit{The Dynamical Theory of Gases}, Cambridge, 1916.} +we may think of the state of the +system at any instant as determined by the position of a point plotted +in a $2n$-dimensional space. Suppose now we had a large number of +systems of the same structure but differing in state, then for each +system we should have at any instant a corresponding point in our +$2n$-dimensional space, and as the systems changed their state, in the +manner required by the laws of motion, the points would describe +stream lines in this space. + +\Subsubsection{93}{Liouville's Theorem.} Suppose now that the points were +originally distributed in the generalized space with the uniform +density~$\rho$. Then it can be shown by familiar methods that, just as +in the classical mechanics, the density of distribution remains uniform. + +Take, for example, some particular cubical element of our generalized +space $d\phi_1\, d\phi_2\, d\phi_3 \dots d\psi_1\, d\psi_2\, d\psi_3\dots$. The density of distribution +will evidently remain uniform if the number of points +entering any such cube per second is equal to the number leaving. +Consider now the two parallel bounding surfaces of the cube which +are perpendicular to the $\phi_1$~axis, one cutting the axis at the point~$\phi_1$ +and the other at the point~$\phi_1 + d\phi_1$. The area of each of these +surfaces is $d\phi_2\, d\phi_3\dots d\psi_1\, d\psi_2\, d\psi_3\dots$, and hence, if $\dot{\phi}_1$~is the component +of velocity which the points have parallel to the $\phi_1$~axis, and $\dfrac{\partial\dot{\phi}_1}{\partial\phi_1}$~is +the rate at which this component is changing as we move along the +axis, we may obviously write the following expression for the difference +between the number of points leaving and entering per second +through these two parallel surfaces +\[ +\rho\left[\left(\frac{\partial\dot{\phi}_1}{\partial\phi_1}\right) d\phi_1\right] +d\phi_2\, d\phi_3\, \dots d\psi_1\, d\psi_2\, d\psi_3 \cdots + = \rho\, \frac{\partial\dot{\phi}_1}{\partial\phi_1}\, dV. +\] + +Finally, considering all the pairs of parallel bounding surfaces, we +%% -----File: 118.png---Folio 104------- +find for the total decrease per second in the contents of the element +\[ +\rho\left( + \frac{\partial\dot{\phi}_1}{\partial\phi_1} + + \frac{\partial\dot{\phi}_2}{\partial\phi_2} + + \frac{\partial\dot{\phi}_3}{\partial\phi_3} + \cdots + + \frac{\partial\dot{\psi}_1}{\partial\psi_1} + + \frac{\partial\dot{\psi}_2}{\partial\psi_2} + + \frac{\partial\dot{\psi}_3}{\partial\psi_3} + \cdots\right) dV. +\] +But the motions of the points are necessarily governed by the Hamiltonian +equations~(113) given above, and these obviously lead to the +relations +\begin{align*} +& \frac{\partial\dot{\phi}_1}{\partial\phi_1} ++ \frac{\partial\dot{\psi}_1}{\partial\psi_1} = 0, \\ +& \frac{\partial\dot{\phi}_2}{\partial\phi_2} ++ \frac{\partial\dot{\psi}_2}{\partial\psi_2} = 0\DPtypo{.}{,} \\ +& \text{etc.} +\end{align*} +So that our expression for the change per second in the number of +points in the cube becomes equal to zero, the necessary requirement +for preserving uniform density. + +This maintenance of a uniform distribution means that there is +no tendency for the points to crowd into any particular region of the +generalized space, and hence if we start some one system going and +plot its state in our generalized space, we may \emph{assume} that, after an +indefinite lapse of time, the point is equally likely to be in any one of +the little elements~$dV$. \emph{In other words, the different states of a system, +which we can specify by stating the region $d\phi_1\, d\phi_2\, d\phi_3 \dots d\psi_1\, d\psi_2\, d\psi_3 \dots$ +in which the values of the \DPchg{coordinates}{coördinates} and momenta of the system fall, +are all equally likely to occur.}\footnote + {The criterion here used for determining whether or not the states are equally + liable to occur is obviously a necessary requirement, although it is not so evident + that it is a sufficient requirement for equal probability.} + +\Subsubsection{94}{A System of Particles.} Consider now a system containing $N_a$~particles +which have the mass~$m_a$ when at rest, $N_b$~particles which +have the mass~$m_b$, $N_c$~particles which have the mass~$m_c$, etc. If at +any given instant we specify the particular differential element +$dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$ which contains the coördinates $x$,~$y$,~$z$, and the +corresponding momenta $\psi_x$,~$\psi_y$,~$\psi_z$ for \emph{each} particle, we shall thereby +completely determine what Planck\footnote + {Planck, \textit{Wärmestrahlung}, Leipzig, 1913.} +has well called the \emph{microscopic} +state of the system, and by the previous paragraph any microscopic +%% -----File: 119.png---Folio 105------- +state of the system in which we thus specify the six-dimensional +position of each particle is just as likely to occur as any other microscopic +state. + +It must be noticed, however, that many of the possible microscopic +states which are determined by specifying the six-dimensional +position of each individual particle are in reality completely identical, +since if all the particles having a given mass~$m_a$ are alike among themselves, +it makes no difference which particular one of the various +available identical particles we pick out to put into a specified range +$dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$. + +For this reason we shall usually be interested in specifying the +\emph{statistical} state\footnote + {What we have here defined as the \emph{statistical} state is what Planck calls the + \emph{macroscopic} state of the system. The word macroscopic is unfortunate, however, in + implying a less minute observation as to the size of the elements $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$ +in which the representative points are found.} +of the system, for which purpose we shall merely +state the number of particles of a given kind which have coördinates +falling in a given range $dx\, dy\,dz\, d\psi_x\, d\psi_y\, d\psi_z$. We see that corresponding +to any given statistical state there will be in general a +large number of microscopic states. + +\Subsubsection{95}{Probability of a Given Statistical State.} We shall now be +particularly interested in the probability that the system of particles +will actually be in some specified \emph{statistical} state, and since Liouville's +theorem has justified our belief that all \emph{microscopic} states are +equally likely to occur, we see that the probability of a given statistical +state will be proportional to the number of microscopic states +which correspond to it. + +For the system under consideration let a particular statistical +state be specified by stating that ${N_a}'$,~${N_a}''$, ${N_a}''', \cdots$, ${N_b}'$,~${N_b}''$, ${N_b}''', \cdots$,~etc., are the number of particles of the corresponding masses +$m_a$,~$m_b$,~etc., which fall in the specified elementary regions $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$, Nos.\ $1a$,~$2a$, $3a, \cdots$, $1b$,~$2b$, $3b, \cdots$,~etc. By familiar +methods of calculation it is evident that the number of arrangements +by which the particular distribution of particles can be effected, +that is, in other words, the number of microscopic states,~$W$, which +correspond to the given statistical state, is given by the expression +\[ +%[** TN: Modernized factorial notation] +W = \frac{N_a!\, N_b!\, N_c! \cdots} + {{N_a}'!\, {N_a}''!\, {N_a}'''! \cdots + {N_b}'!\, {N_b}''!\, {N_b}'''! \cdots} +\] +%% -----File: 120.png---Folio 106------- +and this number~$W$ is proportional to the probability that the system +will be found in the particular statistical state considered. + +If now we assume that each of the regions +\[ +dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z,\ +\text{Nos.}\ 1a,\ 2a,\ 3a,\ \cdots,\ 1b,\ 2b,\ 3b,\ \cdots\ \text{etc.} +\] +is great enough to contain a large number of particles,\footnote + {The idea of successive orders of infinitesimals which permit the differential + region $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$, to contain a large number of particles is a familiar one in + mathematics.} +we may +apply the Stirling formula +\[ +N! = \sqrt{2\pi\, N} \left(\frac{N}{\epsilon}\right)^N +\] +for evaluating $N_a!$,~$N_b!$,~etc., and omitting negligible terms, shall +obtain for~$\log W$ the result +\begin{align*} +\log W &= -N_a \left( + \frac{{N_a}'}{N_a} \log\frac{{N_a}'}{N_a} + + \frac{{N_a}''}{N_a} \log\frac{{N_a}''}{N_a} + + \frac{{N_a}'''}{N_a} \log\frac{{N_a}'''}{N_a} + \cdots\right) \\ + &\quad -N_b\left( + \frac{{N_b}'}{N_b} \log\frac{{N_b}'}{N_b} + + \frac{{N_b}''}{N_b} \log\frac{{N_b}''}{N_b} + + \frac{{N_b}'''}{N_b} \log\frac{{N_b}'''}{N_b} + \cdots\right),\\ + &\quad\text{etc.} +\end{align*} + +For simplicity let us denote the ratios $\dfrac{{N_a}'}{N_a}$, $\dfrac{{N_a}''}{N_a}$,~etc., by the +symbols ${w_a}'$, ${w_a}''$,~etc. These quantities ${w_a}'$, ${w_a}''$,~etc., are evidently +the probabilities, in the case of this particular statistical state, +that any given particle~$m_a$ will be found in the respective regions +Nos.\ $1a$,~$2a$,~etc. + +We may now write +\[ +\log W = -N_a\Sum w_a\log w_a - N_b\Sum w_b\log w_b -{}, \text{ etc.}, +\] +where the summation extends over all the regions Nos.\ $1a$,~$2a$, $\cdots +1b$, $2b$,~etc. + +\Subsubsection{96}{Equilibrium Relations.} Let us now suppose that the system +of particles is contained in an enclosed space and has the definite +energy content~$E$. Let us find the most probable distribution of the +particles. For this the necessary condition will be +\begin{multline*} +\delta\log W = -N_a\Sum (\log w_a + 1)\, \delta w_a \\ + -N_b\Sum (\log w_b + 1)\, \delta w_b \cdots = 0. +\Tag{114} +\end{multline*} +In carrying out our variation, however, the number of particles of +%% -----File: 121.png---Folio 107------- +each kind must remain constant so that we have the added relations +\[ +\Sum \delta w_a=0, \qquad +\Sum \delta w_b=0, \qquad \text{etc.} +\Tag{115} +\] +Finally, since the energy is to have a definite value~$E$, it must also +remain constant in the variation, which will provide still a further +relation. Since the energy of a particle will be a definite function of +its position and momentum,\footnote + {We thus exclude from our considerations systems in which the potential energy + depends appreciably on the \emph{relative} positions of the independent particles.} +let us write the energy of the system +in the form +\[ +E = N_a \Sum w_a E_a + N_b \Sum w_b E_b + \cdots, +\] +where $E_a$ is the energy of a particle in the region $1a$,~etc. + +Since in carrying out our variation the energy is to remain constant, +we have the relation +\[ +E = N_a \Sum E_a\, \delta w_a + + N_b \Sum E_b\, \delta w_b + \cdots = 0. +\Tag{116} +\] + +Solving the simultaneous equations (114),~(115),~(116) by familiar +methods we obtain +\begin{align*} +&\log w_a + 1 + \lambda E_a + \mu_b = 0, \\ +&\log w_b + 1 + \lambda E_b + \mu_b = 0, \\ +&\text{etc.}, +\end{align*} +where $\lambda$,~$\mu_a$, $\mu_b$,~etc., are undetermined constants. (It should be +specially noticed that $\lambda$~is the same constant in each of the series of +equations.) + +Transforming we have +\[ +\begin{aligned} +& w_a = \alpha_a\, e^{-hE_a}, \\ +& w_b = \alpha_b\, e^{-hE_b}, \\ +& \text{etc.}, +\end{aligned} +\Tag{117} +\] +as the expressions which determine the chance that a given particle +of mass $m_a$,~$m_b$,~etc., will fall in a given region $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$, +when we have the distribution of maximum probability. It should +be noticed that~$h$, which corresponds to the~$\lambda$ of the preceding equations, +is the same constant in all of the equations, while $\alpha_a$,~$\alpha_b$,~etc., +are different constants, depending on the mass of the particles $m_a$,~$m_b$,~etc. +%% -----File: 122.png---Folio 108------- + +\Subsubsection{97}{The Energy as a Function of the Momentum.} $E_a$,~$E_b$,~etc., +are of course functions of $x$,~$y$,~$z$, $\psi_x$,~$\psi_y$,~$\psi_z$. Let us now obtain an +expression for~$E_a$ in terms of these quantities. If there is no external +field of force acting, the energy of a particle~$E_a$ will be independent +of $x$,~$y$, and~$z$, and will be determined entirely by its velocity and +mass. In accordance with the theory of relativity we shall have\footnote + {This expression is that for the total energy of the particle, including that internal energy~$m_0 c^2$ + which, according to relativity theory, the particle has when it is at rest. (See \Secref{75}.) + It would be just as correct to substitute for~$E_a$ in equation~(117) the value of the kinetic energy + $m_a c^2 \Biggl(\dfrac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}- 1 \Biggr)$ + instead of the total energy $\dfrac{m_ac^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, + since the two differ merely by a constant~$m_a c^2$ which would be taken care of by assigning a suitable value to~$\alpha_a$.} +\[ +E_a = \frac{m_ac^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\Tag{118} +\] +where $m_a$ is the mass of the particle at rest. + +Let us now express $E_a$ as a function of $\psi_x$,~$\psi_y$,~$\psi_z$. + +We have from our equations (105)~and~(98), which were used for +defining momentum +\begin{align*} +\psi_x + &= \frac{\partial}{\partial\dot{x}}\, m_a + \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\right)\displaybreak[0] \\ + &= \frac{\partial}{\partial\dot{x}}\, m_a + \left(1 - \sqrt{1 - \frac{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}{c^2}}\right)\displaybreak[0] \\ + &= \frac{m_0\dot{x}}{\sqrt{1 - \smfrac{u^2}{c^2}}}. +\end{align*} +Constructing the similar expressions for $\psi_y$~and~$\psi_z$ we may write the +relation +\[ +\psi^2 = \psi_x^2 + \psi_y^2 + \psi_z^2 + = \frac{m^2_a (\dot{x}^2 + \dot{y}^2 + \dot{z}^2)}{1 - \smfrac{u^2}{c^2}} + = \frac{m^2_au^2}{1 - \smfrac{u^2}{c^2}}, +\Tag{119} +\] +which also defines~$\psi^2$. +%% -----File: 123.png---Folio 109------- + +By simple transformations and the introduction of equation~(118) +we obtain the desired relation +\[ +E_a = c\sqrt{\psi^2 + {m_a}^2c^2}. +\Tag{120} +\] + +\Subsubsection{98}{The Distribution Law.} We may now rewrite equations~(117) +in the form +\[ +\begin{aligned} +& w_a = \alpha_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}, \\ +& w_b = \alpha_b\, e^{-hc\sqrt{\psi^2 + {m_b}^2c^2}}, \\ +& \text{etc.} +\end{aligned} +\Tag{121} +\] + +These expressions determine the probability that a given particle +of mass $m_a$,~$m_b$,~etc.\DPtypo{}{,} will fall in a given region $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$, and +correspond to Maxwell's distribution law in ordinary mechanics. We +see that these probabilities are independent of the position $x$,~$y$,~$z$\footnote + {This is true only when, as assumed, no external field of force is acting.} +but dependent on the momentum. + +$\alpha_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}$ is the probability that a given particle will fall in a +particular six-dimensional cube of volume $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$. Let us +now introduce, for convenience, a new quantity $a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}$ which +will be the probability per \emph{unit} volume that a given particle will have +the six dimensional location in question, the constants $\alpha_a$~and~$a_a$ +standing in the same ratio as the volumes $dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z$ and unity. + +We may then write +\begin{alignat*}{2} +w_a &= \alpha_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}} & + &= a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, + dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z \\ +w_b &= \alpha_b\, e^{-hc\sqrt{\psi^2 + {m_b}^2c^2}} & + &= a_b\, e^{-hc\sqrt{\psi^2 + {m_b}^2c^2}}\, + dx\, dy\, dz\, d\psi_x\, d\psi_y\, d\psi_z +\end{alignat*} +etc. + +Since every particle must have components of momentum lying +between minus and plus infinity, and lie somewhere in the whole +volume~$V$ occupied by the mixture, we have the relation +\[ +V \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} + a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, d\psi_x\, d\psi_y\, d\psi_z = 1. +\Tag{122} +\] + +It is further evident that the average value of any quantity~$A$ +which depends on the momentum of the particles is given by the +%% -----File: 124.png---Folio 110------- +expression +\[ +[A]_{\text{av.}} + = V \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} + a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\,A \, d\psi_x\, d\psi_y\, d\psi_z, +\Tag{123} +\] +where $A$ is some function of $\psi_x$,~$\psi_y$, and~$\psi_z$. + +\Subsubsection{99}{Polar Coördinates.} We may express relations corresponding +to (122)~and~(123) more simply if we make use of polar coördinates. +Consider instead of the elementary volume $d\psi_x\, d\psi_y\, d\psi_z$ the volume +$\psi^2\sin\theta\, d\theta\, d\phi\, d\psi$ expressed in polar coördinates, where +\[ +\psi^2 = {\psi_x}^2 + {\psi_y}^2 + {\psi_z}^2. +\] + +The probability that a particle~$m_a$ will fall in the region +\[%[** TN: Displaying to avoid bad line break] +dx\, dy\, dz\, \psi^2 \sin\theta\, d\theta\, d\phi\, d\psi +\] +will be +\[ +a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, + dx\, dy\, dz\, \psi^2 \sin\theta\, d\theta\, d\phi\, d\psi, +\] +and since each particle must fall somewhere in the space $x\:y\:z\: \psi_x\: \psi_y\: \psi_z$ +we shall have corresponding to~(122) the relation +\[ +\begin{gathered} +V \int_{0}^{\pi}\int_{0}^{2\pi}\int_{0}^{\infty} + a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, \psi^2 \sin\theta\, + d\theta\, d\phi\, d\psi = 1, \\ +4\pi V \int_{0}^{\infty} a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, \psi^2\, d\psi = 1. +\end{gathered} +\Tag{124} +\] +Corresponding to equation~(123), we also see that the average value +of any quantity~$A$, which is dependent on the momentum of the +molecules of mass~$m_a$, will be given by the expression +\[ +[A]_{\text{av.}} + = 4\pi V \int_{0}^{\infty} a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, + A\, \psi^2\, d\psi. +\Tag{125} +\] + +\Subsubsection{100}{The Law of Equipartition.} We may now obtain a law which +corresponds to that of the equipartition of \textit{vis~viva} in the classical +mechanics. Considering equation~(124) let us integrate by parts, we +obtain +\begin{multline*} +\left[ 4\pi V a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, \frac{\psi^3}{3}\right]_{\psi=0}^{\psi=\infty} \\ +-4\pi V\int_0^{\infty} \frac{\psi^3}{3}\, a_a\, + e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\,(-hc)\, + \frac{\psi}{\sqrt{\psi^2 + {m_a}^2c^2}}\, d\psi = 1. +\end{multline*} +%% -----File: 125.png---Folio 111------- +Substituting the limits into the first term we find that it becomes +zero and may write +\[ +4\pi V\int_0^{\infty} a_a\, e^{-hc\sqrt{\psi^2 + {m_a}^2c^2}}\, + \frac{\psi^2c}{\sqrt{\psi^2 + {m_a}^2c^2}}\, \psi^2\, d\psi = \frac{3}{h}. +\] + +But by equation~(125) the left-hand side of this relation is the +average value of $\dfrac{\psi^2c}{\sqrt{\psi^2 + {m_a}^2c^2}}$ for the particles of mass~$m_a$. We have +\[ +\left[ \frac{\psi^2c}{\sqrt{\psi^2 + {m_a}^2c^2}}\right]_{\text{av.}} + = \frac{3}{h}. +\] +Introducing equation~(119) which defines~$\psi^2$, we may transform this +expression into +\[ +\Biggl[\frac{m_au^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggl]_{\text{av.}} + = \frac{3}{h}. +\Tag{126} +\] + +Since we have shown that $h$~is independent of the mass of the +particles, \emph{we see that the average value of $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is the same for particles +of all different masses}. This is the principle in relativity mechanics +that corresponds to the law of the equipartition of \textit{vis~viva} in the +classical mechanics. Indeed, for low velocities the above expression +reduces to~$m_0 u^2$, the \textit{vis~viva} of Newtonian mechanics, a fact which +affords an illustration of the general principle that the laws of Newtonian +mechanics are always the limiting form assumed at low velocities +by the more exact formulations of relativity mechanics. + +We may now call attention in passing to the fact that this quantity +$\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$, whose value is the same for particles of different masses, is +not the relativity expression for kinetic energy, which is given rather +by the formula $c^2\Biggl[\dfrac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} - m_0\Biggr]$. So that in relativity mechanics +%% -----File: 126.png---Folio 112------- +the principle of the equipartition of energy is merely an approximation. +We shall later return to this subject. + +\Subsubsection{101}{Criterion for Equality of Temperature.} For a system of particles +of masses $m_a$,~$m_b$,~etc., enclosed in the volume~$V$, and having the +definite energy content~$E$, we have shown that +\[ +4\pi V\, a_a\, e^{-hc\sqrt{\psi^2+{m_a}^2c^2}}\, \psi^2\, d\psi +\] +and +\[ +4\pi V\, a_b\, e^{-hc\sqrt{\psi^2+{m_b}^2c^2}}\, \psi^2\, d\psi +\] +are the respective probabilities that given particles of mass~$m_a$ or +mass~$m_b$ will have momenta between $\psi$~and~$\psi + d\psi$. Suppose now +we consider a differently arranged system in which we have $N_a$~particles +of mass~$m_a$ by themselves in a space of volume~$V_a$ and $N_b$~particles +of mass~$m_b$ in a contiguous space of volume~$V_b$, separated +from~$V_a$ by a partition which permits a transfer of energy, and let +the total energy of the double system be, as before, a definite quantity~$E$ +(the energy content of the partition being taken as negligible). +Then, by reasoning entirely similar to that just employed, we can +obviously show that +\[ +4\pi V_a\, a_a\, e^{-hc\sqrt{\psi^2+{m_a}^2c^2}}\, \psi^2\, d\psi +\] +and +\[ +4\pi V_b\, a_b\, e^{-hc\sqrt{\psi^2+{m_b}^2c^2}}\, \psi^2\, d\psi +\] +are now the respective probabilities that given particles of mass~$m_a$ +or mass~$m_b$ will have momenta between $\psi$~and~$\psi + d\psi$, the only +changes in the expressions being the substitution of the volumes +$V_a$~and~$V_b$ in the place of the one volume~$V$. Furthermore, this +distribution law will evidently lead as before to the equality of the +average values of +\[ +\frac{m_au^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\qquad \text{and}\qquad +\frac{m_bu^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}. +\] +Since, however, the spaces containing the two kinds of particles are in +thermal contact, their temperature is the same. Hence we find that +\emph{the equality of the average values of $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is the necessary condition for +equality of temperature}. +%% -----File: 127.png---Folio 113------- + +\emph{The above distribution law also leads to the important corollary that +for any given system of particles at a definite temperature the momenta +and hence the total energy content is independent of the volume.} + +We may now proceed to the derivation of relations which will +permit us to show that the important quantity $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is directly +proportional to the temperature as measured on the absolute thermodynamic +temperature scale. + +\Subsubsection{102}{Pressure Exerted by a System of Particles.} We first need +to obtain an expression for the pressure exerted by a system of $N$~particles +enclosed in the volume~$V$. Consider an element of surface~$dS$ +perpendicular to the $X$~axis, and let the pressure acting on it be~$p$. +The total force which the element~$dS$ exerts on the particles that +impinge will be~$p\, dS$, and this will be equal to the rate of change of +the momenta in the $X$~direction of these particles.\footnote + {The system is considered dilute enough for the mutual attractions of the + particles to be negligible in their effect on the external pressure.} + +Now by equation~(122) the total number of particles having +momenta between $\psi_x$~and~$\psi_x + d\psi_x$ in the \emph{positive} direction is +\[ +NV \int_{\psi_x}^{\psi_x + d\psi_x}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} + a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, d\psi_x\, d\psi_y\, d\psi_z. +\] +But $\dot{x}\, dS$ gives us the volume which contains the number of particles +having momenta between $\psi_x$~and~$\psi_x + d\psi_x$ which will reach~$dS$ in a +second. Hence the number of such particles which impinge per +second will be +\[ +NV\, \frac{\dot{x}\, dS}{V} + \int_{\psi_x}^{\psi_x + d\psi_x}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} + a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, d\psi_x\, d\psi_y\, d\psi_z. +\] +and their change in momentum, allowing for the effect of the rebound, +will be +\[ +2N\, dS \int_{\psi_x}^{\psi_x + d\psi_x}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} + a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi_x\, \dot{x}\, d\psi_x\, d\psi_y\, d\psi_z. +\] +Finally, the total change in momentum per second for all particles +can be found by integrating for all possible positive values of~$\psi_x$. +%% -----File: 128.png---Folio 114------- +Equating this to the total force~$p\, dS$ we have +\[ +p\, dS = 2N\, dS + \int_{0}^{\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} + a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi_x\, \dot{x}\, d\psi_x\, d\psi_y\, d\psi_z. +\] +Cancelling~$dS$, multiplying both sides of the equation by the volume~$V$, +changing the limits of integration and substituting $\dfrac{m_0\dot{x}}{\sqrt{1 + \smfrac{u^2}{c^2}}}$ for~$\psi_x$, +we have +\[ +pV = NV + \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} + a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, + \frac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, d\psi_x\, d\psi_y\, d\psi_z. +\] +But this by equation~(123) reduces to +\[ +pV = N \Biggl[\frac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}} +\] +or, since +\[ +\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = \frac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}} + + \frac{m_0\dot{y}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}} + + \frac{m_0\dot{z}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +we have from symmetry +\[ +pV = \frac{N}{3}\Biggl[\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Bigg]_{\text{av.}}. +\Tag{127} +\] +Since at a given temperature we have seen that the term in parenthesis +is independent of the volume and the nature of the particles, we see +that the laws of Boyle and Avogadro hold also in relativity mechanics +for a system of particles. + +For slow velocities equation~(127) reduces to the familiar expression +$pV = \dfrac{N}{3}\, (m_0u^2)_{\text{av.}}$. + +\Subsubsection{103}{The Relativity Expression for Temperature.} We are now in +a position to derive the relativity expression for temperature. The +thermodynamic scale of temperature may be defined in terms of the +efficiency of a heat engine. Consider a four-step cycle performed +with a working substance contained in a cylinder provided with a +piston. In the first step let the substance expand isothermally and +%% -----File: 129.png---Folio 115------- +reversibly, absorbing the heat~$Q_2$ from a reservoir at temperature~$T_2$; +in the second step cool the cylinder down at constant volume to~$T_1$; +in the third step compress to the original volume, giving out the +heat~$Q_1$ at temperature~$T_1$, and in the fourth step heat to the original +temperature. Now if the working substance is of such a nature that +the heat given out in the second step could be used for the \emph{reversible} +heating of the cylinder in the fourth step, we may define the absolute +temperatures $T_2$~and~$T_1$ by the relation $\dfrac{T_2}{T_1} = \dfrac{Q_2}{Q_1}$.\footnote + {We have used this cycle for defining the thermodynamic temperature scale + instead of the familiar Carnot cycle, since it avoids the necessity of obtaining an + expression for the relation between pressure and volume in an adiabatic expansion.} + +Consider now such a cycle performed on a cylinder which contains +one of our systems of particles. Since we have shown (\Secref{101}) +that at a definite temperature the energy content of such a +system is independent of the volume, it is evident that our working +substance fulfils the requirement that the heat given out in the second +step shall be sufficient for the reversible heating in the last step. +Hence, in accordance with the thermodynamic scale, we may measure +the temperatures of the two heat reservoirs by the relation $\dfrac{T_2}{T_1} = \dfrac{Q_2}{Q_1}$ +and may proceed to obtain expressions for $Q_2$~and~$Q_1$. + +In order to obtain these expressions we may again make use of the +principle that the energy content at a definite temperature is independent +of the volume. This being true, we see that $Q_2$~and~$Q_1$ +must be equal to the work done in the changes of volume that take +place respectively at $T_2$~and~$T_1$, and we may write the relations +\begin{align*} +Q_2 &= \int_V^{V'} p\, dV\quad \text{(at $T_2$)}, \\ +Q_1 &= \int_V^{V'} p\, dV\quad \text{(at $T_1$)}. +\end{align*} +But equation~(127) provides an expression for~$p$ in terms of~$V$, leading +on integration to the relations +\begin{align*} +Q_2 &= \frac{N}{3}\Biggl[ + \frac{m_0{u_2}^2}{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}} + \Biggr]_{\text{av.}} \log\frac{V'}{V}, \\ +%% -----File: 130.png---Folio 116------- +Q_1 &= \frac{N}{3}\Biggl[ + \frac{m_0{u_1}^2}{\sqrt{1 - \smfrac{{u_1}^2}{c^2}}} + \Biggl]_{\text{av.}} \log\frac{V'}{V}, +\end{align*} +which gives us on division +\[ +\frac{T_2}{T_1} = \frac{Q_2}{Q_1} + = \frac{\Biggl[\smfrac{m_0{u_2}^2}{\sqrt{1 - \smfrac{{u_2}^2}{c^2}}}\Biggr]_{\text{av.}}} + {\Biggl[\smfrac{m_0{u_1}^2}{\sqrt{1 - \smfrac{{u_1}^2}{c^2}}}\Biggr]_{\text{av.}}}. +\] + +\emph{We see that the absolute temperature measured on the thermodynamic +scale is proportional to the average value of $\dfrac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}$.} + +We may finally express our temperature in the same units customarily +employed by comparing equation~(127) +\[ +pV = \frac{N}{3}\Biggl[\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}}, +\] +with the ordinary form of the gas law +\[ +pV = nRT, +\] +where $n$~is the number of mols of gas present. + +We evidently obtain +\[ +\begin{aligned} +nRT &= \frac{N}{3} \Biggl[\frac{m_0 u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}},\\ +T &= \frac{N}{3nR}\Biggl[\frac{m_0 u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}} + = \frac{1}{3k} \Biggl[\frac{m_0 u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}}, +\end{aligned} +\Tag{128} +\] +where the quantity $\dfrac{nR}{N}$, which may be called the gas constant for a +single molecule, has been denoted, as is customary, by the letter~$k$. +%% -----File: 131.png---Folio 117------- +Remembering the relation $\Biggl[\dfrac{m_0\dot{x}^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{av.}} = \dfrac{3}{h}$, we have +\[ +kT = \frac{1}{h}. +\Tag{129} +\] + +\Subsubsection{104}{The Partition of Energy.} We have seen that our new equipartition +law precludes the possibility of an exact equipartition of +energy. It becomes very important to see what the average energy +of a particle of a given mass does become at any temperature. + +Equation~(125) provides a general expression for the average value +of any property of the particles. For the average value of the energy +$c\sqrt{\psi^2 + {m_0}^2c^2}$ of particles of mass~$m_0$ (see equation~120) we shall have +\[ +[E]_{\text{av.}} + = 4\pi V \int_0^{\infty} a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, + c\, \sqrt{\psi^2 + {m_0}^2c^2}\, \psi^2\, d\psi. +\] +The unknown constant~$a$ may be eliminated with the help of the +relation~(124) +\[ +4\pi V \int_0^{\infty} a\, e^{-hc\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi^2\, d\psi = 1 +\] +and for~$h$ we may substitute the value given by~(129), which gives us +the desired equation +\[ +[E]_{\text{av.}} + = \frac{\ds\int_0^{\infty} e^{-(c/kT)\sqrt{\psi^2 + {m_0}^2c^2}}\, c\, \sqrt{\psi^2 + {m_0}^2c^2}\, \psi^2\, d\psi} + {\ds\int_0^{\infty} e^{-(c/kT)\sqrt{\psi^2 + {m_0}^2c^2}}\, \psi^2\, d\psi}. +\Tag{130} +\] + +\Subsubsection{105}{Partition of Energy for Zero Mass.} Unfortunately, no general +method for the evaluation of this expression seems to be available. +For the particular case that the mass~$m_0$ of the particles approaches +zero compared to the momentum, the expression reduces to +\[ +[E]_{\text{av.}} + = \frac{c\ds\int_0^{\infty} e^{-(c\psi /kT)}\, \psi^3\, d\psi} + { \ds\int_0^{\infty} e^{-(c\psi /kT)}\, \psi^2\, d\psi} +\] +%% -----File: 132.png---Folio 118------- +in terms of integrals whose values are known. Evaluating, we obtain +\[ +[E]_{\text{av.}} = 3kT. +\] +For the total energy of $N$ such particles we obtain +\[ +E = 3NkT, +\] +and introducing the relation $k = \dfrac{nR}{N}$ by which we defined~$k$ we have +\[ +E = 3nRT +\Tag{131} +\] +as the expression for the energy of $n$~mols of particles if their value of~$m_0$ +is small compared with their momentum. + +It is instructive to compare this with the ordinary expression of +Newtonian mechanics +\[ +E = \frac{3}{2}\, nRT, +\] +which undoubtedly holds when the masses are so large and the velocities +so small that no appreciable deviations from the laws of Newtonian +mechanics are to be expected. We see that for particles of +very small mass the average kinetic energy at any temperature is +twice as large as that for large particles at the same temperature. +It is also interesting to note that in accordance with equation~(131) +a mol of particles which approach zero mass at the absolute zero, +would have a mass of +\[ +\frac{3 × 8.31 × 10^{7} × 300}{10^{21}} = 7.47 × 10^{-11} +\] +grams at room temperature ($300°$~absolute). This suggests a field +of fascinating if profitless speculation. + +%[** TN: ToC entry reads "Approximate Partition for Particles of any Mass"] +\Subsubsection{106}{Approximate Partition of Energy for Particles of any Desired +Mass.} For particles of any desired mass we may obtain an approximate +idea of the relation between energy and temperature by expanding +the expression for kinetic energy into a series. For the average +kinetic energy of a particle we have +\[ +[K]_{\text{av.}} + = c^2\Biggl[\frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} - m_0\Biggr]_{\text{av.}}. +\] +%% -----File: 133.png---Folio 119------- +Expanding into a series we obtain for the total kinetic energy of $N$~particles +\[ +K = Nm_0\left( + \frac{1}{2}\, \vc{u}^2 + \frac{3}{8}\, \frac{\vc{u}^4}{c^2} + + \frac{15}{48}\, \frac{\vc{u}^6}{c^4} + + \frac{105}{384}\, \frac{\vc{u}^8}{c^6} + \cdots\right), +\Tag{132} +\] +where $\vc{u}^2$, $\vc{u}^4$,~etc., are the average values of $u^2$,~$u^4$,~etc., for the individual +particles. + +To determine approximately how the value of~$K$ varies with the +temperature we may also expand our expression~(128) for temperature, +\[ +T = \frac{1}{3k}\Biggl[\frac{m_0u^2}{\sqrt{1 - \smfrac{u^2}{c^2}}}\Biggr]_{\text{Av.}}, +\] +into a series; we obtain +{\small% +\[ +\frac{3}{2}\, kNT = \frac{3}{2}\, nRT + = Nm_0 \left( + \frac{1}{2}\, \vc{u}^2 + \frac{1}{4}\, \frac{\vc{u}^4}{c^2} + + \frac{3}{16}\, \frac{\vc{u}^6}{c^4} + + \frac{15}{96}\, \frac{\vc{u}^8}{c^6} + \cdots\right). +\Tag{133} +\]}% +Combining expressions (132)~and~(133) by subtraction and transposition, +we obtain +\[ +K = \frac{3}{2}\, nRT + + Nm_0 \left( + \frac{1}{8}\, \frac{\vc{u}^4}{c^2} + + \frac{1}{8}\, \frac{\vc{u}^6}{c^4} + + \frac{15}{128}\, \frac{\vc{u}^8}{c^6} + \cdots\right). +\Tag{134} +\] +For the case of velocities low enough so that $\vc{u}^4$~and higher powers +can be neglected, this reduces to the familiar expression of Newtonian +mechanics,~$K = \dfrac{3}{2}\, nRT$. + +In case we neglect in expression~(134) powers higher than~$\vc{u}^4$ we +have the approximate relation +\[ +\frac{Nm_0 \vc{u}^4}{8c^2} + = \frac{1}{2Nm_0c^2} \left(\frac{Nm_0\vc{u}^2}{2}\right)^2, +\] +the left-hand term really being the larger, since the average square of a +quantity is greater than the square of its average. Since $\left(\dfrac{Nm_0\vc{u}^2}{2}\right)^2$ +is approximately equal to $\left(\dfrac{3}{2}\, nRT\right)^2$, we may write the approximation +%% -----File: 134.png---Folio 120------- +\[ +K = \frac{3}{2}\, nRT + \frac{1}{2Nm_0c^2} \left(\frac{3}{2}\, nRT\right)^2, +\] +or, noting that $N m_0 = M$, the total mass of the system at the absolute +zero, we have +\[ +K = \frac{3}{2}\, nRT + \frac{9}{8}\, \frac{n^2R^2}{Mc^2}\, T^2. +\] +If we use the erg as our unit of energy, $R$~will be~$8.31 × 10^7$; expressing +velocities in centimeters per second, $c^2$~will be~$10^{21}$, and $M$~will be the +mass of the system in grams. + +For one mol of a monatomic gas we should have in ergs +\[ +K = 12.4 × 10^7T + \frac{7.77}{M}\, 10^{-6}\, T^2. +\] + +In the case of the electron $M$~may be taken as approximately +$1/1800$. At room temperature the second term of our equation would +be entirely negligible, being only $3.5 × 10^{-6}$~per cent of the first, and +still be only $3.5 × 10^{-4}$~per cent in a fixed star having a temperature of +$30,000°$. Hence at all ordinary temperatures we may expect the +law of the equipartition of energy to be substantially exact for particles +of mass as small as the electron. + +Our purpose in carrying through the calculations of this chapter +has been to show that a very important and interesting problem in +the classical mechanics can be handled just as easily in the newer +mechanics, and also to point out the nature of the modifications in +existing theory which will have to be introduced if the later developments +of physics should force us to consider equilibrium relations for +particles of mass much smaller than that of the electron. + +We may also call attention to the fact that we have here considered +a system whose equations of motion agree with the principles +of dynamics and yet do not lead to the equipartition of energy. This +is of particular interest at a time when many scientists have thought +that the failure of equipartition in the hohlraum stood in necessary +conflict with the principles of dynamics. +%% -----File: 135.png---Folio 121------- + + +\Chapter{IX}{The Principle of Relativity and the Principle of +Least Action.} +\SetRunningHeads{Chapter Nine.}{Relativity and the Principle of Least Action.} + +It has been shown by the work of Helmholtz, J.~J. Thomson, +Planck and others that the principle of least action is applicable in +the most diverse fields of physical science, and is perhaps the most +general dynamical principle at our disposal. Indeed, for any system +whose future behavior is determined by the instantaneous values of a +number of \DPchg{coordinates}{coördinates} and their time rate of change, it seems possible +to throw the equations describing the behavior of the system into +the form prescribed by the principle of least action. This generality +of the principle of least action makes it very desirable to develop the +relation between it and the principle of relativity, and we shall obtain +in this way the most important and most general method for deriving +the consequences of the theory of relativity. We have already +developed in \Chapref{VII} the particular application of the principle +of least action in the case of a system of particles, and with the help +of the more general development which we are about to present, we +shall be able to apply the principle of relativity to the theories of +elasticity, of thermodynamics and of electricity and magnetism. + +\Subsubsection{107}{The Principle of Least Action.} For our purposes the principle +of least action may be most simply stated by the equation +\[ +\int_{t_1}^{t_2}(\delta H + W)\, dt = 0. +\Tag{135} +\] +This equation applies to any system whose behavior is determined +by the values of a number of independent coördinates $\phi_1\phi_2\phi_3\cdots$ +and their rate of change with the time $\dot{\phi}_1\dot{\phi}_2\dot{\phi}_3\cdots$, and the equation +describes the path by which the system travels from its configuration +at any time~$t_1$ to its configuration at any subsequent time~$t_2$. + +$H$~is the so-called kinetic potential of the system and is a function +of the coördinates and their generalized velocities: +\[ +H = F(\phi_1\phi_2\phi_3\cdots \dot{\phi}_1\dot{\phi}_2\dot{\phi}_3\cdots). +\Tag{136} +\] +%% -----File: 136.png---Folio 122------- +$\delta H$~is the variation of~$H$ at any instant corresponding to a slightly +displaced path by which the system might travel from the same +initial to the same final state in the same time interval, and $W$~is the +external work corresponding to the variation~$\delta$ which would be done +on the system by the external forces if at the instant in question the +system should be displaced from its actual configuration to its configuration +on the displaced path. Thus +\[ +W = \Phi_1\, \delta\phi_1 + + \Phi_2\, \delta\phi_2 + + \Phi_3\, \delta\phi_3 + \cdots, +\Tag{137} +\] +where $\Phi_1$, $\Phi_2$,~etc., are the so-called generalized external forces which +act in such a direction as to increase the values of the corresponding +coördinates. + +The form of the function which determines the kinetic potential~$H$ +depends on the particular nature of the system to which the principle +of least action is being applied, and it is one of the chief tasks of +general physics to discover the form of the function in the various +fields of mechanical, electrical and thermodynamic investigation. +As soon as we have found out experimentally what the form of~$H$ is +for any particular field of investigation, the principle of least action, +as expressed by equation~(135), becomes the basic equation for the +mathematical development of the field in question, a development +which can then be carried out by well-known methods. + +The special task for the theory of relativity will be to find a general +relation applicable to any kind of a system, which shall connect the +value of the kinetic potential~$H$ as measured with respect to a set of +coördinates~$S$ with its value~$H'$ as measured with reference to another +set of coördinates~$S'$ which is in motion relative to~$S$. This relation +will of course be of such a nature as to agree with the principle of the +relativity of motion, and in this way we shall introduce the principle +of relativity at the very start into the fundamental equation for all +fields of dynamics. + +Before proceeding to the solution of that problem we may put +the principle of least action into another form which is sometimes +more convenient, by obtaining the equations for the motion of a +system in the so-called Lagrangian form. + +\Subsubsection{108}{The Equations of Motion in the Lagrangian Form.} To obtain +the equations of motion in the Lagrangian form we may evidently +%% -----File: 137.png---Folio 123------- +rewrite our fundamental equation~(135) in the form +\[ +\begin{aligned} +\int_{t_1}^{t_2} \biggl( + \frac{\partial H}{\partial\phi_1}\, \delta\phi_1 + + \frac{\partial H}{\partial\phi_2}\, \delta\phi_2 + \cdots + &+ \frac{\partial H}{\partial\dot{\phi}_1}\, \delta\dot{\phi}_1 + + \frac{\partial H}{\partial\dot{\phi}_2}\, \delta\dot{\phi}_2 + \cdots \\ + &+ \Phi_1\, \delta\phi_1 + \Phi_2\, \delta\phi_2 + \cdots\biggr) dt = 0 +\end{aligned} +\Tag{138} +\] + +We have now, however, +\[ +\delta\dot\phi_1 = \frac{d}{dt}(\delta\phi_1), \qquad +\delta\dot\phi_2 = \frac{d}{dt}(\delta\phi_2), \qquad \text{etc.,} +\] +which gives us +\begin{align*} +\int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_1}\, \delta\dot\phi_1\, dt + &= \int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_1}\, + \frac{d}{dt}(\delta\phi_1)\, dt \\ + &= \left[\frac{\partial H}{\partial\dot\phi_1}\, \delta\phi_1\right]_{t_1}^{t_2} + - \int_{t_1}^{t_2} \delta\phi_1\, + \frac{d}{dt} \left(\frac{\partial H}{\partial\dot\phi_1}\right) dt, +\end{align*} +or, since $\delta\phi_1$, $\delta\phi_2$,~etc., are by hypothesis zero at times $t_1$~and~$t_2$, we +obtain +\begin{align*} +&\int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_1}\, \delta\dot\phi_1 + = - \int_{t_1}^{t_2} \frac{d}{dt} + \left(\frac{\partial H}{\partial\dot\phi_1}\right) \delta\phi_1\, dt, \\ +&\int_{t_1}^{t_2} \frac{\partial H}{\partial\dot\phi_2}\, \delta\dot\phi_2 + = - \int_{t_1}^{t_2} \frac{d}{dt} + \left(\frac{\partial H}{\partial\dot\phi_2}\right) \delta\phi_2\, dt, \\ +&\text{etc.} +\end{align*} +On substituting these expressions in~(138) we obtain +\begin{multline*} + \int_{t_1}^{t_2} \left[ + \left(\frac{\partial H}{\partial\phi_1} + - \frac{d}{dt} \left(\frac{\partial H}{\partial\dot\phi_1}\right) + + \Phi_1\right) \right. \delta\phi_1 \\ + \left. + \left(\frac{\partial H}{\partial\phi_2} + - \frac{d}{dt} \left(\frac{\partial H}{\partial\dot\phi_2}\right) + + \Phi_2\right) \delta\phi_2 + \cdots \right] dt = 0\DPtypo{}{,} +\end{multline*} +and since the variations of $\phi_1$, $\phi_2$,~etc., are entirely independent and +the limits of integration $t_1$~and~$t_2$ are entirely at our disposal, this +equation will be true only when each of the following equations is +true. And these are the equations of motion in the desired Lagrangian +%% -----File: 138.png---Folio 124------- +form, +\[ +\begin{aligned} +&\frac{d}{dt}\, \frac{\partial H}{\partial\dot{\phi}_1} + - \frac{\partial H}{\partial \phi_1} = \Phi_1, \\ +&\frac{d}{dt}\, \frac{\partial H}{\partial\dot{\phi}_2} + - \frac{\partial H}{\partial \phi_2} = \Phi_2, \\ +&\text{etc.} +\end{aligned} +\Tag{139} +\] + +In these equations $H$ is the kinetic potential of a system whose +state is determined by the generalized coördinates $\phi_1$,~$\phi_2$,~etc., and +their time derivatives $\dot{\phi}_1$,~$\dot{\phi}_2$~etc., where $\Phi_1$,~$\Phi_2$,~etc., are the generalized +external forces acting on the system in such a sense as to tend +to \emph{increase} the values of the corresponding generalized coördinates. + +\Subsubsection{109}{Introduction of the Principle of Relativity.} Let us now investigate +the relation between our dynamical principle and the principle +of the relativity of motion. To do this we must derive an equation +for transforming the kinetic potential~$H$ for a given system +from one set of \DPchg{coordinates}{coördinates} to another. In other words, if $S$~and~$S'$ +are two sets of reference axes, $S'$~moving past~$S$ in the $X$\DPchg{-}{~}direction +with the velocity~$V$, what will be the relation between $H$~and~$H'$, +the values for the kinetic potential of a given system as measured +with reference to $S$~and~$S'$? + +It is evident from the theory of relativity that our fundamental +equation~(135) must hold for the behavior of a given system using +either set of \DPchg{coordinates}{coördinates} $S$~or~$S'$, so that both of the equations +\[ +\int_{t_1}^{t_2} (\delta H + W)\, dt = 0\qquad\text{and}\qquad +\int_{{t_1}'}^{{t_2}'} (\delta H' + W')\, dt' = 0\DPtypo{}{,} \Tag{140} +\] +or +\[ +\int_{t_1}^{t_2} (\delta H + W)\, dt + = \int_{{t_1}'}^{{t_2}'} (\delta H' + W')\, dt' = 0\DPtypo{}{,} +\] +must hold for a given process, where it will be necessary, of course, +to choose the limits of integration $t_1$~and~$t_2$, ${t_1}'$~and~${t_2}'$ wide enough +apart so that for both sets of coördinates the varied motion will be +completed within the time interval. Since we shall find it possible +now to show that in general $\ds\int W\, dt = \int W'\, dt'$, we shall be able to +obtain from the above equations a simple relation between $H$~and~$H'$. + +%[** TN: Bold symbols in original] +\Subsubsection{110}{Relation between $\int W\, dt$ and $\int W'\, dt'$.} To obtain the desired +%% -----File: 139.png---Folio 125------- +proof we must call attention in the first place to the fact that all +kinds of force which can act at a given point must be governed by +the same transformation equations when changing from system~$S$ to +system~$S'$. This arises because when two forces of a different nature +are of such a magnitude as to exactly balance each other and produce +no acceleration for measurements made with one set of coördinates +they must evidently do so for any set of coördinates (see Chapter~IV, %[** TN: Not a useful cross-reference] +\Secref{42}). Since we have already found transformation equations +for the force acting at a point, in our consideration of the dynamics +of a particle, we may now use these expressions in general for the +evaluation~$\int W'\, dt'$. + +$W'$ is the work which would be done by the external forces if at +any instant~$t'$ we should displace our system from its actual configuration +to the simultaneous configuration on the displaced path. +Hence it is evident that $\int W'\, dt\DPtypo{}{'}$~will be equal to a sum of terms of the +type +\[ +%[** TN: Subscripts y and z misprinted (not as subscripts) in original] +\int ({F_x}'\, \delta x' + {F_y}'\, \delta y' + {F_z}'\, \delta z')\, dt', +\] +where ${F_x}'$,~${F_y}'$,~${F_z}'$, is the force acting at a given point of the system +and $\delta x'$,~$\delta y'$,~$\delta z'$ are the displacements necessary to reach the corresponding +point on the displaced path, all these quantities being +measured with respect to~$S'$. + +Into this expression we may substitute, however, in accordance +with equations (61),~(62),~(63) and~(13), the values +\[ +\begin{aligned} +{F_x}' &= F_x - \frac{\dot{y}V}{c^2}\, \frac{1}{1 - \smfrac{\dot{x}V}{c^2}}\, F_y + - \frac{\dot{z}V}{c^2}\, \frac{1}{1 - \smfrac{\dot{x}V}{c^2}}\, F_z, \\ +{F_y}' &= \frac{F_y\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}, \\ +{F_z}' &= \frac{F_z\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}, \\ +dt' &= \kappa \left(1 - \frac{\dot{x}V}{c^2}\right) dt. +\end{aligned} +\Tag{141} +\] +%% -----File: 140.png---Folio 126------- + +We may also make substitutions for $\delta x'$,~$\delta y'$ and~$\delta z'$ in terms of +$\delta x$,~$\delta y$ and~$\delta z$, but to obtain transformation equations for these quantities +is somewhat complicated owing to the fact that positions on the +actual and displaced path, which are simultaneous when measured +with respect to~$S'$, will not be simultaneous with respect to~$S$. We +have denoted by~$t'$ the time in system~$S'$ when the point on the \emph{actual} +path has the position $x'$,~$y'$,~$z'$ and simultaneously the point on the +\emph{displaced} path has the position $(x'+ \delta x')$, $(y' + \delta y')$, $(z' + \delta z')$, +when measured in system~$S'$, or by our fundamental transformation +equations (9),~(10) and~(11) the positions $\kappa (x' + Vt')$,~$y'$,~$z'$ and +$\kappa \bigl([x' + \delta x'] + Vt'\bigr)$, $(y'+ \delta y')$, $(z'+\delta z')$ when measured in system~$S$. +If now we denote by $t_A$~and~$t_D$ the corresponding times in system~$S$ +we shall have, by our fundamental transformation equation~(12), +\begin{align*} +t_A &= \kappa \left(t' + \frac{Vx'}{c^2}\right), \\ +t_D &= \kappa \left(t' + \frac{V}{c^2}[x' + \delta x']\right), +\end{align*} +and we see that in system~$S$ the point has reached the displaced +position at a time later than that of the actual position by the amount +\[ +t_D - t_A = \frac{\kappa V}{c^2}\, \delta x', +\] +and, since during this time-interval the displaced point would have +moved, neglecting higher-order terms, the distances +\[ +\dot{x}\, \frac{\kappa V}{c^2}\, \delta x', \qquad +\dot{y}\, \frac{\kappa V}{c^2}\, \delta x', \qquad +\dot{z}\, \frac{\kappa V}{c^2}\, \delta x', +\] +these quantities must be subtracted from the coördinates of the +displaced point in order to obtain a position on the displaced path +which will be simultaneous with~$t_A$ as measured in system~$S$. We +obtain for the simultaneous position on the displaced path +\begin{gather*} +\kappa \bigl([x' + \delta x'] + Vt'\bigr) + - \kappa\, \frac{\dot{x}V}{c^2}\, \delta x', \qquad +y' + \delta y' - \kappa\, \frac{\dot{x}V}{c^2}\, x', \\ +z' + \delta z' - \kappa\, \frac{\dot{z}V}{c^2}\, \delta x', +\end{gather*} +%% -----File: 141.png---Folio 127------- +and for the corresponding position on the actual path +\[ +\kappa (x' + Vt'), \quad y', \quad z', +\] +and obtain by subtraction +\[ +\begin{aligned} +\delta x &= \kappa \left(1 - \frac{\dot{x}V}{c^2}\right) \delta x', \\ +\delta y &= \delta y' - \kappa \frac{\dot{y}V}{c^2}\, \delta x', \\ +\delta z &= \delta z' - \kappa \frac{\dot{z}V}{c^2}\, \delta x'. +\end{aligned} +\Tag{142} +\] +Substituting now these equations, together with the other transformation +equations~(141), in our expression we obtain +\[ +\begin{aligned} +\int ({F_x}'\, \delta x' &+ {F_y}'\, \delta y' + {F_z}'\, \delta z')\, dt' \\ + &= \int \Biggl(\Biggl[ + F_x - \frac{\dot{y}V}{c^2}\, \frac{F_y}{1 - \smfrac{\dot{x}V}{c^2}} + - \frac{\dot{z}V}{c^2}\, \frac{F_z}{1 - \smfrac{\dot{x}V}{c^2}} + \Biggr] +%[** TN: \rlap hack to get equation number centered] + \rlap{$\ds \frac{\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}\, \delta x$} \\ + &+ \frac{\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}\, F_y \Biggl[ + \delta y + \frac{\dot{y}V/c^2}{1 - \smfrac{\dot{x}V}{c^2}}\, \delta x + \Biggr] \\ + &+ \frac{\kappa^{-1}}{1 - \smfrac{\dot{x}V}{c^2}}\, F_z \Biggl[ + \delta z + \frac{\dot{z}V/c^2}{1 - \smfrac{\dot{x}V}{c^2}}\, \delta x + \Biggr]\Biggr) \kappa \left(1 - \frac{\dot{x}V}{c^2}\right) \\ + &= \int(F_x\, \delta x + F_y\, \delta y + F_z\, \delta z)\, dt'. +\end{aligned} +\Tag{143} +\] +We thus see that we must always have the general equality +\[ +\int W'\, dt' = \int W\, dt. +\Tag{144} +\] + +\Subsubsection{111}{Relation between $H'$~and~$H$.} Introducing this equation into +our earlier expression~(140) we obtain as a general relation between +$H'$~and~$H$ +\[ +\int \delta H'\, dt' = \int \delta H\, dt. +\Tag{145} +\] + +Restricting ourselves to systems of such a nature that we can +%% -----File: 142.png---Folio 128------- +assign them a definite velocity $u = \dot{x}\vc{i} + \dot{y}\vc{j} + \dot{z}\vc{k}$, we can rewrite +this expression in the following form, where by $H_D$~and~$H_A$ we denote +the values of the kinetic potential respectively on the displaced and +actual paths +\begin{align*} +\int \delta H'\, dt' = \int {H_D}'\, dt' + &- \int {H_A}'\, dt' + = \int {H_D}' \kappa + \left(1 - \frac{(\dot{x} + \delta \dot{x})V}{c^2}\right) dt \\ + &- \int {H_A}' \kappa + \left(1 - \frac{\dot{x}V}{c^2}\right) dt + = \int H_D\, dt - \int H_A\, dt, +\end{align*} +and hence obtain for such systems the simple expression +\[ +H' = \frac{H}{\kappa \left(1 - \smfrac{\dot{x}V}{c^2}\right)}. +\] +Noting the relation between $\sqrt{1 - \dfrac{{u'}^2}{c^2}}$ and $\sqrt{1 - \dfrac{u^2}{c^2}}$ given in equation~(17), +this can be rewritten +\[ +\frac{H'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}} + = \frac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\Tag{146} +\] +and this is the expression which we shall find most useful for our +future development of the consequences of the theory of relativity. +Expressing the requirement of the equation in words we may say +that the theory of relativity requires an invariance of $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ in the +Lorentz transformation. + +\Paragraph{112.} As indicated above, the use of this equation is obviously +restricted to systems moving with some perfectly definite velocity~$\vc{u}$. +Systems satisfying this condition would include particles, infinitesimal +portions of continuous systems, and larger systems in a steady state. + +\Paragraph{113.} Our general method of procedure in different fields of investigation +will now be to examine the expression for kinetic potential +which is known to hold for the field in question, provided the velocities +involved are low and by making slight alterations when necessary, +%% -----File: 143.png---Folio 129------- +see if this expression can be made to agree with the requirements of +equation~(146) without changing its value for low velocities. Thus +it is well known, for example, that, in the case of low velocities, for a +single particle acted on by external forces the kinetic potential may +be taken as the kinetic energy~$\frac{1}{2}m_0u^2$. For relativity mechanics, as +will be seen from the developments of \Chapref{VII}, we may take for +the kinetic potential, $-m_0c^2 \sqrt{1 - \dfrac{u^2}{c^2}}$, an expression which, except for +an additive constant, becomes identical with~$\frac{1}{2}m_0u^2$ at low velocities, +and which at all velocities agrees with equation~(146). +%% -----File: 144.png---Folio 130------- + + +\Chapter{X}{The Dynamics of Elastic Bodies.} +\SetRunningHeads{Chapter Ten.}{Dynamics of Elastic Bodies.} + +We shall now treat with the help of the principle of least action +the rather complicated problem of the dynamics of continuous elastic +media. Our considerations will \emph{extend} the appreciation of the intimate +relation between mass and energy which we found in our treatment +of the dynamics of a particle. We shall also be able to show +that the dynamics of a particle may be regarded as a special case +of the dynamics of a continuous elastic medium, and to apply our +considerations to a number of other important problems. + +\Subsubsection{114}{On the Impossibility of Absolutely Rigid Bodies.} In the +older treatises on mechanics, after considering the dynamics of a +particle it was customary to proceed to a discussion of the dynamics +of rigid bodies. These rigid bodies were endowed with definite and +\DPtypo{nu}{un}changeable size and shape and hence were assigned five degrees +of freedom, since it was necessary to state the values of five variables +completely to specify their position in space. As pointed out by +Laue, however, our newer ideas as to the velocity of light as a limiting +value will no longer permit us to conceive of a continuous body as +having only a finite number of degrees of freedom. This is evident +since it is obvious that we could start disturbances simultaneously +at an indefinite number of points in a continuous body, and as these +disturbances cannot spread with infinite velocity it will be necessary +to give the values of an infinite number of variables in order completely +to specify the succeeding states of the system. For our newer +mechanics the nearest approach to an absolutely rigid body would +of course be one in which disturbances are transmitted with the +velocity of light. Since, then, the theory of relativity does not +permit rigid bodies we may proceed at once to the general theory of +deformable bodies. + + +\Section[I]{Stress and Strain.} + +\Subsubsection{115}{Definition of Strain.} In the more familiar developments of +the theory of elasticity it is customary to limit the considerations to +%% -----File: 145.png---Folio 131------- +the case of strains small enough so that higher powers of the displacements +can be neglected, and this introduces considerable simplification +into a science which under any circumstances is necessarily +one of great complication. Unfortunately for our purposes, we +cannot in general introduce such a simplification if we wish to apply +the theory of relativity, since in consequence of the Lorentz shortening +a body which appears unstrained to one observer may appear tremendously +compressed or elongated to an observer moving with a +different velocity. The best that we can do will be arbitrarily to +choose our state of zero deformation such that the strains will be +small when measured in the particular system of coördinates $S$ in +which we are specially interested. + +A theory of strains of any magnitude was first attempted by +Saint-Venant and has been amplified and excellently presented by +Love in his \textit{Treatise on the Theory of Elasticity}, Appendix to Chapter~I. +In accordance with this theory, the strain at any point in a body is +completely determined by six component strains which can be defined +by the following equations, wherein $(u, v, w)$~is the displacement of a +point having the unstrained position $(x, y, z)$: +%[** TN: Setting as two groups, both numbered (148), to permit a page break] +\begin{align*} +&\begin{aligned} +\epsilon_{xx} &= \frac{\partial u}{\partial x} + + \tfrac{1}{2} \left\{ + \left(\frac{\partial u}{\partial x}\right)^2 + + \left(\frac{\partial v}{\partial x}\right)^2 + + \left(\frac{\partial w}{\partial x}\right)^2 \right\}, \\ +% +\epsilon_{yy} &= \frac{\partial y}{\partial v} + + \tfrac{1}{2} \left\{ + \left(\frac{\partial u}{\partial y}\right)^2 + + \left(\frac{\partial v}{\partial y}\right)^2 + + \left(\frac{\partial w}{\partial y}\right)^2 \right\}, \\ +% +\epsilon_{zz} &= \frac{\partial w}{\partial z} + + \tfrac{1}{2} \left\{ + \left(\frac{\partial u}{\partial z}\right)^2 + + \left(\frac{\partial v}{\partial z}\right)^2 + + \left(\frac{\partial w}{\partial z}\right)^2 \right\}, +\end{aligned} +\Tag{148} +\displaybreak[0] \\ +% +&\begin{aligned} +\epsilon_{yz} &= \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z} + + \frac{\partial u}{\partial y}\, \frac{\partial u}{\partial z} + + \frac{\partial v}{\partial y}\, \frac{\partial v}{\partial z} + + \frac{\partial w}{\partial y}\, \frac{\partial w}{\partial z}, \\ +% +\epsilon_{xz} &= \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} + + \frac{\partial u}{\partial x}\, \frac{\partial u}{\partial z} + + \frac{\partial v}{\partial x}\, \frac{\partial v}{\partial z} + + \frac{\partial w}{\partial x}\, \frac{\partial w}{\partial z}, \\ +% +\epsilon_{xy} &= \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} + + \frac{\partial u}{\partial x}\, \frac{\partial u}{\partial y} + + \frac{\partial v}{\partial x}\, \frac{\partial v}{\partial y} + + \frac{\partial w}{\partial x}\, \frac{\partial w}{\partial y}. +\end{aligned} +\Tag{148} +\end{align*} + +It will be seen that these expressions for strain reduce to those +familiar in the theory of small strains if such second-order quantities as +$\left(\dfrac{\partial u}{\partial x}\right)^2$ or $\dfrac{\partial u}{\partial y}\,\dfrac{\partial u}{\partial z}$ can be neglected. +%% -----File: 146.png---Folio 132------- + +\Paragraph{116.} A physical significance for these strain components will be +obtained if we note that it can be shown from geometrical considerations +that lines which are originally parallel to the axes have, when +strained, the elongations +\[ +\begin{aligned} +e_x &= \sqrt{1 + 2\epsilon_{xx}} - 1, \\ +e_y &= \sqrt{1 + 2\epsilon_{yy}} - 1, \\ +e_z &= \sqrt{1 + 2\epsilon_{zz}} - 1, +\end{aligned} +\Tag{149} +\] +and that the angles between lines originally parallel to the axes are +given in the strained condition by the expressions +\[ +\begin{aligned} +\cos \theta_{yz} + &= \frac{\epsilon_{yz}} + {\sqrt{1 + 2\epsilon_{yy}}\, \sqrt{1 + 2\epsilon_{zz}}}, \\ +\cos \theta_{xz} + &= \frac{\epsilon_{xz}} + {\sqrt{1 + 2\epsilon_{xx}}\, \sqrt{1 + 2\epsilon_{zz}}}, \\ +\cos \theta_{xy} + &= \frac{\epsilon_{xy}} + {\sqrt{1 + 2\epsilon_{xx}}\, \sqrt{1 + 2\epsilon_{yy}}}, +\end{aligned} +\Tag{150} +\] + +Geometrical considerations are also sufficient to show that in +case the strain is a simple elongation of amount~$e$ the following equation +will be true: +\[ +\frac{\epsilon_{xx}}{l^2} = +\frac{\epsilon_{yy}}{m^2} = +\frac{\epsilon_{zz}}{n^2} = +\frac{\epsilon_{yz}}{2mn} = +\frac{\epsilon_{xz}}{2ln} = +\frac{\epsilon_{xy}}{2lm} = e + \tfrac{1}{2}e^2, +\Tag{151} +\] +where $l$,~$m$,~$n$ are the cosines which determine the direction of the +elongation. + +\Subsubsection{117}{Definition of Stress.} We have just considered the expressions +for the strain at a given point in an elastic medium; we may +now define stress in terms of the work done in changing from one +state of strain to another. Considering the material contained in +\emph{unit volume when the body is unstrained}, we may write, for the work +done by this material on its surroundings when a change in strain +takes place, +%% -----File: 147.png---Folio 133------- +\[ +\begin{aligned} +\delta W = -\delta E + = t_{xx}\, \delta\epsilon_{xx} + + t_{yy}\, \delta\epsilon_{yy} + &+ t_{zz}\, \delta\epsilon_{zz} \\ + &+ t_{yz}\, \delta\epsilon_{yz} + + t_{xz}\, \delta\epsilon_{xz} + + t_{xy}\, \delta\epsilon_{xy}, +\end{aligned} +\Tag{152} +\] +and this equation serves to define the stresses $t_{xx}$,~$t_{yy}$,~etc. In case +the strain varies from point to point we must consider of course the +work done \textit{per}~unit volume of the unstrained material. In case the +strains are small it will be noticed that the stresses thus defined are +identical with those used in the familiar theories of elasticity. + +\Subsubsection{118}{Transformation Equations for Strain.} We must now prepare +for the introduction of the theory of relativity into our considerations, +by determining the way the strain at a given point~$P$ appears to observers +moving with different velocities. Let the point~$P$ in question +be moving with the velocity $\vc{u} = x\vc{i} + y\vc{j} + z\vc{k}$ as measured in system~$S$. +Since the state of zero deformation from which to measure +strains can be chosen perfectly arbitrarily, let us for convenience +take the strain as zero as measured in system~$S$, giving us +\[ +\epsilon_{xx} = +\epsilon_{yy} = +\epsilon_{zz} = +\epsilon_{yz} = +\epsilon_{xz} = +\epsilon_{xy} = 0. +\Tag{153} +\] +What now will be the strains as measured by an observer moving +along with the point~$P$ in question? Let us call the system of coördinates +used by this observer~$S°$. It is evident now from our considerations +as to the shape of moving systems presented in \Chapref{V} that +in system~$S°$ the material in the neighborhood of the point in question +will appear to have been elongated in the direction of motion in the +ratio of $1 : \sqrt{1 - \dfrac{u^2}{c^2}}$. Hence in system~$S°$ the strain will be an elongation +\[ +e = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} - 1 +\Tag{154} +\] +in the line determined by the direction cosines +\[ +l = \frac{\dot{x}}{u},\qquad +m = \frac{\dot{y}}{u},\qquad +\DPtypo{u}{n} + = \frac{\dot{z}}{u}. +\Tag{155} +\] + +We may now calculate from this elongation the components of +strain by using equation~(151). We obtain +%% -----File: 148.png---Folio 134------- +{\small%[** TN: Setting on two lines, not six] +\[ +\begin{aligned} +%[** TN: \llap coaxes equation to the left without crowding the tag] +\llap{$\epsilon°$}_{xx} + &= \frac{\dot{x}^2}{2c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],& +{\epsilon°}_{yy} + &= \frac{\dot{y}^2}{2c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],& +{\epsilon°}_{zz} + &= \frac{\dot{z}^2}{2c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],\\ +% +\llap{$\epsilon°$}_{yz}%[** See above] + &= \ \frac{\dot{y}\dot{z}}{c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],& +{\epsilon°}_{xz} + &= \ \frac{\dot{x}\dot{z}}{c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr],& +{\epsilon°}_{xy} + &= \ \frac{\dot{x}\dot{y}}{c^2}\Biggl[\frac{1}{1 - \smfrac{u^2}{c^2}}\Biggr], +\end{aligned} +\Tag{156} +\]}% +and these are the desired equations for the strains at the point~$P$, +the accent~$°$ indicating that they are measured with reference to a +system of coördinates~$S°$ moving along with the point itself. + +\Subsubsection{119}{Variation in the Strain.} We shall be particularly interested +in the variation in the strain as measured in~$S°$ when the velocity +experiences a small variation~$\delta\vc{u}$, the strains remaining zero as measured +in~$S$. For the sake of simplicity let us choose our coördinates +in such a way that the $X$\DPchg{-}{~}axis is parallel to the original velocity, so +that our change in velocity will be from $\vc{u} = \dot{x}\vc{i}$ to +\[ +\vc{u} + \delta\vc{u} + = (\dot{x} + \delta\dot{x})\, \vc{i} + + \delta\dot{y}\, \vc{j} + \delta\dot{z}\, \vc{k}. +\] +Taking $\delta\vc{u}$~small enough so that higher orders can be neglected, and +noting that $\dot{y} = \dot{z} = 0$, we shall then have, from equations~(156), +%% -----File: 149.png---Folio 135------- +\[ +\begin{aligned} +\delta{\epsilon°}_{xx} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta\dot{x},& +\delta{\epsilon°}_{yy} &= 0, \\ +\delta{\epsilon°}_{zz} &= 0, & +\delta{\epsilon°}_{yz} &= 0, \\ +\delta{\epsilon°}_{xz} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta\dot{z},&\qquad +\delta{\epsilon°}_{xy} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta\dot{y}. +\end{aligned} +\Tag{157} +\] + +We shall also be interested in the variation in the strain as measured +in~$S°$ produced by a variation in the strain as measured in~$S$. Considering +again for simplicity that the $X$\DPchg{-}{~}axis is parallel to the motion +of the point, we must calculate the variation produced in ${\epsilon°}_{xx}$,~${\epsilon°}_{yy}$,~etc., +by changing the values of $\epsilon_{xx}$,~$\epsilon_{yy}$,~etc., from zero to $\delta\epsilon_{xx}$,~$\delta\epsilon_{yy}$,~etc. + +The variation~$\delta\epsilon_{xx}$ will produce a variation in~${\epsilon°}_{xx}$ whose amount +can be calculated as follows: By equations~(149) a line which has unit +length and is parallel to the $X$\DPchg{-}{~}axis in the unstrained condition will +have when strained the length $\sqrt{1 + 2\epsilon_{xx}}$ when measured in system~$S$ +and $\sqrt{1 + 2{\epsilon°}_{xx}}$ when measured in system~$S°$. Since the strain in +system~$S$ is small, the line remains sensibly parallel to the $X$\DPchg{-}{~}axis, +which is also the direction of motion, and these quantities will be +connected in accordance with the Lorentz shortening by the equation +\[ +\sqrt{1 + 2\epsilon_{xx}} + = \sqrt{1 - \frac{u^2}{c^2}}\, + \sqrt{1 + 2{\epsilon°}_{xx}}. +\Tag{158} +\] +Carrying out now our variation~$\delta\epsilon_{xx}$, neglecting~$\epsilon_{xx}$ in comparison +with larger quantities and noting that except for second order quantities, +\[ +\sqrt{1 + 2{\epsilon°}_{xx}} = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} +\Tag{159} +\] +we obtain +\[ +\delta{\epsilon°}_{xx} + = \frac{\delta\epsilon_{xx}}{\left(1 - \smfrac{u^2}{c^2}\right)}. +\Tag{160} +\] + +Since the variations $\delta\epsilon_{yy}$,~$\delta\epsilon_{zz}$,~$\delta\epsilon_{yz}$ affect only lines which are at +right angles to the direction of motion, we may evidently write +\[ +\delta{\epsilon°}_{yy} = \delta\epsilon_{yy}, \qquad +\delta{\epsilon°}_{zz} = \delta\epsilon_{zz}, \qquad +\delta{\epsilon°}_{yz} = \delta\epsilon_{yz}. +\Tag{161} +\] +%% -----File: 150.png---Folio 136------- +To calculate $\delta{\epsilon°}_{xz}$ we may note that in accordance with equations~(150) +we must have +\begin{align*} +\cos \theta_{xz} + &= \frac{\epsilon_{xz}} + {\sqrt{1 + 2\epsilon_{xx}}\, \sqrt{1+2\epsilon_{zz}}},\\ +\cos {\theta°}_{xz} + &= \frac{{\epsilon°}_{xz}} + {\sqrt{1 + 2{\epsilon°}_{xx}}\, \sqrt{1 + 2{\epsilon°}_{zz}}}, +\end{align*} +where $\theta_{xz}$~is the angle between lines which in the unstrained condition +are parallel to the $X$~and~$Z$ axes respectively. In accordance with +the Lorentz shortening, however, we shall have +\[ +\cos \theta_{xz} = \sqrt{1 - \frac{u^2}{c^2}} \cos {\theta°}_{xz}. +\] +Introducing this relation, remembering that $\epsilon_{xx} = {\epsilon°}_{zz} = 0$, and +noting equation~(159), we obtain +\begin{align*} +\delta {\epsilon°}_{xz} + &= \frac{\delta\epsilon_{xz}}{\left(1 - \smfrac{u^2}{x^2}\right)}, +\Tag{162} \\ +\intertext{and similarly} +\delta {\epsilon°}_{xy} + &= \frac{\delta\epsilon_{xy}}{\left(1 - \smfrac{u^2}{x^2}\right)}. +\Tag{163} +\end{align*} + +We may now combine these equations (160),~(161),~(162) and~(163) +with those for the variation in strain with velocity and obtain +the final set which we desire: +\[ +\begin{aligned} +\delta {\epsilon°}_{xx} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta \dot{x} + + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \delta\epsilon_{xx}, \\ +\delta {\epsilon°}_{yy} &= \delta\epsilon_{yy}, \\ +\delta {\epsilon°}_{zz} &= \delta\epsilon_{zz}, \\ +\delta {\epsilon°}_{yz} &= \delta\epsilon_{yz}, \\ +\delta {\epsilon°}_{xz} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta \dot{z} + + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \delta\epsilon_{xz}, \\ +\delta {\epsilon°}_{xy} + &= \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, + \frac{\dot{x}}{c^2}\, \delta \dot{y} + + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \delta\epsilon_{xy}. +\end{aligned} +\Tag{164} +\] +%% -----File: 151.png---Folio 137------- + +These equations give the variation in the strain measured in +system~$S°$ at a point~$P$ moving in the $X$~direction with velocity~$u$, +provided the strains are negligibly small as measured in~$S$. + + +\Section[II]{Introduction of the Principle of Least Action.} + +\Subsubsection{120}{The Kinetic Potential for an Elastic Body.} We are now in +a position to develop the mechanics of an elastic body with the help +of the principle of least action. In Newtonian mechanics, as is well +known, the kinetic potential for unit volume of material at a given +point~$P$ in an elastic body may be put equal to the density of kinetic +energy minus the density of potential energy, and it is obvious that +our choice for kinetic potential must reduce to that value at low +velocities. Our choice of an expression for kinetic potential is furthermore +limited by the fundamental transformation equation for kinetic +potential which we found in the last chapter +\[ +\frac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = \frac{H'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}. +\Tag{146} +\] + +Taking these requirements into consideration, we may write for +the kinetic potential per unit volume of the material at a point~$P$ +moving with the velocity~$\vc{u}$ the expression +\[ +H = -E° \sqrt{1 - \frac{u^2}{c^2}}, +\] +where $E°$~is the energy as measured in system~$S°$ of the amount of +material which in the unstrained condition (\ie, as measured in +system~$S$) is contained in unit volume. + +The above expression obviously satisfies our fundamental transformation +equation~(146) and at low velocities reduces in accordance +with the requirements of Newtonian mechanics to +\[ +H = \tfrac{1}{2} m° u^2 - E°, +\] +provided we introduce the substitution made familiar by our previous +work, $m° = \dfrac{E°}{c^2}$. +%% -----File: 152.png---Folio 138------- + +\Subsubsection{121}{Lagrange's Equations.} Making use of this expression for the +kinetic potential in an elastic body, we may now obtain the equations +of motion and stress for an elastic body by substituting into Lagrange's +equations~(139) Chapter~IX\@. %[** TN: Not a useful cross-reference.] + +Considering the material at the point~$P$ contained in unit volume +in the unstrained condition, we may choose as our generalized coördinates +the six component strains $\epsilon_{xx}$,~$\epsilon_{yy}$,~etc., with the corresponding +stresses $-t_{xx}$,~$-t_{yy}$,~etc., as generalized forces, and the +three coördinates $x$,~$y$,~$z$ which give the position of the point with the +corresponding forces $F_x$,~$F_y$ and~$F_z$. + +It is evident that the kinetic potential will be independent of +the time derivatives of the strains, and if we consider cases in which +$E°$~is independent of position, the kinetic potential will also be independent +of the absolute magnitudes of the coördinates $x$,~$y$ and~$z$. +Substituting in Lagrange's equations~(139), we then obtain +\[ +\left. +\begin{aligned} +-\frac{\partial}{\partial \epsilon_{xx}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{xx}, \\ +-\frac{\partial}{\partial \epsilon_{yy}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{yy}, \\ +-\frac{\partial}{\partial \epsilon_{zz}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{zz}, \\ +-\frac{\partial}{\partial \epsilon_{yz}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{yz}, \\ +-\frac{\partial}{\partial \epsilon_{xz}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{xz}, \\ +-\frac{\partial}{\partial \epsilon_{xy}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -t_{xy}, +\end{aligned} +\right\} +\Tag{165} +\] +\[ +\left. +\begin{aligned} +\frac{d}{dt}\, \frac{\partial}{\partial \dot{x}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_x, \\ +\frac{d}{dt}\, \frac{\partial}{\partial \dot{y}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_y, \\ +\frac{d}{dt}\, \frac{\partial}{\partial \dot{z}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_z. +\end{aligned} +\right\} +\Tag{166} +\] +%% -----File: 153.png---Folio 139------- + +We may simplify these equations, however; by performing the +indicated differentiations and making suitable substitutions, we have +\[ +\frac{\partial {E°}_{xx}}{\partial \epsilon_{xx}} + = \frac{\partial {E°}_{xx}}{\partial {\epsilon°}_{xx}}\, + \frac{\partial {\epsilon°}_{xx}}{\partial \epsilon_{xx}}. +\] +But in accordance with equation~(152) we may write +\[ +\frac{\partial {E°}_{xx}}{\partial {\epsilon°}_{xx}} = -{t°}_{xx} +\] +and from equations~(164) we may put +\[ +\frac{\partial {\epsilon°}_{xx}}{\partial \epsilon_{xx}} + = \frac{1}{1 - \smfrac{u^2}{c^2}}. +\] +Making the substitutions in the first of the Lagrangian equations we +obtain +\[ +t_{xx} = -\frac{\partial}{\partial\epsilon_{xx}} + \left(E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) + = {t°}_{xx} \frac{1}{1 - \smfrac{u^2}{c^2}} \sqrt{1 - \frac{u^2}{c^2}} + = \frac{{t°}_{xx}}{\sqrt{1 - \frac{u^2}{c^2}}}. +\] + +\Subsubsection{122}{Transformation Equations for Stress.} Similar substitutions +can be made in all the equations of stress, and we obtain as our set +of transformation equations +\[ +\begin{aligned} +t_{xx} &= \frac{{t°}_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, & +t_{yy} &= \sqrt{1 - \smfrac{u^2}{c^2}}\, {t°}_{yy}, & +t_{zz} &= \sqrt{1 - \smfrac{u^2}{c^2}}\, {t°}_{zz}, \\ +t_{yx} &= \sqrt{1 - \smfrac{u^2}{c^2}}\, {t°}_{yx}, & +t_{xz} &= \frac{{t°}_{xz}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, & +t_{xy} &= \frac{{t°}_{xy}}{\sqrt{1 - \smfrac{u^2}{c^2}}}. +\end{aligned} +\Tag{167} +\] + +%[** TN: Bold symbol in original] +\Subsubsection{123}{Value of $E^\circ$.} With the help of these transformation equations +for stress we may calculate the value of~$E°$, the energy content, as +measured in system~$S°$, of material which in the unstrained condition +is contained in unit volume. + +Consider unit volume of the material in the unstrained condition +and call its energy content~$w°°$. Give it now the velocity $u = \dot{x}$, +keeping its state of strain unchanged in system~$S$. Since the \emph{strain} +%% -----File: 154.png---Folio 140------- +is not changing in system~$S$, the stresses $t_{xx}$,~etc., will also be constant +in system~$S$. In system~$S°$, however, the component strain will +change in accordance with equations~(156) from zero to +\[ +{\epsilon°}_{xx} + = \frac{\dot{x}^2}{2c^2}\, \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}, +\] +and the corresponding stress will be given at any instant by the +expression just derived, +\[ +{t°}_{xx} = t_{xx} \sqrt{1 - \frac{u^2}{c^2}}, +\] +$t_{xx}$ being, as we have just seen, a constant. We may then write for~$E°$ +the expression +\[ +E° = w°° - t_{xx} \int_0^w \sqrt{1 - \frac{u^2}{c^2}}\, + d\Biggl[\frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, + \frac{\dot{x}}{2c^2}\Biggr]. +\] +Noting that $u = \dot{x}$ we obtain on integration, +\[ +E° = w°° + t_{xx} - \frac{t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}} +\Tag{168} +\] +as the desired expression for the energy as measured in system~$S°$ +contained in the material which in system~$S$ is unstrained and has +unit volume. + +\Subsubsection{124}{The Equations of Motion in the Lagrangian Form.} We are +now in a position to simplify the three Lagrangian equations~(166) +for $F_x$,~$F_y$ and~$F_z$. Carrying out the indicated differentiation we have +\[ +F_x = \frac{d}{dt}\, \frac{\partial}{\partial \dot{x}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\right) + = \frac{d}{dt} \Biggl[ + \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2} + - \sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial\dot{x}} + \Biggr], +\] +and introducing the value of~$E°$ given by equation~(168) we obtain +\[ +F_x = \frac{d}{dt} \Biggl[ + \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}\Biggr]. +\Tag{169} +\] +%% -----File: 155.png---Folio 141------- +Simple calculations will also give us values for $F_y$~and~$F_z$. We have +from~(166) +\[ +F_y = \frac{d}{dt}\, \frac{\partial}{\partial \dot{y}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\right) + = \frac{d}{dt} \Biggl[ + \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{y}}{c^2} + - \sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial\dot{y}} + \Biggr]. +\] +But since we have adapted our considerations to cases in which the +direction of motion is along the $X$\DPchg{-}{~}axis, we have $\dot{y} = 0$; furthermore +we may substitute, in accordance with equations (152),~(157) and~(167), +\[ +\frac{\partial E°}{\partial \dot{y}} + = \frac{\partial E°}{\partial {\epsilon°}_{xy}}\, + \frac{\partial {\epsilon°}_{xy}}{\partial \dot{y}} + = -{t°}_{xy}\, \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, + \frac{\dot{x}}{c^2} + = \frac{-t_{xy}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2}. +\] +We thus obtain as our three equations of motion +\[ +\begin{aligned} +F_x &= \frac{d}{dt}\Biggl[ + \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2} + \Biggr], \\ +F_y &= \frac{d}{dt} \left(t_{xy}\, \frac{\dot{x}}{c^2}\right), \\ +F_z &= \frac{d}{dt} \left(t_{xz}\, \frac{\dot{x}}{c^2}\right). +\end{aligned} +\Tag{170} +\] +In these equations the quantities $F_x$,~$F_y$ and~$F_z$ are the components +of force acting on a particular system, namely that quantity of material +which at the instant in question has unit volume. Since the volume +of this material will in general be changing, $F_x$,~$F_y$ and~$F_z$ do not give +us the force per unit volume as usually defined. If we represent, +however, by $f_x$,~$f_y$ and~$f_z$ the components of force per unit volume, +we may rewrite these equations in the form +\[ +\begin{aligned} +F_x\, \delta V &= \frac{d}{dt}\Biggl[ + \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}\, \delta V\Biggr], \\ +F_y\, \delta V &= \frac{d}{dt}\left(t_{xy}\, \frac{\dot{x}}{c^2}\, \delta V\right),\\ +F_z\, \delta V &= \frac{d}{dt}\left(t_{xz}\, \frac{\dot{x}}{c^2}\, \delta V\right), +\end{aligned} + \Tag{171} +\] +%% -----File: 156.png---Folio 142------- +where by $\delta V$ we mean a small element of volume at the point in +question. + +\Subsubsection{125}{Density of Momentum.} Since we customarily define force as +equal to the time rate of change of momentum, we may now write for +the density of momentum~$\vc{g}$ at a point in an elastic body which is +moving in the $X$~direction with the velocity $u = \dot{x}$ +\[ +\vc{g}_x = \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}, \qquad +\vc{g}_y = t_{xy}\, \frac{\dot{x}}{c^2}, \qquad +\vc{g}_y = t_{xy}\, \frac{\dot{x}}{c^2}. +\Tag{172} +\] + +It is interesting to point out that there are components of momentum +in the $Y$~and~$Z$ directions in spite of the fact that the material +at the point in question is moving in the $X$~direction. We shall +later see the important significance of this discovery. + +\Subsubsection{126}{Density of Energy.} It will be remembered that the forces +whose equations we have just obtained are those acting on unit +volume of the material as measured in system~$S$, and hence we are +now in a position to calculate the energy density of our material. +Let us start out with unit volume of our material at rest, with the +energy content~$w°°$ and determine the work necessary to give it the +velocity $u = \dot{x}$ without change in stress or strain. Since the only +component of force which suffers displacement is~$F_x$, we have +\[ +\begin{aligned} +%[** TN: Commas present in original, arguably serve a grammatical purpose] +w &= w°° + \int_0^u \frac{d}{dt} \Biggl[ + \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}\Biggr] \dot{x}\, dt, \\ + &= w°° + (w°° + t_{xx}) + \int_0^u \dot{x}\, d\Biggl[ + \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2}\Biggr], \\ + &= \Biggl\{\frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}} + - t_{xx}\Biggr\} +\end{aligned} +\Tag{173} +\] +as an expression for the energy density of the elastic material. + +\Subsubsection{127}{Summary of Results Obtained from the Principle of Least +Action.} We may now tabulate for future reference the results obtained +from the principle of least action. +%% -----File: 157.png---Folio 143------- + +At a given point in an elastic medium which is moving in the $X$~direction +with the velocity $u = \dot{x}$, we have for the components of +stress +\[ +\begin{aligned} +t_{xx} &= \frac{{t°}_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, & +t_{yy} &= \sqrt{1 - \frac{u^2}{c^2}}\, {t°}_{yy}, & +t_{zz} &= \sqrt{1 - \frac{u^2}{c^2}}\, {t°}_{zz}, \\ +t_{yz} &= \sqrt{1 - \frac{u^2}{c^2}}\, {t°}_{yz}, & +t_{xz} &= \frac{{t°}_{xz}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, & +t_{xy} &= \frac{{t°}_{xz}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\end{aligned} +\Tag{167} +\] +For the density of energy at the point in question we have +\[ +w = \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}} - t_{xx}. +\Tag{173} +\] +For the density of momentum we have +\[ +\vc{g}_{x} + = \frac{w°° + t_{xx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}, \qquad +\vc{g}_y = t_{xy}\, \frac{\dot{x}}{c^2}, \qquad +\vc{g}_z = t_{xz}\, \frac{\dot{x}}{c^2}. +\Tag{172} +\] + + +\Section[III]{Some Mathematical Relations.} + +Before proceeding to the applications of these results which we +have obtained from the principle of least action, we shall find it desirable +to present a number of mathematical relations which will +later prove useful. + +\Subsubsection{128}{The Unsymmetrical Stress Tensor $\vc{t}$.} We have defined the +components of stress acting at a point by equation~(152) +\[ +\delta W + = t_{xx}\, \delta\epsilon_{xx} + + t_{yy}\, \delta\epsilon_{yy} + + t_{zz}\, \delta\epsilon_{zz} + + t_{yz}\, \delta\epsilon_{yz} + + t_{xz}\, \delta\epsilon_{xz} + + t_{xy}\, \delta\epsilon_{xy}, +\] +where $\delta W$~is the work which accompanies a change in strain and is +performed on the surroundings by the amount of material which was +contained in unit volume in the unstrained state. Since for convenience +we have taken as our state of zero strain the condition of +the body as measured in system~$S$, it is evident that the components +$t_{xx}$,~$t_{yy}$,~etc., may be taken as the forces acting on the faces of a unit +cube of material at the point in question, the first letter of the subscript +%% -----File: 158.png---Folio 144------- +indicating the direction of the force and the second subscript +the direction of the normal to the face in question. + +Interpreting the components of stress in this fashion, we may +now add three further components and obtain a complete tensor +\[ +\vc{t} = \left\{ +\begin{matrix} +t_{xx} & t_{xy} & t_{xz} \\ +t_{yx} & t_{yy} & t_{yz} \\ +t_{zx} & t_{zy} & t_{zz} +\end{matrix} +\right. +\Tag{174} +\] + +The three new components $t_{yx}$,~$t_{zx}$,~$t_{zy}$ are forces acting on the +unit cube, in the directions and on the faces indicated by the subscripts. +A knowledge of their value was not necessary for our developments +of the consequences of the principle of least action, since it was +possible to obtain an expression for the work accompanying a change +in strain without their introduction. We shall find them quite important +for our later considerations, however, and may proceed to +determine their value. + +$t_{yz}$ is the force acting in the $Y$~direction tangentially to a face of +the cube perpendicular to the $X$\DPchg{-}{~}axis, and measured with a system +of coördinates~$S$. Using a system of \DPchg{coordinates}{coördinates}~$S°$ which is stationary +with respect to the point in question, we should obtain, for the measurement +of this force, +\[ +{t°}_{yx} = \frac{t_{yx}}{\sqrt{1 - \smfrac{u^2}{c^2}}} +\] +in accordance with our transformation equation for force~(62), Chapter~VI\@. %[** TN: Not a useful cross-reference] +Similarly we shall have the relation +\[ +{t°}_{xy} = t_{xy}. +\] +In accordance with the elementary theory of elasticity, however, the +forces ${t°}_{yx}$~and~${t°}_{xy}$ which are measured by an observer moving with +the body will be connected by the relation +\[ +{t°}_{xy} = \frac{{t°}_{yx}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +${t°}_{xy}$ being larger than~${t°}_{yx}$ in the ratio of the areas of face upon which +they act. Combining these three equations, and using similar methods +%% -----File: 159.png---Folio 145------- +for the other quantities, we can obtain the desired relations +\[ +t_{yx} = \left(1 - \frac{u^2}{c^2}\right) t_{xy}, \qquad +t_{zx} = \left(1 - \frac{u^2}{c^2}\right) t_{xz}, \qquad +t_{zy} = t_{yz}. +\Tag{175} +\] +We see that $\vc{t}$~is an unsymmetrical tensor. + +\Subsubsection{129}{The Symmetrical Tensor $\vc{p}$.} Besides this unsymmetrical tensor~$\vc{t}$ +we shall find it desirable to define a further tensor~$\vc{p}$ by the +equation +\[ +\vc{p} = \vc{t} + \vc{gu}. +\Tag{176} +\] + +We shall call $\vc{gu}$ the tensor product of $\vc{g}$~and~$\vc{u}$ and may indicate +tensor products in general by a simple juxtaposition of vectors. $\vc{gu}$~is +itself a tensor with components as indicated below: +\[ +\vc{gu} = \left\{ +\begin{matrix} +g_xu_x & g_xu_y & g_xu_z, \\ +g_yu_x & g_yu_y & g_yu_z, \\ +g_zu_x & g_zu_y & g_zu_z. +\end{matrix} +\right. +\Tag{177} +\] + +Unlike $\vc{t}$, $\vc{p}$~will be a symmetrical tensor, since we may show, by +substitution of the values for $\vc{g}$~and~$\vc{u}$ already obtained, that +\[ +p_{yx} = p_{xy}, \qquad +p_{zx} = p_{xz}, \qquad +p_{zy} = p_{yz}. +\Tag{178} +\] +Consider for example the value of~$p_{yx}$; we have from our definition +\[ +p_{yx} = t_{yx} + g_{y}u_{x}, +\] +and by equations (175)~and~(172) we have +\[ +t_{xy} = \left(1 - \frac{u^2}{c^2}\right) t_{xy},\qquad +g_y = t_{xy}\, \frac{u_x}{c^2}, +\] +and hence by substitution obtain +\[ +p_{yx} = t_{xy}. +\] +We also have, however, by definition +\[ +p_{xy} = t_{xy} + g_xu_y, +\] +and since for the case we are considering $u_y = 0$, we arrive at the +equality +\[ +p_{xy} = p_{yx}. +\] +The other equalities may be shown in a similar way. +%% -----File: 160.png---Folio 146------- + +\Subsubsection{130}{Relation between $\divg\vc{t}$~and~$\vc{t}_n$.} At a given point~$P$ in our +elastic body we shall define the divergence of the tensor~$\vc{t}$ by the equation +\[ +\begin{aligned} +\divg\vc{t} + &= \left(\frac{\partial t_{xx}}{\partial x} + + \frac{\partial t_{xy}}{\partial y} + + \frac{\partial t_{xz}}{\partial z}\right) \vc{i} \\ + &+ \left(\frac{\partial t_{yx}}{\partial x} + + \frac{\partial t_{yy}}{\partial y} + + \frac{\partial t_{yz}}{\partial z}\right) \vc{j} \\ + &+ \left(\frac{\partial t_{zx}}{\partial x} + + \frac{\partial t_{zy}}{\partial y} + + \frac{\partial t_{zz}}{\partial z}\right) \vc{k}, +\end{aligned} + \Tag{179} +\] +where $\vc{i}$,~$\vc{j}$ and~$\vc{k}$ are unit vectors parallel to the axes, $\divg\vc{t}$~thus being +an ordinary vector. It will be seen that $\divg\vc{t}$~is the elastic force +acting per unit volume of material at the point~$P$. + +Considering an element of surface~$dS$, we shall define a further +vector~$\vc{t}_n$ by the equation +\[ +\begin{aligned} +\vc{t}_n + &= (t_{xx}\cos\alpha + t_{xy}\cos\beta + t_{xz}\cos\gamma)\, \vc{i} \\ + &+ (t_{yx}\cos\alpha + t_{yy}\cos\beta + t_{yz}\cos\gamma)\, \vc{j} \\ + &+ (t_{zx}\cos\alpha + t_{zy}\cos\beta + t_{zz}\cos\gamma)\, \vc{k}, +\end{aligned} +\Tag{180} +\] +where $\cos \alpha$,~$\cos \beta$ and~$\cos \gamma$ are the direction cosines of the inward-pointing +normal to the element of surface~$dS$. + +Considering now a definite volume~$V$ enclosed by the surface~$S$ +it is evident that $\divg\vc{t}$~and~$\vc{t}_n$ will be connected by the relation +\[ +-\int \divg\vc{t}\, dV = \int_0 \vc{t}_n\, dS, +\Tag{181} +\] +where the symbol~$0$ indicates that the integration is to be taken over +the whole surface which encloses the volume~$V$. This equation is +of course merely a direct application of Gauss's formula, which states +in general the equality +{\small%[** TN: Not breaking] +\[ +-\int \left( + \frac{\partial P}{\partial x} + + \frac{\partial Q}{\partial y} + + \frac{\partial R}{\partial z}\right) dV + = \int_0 (P\cos \alpha + Q\cos \beta + R\cos \gamma)\, dS, +\Tag{182} +\]}% +where $P$,~$Q$ and~$R$ may be any functions of $x$,~$y$ and~$z$. +%% -----File: 161.png---Folio 147------- + +We shall also find use for a further relation between $\divg\vc{t}$~and~$\vc{t}_n$. +Consider a given point of reference~$O$, and let $\vc{r}$~be the radius vector +to any point~$P$ in the elastic body; we can then show with the help +of Gauss's Formula~(182) that +\begin{multline*} +-\int (\vc{r} × \divg\vc{t})\, dV = \int_0 (\vc{r} × \vc{t}_n)\, dS \\ +-\int \bigl[(t_{yz}-t_{zy})\vc{jk} + + (t_{xz}-t_{zx})\vc{ik} + + (t_{xy}-t_{yx})\vc{ij}\bigr]\, dV, +\end{multline*} +where $×$~signifies as usual the outer product. Taking account of +equations (172)~and~(175) this can be rewritten +\[ +-\int (\vc{r} × \divg\vc{t})\, dV + = \int_0 (\vc{r} × \vc{t}_n)\, dS - \int(\vc{u} × \vc{g})\, dV. +\Tag{183} +\] + +\Subsubsection{131}{The Equations of Motion in the Eulerian Form.} We saw in +\DPchg{sections (\Secnumref{124})~and~(\Secnumref{125})}{Sections \Secnumref{124}~and~\Secnumref{125}} that the equations of motion in the Lagrangian +form might be written +\[ +\vc{f}\, \delta V = \frac{d}{dt} (\vc{g}\, \delta V), +\] +where $\vc{f}$~is the density of force acting at any point and $\vc{g}$~is the density +of momentum. + +Provided that there are no external forces acting and $\vc{f}$~is produced +solely by the elastic forces, our definition of the divergence of a +tensor will now permit us to put +\[ +\vc{f} = - \divg\vc{t}, +\] +and write for our equation of motion +\[ +(-\divg\vc{t})\, \delta V + = \frac{d}{dt} (\vc{g}\, \delta V) + = \delta V\, \frac{d\vc{g}}{dt} + \vc{g}\, \frac{d(\delta V)}{dt}. +\] +Expressing $\dfrac{d\vc{g}}{dt}$ in terms of partial differentials, and putting +\[ +\frac{d(\delta V)}{dt} = \delta V \divg\vc{u} +\] +we obtain +\[ +-\divg \vc{t} = \left(\frac{\partial \vc{g}}{\partial t} + + u_x\, \frac{\partial \vc{g}}{\partial x} + + u_y\, \frac{\partial \vc{g}}{\partial y} + + u_z\, \frac{\partial \vc{g}}{\partial z}\right) + + \vc{g} \divg\vc{u}. +\] +%% -----File: 162.png---Folio 148------- +Our symmetrical tensor~$\vc{p}$, however, was defined by the equation~(176) +\[ +\vc{p} = \vc{t} + \vc{gu}, +\] +and hence we may now write our equations of motion in the very +beautiful Eulerian form +\[ +-\divg\vc{p} = \frac{\partial \vc{g}}{\partial t}. +\Tag{184} +\] + +We shall find this simple form for the equations of motion very +interesting in connection with our considerations in the last chapter. + + +\Section[IV]{Applications of the Results.} + +We may now use the results which we have obtained from the +principle of least action to elucidate various problems concerning +the behavior of elastic bodies. + +\Subsubsection{132}{Relation between Energy and Momentum.} In our work on +the dynamics of a particle we found that the mass of a particle was +equal to its energy divided by the square of the velocity of light, and +hence have come to expect in general a necessary relation between +the existence of momentum in any particular direction and the transfer +of energy in that same direction. We find, however, in the case +of elastically stressed bodies a somewhat more complicated state of +affairs than in the case of particles, since besides the energy which is +transported bodily by the motion of the medium an additional quantity +of energy may be transferred through the medium by the action +of the forces which hold it in its state of strain. Thus, for example, +in the case of a longitudinally compressed rod moving parallel to its +length, the forces holding it in its state of longitudinal compression +will be doing work at the rear end of the rod and delivering an equal +quantity of energy at the front end, and this additional transfer of +energy must be included in the calculation of the momentum of the +bar. + +As a matter of fact, an examination of the expressions for momentum +which we obtained from the principle of least action will show +the justice of these considerations. For the density of momentum +in the $X$~direction we obtained the expression +\[ +g_x = (w + t_{xx})\, \frac{\dot{x}}{c^2}, +\] +%% -----File: 163.png---Folio 149------- +and we see that in order to calculate the momentum in the $X$~direction +we must consider not merely the energy~$w$ which is being bodily +carried along in that direction with the velocity~$\dot{x}$, but also must take +into account the additional flow of energy which arises from the +stress~$t_{xx}$. As we have already seen in \Secref{128}, this stress~$t_{xx}$ can +be thought of as resulting from forces which act on the front and +rear faces of a centimeter cube of our material. Since the cube is +moving with the velocity~$\dot{x}$, the force on the rear face will do the +work $t_{xx}\dot{x}$~per second and this will be given up at the forward face. +We thus have an additional density of energy-flow in the $X$~direction +of the magnitude~$t_{xx}\dot{x}$ and hence a corresponding density of momentum~$\dfrac{t_{xx}\dot{x}}{c^2}$. + +Similar considerations explain the interesting occurrence of components +of momentum in the $Y$~and~$Z$ directions, +\[ +g_y = t_{xy}\, \frac{\dot{x}}{c^2},\qquad +g_z = t_{xz}\, \frac{\dot{x}}{c^2}, +\] +in spite of the fact that the material involved is moving in the $X$~direction. +The stress~$t_{xy}$, for example, can be thought of as resulting +from forces which act tangentially in the $X$~direction on the pair of +faces of our unit cube which are perpendicular to the $Y$~axis. Since +the cube is moving in the $X$~direction with the velocity~$\dot{x}$, we shall +have the work~$t_{xy}\dot{x}$, done at one surface per second and transferred to +the other, and the resulting flow of energy in the $X$~direction is accompanied +by the corresponding momentum~$\dfrac{t_{xy}\dot{x}}{c^2}$. + +\Subsubsection{133}{The Conservation of Momentum.} It is evident from our +previous discussions that we may write the equation of motion for +an elastic medium in the form +\[ +\vc{f}\, \delta V = \frac{d(\vc{g}\, \delta V)}{dt}, +\] +where $\vc{g}$~is the density of momentum at any given point and $\vc{f}$~is the +force acting per unit volume of material. We have already obtained, +from the principle of least action, expressions~(172) which permit +the calculation of~$\vc{g}$ in terms of the energy density, stress and velocity +at the point in question, and our present problem is to discuss somewhat +further the nature of the force~$\vc{f}$. +%% -----File: 164.png---Folio 150------- + +We shall find it convenient to analyze the total force per unit +volume of material~$\vc{f}$ into those external forces~$\vc{f}_{\text{ext.}}$ like gravity, which +are produced by agencies outside of the elastic body and the internal +force~$\vc{f}_{\text{int.}}$ which arises from the elastic interaction of the parts of the +strained body itself. It is evident from the way in which we have +defined the divergence of a tensor~(179) that for this latter we may +write +\[ +\vc{f}_{\text{int.}} = -\divg\vc{t}. +\Tag{185} +\] +Our equation of motion then becomes +\[ +(\vc{f}_{\text{ext.}} - \divg\vc{t})\, \delta V + = \frac{d(\vc{g}\, \delta V)}{dt}, +\Tag{186} +\] +or, integrating over the total volume of the elastic body, +\[ +\int \vc{f}_{\text{ext.}}\, dV - \int \divg\vc{t}\, dV + = \frac{d}{dt} \int \vc{g}\, dV + = \frac{d\vc{G}}{dt}, +\Tag{187} +\] +where $\vc{G}$ is the total momentum of the body. With the help of the +purely analytical relation~(181) we may transform the above equation +into +\[ +\int \vc{f}_{\text{ext.}}\, dV + \int \vc{t}_n\, dS = \frac{d\vc{G}}{dt}, +\Tag{188} +\] +where $\vc{t}_n$~is defined in accordance with~(180) so that the integral +$\ds\int_{0} \vc{t}_n\, dS$ becomes the force exerted by the surroundings on the surface +of the elastic body. + +In the case of an isolated system both $\vc{f}_{\text{ext.}}$~and~$\vc{t}_n$ would evidently +be equal to zero and we have the principle of the conservation of +momentum. + +\Subsubsection{134}{The Conservation of Angular Momentum.} Consider the +%[** TN: O and P in next line are boldface in the original] +radius vector~$\vc{r}$ from a point of reference~$O$ to any point~$P$ in an elastic +body; then the angular momentum of the body about~$O$ will be +\[ +\vc{M} = \int (\vc{r} × \vc{g})\, dV, +\] +and its rate of change will be +\[ +\frac{d\vc{M}}{dt} + = \int \left(\vc{r} × \frac{d\vc{g}}{dt}\right) dV + + \int \left(\frac{d\vc{r}}{dt} × \vc{g}\right) dV. +\Tag{189} +\] +%% -----File: 165.png---Folio 151------- +Substituting equation~(186), this may be written +\[ +\frac{d\vc{M}}{dt} + = \int (\vc{r} × \vc{f}_{\text{ext.}})\, dV + - \int (\vc{r} × \divg\vc{t})\, dV + \int (\vc{u} × \vc{g})\, dV, +\] +or, introducing the purely mathematical relation~(183) we have, +\[ +\frac{d\vc{M}}{dt} + = \int (\vc{r} × \vc{f}_{\text{ext.}})\, dV + + \int_{0} (\vc{r} × \vc{t}_n)\, dS. +\Tag{190} +\] +We see from this equation that the rate of change of the angular +momentum of an elastic body is equal to the moment of the external +forces acting on the body plus the moment of the surface forces. + +In the case of an isolated system this reduces to the important +principle of the conservation of angular momentum. + +\Subsubsection{135}{Relation between Angular Momentum and the Unsymmetrical +Stress Tensor.} The fact that at a point in a strained elastic medium +there may be components of momentum at right angles to the motion +of the point itself, leads to the interesting conclusion that even in a +state of steady motion the angular momentum of a strained body +will in general be changing. + +This is evident from equation~(189), in the preceding section, +which may be written +\[ +\frac{d\vc{M}}{dt} + = \int \left(\vc{r} × \frac{d\vc{g}}{dt}\right) dV + + \int(\vc{u} × \vc{g})\, dV. +\Tag{191} +\] +In the older mechanics velocity~$\vc{u}$ and momentum~$\vc{g}$ were always in +the same direction so that the last term of this equation became zero. +%[** TN: Awkward grammar/repeated verb in the original.] +In our newer mechanics, however, we have found~(172) components +of momentum at right angles to the velocity and \emph{hence even for a body +moving in a straight line with unchanging stresses and velocity we find +that the angular momentum is increasing at the rate +\[ +\frac{d\vc{M}}{dt} = \int (\vc{u} × \vc{g})\, dV, +\Tag{192} +\] +and in order to maintain the body in its state of uniform motion we +must apply external forces with a turning moment of this same amount}. + +The presence of this increasing angular momentum in a strained +body arises from the unsymmetrical nature of the stress tensor, the integral +$\int (\vc{u} × \vc{g})\, dV$ being as a matter of fact exactly equal to the integral +%% -----File: 166.png---Folio 152------- +over the same volume of the turning moments of the unsymmetrical +components of the stress. Thus, for example, if we have a body moving +in the $X$~direction with the velocity $\vc{u} = \dot{x}\vc{i}$ we can easily see from +equations (172)~and~(175) the truth of the equality +\[ +(\vc{u} × \vc{g}) + = \bigl[(t_{yz} - t_{zy})\, \vc{jk} + + (t_{xz} - t_{zx})\, \vc{ik} + + (t_{xy} - t_{yx})\, \vc{ij}\bigr]. +\] + +\Subsubsection{136}{The Right-Angled Lever.} An interesting example of the +\begin{wrapfigure}{l}{2.125in} + \Fig{14} + \Input[2in]{166} +\end{wrapfigure} +principle that in general a turning +moment is needed for the uniform +translatory motion of a strained body +is seen in the apparently paradoxical +case of the right-angled lever. + +Consider the right-angled lever +shown in \Figref{14}. This lever is stationary +with respect to a system of +coördinates~$S°$. Referred to~$S°$ the +two lever arms are equal in length: +\[ +{l_1}° = {l_2}°, +\] +and the lever is in equilibrium under the action of the equal forces +\[ +{F_1}° = {F_2}°. +\] + +Let us now consider the equilibrium as it appears, using a system +of coördinates~$S$ with reference to which the lever is moving in $X$~direction +with the velocity~$V$. Referred to this new system of coördinates +the length~$l_1$ of the arm which lies in the $Y$~direction will be +the same as in system~$S°$, giving us +\[ +l_1 = {l_1}°. +\] +But for the other arm which lies in the direction of motion we shall +have, in accordance with the Lorentz shortening, +\[ +l_2 = {l_2}° \sqrt{1 - \frac{V^2}{c^2}}. +\] +For the forces $F_1$~and~$F_2$ we shall have, in accordance with our equations +%% -----File: 167.png---Folio 153------- +for the transformation of force (61)~and~(62), +\begin{align*} +F_1 &= {F_1}°, \\ +F_2 &= {F_2}° \sqrt{1 - \frac{V^2}{c^2}}. +\end{align*} +We thus obtain for the moment of the forces around the pivot~$B$ +\[ +F_1l_1 - F_2l_2 + = {F_1}° {l_1}° + - {F_2}° {l_2}° \left(1 - \frac{V^2}{c^2}\right) + = {F_1}°{l_1}°\, \frac{V^2}{c^2}, + = F_1l_1\, \frac{V^2}{c^2}, +\] +and are led to the remarkable conclusion that such a moving lever +will be in equilibrium only if the external forces have a definite turning +moment of the magnitude given above. + +The explanation of this apparent paradox is obvious, however, +in the light of our previous discussion. In spite of the fact that the +lever is in uniform motion in a straight line, its angular momentum +is continually increasing owing to the fact that it is elastically strained, +and it can be shown by carrying out the integration indicated in +equation~(192) that the rate of change of angular momentum is as a +matter of fact just equal to the turning moment $F_1l_1\, \dfrac{V^2}{c^2}$. + +This necessity for a turning moment $F_1l_1\, \dfrac{V^2}{c^2}$ can also be shown +directly from a consideration of the energy flow in the lever. Since +the force~$F_1$ is doing the work $F_1V$~per second at the point~$A$, a stream +of energy of this amount is continually flowing through the lever +from~$A$ to the pivot~$B$. In accordance with our ideas as to the relation +between energy and mass, this new energy which enters at~$A$ each +second has the mass~$\dfrac{F_1V}{c^2}$, and hence each second the angular momentum +of the system around the point~$B$ is increased by the amount +\[ +\frac{F_1V}{c^2}\, Vl_1 = F_1l_1\, \frac{V^2}{c^2}. +\] +We have already found, however, exactly this same expression for +the moment of the forces around the pivot~$B$ and hence see that they +are of just the magnitude necessary to keep the lever from turning, +thus solving completely our apparent paradox. +%% -----File: 168.png---Folio 154------- + +\Subsubsection{137}{Isolated Systems in a Steady State.} Our considerations have +shown that the density of momentum is equal to the density of energy +flow divided by the square of the velocity of light. If we have a +system which is in a steady internal state, and is either isolated or +merely subjected to an external pressure with no components of force +tangential to the bounding surface, it is evident that the resultant +flow of energy for the whole body must be in the direction of motion, +and hence for these systems momentum and velocity will be in the +same direction without the complications introduced by a transverse +energy flow. + +Thus for an \emph{isolated} system in a steady \emph{internal} state we may +write, +\[ +\vc{G} = \frac{E}{c^2}\, \vc{u} + = \frac{\smfrac{E°}{c^2}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{u}. +\Tag{193} +\] + +\Subsubsection{138}{The Dynamics of a Particle.} It is important to note that +particles are interesting examples of systems in which there will +obviously be no transverse component of energy flow since their +infinitesimal size precludes the action of tangential surface forces. +We thus see that the dynamics of a particle may be regarded as a +special case of the more general dynamics which we have developed +in this chapter, the equation of motion for a particle being +\[ +\vc{F} = \frac{d}{dt} \left[ + \frac{\smfrac{E°}{c^2}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\,\vc{u} + \right] + = \frac{d}{dt} \Biggl[ + \frac{m°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\vc{u}\Biggr], +\] +in agreement with the work of \Chapref{VI}. + +\Subsubsection{139}{Conclusion.} We may now point out in conclusion the chief +results of this chapter. With the help of Einstein's equations for +spatial and temporal considerations, we have developed a set of +transformation equations for the strain in an elastic body. Using the +components of strain and velocity as generalized coördinates, we then +introduced the principle of least action, choosing a form of function +%% -----File: 169.png---Folio 155------- +for kinetic potential which agrees at low velocities with the choice +made in the older theories of elasticity and at all velocities agrees +with the requirements of the principle of relativity. Using the +Lagrangian equations, we were then able to develop all that is necessary +for a complete theory of elasticity. + +The most important consequence of these considerations is an +extension in our ideas as to the relation between momentum and +energy. We find that the density of momentum in any direction +must be placed equal to the total density of energy flow in that same +direction divided by the square of the velocity of light; and we find +that we must include in our density of energy flow that transferred +through the elastic body by the forces which hold it in its state of +strain and suffer displacement as the body moves. This involves in +general a flow of energy and hence momentum at right angles to the +motion of the body itself. + +At present we have no experiments of sufficient accuracy so that +we can investigate the differences between this new theory of elasticity +and the older ones, and hence of course have found no experimental +contradiction to the new theory. It will be seen, however, from the +expressions for momentum that even at low velocities the consequences +of this new theory will become important as soon as we +run across elastic systems in which very large stresses are involved. +It is also important to show that a theory of elasticity can be developed +which agrees with the requirements of the theory of relativity. +In fairness, it must, however, be pointed out in conclusion that since +our expression for kinetic potential was not absolutely uniquely determined +there may also be other theories of elasticity which will agree +with the principle of relativity and with all the facts as now known. +%% -----File: 170.png---Folio 156------- + + +\Chapter{XI}{The Dynamics of a Thermodynamic System.} +\SetRunningHeads{Chapter Eleven.}{Dynamics of a Thermodynamic System.} + +We may now use our conclusions as to the relation between the +principle of least action and the theory of relativity to obtain information +as to the behavior of thermodynamic systems in motion. + +\Subsubsection{140}{The Generalized Coördinates and Forces.} Let us consider a +thermodynamic system whose state is defined by the \emph{generalized +coördinates} volume~$v$, entropy~$S$ and the values of $x$,~$y$ and~$z$ which +determine its position. Corresponding to these coördinates we shall +have the generalized external forces, the negative of the pressure,~$-p$, +temperature,~$T$, and the components of force, $F_x$,~$F_y$ and~$F_z$. +These generalized coördinates and forces are related to the energy +change~$\delta E$ accompanying a small displacement~$\delta$, in accordance with +the equation +\[ +\delta E = -\delta W + = -p\, \delta v + T\, \delta S + + F_x\, \delta x + F_y\, \delta y + F_z\, \delta z. +\Tag{194} +\] + +\Subsubsection{141}{Transformation Equation for Volume.} Before we can apply +the principle of least action we shall need to have transformation +equations for the generalized coördinates, volume and entropy. + +In accordance with the Lorentz shortening, we may write the +following expression for the volume~$v$ of the system in terms of~$v°$ as +measured with a set of axes~$S°$ with respect to which the system is +stationary: +\[ +v = v° \sqrt{1 - \frac{u^2}{c^2}} + = v° \sqrt{1 - \frac{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}{c^2}}, +\] +where $u$ is the velocity of the system. + +By differentiation we may obtain expressions which we shall find +useful, +\begin{align*} +\frac{\partial v°}{\partial v} + &= \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\Tag{195}\displaybreak[0] \\ +\frac{\partial v°}{\partial \dot{x}} + &= \frac{v}{\left(1 - \smfrac{u^2}{c^2}\right)^{\frac{3}{2}}}\, + \frac{\dot{x}}{c^2} + = \frac{v°}{\left(1 - \smfrac{u^2}{c^2}\right)}\, + \frac{\dot{x}}{c^2}. +\Tag{196} +\end{align*} +%% -----File: 171.png---Folio 157------- + +\Subsubsection{142}{Transformation Equation for Entropy.} As for the entropy +of a thermodynamic system, this is a quantity which must appear +the same to all observers regardless of their motion. This invariance +of entropy is a direct consequence of the close relation between the +entropy of a system in a given state and the probability of that state. +Let us write, in accordance with the Boltzmann-Planck ideas as to +the interdependence of these quantities, +\[ +S = k\log W, +\] +where $S$ is the entropy of the system in the state in question, $k$~is a +universal constant, and $W$~the probability of having a microscopic +arrangement of molecules or other elementary constituent parts which +corresponds to the desired thermodynamic state. Since this probability +is evidently independent of the relative motion of the observer +and the system we see that the entropy of a system~$S$ must be an +invariant and may write +\[ +S = S°. +\Tag{197} +\] + +\Subsubsection{143}{Introduction of the Principle of Least Action. The Kinetic +Potential.} We are now in a position to introduce the principle of +least action into our considerations by choosing a form of function +for the kinetic potential which will agree at low velocities with the +familiar principles of thermodynamics and will agree at all velocities +with the requirements of the theory of relativity. + +If we use volume and entropy as our generalized coördinates, these +conditions are met by taking for kinetic potential the expression +\[ +H = -E° \sqrt{1 - \frac{u^2}{c^2}}. +\Tag{198} +\] + +This expression agrees with the requirements of the theory of +relativity that $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ shall be an invariant (see \Secref{111}) and +at low velocities reduces to $H = -E$, which with our choice of +coördinates is the familiar form for the kinetic potential of a thermodynamic +system. +%% -----File: 172.png---Folio 158------- + +It should be noted that this expression for the kinetic potential +of a thermodynamic system applies of course only provided we pick +out volume~$v$ and entropy~$S$ as generalized coördinates. If, following +Helmholtz, we should think it more rational to take $v$ as one coördinate +and a quantity~$\theta$ whose time derivative is equal to temperature, +$\dot{\theta} = T$, as the other coördinate, we should obtain of course a different +expression for the kinetic potential; in fact should have under those +circumstances +\[ +H = (E° - T° S°) \sqrt{1 - \frac{u^2}{c^2}}. +\] +Using this value of kinetic potential, however, with the corresponding +coördinates we should obtain results exactly the same as those which +we are now going to work out with the help of the other set of coördinates. + +\Subsubsection{144}{The Lagrangian Equations.} Having chosen a form for the +kinetic potential we may now substitute into the Lagrangian equations~(139) +and obtain the desired information with regard to the +behavior of thermodynamic systems. + +Since we shall consider cases in which the energy of the system is +independent of the position in space, the kinetic potential will be +independent of the coördinates $x$,~$y$ and~$z$, depending only on their +time derivatives. Noting also that the kinetic potential is independent +of the time derivatives of volume and entropy, we shall +obtain the Lagrangian equations in the simple form +\[ +\begin{aligned} +-\frac{\partial}{\partial v} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= -p, \\ +-\frac{\partial}{\partial S} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= T, \\ +\frac{d}{dt}\, \frac{\partial}{\partial\dot{x}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_x, \\ +\frac{d}{dt}\, \frac{\partial}{\partial\dot{y}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_y, \\ +\frac{d}{dt}\, \frac{\partial}{\partial\dot{z}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) &= F_z. +\end{aligned} +\Tag{199} +\] +%% -----File: 173.png---Folio 159------- + +\Subsubsection{145}{Transformation Equation for Pressure.} We may use the first +of these equations to show that the pressure is a quantity which +appears the same to all observers regardless of their relative motion. +We have +\[ +p = \frac{\partial}{\partial v} \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\right) + = -\sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial v} + = -\sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial v°}\, + \frac{\partial v°}{\partial v}. +\] +But, in accordance with equation~(194), $p° = -\dfrac{\partial E°}{\partial v°}$, and in accordance +with equation~(195), +\[ +\frac{\partial v°}{\partial v} = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +which gives us the desired relation +\[ +p = p°. +\Tag{200} +\] + +Defining pressure as force per unit area, this result will be seen +to be identical with that which is obtained from the transformation +equations for force and area which result from our earliest considerations. + +\Subsubsection{146}{Transformation Equation for Temperature.} The second of +the Lagrangian equations~(199) will provide us information as to +measurements of temperature made by observers moving with different +velocities. We have +\[ +T = \frac{\partial}{\partial S} + \left(E° \sqrt{1 - \frac{u^2}{c^2}}\right) + = \sqrt{1 - \frac{u^2}{c^2}}\, \frac{\partial E°}{\partial S°}\, + \frac{\partial S°}{\partial S}. +\] +But, in accordance with equation~(194), $\dfrac{\partial E°}{\partial S°} = T°$ and in accordance +with~(197) $\dfrac{\partial S°}{\partial S} = 1$. We obtain as our transformation equation, +\[ +T = T° \sqrt{1 - \frac{u^2}{c^2}}, +\Tag{201} +\] +and see that the quantity $\dfrac{T}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is an invariant for the Lorentz +transformation\DPchg{}{.} +%% -----File: 174.png---Folio 160------- + +\Subsubsection{147}{The Equations of Motion for Quasistationary Adiabatic +Acceleration.} Let us now turn our attention to the last three of the +Lagrangian equations. These are the equations for the motion of a +thermodynamic system under the action of external force. It is +evident, however, that these equations will necessarily apply only +to cases of quasistationary acceleration, since our development of +the principle of least action gave us an equation for kinetic potential +which was true only for systems of infinitesimal extent or large systems +in a steady internal state. It is also evident that we must confine our +considerations to cases of adiabatic acceleration, since otherwise the +value of~$E°$ which occurs in the expression for kinetic potential might +be varying in a perfectly unknown manner. + +The Lagrangian equations for force may be advantageously transformed. +We have +\begin{align*} +F_x &= \frac{d}{dt}\, \frac{\partial}{\partial \dot{x}} + \left(-E° \sqrt{1 - \frac{u^2}{c^2}}\;\right) + = \frac{d}{dt} \Biggl[ + \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2} + - \sqrt{1 - \frac{u^2}{c^2}}\, + \frac{\partial E°}{\partial \dot{x}}\Biggr] \\ + &= \frac{d}{dt}\Biggl\{ + \frac{E°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \frac{\dot{x}}{c^2} + - \sqrt{1 - \frac{u^2}{c^2}} + \left(\frac{\partial E°}{\partial v°}\, + \frac{\partial v°}{\partial \dot{x}} + + \frac{\partial E°}{\partial S°}\, + \frac{\partial S°}{\partial \dot{x}}\right)\Biggr\}. +\end{align*} +But by equations (194),~(196) and~(197) we have +\[ +\frac{\partial E°}{\partial v°} = -p°, \qquad +\frac{\partial v°}{\partial \dot{x}} + = \frac{v°}{\left(1 - \smfrac{u^2}{c^2}\right)}\, + \frac{\dot{x}}{c^2}, \qquad\text{and}\qquad +\frac{\partial S°}{\partial \dot{x}} = 0. +\] +We obtain +\[ +F_x = \frac{d}{dt}\Biggl\{ + \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\dot{x}}{c^2}\Biggr\}. +\Tag{202} +\] + +Similar equations may be obtained for the components of force in +the $Y$~and~$Z$ directions and these combined to give the vector equation +\[ +\vc{F} = \frac{d}{dt} \Biggl\{ + \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}\Biggr\}. +\Tag{203} +\] +%% -----File: 175.png---Folio 161------- + +This is the fundamental equation of motion for the dynamics of a +thermodynamic system. + +\Subsubsection{148}{The Energy of a Moving Thermodynamic System.} We may +use this equation to obtain an expression for the energy of a moving +thermodynamic system. If we adiabatically accelerate a thermodynamic +system in the direction of its motion, its energy will increase +both because of the work done by the force +\[ +\vc{F} = \frac{d}{dt} \Biggl\{ + \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}\Biggr\} +\] +which produces the acceleration and because of the work done by the +pressure $p = p°$ which acts on a volume which is continually diminishing +as the velocity~$u$ increases, in accordance with the expression +$v = v° \sqrt{1 - \dfrac{u^2}{c^2}}$. Hence we may write for the total energy +\begin{align*} +E &= E° + \int_0^u \frac{d}{dt}\Biggl\{ + \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}\Biggr\} \vc{u}\, dt + + p° v° \left(1 - \sqrt{1 - \frac{u^2}{c^2}}\:\right)\DPchg{}{,} \\ +E &= \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}} + - p° v° \sqrt{1 - \frac{u^2}{c^2}} + = \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}} - pv. +\Tag{204} +\end{align*} + +\Subsubsection{149}{The Momentum of a Moving Thermodynamic System.} We +may compare this expression for the energy of a thermodynamic +system with the following expression for momentum which is evident +from the equation~(203) for force: +\[ +\vc{G} = \frac{E° + p° v°}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}. +\Tag{205} +\] + +We find again, as in our treatment of \Chapnumref[X]{elastic bodies} presented +in the last chapter, that the momentum of a moving system may be +calculated by taking the \emph{total} flow of energy in the desired direction +%% -----File: 176.png---Folio 162------- +and dividing by~$c^2$. Thus, comparing equations (204)~and~(205), +we have +\[ +\vc{G} = \frac{E}{c^2}\, \vc{u} + \frac{pv}{c^2}\, \vc{u}, +\Tag{206} +\] +where the term $\dfrac{E}{c^2}\, \vc{u}$ takes care of the energy transported bodily along +by the system and the term $\dfrac{pv}{c^2}\, \vc{u}$ takes care of the energy transferred +in the $\vc{u}$~direction by the action of the external pressure on the rear +and front end of the moving system. + +\Subsubsection{150}{The Dynamics of a Hohlraum.} As an application of our considerations +we may consider the dynamics of a hohlraum, since a +hohlraum in thermodynamic equilibrium is of course merely a special +example of the general dynamics which we have just developed. The +simplicity of the hohlraum and its importance from a theoretical +point of view make it interesting to obtain by the present method the +same expression for momentum that can be obtained directly but +with less ease of calculation from electromagnetic considerations. + +As is well known from the work of Stefan and Boltzmann, the +energy content~$E°$ and pressure~$p°$ of a hohlraum at rest and in thermodynamic +equilibrium are completely determined by the temperature~$T°$ +and volume~$v°$ in accordance with the equations +\begin{align*} +E° &= av° {T°}^4, \\ +p° &= \frac{a}{3}\, {T°}^4, +\end{align*} +where $a$~is the so-called Stefan's constant. + +Substituting these values of $E°$~and~$p°$ in the equation for the +motion of a thermodynamic system~(203), we obtain +\[ +\vc{F} = \frac{d}{dt}\Biggl[ + \frac{4}{3}\, \frac{av° {T°}^4}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}\Biggr] + = \frac{d}{dt}\Biggl[ + \frac{4}{3}\, \frac{avT^4}{\left(1 - \smfrac{u^2}{c^2}\right)^3}\, + \frac{\vc{u}}{c^2}\Biggr] +\Tag{207} +\] +as the equation for the quasistationary adiabatic acceleration of a +%% -----File: 177.png---Folio 163------- +hohlraum. In view of this equation we may write for the momentum +of a hohlraum the expression +\[ +\vc{G} = \frac{4}{3}\, \frac{av°{T°}^4}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, + \frac{\vc{u}}{c^2}. +\Tag{208} +\] + +It is a fact of significance that our dynamics leads to a result for +the momentum of a hohlraum which had been adopted on the ground +of electromagnetic considerations even without the express introduction +of relativity theory. +%% -----File: 178.png---Folio 164------- + + +\Chapter{XII}{Electromagnetic Theory.} +\SetRunningHeads{Chapter Twelve.}{Electromagnetic Theory.} + +The Einstein theory of relativity proves to be of the greatest +significance for electromagnetics. On the one hand, the new electromagnetic +theory based on the first postulate of relativity obviously +accounts in a direct and straightforward manner for the results of the +Michelson-Morley experiment and other unsuccessful attempts to +detect an ether drift, and on the other hand also accounts just as +simply for the phenomena of moving dielectrics as did the older +theory of a stationary ether. Furthermore, the theory of relativity +provides considerably simplified methods for deriving a great many +theorems which were already known on the basis of the ether theory, +and gives us in general a clarified insight into the nature of electromagnetic +action. + +\Subsubsection{151}{The Form of the Kinetic Potential.} In \Chapref{IX} we investigated +the general relation between the principle of least action +and the theory of the relativity of motion. We saw that the development +of any branch of dynamics would agree with the requirements +of relativity provided only that the kinetic potential~$H$ has such a form +that the quantity $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ is an invariant for the Lorentz transformation. +Making use of this discovery we have seen the possibility +of developing the dynamics of a particle, the dynamics of an elastic +body, and the dynamics of a thermodynamic system, all of them in +forms which agree with the theory of relativity by merely introducing +slight modifications into the older expressions for kinetic potential in +such a way as to obtain the necessary invariance for $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$. +In the case of electrodynamics, however, on account of the closely +interwoven historical development of the theories of electricity and +relativity, we shall not find it necessary to introduce any modification +%% -----File: 179.png---Folio 165------- +in the form of the kinetic potential, but may take for~$H$ the following +expression, which is known to lead to the familiar equations of the +Lorentz electron theory +\[ +H = \int dV \left\{\frac{\vc{e}^2}{2} + \frac{\curl \vc{\phi})^2}{2} + - \vc{\phi} · \left(\frac{\dot{\vc{e}}}{c} + \rho\, \frac{\vc{u}}{c}\right) +\right\}, +\Tag{209} +\] +where the integration is to extend over the whole volume of the +system~$V$, $\vc{e}$~is the intensity of the electric field at the point in question, +$\vc{\phi}$~is the value of the vector potential, $\rho$~the density of charge and $\vc{u}$~its +velocity.\footnote + {Strictly speaking this expression for kinetic potential is not quite correct, + since kinetic potential must have the dimensions of energy. To complete the equation + and give all the terms their correct dimensions, we could multiply the first term + by the dielectric inductivity of free space~$\epsilon$, and the last two terms by the magnetic + permeability~$\mu$. Since, however, $\epsilon$~and~$\mu$ have the numerical value unity with the + usual choice of units, we shall not be led into error in our particular considerations + if we omit these factors.} + +Let us now show that the expression which we have chosen for +kinetic potential does lead to the familiar equations of the electron +theory. + +\Subsubsection{152}{The Principle of Least Action.} If now we denote by~$\vc{f}$ the +force per unit volume of material exerted by the electromagnetic +action it is evident that we may write in accordance with the principle +of least action~(135) +\[ +\int dt\, dV \left[\delta \left\{ + \frac{\vc{e}^2}{2} + \frac{(\curl \vc{\phi})^2}{2} + - \vc{\phi} · \left(\frac{\dot{\vc{e}}}{c} + \rho\, \frac{\vc{u}}{c}\right) + \right\} + \vc{f}· \delta\vc{r} \right] = 0, +\Tag{210} +\] +where $\delta\vc{r}$ is the variation in the radius vector to the particle under +consideration, and where the integration is to be taken over the +whole volume occupied by the system and between two instants of +time $t_1$~and~$t_2$ at which the actual and displaced configurations of the +system coincide. + +\Subsubsection{153}{The Partial Integrations.} In order to simplify this equation, +we shall need to make use of two results which can be obtained by +partial integrations with respect to time and space respectively. + +Thus we may write +\[ +\int_{t_1}^{t_2} dt\, (a\, \dot{\delta b}) + = \int_{t_1}^{t_2} a\, d(\delta b) + = [a\, \delta b]_{t_1}^{t_2} + - \int_{t_1}^{t_2} dt \left(\frac{da}{dt}\, \delta b\right), +\] +%% -----File: 180.png---Folio 166------- +or, since the displaced and actual motions coincide at $t_1$~and~$t_2$, +\[ +\int dt\, (a\, \dot{\delta b}) + = -\int dt \left(\frac{da}{dt}\, \delta b\right)\DPtypo{}{.} +\Tag{211} +\] +We may also write +\[ +\int dV \left(a\, \frac{db}{dx}\right) + = \int dy\, dz\, (a\, db) + = \int dy\, dz\, [ab]_{x=-\infty}^{x=+\infty} + - \int dV \left(b\, \frac{da}{dx}\right), +\] +or, since we are to carry out our integrations over the whole volume +occupied by the system, we shall take our functions as zero at the +limits of integration and may write +\[ +\int dV \left(a\, \frac{db}{dx}\right) + = -\int dV \left(b\, \frac{da}{dx}\right). +\Tag{212} +\] +Since similar considerations apply to derivatives with respect to the +other variables $y$~and~$z$, we can also obtain +\begin{gather*} +\int dV\, a \divg\vc{b} = -\int dV\, \vc{b} · \grad a, +\Tag{213} \\ +\int dV\, \vc{a} · \curl\vc{b} = \int dV\, \vc{b} · \curl\vc{a}. +\Tag{214} +\end{gather*} + +\Subsubsection{154}{Derivation of the Fundamental Equations of Electromagnetic +Theory.} {\stretchyspace% +Making use of these purely mathematical relationships we +are now in a position to develop our fundamental equation~(210). +Carrying out the indicated variation, noting that $\delta \vc{u} = \dfrac{d(\delta\vc{r})}{dt}$ and +making use of (211)~and~(214) we easily obtain} +\[ +\begin{aligned} +\int dt\, dV \Biggl[ + \left\{\vc{e} + \frac{1}{c}\, \frac{\partial\vc{\phi}}{\partial t}\right\} + · \delta\vc{e} + &+ \left\{\curl\curl\vc{\phi} - \left(\frac{\dot{\vc{e}}}{c} + + \rho\, \frac{\vc{u}}{c}\right) \right\} · \delta\vc{\phi} \\ + &\qquad\qquad + - \frac{\vc{\phi}}{c} · \delta(\rho\vc{u}) + \vc{f}· \delta\vc{r}\Biggr] + = 0. +\end{aligned} +\Tag{215} +\] + +In developing the consequences of this equation, it should be +noted, however, that the variations are not all of them independent; +thus, since we shall define the density of charge by the equation +\[ +\rho = \divg\vc{e}, +\Tag{216} +\] +it is evident that it will be necessary to preserve the truth of this +equation in any variation that we carry out. This can evidently be +%% -----File: 181.png---Folio 167------- +done if we add to our equation~(215) the expression +\[ +\int dt\, dV\, \psi[\delta\rho - \divg\delta\vc{e}] = 0, +\] +where $\psi$~is an undetermined scalar multiplier. We then obtain with +the help of~(213) +{\small% +%[** TN: Re-breaking] +\[ +\begin{aligned} +&\int dt\, dV \Biggl[ + \left\{\vc{e} + \frac{1}{c}\, \frac{\partial\vc{\phi}}{\partial t} + + \grad\psi \right\} · \delta\vc{e} \\ ++& \left\{\curl\curl\vc{\phi} - \left(\frac{\dot{\vc{e}}}{c} + + \rho\, \frac{\vc{u}}{c}\right)\right\} · \delta\vc{\phi} + - \frac{\vc{\phi}}{c} · \delta(\rho\vc{u}) + \psi\, \delta\rho + + \vc{f} · \delta\vc{r}\Biggr] = 0, +\end{aligned} +\Tag{217} +\]}% +and may now treat the variations $\delta \vc{e}$~and~$\delta\vc{\phi}$ as entirely independent +of the others; we must then have the following equations true +\begin{gather*} +\vc{e} = -\frac{1}{c}\, \frac{\partial \vc{\phi}}{\partial t} - \grad \psi, +\Tag{218} \\ +\curl\curl\vc{\phi} = \frac{\dot{\vc{e}}}{c} + \frac{\rho\vc{u}}{c}, +\Tag{219} +\end{gather*} +and have thus derived from the principle of least action the fundamental +equations of modern electron theory. We may put these in +their familiar form by defining the magnetic field strength~$\vc{h}$ by the +equation +\[ +\vc{h} = \curl\vc{\phi}\DPtypo{}{.} +\Tag{220} +\] +We then obtain from~(219) +\begin{align*}%[** TN: Next four equations not aligned in original] +\curl\vc{h} &= \frac{1}{c}\, \frac{\partial\vc{e}}{\partial t} + + \rho\, \frac{\vc{u}}{c}, +\Tag{221} \\ +\intertext{and, noting the mathematical identity $\curl\grad\psi = 0$, we obtain +from (218)} +\curl\vc{e} &= -\frac{1}{c}\, \frac{\partial\vc{h}}{\partial t}. +\Tag{222} \\ +\intertext{We have furthermore by definition~(216)} +\divg\vc{e} &= \rho, +\Tag{223} \\ +\intertext{and noting equation~(220) may write the mathematical identity} +\divg\vc{h} &= 0. +\Tag{224} +\end{align*} +%% -----File: 182.png---Folio 168------- + +These four equations~\DPchg{(221--4)}{(221)--(224)} are the familiar expressions which +have been made the foundation of modern electron theory. They +differ from Maxwell's original four field equations only by the introduction +in (221)~and~(223) of terms which arise from the density of +charge~$\rho$ of the electrons, and reduce to Maxwell's set in free space. + +\Paragraph{155.} We have not yet made use of the last three terms in the +fundamental equation~(217) which results from the principle of least +action. As a matter of fact, it can be shown that these terms can be +transformed into the expression +\[ +\int dt\, dV \left[ + \frac{\rho}{c}\, \frac{\partial\vc{\phi}}{\partial t} + - \frac{\rho}{c}\, [\vc{u} × \curl\vc{\phi}]^* + + \rho \grad\psi + \vc{f}\right] · \delta\vc{r}, +\Tag{225} +\] +and hence lead to the familiar fifth fundamental equation of modern +electron theory, +\begin{align*} +\vc{f} &= \rho \left\{-\frac{\partial\vc{\phi}}{c\partial t} + - \grad\psi + \left[\frac{\vc{u}}{c} × \curl\vc{\Phi}\right]^*\right\}, \\ +\vc{f} &= \rho \left\{\vc{e} + \left[\frac{\vc{u}}{c} × \vc{h}\right]^*\right\}. +\Tag{226} +\end{align*} +The transformation of the last three terms of~(217) into the form +given above~(225) is a complicated one and it has not seemed necessary +to present it here since in a later paragraph we shall show the +possibility of deriving the fifth fundamental equation of the electron +theory~(226) by combining the four field equations~\DPchg{(221--4)}{(221)--(224)} with the +transformation equations for force already obtained from the principle +of relativity. The reader may carry out the transformation himself, +however, if he makes use of the partial integrations which we have +already obtained, notes that in accordance with the principle of the +conservation of electricity we must have $\delta\rho = - \divg\rho\, \delta\vc{r}$ and notes +that $\delta\vc{u} = \dfrac{d(\delta\vc{r})}{dt}$, where the differentiation $\smash{\dfrac{d}{dt}}\rule{0pt}{12pt}$ indicates that we are +following some particular particle in its motion, while the differentiation +$\dfrac{\partial}{\partial t}$ occurring in $\dfrac{\partial\vc{\phi}}{\partial t}$ indicates that we intend the rate of change +at some particular stationary point. + +\Subsubsection{156}{The Transformation Equations for $\vc{e}$,~$\vc{h}$ and~$\rho$.} We have thus +shown the possibility of deriving the fundamental equations of modern +%% -----File: 183.png---Folio 169------- +electron theory from the principle of least action. We now wish to +introduce the theory of relativity into our discussions by presenting +a set of equations for transforming measurements of $\vc{e}$,~$\vc{h}$ and~$\rho$ from +one set of space-time coördinates~$S$ to another set~$S'$ moving past~$S$ +in the $X$\DPchg{-}{~}direction with the velocity~$V$. This set of equations is as +follows: +\begin{gather*} %[** TN: Set equation groups on one line each] +\begin{alignat*}{3} +{e_x}' &= e_x, \qquad & +{e_y}' &= \kappa \left(e_y - \frac{V}{c}h_z\right),\qquad & +{e_z}' &= \kappa \left(e_z + \frac{V}{c}h_y\right), \Tag{227}\displaybreak[0] \\ +{h_x}' &= h_x, & +{h_y}' &= \kappa \left(h_y + \frac{V}{c}e_z\right), & +{h_z}' &= \kappa \left(h_z - \frac{V}{c}e_y\right), \Tag{228} +\end{alignat*} \displaybreak[0] \\ +\rho' = \rho\kappa \left(1 - \frac{u_zV}{c^2}\right), \Tag{229} +\end{gather*} +where $\kappa$ has its customary significance $\dfrac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}$. +\bigskip%[** Explicit space] + +As a matter of fact, this set of transformation equations fulfills +all the requirements imposed by the theory of relativity. Thus, in +the first place, it will be seen, on development, that these equations +are themselves perfectly symmetrical with respect to the primed and +unprimed quantities except for the necessary change from $+V$~to~$-V$. +In the second place, it will be found that the substitution of +these equations into our five fundamental equations for electromagnetic +theory \DPchg{(221--2--3--4--6)}{(221), (222), (223), (224), (226)} will successfully transform them +into an entirely similar set with primed quantities replacing the +unprimed ones. And finally it can be shown that these equations +agree with the general requirement derived in \Chapref{IX} that the +%% -----File: 184.png---Folio 170------- +quantity $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ shall be an invariant for the Lorentz transformation. + +To demonstrate this important invariance of $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ we may +point out that by introducing equations (220),~(221) and~(214), our +original expression for kinetic potential +\[ +H = \int dV \left\{ + \frac{\vc{e}^2}{2} + \frac{(\curl\vc{\phi})^2}{2} + - \vc{\phi} · \left(\frac{\dot{\vc{e}}}{c} + \rho\, \frac{\vc{u}}{c}\right) + \right\} +\] +can easily be shown equal to +\[ +\int dV \left(\frac{\vc{e}^2}{2} - \frac{\vc{h}^2}{2}\right), +\Tag{230} +\] +and, noting that our fundamental equations for space and time provide +us with the relation +\[ +\frac{dV}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = \frac{dV'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}, +\] +we can easily show that our transformation equations for $\vc{e}$~and~$\vc{h}$ do +lead to the equality +\[ +\frac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}} + = \frac{H'}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}. +\] + +We thus know that our development of the fundamental equations +for electromagnetic theory from the principle of least action is indeed +in complete accordance with the theory of relativity, since it conforms +with the general requirement which was found in \Chapref{IX} to be +imposed by the theory of relativity on all dynamical considerations. + +\Subsubsection{157}{The Invariance of Electric Charge.} As to the significance of +the transformation equations which we have presented for $\vc{e}$,~$\vc{h}$ and~$\rho$, +we may first show, in accordance with the last of these equations, +that a given electric charge will appear the same to all observers no +matter what their relative motion. +%% -----File: 185.png---Folio 171------- + +To demonstrate this we merely have to point out that, by introducing +equation~(17), we may write our transformation equation +for~$\rho$~(229) in the form +\[ +\frac{\rho'}{\rho} + = \frac{\sqrt{1 - \smfrac{u^2}{c^2}}}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}}, +\] +which shows at once that the two measurements of density of charge +made by $O$~and~$O'$ are in exactly the same ratio as the corresponding +measurements for the Lorentz shortening of the charged body, so +that the total charge will evidently measure the same for the two +observers. + +We might express this invariance of electric charge by writing the +equation +\[ +Q' = Q. +\Tag{231} +\] + +It should be noted in passing that this result is in entire accord +with the whole modern development of electrical theory, which lays +increasing stress on the fundamentality and indivisibility of the +electron as the natural unit quantity of electricity. On this basis +the most direct method of determining the charge on an electrified +body would be to count the number of electrons present and this +number must obviously appear the same both to observer~$O$ and +observer~$O'$.\footnote + {A similar invariance of electric charge has been made fundamental in the + author's development of the theory of similitude (\ie, the theory of the relativity + of size). See for example \textit{Phys.\ Rev}., vol.~3, p.~244 (1914).} + +\Subsubsection{158}{The Relativity of Magnetic and Electric Fields.} As to the +significance of equations (227)~and~(228) for transforming the values +of the electric and magnetic field strengths from one system to another, +we see that at a given point in space we may distinguish between the +electric vector $\vc{e} = e_x\, \vc{i} + e_y\, \vc{j} + e_z\, \vc{k}$ as measured by our original +observer~$O$ and the vector $\vc{e}' = {e_x}'\, \vc{i} + {e_y}'\, \vc{j} + {e_z}'\, \vc{k}$ as measured in +units of his own system by an observer~$O'$ who is moving past~$O$ with +the velocity~$V$ in the $X$\DPchg{-}{~}direction. Thus if $O$~finds in an unvarying +electromagnetic field that $Q\vc{e}$~is the force on a small test charge~$Q$ +which is stationary with respect to his system, $O'$~will find experimentally +%% -----File: 186.png---Folio 172------- +for a similar test charge that moves along with him a value +for the force~$Q\vc{e}'$, where $\vc{e}'$~can be calculated from with the help of +these equations~(227). Similar remarks would apply to the forces +which would act on magnetic poles. + +These considerations show us that we should now use caution in +speaking of a pure electrostatic or pure magnetic field, since the +description of an electromagnetic field is determined by the particular +choice of coördinates with reference to which the field is measured. + +\Subsubsection{159}{Nature of Electromotive Force.} We also see that the ``electromotive'' +force which acts on a charge moving through a magnetic +field finds its interpretation as an ``electric'' force provided we make +use of a system of coördinates which are themselves stationary with +respect to the charge. Such considerations throw light on such questions, +for example, as to the seat of the ``electromotive'' forces in +``homopolar'' electric dynamos where there is relative motion of a +conductor and a magnetic field. + + +\Subsection{Derivation of the Fifth Fundamental Equation.} + +\Paragraph{160.} We may now make use of this fact that the forces acting on +a moving charge of electricity may be treated as purely electrostatic, +by using a set of coördinates which are themselves moving along with +the charge, to derive the fifth fundamental equation of electromagnetic +theory. + +Consider an electromagnetic field having the values $\vc{e}$~and~$\vc{h}$ for +the electric and magnetic field strengths at some particular point. +What will be the value of the electromagnetic force~$\vc{f}$ acting per +unit volume on a charge of density~$\rho$ which is passing through the +point in question with the velocity~$\vc{u}$? + +To solve the problem take a system of coördinates~$S'$ which itself +moves with the same velocity as the charge, for convenience letting +the $X$\DPchg{-}{~}axis coincide with the direction of the motion of the charge. +Since the charge of electricity is stationary with respect to this system, +the force acting on it as measured in units of this system will be by +definition equal to the product of the charge by the strength of the +electric field as it appears to an observer in this system, so that we may +write +\[ +\vc{F} = Q'\vc{e}', +\] +%% -----File: 187.png---Folio 173------- +or +\[ +{F_x}' = Q'{e_x}', \qquad +{F_y}' = Q'{e_y}', \qquad +{F_z}' = Q'{e_z}'. +\] +For the components of the electrical field ${e_x}'$,~${e_y}'$,~${e_z}'$, we have just +obtained the transformation equations~(227), while in our earlier +dynamical considerations in \Chapref{VI} we obtained transformation +equations (61),~(62), and~(63) for the components of force. Substituting +above and bearing in mind that $u_x = V$, $u_y = u_z = 0$, and +that $Q' = Q$, we obtain on simplification +\begin{align*} +F_x &= Q e_x, \\ +F_y &= Q \left(e_y - \frac{u_x}{c}h_z\right), \\ +F_z &= Q \left(e_z - \frac{u_x}{c}h_y\right), +\end{align*} +which in vectorial form gives us the equation +\[ +\vc{F} = Q \left(\vc{e} - \frac{1}{c}\, [\vc{u} × \vc{h}]^*\right) +\] +or for the force per unit volume +\[ +\vc{f} = \rho \left(\vc{e} + \frac{1}{c}\, [\vc{u} × \vc{h}]^*\right). +\Tag{226} +\] + +This is the well-known fifth fundamental equation of the Maxwell-Lorentz +theory of electromagnetism. We have already indicated the +method by which it could be derived from the principle of least action. +This derivation, however, from the transformation equations, provided +by the theory of relativity, is particularly simple and attractive. + + +\Subsection{Difference between the Ether and the Relativity Theories of Electromagnetism.} + +\Paragraph{161.} In spite of the fact that we have now found five equations +which can be used as a basis for electromagnetic theory which agree +with the requirements of relativity and also have exactly the same +form as the five fundamental equations used by Lorentz in building +up the stationary ether theory, it must not be supposed that the +relativity and ether theories of electromagnetism are identical. Although +the older equations have exactly the same form as the ones +which we shall henceforth use, they have a different interpretation, +since our equations are true for measurements made with the help +of any non-accelerated set of coördinates, while the equations of +%% -----File: 188.png---Folio 174------- +Lorentz were, in the first instance, supposed to be true only for measurements +which were referred to a set of coördinates which were +stationary with respect to the assumed luminiferous ether. Suppose, +for example, we desire to calculate with the help of equation~(226), +\[ +\vc{t} = \rho \left(\vc{e} + \frac{1}{\vc{c}}\, [\vc{u} × \vc{h}]^*\right), +\] +the force acting on a charged body which is moving with the velocity~$\vc{u}$; +we must note that for the stationary ether theory, $\vc{u}$~must be the +velocity of the charged body through the ether, while for us $\vc{u}$~may be +taken as the velocity past any set of unaccelerated coördinates, provided +$\vc{e}$~and~$\vc{h}$ are measured with reference to the same set of coördinates. +It will be readily seen that such an extension in the meaning +of the fundamental equations is an important simplification. + +\Paragraph{162.} A word about the development from the theory of a stationary +ether to our present theory will not be out of place. When it was +found that the theory of a stationary ether led to incorrect conclusions +in the case of the Michelson-Morley experiment, the hypothesis +was advanced by Lorentz and Fitzgerald that the failure of that +experiment to show any motion through the ether was due to a contraction +of the apparatus in the direction of its motion through the +ether in the ratio $1 : \sqrt{1 - \dfrac{u^2}{c^2}}$. Lorentz then showed that if all systems +should be thus contracted in the line of their motion through the +ether, and observers moving with such system make use of suitably +contracted meter sticks and clocks adjusted to give what Lorentz +called the ``local time,'' their measurements of electromagnetic +phenomena could be described by a set of equations which have +nearly the same form as the original four field equations which would +be used by a stationary observer. It will be seen that Lorentz was +thus making important progress towards our present idea of the complete +relativity of motion. The final step could not be taken, however, +without abandoning our older ideas of space and time and giving up +the Galilean transformation equations as the basis of kinematics. +It was Einstein who, with clearness and boldness of vision, pointed +out that the failure of the Michelson-Morley experiment, and all +other attempts to detect motion through the ether, is not due to a +%% -----File: 189.png---Folio 175------- +fortuitous compensation of effects but is the expression of an important +general principle, and the new transformation equations for kinematics +to which he was led have not only provided the basis for an \emph{exact} +transformation of the field equations but have so completely revolutionized +our ideas of space and time that hardly a branch of science +remains unaffected. + +\Paragraph{163.} With regard to the present status of the ether in scientific +theory, it must be definitely stated that this concept has certainly +lost both its fundamentality and the greater part of its usefulness, +and this has been brought about by a gradual process which has only +found its culmination in the work of Einstein. Since the earliest +days of the luminiferous ether, the attempts of science to increase the +substantiality of this medium have met with little success. Thus +we have had solid elastic ethers of most extreme tenuity, and ethers +with a density of a thousand tons per cubic millimeter; we have had +quasi-material tubes of force and lines of force; we have had vibratory +gyrostatic ethers and perfect gases of zero atomic weight; but after +every debauch of model-making, science has recognized anew that a +correct mathematical description of the actual phenomena of light +propagation is superior to any of these sublimated material media. +Already for Lorentz the ether had been reduced to the bare function +of providing a stationary system of reference for the measurement of +positions and velocities, and now even this function has been taken +from it by the work of Einstein, which has shown that any unaccelerated +system of reference is just as good as any other. + +To give up the notion of an ether will be very hard for many +physicists, in particular since the phenomena of the interference and +polarization of light are so easily correlated with familiar experience +with wave motions in material elastic media. Consideration will +show us, however, that by giving up the ether we have done nothing +to destroy the periodic or polarizable nature of a light disturbance. +When a plane polarized beam of light is passing through a given +point in space we merely find that the electric and magnetic fields at +that point lie on perpendiculars to the direction of propagation and +undergo regular periodic changes in magnitude. There is no need of +going beyond these actual experimental facts and introducing any +hypothetical medium. It is just as simple, indeed simpler, to say +%% -----File: 190.png---Folio 176------- +that the electric or magnetic field has a certain intensity at a given +point in space as to speak of a complicated sort of strain at a given +point in an assumed ether. + + +\Subsection{Applications to Electromagnetic Theory.} + +\Paragraph{164.} The significant fact that the fundamental equations of the +new electromagnetic theory have the same form as those of Lorentz +makes it of course possible to retain in the structure of modern electrical +theory nearly all the results of his important researches, care +being taken to give his mathematical equations an interpretation in +accordance with the fundamental ideas of the theory of relativity. It +is, however, entirely beyond our present scope to make any presentation +of electromagnetic theory as a whole, and in the following paragraphs +we shall confine ourselves to the proof of a few theorems which +can be handled with special ease and directness by the methods introduced +by the theory of relativity. + +\Subsubsection{165}{The Electric and Magnetic Fields around a Moving Charge.} +Our transformation equations for the electromagnetic field make it +very easy to derive expressions for the field around a point charge in +uniform motion. Consider a point charge~$Q$ moving with the velocity~$V$. +For convenience consider a system of reference~$S$ such that $Q$~is +moving along the $X$\DPchg{-}{~}axis and at the instant in question, $t=0$, let the +charge coincide with the origin of coördinates~$O$. We desire now to +calculate the values of electric field~$\vc{e}$ and the magnetic field~$\vc{h}$ at any +point in space $x$,~$y$,~$z$. + +Consider another system of reference,~$S'$, which moves along with +the same velocity as the charge~$Q$, the origin of coördinates~$O'$\DPchg{,}{} and +the charge always coinciding in position. Since the charge is stationary +with respect to their new system of reference, we shall have +the electric field at any point $x'$,~$y'$,~$z'$ in this system given by the +equations +\begin{align*} +{e_x}' &= \frac{Qx'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \\ +{e_y}' &= \frac{Qy'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, \\ +{e_z}' &= \frac{Qz'}{({x'}^2 + {y'}^2 + {z'}^2)^{3/2}}, +\end{align*} +%% -----File: 191.png---Folio 177------- +while the magnetic field will obviously be zero for measurements made +in system~$S'$, giving us +\[%[** TN: Setting on one line] +{h_x}' = 0, \qquad {h_y}' = 0, \qquad {h_z}' = 0. +\] +Introducing our transformation equations (9),~(10) and~(11) for $x'$,~$y'$ +and~$z'$ and our transformation equations (227)~and~(228) for the +electric and magnetic fields and substituting $t=0$, we obtain for the +values of $\vc{e}$~and~$\vc{h}$ in system~$S$ at the instant when the charge passes +through the point~$O$, +\begin{align*} +e_x &= \frac{Q\kappa x}{(\kappa^2x^2 + y^2 + z^2)^{3/2}} + = \frac{Q \left(1 - \smfrac{V^2}{c^2}\right) x} + {\left[x^2 + \left(1 - \smfrac{V^2}{c^2}\right)(y^2 + z^2)\right]^{3/2}}, +\displaybreak[0] \\ +e_y &= \frac{Q\kappa y}{(\kappa^2x^2 + y^2 + z^2)^{3/2}} + = \frac{Q \left(1 - \smfrac{V^2}{c^2}\right) y} + {\left[x^2 + \left(1 - \smfrac{V^2}{c^2}\right)(y^2 + z^2)\right]^{3/2}}, +\displaybreak[0] \\ +e_z &= \frac{Q\kappa z}{(\kappa^2x^2 + y^2 + z^2)^{3/2}} + = \frac{Q \left(1 - \smfrac{V^2}{c^2}\right) z} + {\left[x^2 + \left(1 - \smfrac{V^2}{c^2}\right)(y^2 + z^2)\right]^{3/2}}, +\displaybreak[0] \\ +h_x &= 0, \\ +h_y &= -\frac{V}{c}\, e_z,\\ +h_z &= \frac{V}{c}\, e_y, +\end{align*} +or, putting $s$ for the important quantity $\sqrt{x^2 + \left(1 - \dfrac{V^2}{c^2}\right)(y^2 + z^2)}$ +and writing the equations in the vectorial form where we put +\[ +\vc{r} = (x\, \vc{i} + y\, \vc{j} + z\, \vc{k}), +\] +we obtain the familiar equations for the field around a point charge +%% -----File: 192.png---Folio 178------- +in uniform motion with the velocity $u=V$ in the $X$\DPchg{-}{~}direction +\begin{gather*} +\vc{e} = Q\, \frac{\left(1 - \smfrac{u^2}{c^2}\right)\vc{r}}{s^3}, +\Tag{232} \\ +\vc{h} = \frac{1}{c}\, [\vc{u} × \vc{e}]\DPtypo{.^*}{^*.} +\Tag{233} +\end{gather*} + +\Subsubsection{166}{The Energy of a Moving Electromagnetic System.} Our +transformation equations will permit us to obtain a very important +expression for the energy of an isolated electromagnetic system in +terms of the velocity of the system and the energy of the same system +as it appears to an observer who is moving along with it. + +Consider a physical system surrounded by a shell which is impermeable +to electromagnetic radiation. This system is to be thought +of as consisting of the various mechanical parts, electric charges and +electromagnetic fields which are inside of the impermeable shell. +The system is free in space, except that it may be acted on by external +electromagnetic fields, and its energy content thus be changed. + +Let us now equate the increase in the energy of the system to the +work done by the action of the external field on the electric charges +in the system. Since the force which a magnetic field exerts on a +charge is at right angles to the motion of the charge it does no work +and we need to consider only the work done by the external electric +field and may write for the increase in the energy of the system +\[ +\Delta E %[** TN: Textstyle integral in original] + = \iiiint \rho(e_xu_x + e_yu_y + e_zu_z)\, dx\, dy\, dz\, dt, +\Tag{234} +\] +where the integration is to be taken over the total volume of the +system and over any time interval in which we may be interested. + +Let us now transform this expression with the help of our transformation +equations for the electric field~(227) for electric charge~(229), +and for velocities \DPchg{(14--15--16)}{(14), (15), (16)}. Noting that our fundamental +equations for kinematic quantities give us $dx\, dy\, dz\, dt = dx'\, dy'\, dz'\, dt'$, +we obtain +\begin{align*} +\Delta E &= \kappa \iiiint + \rho'({e_x}'{u_x}' + {e_y}'{u_y}' + {e_z}'{u_z}')\, dx'\, dy'\, dz'\, dt' \\ + &\quad + + \kappa V \iiiint \rho'\left( + {e_x}' + \frac{{u_y}'}{c}\, {h_z}' - \frac{{u_z}'}{c}\, {h_y}' + \right) dx'\, dy'\, dz'\, dt'. +\end{align*} +%% -----File: 193.png---Folio 179------- + +Consider now a system which \emph{both at the beginning and end of our +time interval is free from the action of external forces}; we may then +rewrite the above equation for this special case in the form +\[ +\Delta E = \kappa \Delta E' + + \kappa V \int \Sum {F_x}'\, dt', +\] +where, in accordance with our earlier equation~(234), $\Delta E'$~is the increase +in the energy of the system as it appears to observer~$O'$ and $\Sum {F_x}'$ +is the total force acting on the system in $X$\DPchg{-}{~}direction as measured +by~$O'$. + +The restriction that the system shall be unacted on by external +forces both at the beginning and end of our time interval is necessary +because it is only under those circumstances that an integration +between two values of~$t$ can be considered as an integration between +two definite values of~$t'$, simultaneity in different parts of the system +not being the same for observers $O$~and~$O'$. + +We may now apply this equation to a specially interesting case. +Let the system be of such a nature that we can speak of it as being +at rest with respect to~$S'$, meaning thereby that all the mechanical +parts have low velocities with respect to~$S'$ and that their center of +gravity moves permanently along with~$S'$. Under these circumstances +we may evidently put $\int\Sum {F_x}'\, dt' = 0$ and may write the +above equation in the form +\begin{align*} +\Delta E &= \frac{\Delta E_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}, \\ +\intertext{or} +\frac{\partial \Delta E}{\partial E_0} + &= \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\end{align*} +where $u$~is the velocity of the system, and $E°$~is its energy as measured +by an observer moving along with it. The energy of a system which +is \emph{unacted on by external forces} is thus a function of two variables, its +energy~$E_0$ as measured by an observer moving along with the system +and its velocity~$u$. +%% -----File: 194.png---Folio 180------- + +We may now write +\[ +E = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, E_0 + \phi(u) + \text{const.}, +\] +where $\phi(u)$ represents the energy of the system which depends solely +on the velocity of the system and not on the changes in its $E_0$~values. +$\phi(u)$~will thus evidently be the kinetic energy of the mechanical masses +in the system which we have already found~(82) to have the value +$\dfrac{m_0c^2}{\sqrt{1 - \smfrac{u^2}{c^2}}} - m_0c^2$ where $m_0$~is to be taken as the total mass of the +mechanical part of our system when at rest. We may now write +\[ +E = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, (m_0c^2 + E_0) + - m_0c^2 + \text{const.} +\] +Or, assuming as before that the constant is equal to~$m_0c^2$, which will +be equivalent to making a system which has zero energy also have +zero mass, we obtain +\[ +E = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, (m_0c^2 + E_0), +\Tag{235} +\] +which is the desired expression for the energy of an isolated system +which may contain both electrical and mechanical parts. + +\Subsubsection{167}{Relation between Mass and Energy.} This expression for the +energy of a system that contains electrical parts permits us to show +that the same relation which we found between mass and energy for +mechanical systems also holds in the case of electromagnetic energy. +Consider a system containing electromagnetic energy and enclosed +by a shell which is impermeable to radiation. Let us apply a force~$\vc{F}$ +to the system in such a way as to change the velocity of the system +without changing its $E_0$~value. We can then equate the work done +per second by the force to the rate of increase of the energy of the +system. We have +\[ +\vc{F} · \vc{u} = \frac{dE}{dt}. +\] +%% -----File: 195.png---Folio 181------- +But from equation~(235) we can obtain a value for the rate of increase +of energy~$\dfrac{dE}{dt}$, giving us +\[ +\vc{F} · \vc{u} + = F_xu_x + F_yu_y + F_zu_z + = \left(m_0 + \frac{E_0}{c^2}\right) + \frac{u\, \smfrac{du}{dt}}{\left(1 - \smfrac{u^2}{c^2}\right)^{\tfrac{3}{2}}}, +\] +and solving this equation for~$\vc{F}$ we obtain +\begin{align*} +\vc{F} &= \frac{d}{dt}\left[ + \frac{\left(m_0 + \smfrac{E_0}{c^2}\right)} + {\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{u} + \right], +\Tag{236} \\ +\intertext{which for low velocities assumes the form} +\vc{F} &= \frac{d}{dt}\left[\left(m_0 + \frac{E_0}{c^2}\right) \vc{u}\right]. +\Tag{237} +\end{align*} + +Examination of these expressions shows that our system which +contains electromagnetic energy behaves like an ordinary mechanical +system with the mass $\left(m_0 + \dfrac{E_0}{c^2}\right)$ at low velocities or $\dfrac{m_0 + \smfrac{E_0}{c^2}}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ at +any desired velocity~$u$. To the energy of the system~$E_0$, part of which +is electromagnetic, we must ascribe the mass~$\dfrac{E_0}{c^2}$ just as we found in +the case of mechanical energy. We realize again that matter and +energy are but different names for the same fundamental entity, +$10^{21}$~ergs of energy having the mass $1$~gram. + + +\Subsection{The Theory of Moving Dielectrics.} + +\Paragraph{168.} The principle of relativity proves to be very useful for the +development of the theory of moving dielectrics. + +It was first shown by Maxwell that a theory of electromagnetic +phenomena in material media can be based on a set of field equations +similar in form to those for free space, provided we introduce besides +the electric and magnetic field strengths, $\vc{E}$~and~$\vc{F}$, two new field vectors, +%% -----File: 196.png---Folio 182------- +the dielectric displacement~$\vc{D}$ and the magnetic induction~$\vc{B}$, and +also the density of electric current in the medium~$\vc{i}$. These quantities +are found to be connected by the four following equations similar in +form to the four field equations for free space: +\begin{align*} +\curl \vc{H} + &= \frac{1}{c} \left(\frac{\partial\vc{D}}{\partial t} + \vc{i}\right), + \Tag{238} \\ +\curl \vc{E} + &= -\frac{1}{c}\, \frac{\partial\vc{B}}{\partial t}, \Tag{239} \\ +\divg \vc{D} &= \rho, + \Tag{240} \\ +\divg \vc{B} &= 0. + \Tag{241} +\end{align*} + +For \emph{stationary homogeneous} media, the dielectric displacement, +magnetic induction and electric current are connected with the +electric and magnetic field strengths by the following equations: +\begin{align*} +\vc{D} &= \epsilon \vc{E}, \Tag{242}\\ +\vc{B} &= \mu \vc{H}, \Tag{243}\\ +\vc{i} &= \sigma \vc{E}, \Tag{244} +\end{align*} +where $\epsilon$~is the dielectric constant, $\mu$~the magnetic permeability and $\sigma$~the +electrical conductivity of the medium in question. + +\Subsubsection{169}{Relation between Field Equations for Material Media and +Electron Theory.} It must not be supposed that the four field equations +\DPchg{(238--241)}{(238)--(241)} for electromagnetic phenomena in \emph{material media} are +in any sense contradictory to the four equations \DPchg{(221--224)}{(221)--(224)} for free +space which we took as the fundamental basis for our development of +electromagnetic theory. As a matter of fact, one of the main achievements +of modern electron theory has been to show that the electromagnetic +behavior of material media can be explained in terms of +the behavior of the individual electrons and ions which they contain, +these electrons and ions acting in accordance with the four fundamental +field equations for free space. Thus our new equations for material +media merely express from a \emph{macroscopic} point of view the statistical +result of the behavior of the individual electrons in the material in +question. $\vc{E}$~and~$\vc{H}$ in these new equations are to be looked upon as +the average values of $\vc{e}$~and~$\vc{h}$ which arise from the action of the +individual electrons in the material, the process of averaging being so +%% -----File: 197.png---Folio 183------- +carried out that the results give the values which a \emph{macroscopic} observer +would actually find for the electric and magnetic forces acting +respectively on a unit charge and a unit pole at the point in question. +These average values, $\vc{E}$~and~$\vc{H}$, will thus pay no attention to the +rapid fluctuations of $\vc{e}$~and~$\vc{h}$ which arise from the action and motion +of the individual electrons, the macroscopic observer using in fact +differentials for time,~$dt$, and space,~$dx$, which would be large from a +microscopic or molecular viewpoint. + +Since from a microscopic point of view $\vc{E}$~and~$\vc{H}$ are not really +the instantaneous values of the field strength at an actual point in +space, it has been found necessary to introduce two new vectors, +electric displacement,~$\vc{D}$, and magnetic induction,~$\vc{B}$, whose time +rate of change will determine the curl of $\vc{E}$~and~$\vc{H}$ respectively. It will +evidently be possible, however, to relate $\vc{D}$~and~$\vc{B}$ to the actual electric +and magnetic fields $\vc{e}$~and~$\vc{h}$ produced by the individual electrons, +and this relation has been one of the problems solved by modern +electron theory, and the field equations \DPchg{(238--241)}{(238)--(241)} for material media +have thus been shown to stand in complete agreement with the most +modern views as to the structure of matter and electricity. For +the purposes of the rest of our discussion we shall merely take these +equations as expressing the experimental facts in stationary or in +moving media. + +\Subsubsection{170}{Transformation Equations for Moving Media.} Since equations +\DPchg{(238 to 241)}{(238) to (241)} are assumed to give a correct description of electromagnetic +phenomena in media whether stationary or moving with +respect to our reference system~$S$, it is evident that the equations +must be unchanged in form if we refer our measurements to a new +system of coördinates~$S'$ moving past~$S$, say, with the velocity~$V$ in the +$X$\DPchg{-}{~}direction. + +As a matter of fact, equations \DPchg{(238 to 241)}{(238) to (241)} can be transformed +into an entirely similar set +\begin{align*} +\curl \vc{H'} + &= \frac{1}{c} \left(\frac{\partial\vc{D'}}{\partial t'} + \vc{i}'\DPtypo{,}{}\right)\DPtypo{}{,} \\ +\curl \vc{E'} &= -\frac{1}{c}\, \frac{\partial\vc{B'}}{\partial t'}, \\ +\divg \vc{D'} &= \rho', \\ +\divg \vc{B'} &= 0, +\end{align*} +%% -----File: 198.png---Folio 184------- +provided we substitute for $x$,~$y$,~$z$ and~$t$ the values of $x'$,~$y'$,~$z'$ and~$t'$ +given by the fundamental transformation equations for space and +time \DPchg{(9~to~12)}{(9)~to~(12)}, and substitute for the other quantities in question the +relations +{\small% +\begin{align*}%[** TN: Re-grouping] +\begin{aligned} +{E_x}' &= E_x, & +{E_y}' &= \kappa \left(E_y - \frac{V}{c} B_z\right), & +{E_z}' &= \kappa \left(E_z + \frac{V}{c} B_y\right), \\ +% +{D_x}' &= D_x, & +{D_y}' &= \kappa \left(D_y - \frac{V}{c} H_z\right), & +{D_z}' &= \kappa \left(D_z + \frac{V}{c} H_y\right), +\end{aligned} +\Tag{245}\displaybreak[0] \\[12pt] +\begin{aligned} +{H_x}' &= H_x, & +{H_y}' &= \kappa \left(H_y + \frac{V}{c} D_z\right), & +{H_z}' &= \kappa \left(H_z - \frac{V}{c} D_y\right), \\ +{B_x}' &= B_x, & +{B_y}' &= \kappa \left(B_y + \frac{V}{c} E_z\right), & +{B_z}' &= \kappa \left(B_z - \frac{V}{c} E_y\right), +\end{aligned} +\Tag{246}\displaybreak[0] \\[12pt] +\begin{gathered} +\rho' = \kappa \left(\rho - \frac{V}{c^2}\, i_x\right),\qquad +{i_x}' = \kappa(i_x - V_\rho), \qquad +{i_y}' = i_y, \qquad +{i_z}' = i_z. +\end{gathered} +\Tag{247} +\end{align*}}% + +It will be noted that for free space these equations will reduce to +the same form as our earlier transformation equations \DPchg{(227~to~229)}{(227)~to~(229)} +since we shall have the simplifications $\vc{D} = \vc{E}$, $\vc{B} = \vc{H}$ and $\vc{i} = \rho \vc{u}$. + +We may also call attention at this point to the fact that our fundamental +%% -----File: 199.png---Folio 185------- +equations for electromagnetic phenomena \DPchg{(238--241)}{(238)--(241)} in dielectric +media might have been derived from the principle of least +action, making use of an expression for kinetic potential which could +be shown equal to $H = \ds\int dV \left(\frac{\vc{E·D}}{2} - \frac{\vc{H}·\vc{B}}{2}\right)$, and it will be noticed +that our transformation equations for these quantities are such as to +preserve that necessary invariance for $\dfrac{H}{\sqrt{1 - \smfrac{u^2}{c^2}}}$ which we found in +\Chapref{IX} to be the general requirement for any dynamical development +which agrees with the theory of relativity. + +\Paragraph{171.} We are now in a position to handle the theory of moving +media. Consider a homogeneous medium moving past a system of +coördinates $S$ in the $X$\DPchg{-}{~}direction with the velocity~$V$; our problem is +to discover relations between the various electric and magnetic +vectors in this medium. To do this, consider a new system of coördinates~$S'$ +also moving past our original system with the velocity~$V$. +Since the medium is stationary with respect to this new system~$S'$ we +may write for measurements referred to~$S'$ in accordance with equations +\DPchg{(242~to~244)}{(242)~to~(244)} the relations +\begin{align*} +\vc{D'} &= \epsilon \vc{E'},\\ +\vc{B'} &= \mu \vc{H'},\\ +\vc{i'} &= \sigma \vc{E'}, +\end{align*} +which, as we have already pointed out, are known experimentally to +be true in the case of \emph{stationary, homogeneous} media. $\epsilon$,~$\mu$ and~$\sigma$ are +evidently the values of dielectric constant, permeability and conductivity +of the material in question, which would be found by an +experimenter with respect to whom the medium is stationary. + +Making use of our transformation equations \DPchg{(245~to~247)}{(245)~to~(247)} we can +obtain by obvious substitutions the following set of relations for +measurements made with respect to the original system of coördinates~$S$: +\begin{align*} +&\begin{aligned} +D_x &= \epsilon E_x, \\ +D_y - \frac{V}{c} H_z + &= \epsilon \left(E_y - \frac{V}{c} B_z\right), \\ +%% -----File: 200.png---Folio 186------- +D_z + \frac{V}{c} H_y + &= \epsilon \left(E_z + \frac{V}{c} B_y\right), +\end{aligned} +\Tag{248} \displaybreak[0] \\[12pt] +&\begin{aligned} +B_x &= \mu H_x, \\ +B_y + \frac{V}{c} E_z + &= \mu\left(H_y + \frac{V}{c} D_z\right), \\ +B_z - \frac{V}{c}E_y + &= \mu\left(H_z - \frac{V}{c} D_y\right), +\end{aligned} +\Tag{249} \displaybreak[0] \\[12pt] +&\begin{aligned} +\kappa (i_x - V_\rho) &= \sigma E_x, \\ +i_y &= \sigma\kappa \left(E_y - \frac{V}{c} B_z\right), \\ +i_z &= \sigma\kappa \left(E_z + \frac{V}{c} B_y\right). +\end{aligned} +\Tag{250} +\end{align*} + +\Subsubsection{172}{Theory of the Wilson Experiment.} The equations which we +have just developed for moving media are, as a matter of fact, in +complete accord with the celebrated experiment of H.~A. Wilson on +moving dielectrics and indeed all other experiments that have been +performed on moving media. + +Wilson's experiment consisted in the rotation of a hollow cylinder +of dielectric, in a magnetic field which was parallel to the axis of the +cylinder. The inner and outer surfaces of the cylinder were covered +with a thin metal coating, and arrangements made with the help of +wire brushes so that electrical contact could be made from these +coatings to the pairs of quadrants of an electrometer. By reversing +the magnetic field while the apparatus was in rotation it was possible +to measure with the electrometer the charge produced by the electrical +displacement in the dielectric. We may make use of our equations +to compute the quantitative size of the effect. +\begin{figure}[hbt] + \begin{center} + \Fig{15} + \Input[3.75in]{200} + \end{center} +\end{figure} +%% -----File: 201.png---Folio 187------- + +Let \Figref{15} represent a cross-section of the rotating cylinder. +Consider a section of the dielectric~$AA$ which is moving perpendicularly +to the plane of the paper in the $X$\DPchg{-}{~}direction with the velocity~$V$. Let +the magnetic field be in the $Y$\DPchg{-}{~}direction parallel to the axis of rotation. +The problem is to calculate dielectric displacement~$D_z$ in the $Z$\DPchg{-}{~}direction. + +Referring to equations~(248) we have +\begin{align*} +D_z + \frac{V}{c} H_y &= \epsilon \left(E_z + \frac{V}{c} B_y\right), \\ +\intertext{and, substituting the value of~$B_y$ given by equations~(249),} +B_y + \frac{V}{c} E_z &= \mu \left(H_y + \frac{V}{c} D_z\right) +\end{align*} +we obtain +\[ +\left(1 - \epsilon\mu\, \frac{V^2}{c^2}\right) D_z + = \epsilon \left(1 - \frac{V^2}{c^2}\right) E_z + + \frac{V}{c}\, (\epsilon\mu - 1)\, H_y, +\] +or, neglecting terms of orders higher than~$\dfrac{V}{c}$, we have +\[ +D_z = \epsilon E_z + \frac{V}{c}\, (\epsilon\mu - 1)\, H_y. +\Tag{251} +\] + +For a substance whose permeability is practically unity such as +Wilson actually used the equation reduces to +\[ +D_z = \epsilon E_z + \frac{V}{c}\, (\epsilon - 1)\, H_y, +\] +and this was found to fit the experimental facts, since measurements +with the electrometer show the surface charge actually to have the +magnitude $D_z$~per square centimeter in accordance with our equation +$\divg D = \rho$. + +It would be a matter of great interest to repeat the Wilson experiment +with a dielectric of high permeability so that we could test the +complete equation~(251). This is of some importance since the +original Lorentz theory led to a different equation, +\[ +D_z = \epsilon E_z + \frac{V}{c}\, (\epsilon - 1)\, \mu H_y. +\] +%% -----File: 202.png---Folio 188------- + + +\Chapter{XIII}{Four-Dimensional Analysis.} +%[** TN: Running head not hyphenated in original] +\SetRunningHeads{Chapter Thirteen.}{Four-Dimensional Analysis.} + +\Paragraph{173.} In the present chapter we shall present a four-dimensional +method of expressing the results of the Einstein theory of relativity, +a method which was first introduced by Minkowski, and in the form +which we shall use, principally developed by Wilson and Lewis. The +point of view adopted\DPtypo{,}{} consists essentially in considering the properties +of an assumed four-dimensional space in which intervals of time are +thought of as plotted along an axis perpendicular to the three Cartesian +axes of ordinary space, the science of kinematics thus becoming +the geometry of this new four-dimensional space. + +The method often has very great advantages not only because it +sometimes leads to considerable simplification of the mathematical +form in which the results of the theory of relativity are expressed, +but also because the analogies between ordinary geometry and the +geometry of this imaginary space often suggest valuable modes of +attack. On the other hand, in order to carry out actual numerical +calculations and often in order to appreciate the physical significance +of the conclusions arrived at, it is necessary to retranslate the results +obtained by this four-dimensional method into the language of ordinary +kinematics. It must further be noted, moreover, that many important +results of the theory of relativity can be more easily obtained +if we do not try to employ this four-dimensional geometry. The +reader should also be on his guard against the fallacy of thinking that +extension in time is of the same nature as extension in space merely +because intervals of space and time can both be represented by +plotting along axes drawn on the same piece of paper. + +\Subsubsection{174}{Idea of a Time Axis.} In order to grasp the method let us +consider a particle constrained to move along a single axis, say~$OX$, +and let us consider a time axis~$OT$ perpendicular to~$OX$. Then the +\emph{position} of the particle at any \emph{instant} of time can be represented by a +point in the $XT$~plane, and its motion as time progresses by a line in +the plane. If, for example, the particle were stationary, its behavior +%% -----File: 203.png---Folio 189------- +in time and space could be represented by a line parallel to the time +axis~$OT$ as shown for example by the line~$ab$ in \Figref{16}. A particle +\begin{figure}[hbt] + \begin{center} + \Fig{16} + \Input[3.5in]{203} + \end{center} +\end{figure} +moving with the uniform velocity $u = \dfrac{dx}{dt}$ could be represented by a +straight line $ac$ making an angle with the time axes, and the kinematical +behavior of an accelerated particle could be represented by a +curved line. + +By conceiving of a \emph{four}-dimensional space we can extend this +method which we have just outlined to include motion parallel to +all three space axes, and in accordance with the nomenclature of +Minkowski might call such a geometrical representation of the space-time +manifold ``the world,'' and speak of the points and lines which +represent the instantaneous positions and the motions of particles as +``world-points'' and ``world-lines.'' + +\Subsubsection{175}{Non-Euclidean Character of the Space.} It will be at once +evident that the graphical method of representing kinematical events +which is shown by \Figref[Figure]{16} still leaves something to be desired. One +of the most important conclusions drawn from the theory of relativity +was the fact that it is impossible for a particle to move with a velocity +greater than that of light, and it is evident that there is nothing in +our plot to indicate that fact, since we could draw a line making any +desired angle with the time axis, up to perpendicularity, and thus +%% -----File: 204.png---Folio 190------- +represent particles moving with any velocity up to infinity, +\[ +u = \frac{\Delta x}{\Delta t} = \infty. +\] +It is also evident that there is nothing in our plot to correspond to +that invariance in the velocity of light which is a cornerstone of the +theory of relativity. Suppose, for example, the line~$OC$, in \Figref{17}, +\begin{figure}[hbt] + \begin{center} + \Fig{17} + \Input[3.75in]{204} + \end{center} +\end{figure} +represents the trajectory of a beam of light with the velocity $\dfrac{\Delta x}{\Delta t} = c$; +there is then nothing so far introduced into our method of plotting +to indicate the fact that we could not equally well make use of another +set of axes~$OX'T'$, inclined to the first and thus giving quite a different +value, $\dfrac{\Delta x'}{\Delta t'}$, to the velocity of the beam of light. + +There are a number of methods of meeting this difficulty and +obtaining the invariance for the four-dimensional expression $x^2 + y^2 ++ z^2 - c^2t^2$ (see \Chapref{IV}) which must characterize our system of +kinematics. One of these is to conceive of a four-dimensional Euclidean +%% -----File: 205.png---Folio 191------- +space with an imaginary time axis, such that instead of plotting +real instants in time along this axis we should plot the quantity +$l = ict$ where $i = \sqrt{-1}$. In this way we should obtain invariance +for the quantity $x^2 + y^2 + z^2 + l^2 = x^2 + y^2 + z^2 - c^2t^2$, since it may +be regarded as the square of the magnitude of an imaginary four-dimensional +radius vector. This method of treatment has been +especially developed by Minkowski, Laue, and Sommerfeld. Another +method of attack, which has been developed by Wilson and Lewis +and is the one which we shall adopt in this chapter, is to use a real +time axis, for plotting the real quantity~$ct$, but to make use of a non-Euclidean +four-dimensional space in which the quantity $(x^2 + y^2 + z^2 +- c^2t^2)$ is itself taken as the square of the magnitude of a radius vector. +This latter method has of course the disadvantages that come from +using a non-Euclidean space; we shall find, however, that these reduce +largely to the introduction of certain rules as to signs. The method +has the considerable advantage of retaining a real time axis which is +of some importance, if we wish to visualize the methods of attack and +to represent them graphically. + +We may now proceed to develop an analysis for this non-Euclidean +space. We shall find this to be quite a lengthy process but at its +completion we shall have a very valuable instrument for expressing +in condensed language the results of the theory of relativity. Our +method of treatment will be almost wholly analytical, and the geometrical +analogies may be regarded merely as furnishing convenient +names for useful analytical expressions. A more geometrical method +of attack will be found in the original work of Wilson and Lewis. + + +\Section[I]{Vector Analysis of the Non-Euclidean Four-Dimensional +Manifold.} + +\Paragraph{176.} Consider a four-dimensional manifold in which the position +of a point is determined by a radius vector +\[ +\vc{r} = (x_1 \vc{k}_1 + x_2 \vc{k}_2 + x_3 \vc{k}_3 + x_4 \vc{k}_4), +\] +where $\vc{k}_1$,~$\vc{k}_2$,~$\vc{k}_3$ and~$\vc{k}_4$ may be regarded as unit vectors along four +mutually perpendicular axes and $x_1$,~$x_2$,~$x_3$, and~$x_4$ as the magnitudes +of the four components of~$\vc{r}$ along these four axes. We may identify +$x_1$,~$x_2$, and~$x_3$ with the three spatial coördinates of a point $x$,~$y$ and~$z$ +%% -----File: 206.png---Folio 192------- +with reference to an ordinary set of space axes and consider~$x_4$ as a +coördinate which specifies the time (multiplied by the velocity of +light) when the occurrence in question takes place at the point~$xyz$. +We have +\[ +x_1 = x,\qquad +x_2 = y,\qquad +x_3 = z,\qquad +x_4 = ct, +\Tag{252} +\] +and from time to time we shall make these substitutions when we +wish to interpret our results in the language of ordinary kinematics. +We shall retain the symbols $x_1$,~$x_2$,~$x_3$, and~$x_4$ throughout our development, +however, for the sake of symmetry. + +\Subsubsection{177}{Space, Time and Singular Vectors.} Our space will differ in +an important way from Euclidean space since we shall consider three +classes of one-vector, space, time and singular vectors. Considering +the coördinates $x_1$,~$x_2$,~$x_3$, and~$x_4$ which determine the end of a radius +vector, \\ +\emph{Space or $\gamma$-vectors} will have components such that +\[ +({x_1}^2 + {x_2}^2 + {x_3}^2) > {x_4}^2, +\] +and we shall put for their magnitude +\[ +s = \sqrt{{x_1}^2 + {x_2}^2 + {x_3}^2 - {x_4}^2}. +\Tag{253} +\] +\emph{Time or $\delta$-vectors} will have components such that +\[ +{x_4}^2 > ({x_1}^2 + {x_2}^2 + {x_3}^2), +\] +and we shall put for their magnitude +\[ +s = \sqrt{{x_4}^2-{x_1}^2- {x_2}^2 - {x_3}^2}. +\Tag{254} +\] +\emph{Singular or $\alpha$-vectors} will have components such that +\[ +({x_1}^2 + {x_2}^2 + {x_3}^2) = {x_4}^2, +\] +and their magnitude will be zero. + +\Subsubsection{178}{Invariance of $x^2 + y^2 + z^2 - c^2t^2$.} Since we shall naturally +consider the magnitude of a vector to be independent of any particular +choice of axes we have obtained at once by our definition of magnitude +for any rotation of axes that invariance for the expression +\[ +({x_1}^2 + {x_2}^2 + {x_3}^2 - {x_4}^2) = (x^2 + y^2 + z^2 - c^2t^2), +\] +%% -----File: 207.png---Folio 193------- +which is characteristic of the Lorentz transformation, and have thus +evidently set up an imaginary space which will be suitable for plotting +kinematical events in accordance with the requirements of the theory +of the relativity of motion. + +\Subsubsection{179}{Inner Product of One-Vectors.} We shall define the inner +product of two one-vectors with the help of the following rules for the +multiplication of unit vectors along the axes +\[ +\vc{k}_1 · \vc{k}_1 = \vc{k}_2 · \vc{k}_2 = \vc{k}_3· \vc{k}_3 = 1,\qquad +\vc{k}_4 · \vc{k}_4 = -1,\qquad \vc{k}_n · \vc{k}_m = 0. +\Tag{255} +\] + +It should be noted, of course, that there is no particular significance +in picking out the product $\vc{k}_4 · \vc{k}_4$ as the one which is negative; +it would be equally possible to develop a system in which the +products $\vc{k}_1 · \vc{k}_1, \vc{k}_2 · \vc{k}_2$ and $\vc{k}_3 · \vc{k}_3$ should be negative and $\vc{k}_4 · \vc{k}_4$ positive. + +The above rules for unit vectors are sufficient to define completely +the inner product provided we include the further requirements that +this product shall obey the \emph{associative law} for a scalar factor and the +\emph{distributive} and \emph{commutative} laws, namely +\[ +\begin{aligned} +(n\vc{a}) · \vc{b} &= n(\vc{a} · \vc{b}) = (\vc{a}· \vc{b})(n), \\ +\vc{a} · \vc{(b+c)} &= \vc{a} · \vc{b} + \vc{a} · \vc{c}, \\ +\vc{a} · \vc{b} &= \vc{b} · \vc{a}. +\end{aligned} +\Tag{256} +\] + +For the inner product of a one-vector by itself we shall have, in +accordance with these rules, +\begin{multline*} +\vc{r} · \vc{r} + = (x_1 \vc{k}_1 + x_2 \vc{k}_2 + x_3 \vc{k}_3 + x_4 \vc{k}_4) + · (x_1 \vc{k}_1 + x_2 \vc{k}_2 + x_3 \vc{k}_3 + x_4 \vc{k}_4) \\ + = (x_1^2 + x_2^2 + x_3^2 - x_4^2) +\Tag{257} +\end{multline*} +and hence may use the following expressions for the magnitudes of +vectors in terms of inner product +\[ +s = \sqrt{ \vc{r} · \vc{r}} \text{ for $\gamma$-vectors},\qquad +s = \sqrt{-\vc{r} · \vc{r}} \text{ for $\delta$-vectors}. +\Tag{258} +\] + +For curved lines we shall define interval along the curve by the +equations +\[ +\begin{aligned} +\int ds &= \int\sqrt { dr · dr} \text{ for $\gamma$-curves}, \\ +\int ds &= \int\sqrt {-dr · dr} \text{ for $\delta$-curves}. +\end{aligned} +\Tag{259} +\] +%% -----File: 208.png---Folio 194------- + +Our rules further show us that we may obtain the space components +of any one vector by taking its inner product with a unit vector +along the desired axis and may obtain the time component by taking +the negative of the corresponding product. Thus +\[ +\begin{aligned} +\vc{r}·\vc{k}_1 + &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_1 = x_1,\\ +\vc{r}·\vc{k}_2 + &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_2 = x_2,\\ +\vc{r}·\vc{k}_3 + &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_3 = x_3,\\ +\vc{r}·\vc{k}_4 + &= (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)·\vc{k}_4 = -x_4.\\ +\end{aligned} +\Tag{260} +\] + +We see finally moreover in general that the inner product of any +pair of vectors will be numerically equal to the product of the magnitude +of either by the projection of the other upon it, the sign depending +on the nature of the vectors involved. + +\Subsubsection{180}{Non-Euclidean Angle.} We shall define the non-Euclidean +angle~$\theta$ between two vectors $\vc{r}_1$~and~$\vc{r}_2$ in terms of their magnitudes +$s_1$~and~$s_2$ by the expressions +\[ +\pm \vc{r}_1·\vc{r}_2 + = (s_1 × \text{projection}\ s_2) + = s_1s_2\cosh\theta, +\Tag{261} +\] +the sign depending on the nature of the vectors in the way indicated +in the preceding section. We note the analogy between this equation +and those familiar in Euclidean vector-analysis, the hyperbolic +\DPtypo{trigonometeric}{trigonometric} functions taking the place of the circular functions +used in the more familiar analysis. + +For the angle between unit vectors $\vc{k}$~and~$\vc{k'}$ we shall have +\[ +\cosh\theta = \pm \vc{k}·\vc{k'}, +\Tag{262} +\] +where the sign must be chosen so as to make $\cosh\theta$ positive, the +plus sign holding if both are $\gamma$-vectors and the minus sign if both are +$\delta$-vectors. + +\Subsubsection{181}{Kinematical Interpretation of Angle in Terms of Velocity.} +At this point we may temporarily interrupt the development of our +four-dimensional analysis to consider a kinematical interpretation of +non-Euclidean angles in terms of velocity. It will be evident from +our introduction that the behavior of a moving particle can be represented +in our four-dimensional space by a $\delta$-curve,\footnote + {It is to be noted that the actual trajectories of particles are all of them represented + by $\delta$-curves since as we shall see $\gamma$-curves would correspond to velocities + greater than that of light.} +each point on +%% -----File: 209.png---Folio 195------- +this curve denoting the position of the particle at a given instant of +time, and it is evident that the velocity of the particle will be determined +by the angle which this curve makes with the axes. + +Let $\vc{r}$ be the radius vector to a given point on the curve and consider +the derivative of~$\vc{r}$ with respect to the interval $s$ along the curve; +we have +\[ +\vc{w} = \frac{d\vc{r}}{ds} + = \frac{dx_1}{ds}\, \vc{k}_1 + + \frac{dx_2}{ds}\, \vc{k}_2 + + \frac{dx_3}{ds}\, \vc{k}_3 + + \frac{dx_4}{ds}\, \vc{k}_4, +\Tag{263} +\] +and this may be regarded as a unit vector tangent to the curve at the +point in question. + +If $\phi$ is the angle between the $\vc{k}_4$~axis and the tangent to the curve +at the point in question, we have by equation~(262) +\[ +\cosh\phi = - \vc{w}·\vc{k}_4 = \frac{dx_4}{ds}; +\] +making the substitutions for $x_1$,~$x_2$,~$x_3$, and~$x_4$, in terms of $x$,~$y$,~$z$ and~$t$ +we may write, however, +\[ +ds = \sqrt{\smash[b]{dx_4^2 - dx_1^2 - dx_2^2 - dx_3^2}} + = \sqrt{1 - \frac{u^2}{c^2}}\, c\, dt, \Tag{264} +\] +which gives us +\[ +\cosh\phi = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} \Tag{265} +\] +and by the principles of hyperbolic trigonometry we may write the +further relations +\begin{gather*} +\sinh\phi = \frac{\smfrac{u}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}, \Tag{266} +\displaybreak[0] \\ +\tanh\phi = \frac{u}{c}. \Tag{267} +\end{gather*} + + +%[** TN: Heading set like a \Section in original] +\Subsection{Vectors of Higher Dimensions} + +\Subsubsection{182}{Outer Products.} We shall define the outer product of two +one-vectors so that it obeys the \emph{associative law} for a scalar factor, the +%% -----File: 210.png---Folio 196------- +\emph{distributive law} and the \emph{anti-commutative law}, namely, +\[ +\begin{aligned} +(n\vc{a}) × \vc{b} &= n(\vc{a} × \vc{b}) = \vc{a} × (n\vc{b}),\\ + \vc{a} × (\vc{b} + \vc{c}) &= \vc{a} × \vc{b} + \vc{a} × \vc{c}\DPchg{}{,}\quad +( \vc{a} + \vc{b}) × \vc{c} = \vc{a} × \vc{c} + \vc{b} × \vc{c}, \\ + \vc{a} × \vc{b} &= -\vc{b}× \vc{a}. +\end{aligned} +\Tag{268} +\] + +From a geometrical point of view, we shall consider the outer +product of two one-vectors to be itself a \emph{two-vector}, namely the parallelogram, +or more generally, the area which they determine. The +sign of the two-vector may be taken to indicate the direction of progression +clockwise or anti-clockwise around the periphery. In order +to accord with the requirement that the area of a parallelogram determined +by two lines becomes zero when they are rotated into the same +direction, we may complete our definition of outer product by adding +the requirement that the outer product of a vector by itself shall be +zero. +\[ +\vc{a} × \vc{a} = 0. +\Tag{269} +\] + +We may represent the outer products of unit vectors along the +chosen axes as follows: +\[ +\begin{aligned} +\vc{k}_1 × \vc{k}_1 &= \vc{k}_2 × \vc{k}_2 = \vc{k}_3 × \vc{k}_3 = \vc{k}_4 × \vc{k}_4 = 0,\\ +\vc{k}_1 × \vc{k}_2 &= -\vc{k}_2 × \vc{k}_1 = \vc{k}_{12} = -\vc{k}_{21},\\ +\vc{k}_1 × \vc{k}_3 &= -\vc{k}_3 × \vc{k}_1 = \vc{k}_{13} = -\vc{k}_{31},\quad \text{etc.},\\ +\end{aligned} +\Tag{270} +\] +where we may regard~$\vc{k}_{12}$, for example, as a unit parallelogram in the +plane~$X_1OX_2$. + +We shall continue to use small letters in Clarendon type for one-vectors +and shall use capital letters in Clarendon type for two-vectors. +The components of a two-vector along the six mutually perpendicular +planes $X_1OX_2$,~$X_1OX_3$,~etc., may be obtained by expressing the one-vectors +involved in terms of their components along the axes and +carrying out the indicated multiplication, thus: +\[ +\begin{aligned} +\vc{A} &= \vc{a} × \vc{b} + = (a_1\vc{k}_1 + a_2\vc{k}_2 + a_3\vc{k}_3 + a_4\vc{k}_4) \\ + &\quad × (b_1\vc{k}_1 + b_2\vc{k}_2 + b_3\vc{k}_3 + b_4\vc{k}_4) \\ + &= (a_1b_2 - a_2b_1)\vc{k}_{12} + + (a_1b_3 - a_3b_1)\vc{k}_{13} + + (a_1b_4 - a_4b_1)\vc{k}_{14} \\ + &\quad + (a_2b_3 - a_3b_2)\vc{k}_{23} + + (a_2b_4 - a_4b_2)\vc{k}_{24} + + (a_3b_4 - a_4b_3)\vc{k}_{34}, +\end{aligned} +\Tag{271} +\] +%% -----File: 211.png---Folio 197------- +or, calling the quantities $(a_1b_2 - a_2b_1)$,~etc., the component magnitudes +of $\vc{A}$,~$A_{12}$,~etc., we may write +\[ +\vc{A} = A_{12}\vc{k}_{12} + A_{13}\vc{k}_{13} + A_{14}\vc{k}_{14} + + A_{23}\vc{k}_{23} + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34}. +\Tag{272} +\] + +The concept of outer product may be extended to include the +idea of vectors of higher number of dimensions than two. Thus the +outer product of three one-vectors, or of a one-vector and a two-vector +will be a three-vector which may be regarded as a \emph{directed} parallelopiped +in our four-dimensional space. The outer product of four one-vectors +will lead to a four-dimensional solid which would have direction +only in a space of more than four dimensions and hence in our case +will be called a pseudo-scalar. The outer product of vectors the +sum of whose dimensions is greater than that of the space considered +will vanish. + +The results which may be obtained from different types of outer +multiplication are tabulated below, where one-vectors are denoted +by small Clarendon type, two-vectors by capital Clarendon type, +three-vectors by Tudor black capitals, and pseudo-scalars by bold face +Greek letters. +{\small% +\begin{align*} %[** TN: Re-breaking] +&\begin{aligned} +\vc{A} + &= \vc{a} × \vc{b} = -\vc{b} × \vc{a} \\ + &= (a_1b_2 - a_2b_1)\vc{k}_{12} + + (a_1b_4 - a_3b_1)\vc{k}_{13} + + (a_1b_4 - a_4b_1)\vc{k}_{14} \\ + &+ (a_2b_3 - a_3b_2)\vc{k}_{23} + + (a_2b_4 - a_4b_2)\vc{k}_{21} + + (a_3b_4 - a_4b_3)\vc{k}_{34}, +\end{aligned} \displaybreak[0] \\[12pt] +&\begin{aligned} +\Alpha + &= \vc{c} × \vc{A} \\ + &= (c_1A_{23} - c_2A_{13} + c_3A_{12})\vc{k}_{123} + + (c_1A_{24} - c_2A_{14} + c_4A_{12})\vc{k}_{124} \\ + &+ (c_1A_{34} - c_2A_{14} + c_4A_{15})\vc{k}_{134} + + (c_2A_{34} - c_3A_{24} + c_4A_{23})\vc{k}_{234} +\end{aligned} +\Tag{273} \displaybreak[0] \\[12pt] +&\begin{aligned} +\vc{\alpha} + &= \vc{d} × \Alpha = -\Alpha × \vc{d} \\ + &= (d_1\Alpha_{234} - d_2\Alpha_{134} + + d_3\Alpha_{124} - d_4\Alpha_{123})\vc{k}_{1234}, \\ +\vc{\alpha} + &= \vc{A} × \vc{B} \\ + &= (A_{12}B_{34} - A_{13}B_{24} + A_{14}B_{23} + A_{23}B_{14} + - A_{24}B_{13} + A_{34}B_{12})\vc{k}_{1234}. +\end{aligned} +\end{align*}}% + +\emph{The signs in these expressions are determined by the general rule +that the sign of any unit vector~$\vc{\bar{k}}_{nmo}$ will be reversed by each transposition +of the order of a pair of adjacent subscripts, thus}: +\[ +k_{abcd} = - k_{bacd} = k_{bcad},\qquad \text{etc.},\ \cdots. +\Tag{274} +\] +%% -----File: 212.png---Folio 198------- + +\Subsubsection{183}{Inner Product of Vectors in General.} We have previously +defined the inner product for the special case of a pair of one-vectors, +in order to bring out some of the important characteristics of our +non-Euclidean space. We may now give a general rule for the inner +product of vectors of any number of dimensions. + +The inner product of any pair of vectors follows the \emph{associative} +law for scalar factors, and follows the \emph{distributive} and \emph{commutative} +laws. + +Since we can express any vector in terms of its components, the +above rules will completely determine the inner product of any pair +of vectors provided that we also have a rule for obtaining the inner +products of the unit vectors determined by the mutually perpendicular +axes. This rule is as follows: Transpose the subscripts of the unit +vectors involved so that the common subscripts occur at the end and +in the same order and cancel these common subscripts. If both the +unit vectors still have subscripts the product is zero; if neither vector +has subscripts the product is unity, and if one of the vectors still has +subscripts that itself will be the product. The sign is to be taken +as that resulting from the transposition of the subscripts (see equation~(274)), unless the subscript~$4$ has been cancelled, when the sign +will be changed. + +For example: +\[ +\begin{aligned} +\vc{k}_{124} · \vc{k}_{34} &= \vc{k}_{12} · \vc{k}_{3} = 0, \\ +\vc{k}_{132} · \vc{k}_{123} &= -\vc{k}_{123} · \vc{k}_{123} = -1, \\ +\vc{k}_{124} · \vc{k}_{42} &= -\vc{k}_{124} · \vc{k}_{24} = \vc{k}_{1}. +\end{aligned} +\Tag{275} +\] + +It is evident from these rules that we may obtain the magnitude +of any desired component of a vector by taking the inner product of +the vector by the corresponding unit vector, it being noticed, of course, +that when the unit vector involved contains the subscript~$4$ we obtain +the negative of the desired component. For example, we may obtain +the $k_{12}$~component of a two-vector as follows: +\[ +\begin{aligned} +A_{12} + = \vc{A} · \vc{k}_{12} + = (A_{12}\vc{k}_{12} &+ A_{13}\vc{k}_{13} + A_{14}\vc{k}_{14} \\ + &+ A_{23}\vc{k}_{23} + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34}) · \vc{k}_{12}. +\end{aligned} +\Tag{276} +\] + +\Subsubsection{184}{The Complement of a Vector.} In an $n$-dimensional space +any $m$-dimensional vector will uniquely determine a new vector of +%% -----File: 213.png---Folio 199------- +dimensions $(n-m)$ which may be called the complement of the +original vector. The complement of a vector may be exactly defined +as the inner product of the original vector with the unit pseudo-scalar +$\vc{k}_{123\cdots n}$. In general, we may denote the complement of a vector +by placing an asterisk~$*$ after the symbol. As an example we may +write as the complement of a two-vector~$\vc{A}$ in our non-Euclidean four-dimensional +space: +\[ +\begin{aligned} +\vc{A}^* &= +\begin{aligned}[t] + \vc{A} · \vc{k}_{1234} + = (A_{12}\vc{k}_{12} &+ A_{13}\vc{k}_{13} + A_{14}\vc{k}_{14} \\ + &+ + A_{23}\vc{k}_{23} + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34}) · \vc{k}_{1234} +\end{aligned} \\ + &= (A_{12}\vc{k}_{34} - A_{13}\vc{k}_{24} - A_{14}\vc{k}_{23} + + A_{23}\vc{k}_{14} + A_{24}\vc{k}_{13} - A_{34}\vc{k}_{12}). +\end{aligned} +\Tag{277} +\] + +\Subsubsection{185}{The Vector Operator, $\Qop$ or Quad.} Analogous to the familiar +three-dimensional vector-operator del, +\[ +\nabla + = \vc{k}_1\, \frac{\partial}{\partial x_1} + + \vc{k}_2\, \frac{\partial}{\partial x_2} + + \vc{k}_3\, \frac{\partial}{\partial x_3}, +\Tag{278} +\] +we may define the four-dimensional vector-operator quad, +\[ +\Qop + = \vc{k}_1\, \frac{\partial}{\partial x_1} + + \vc{k}_2\, \frac{\partial}{\partial x_2} + + \vc{k}_3\, \frac{\partial}{\partial x_3} + - \vc{k}_4\, \frac{\partial}{\partial x_4}. +\Tag{279} +\] + +If we have a scalar or a vector field we may apply these operators +by regarding them formally as one-vectors and applying the rules +for inner and outer multiplication which we have already given. + +Thus if we have a scalar function~$F$ which varies continuously +from point to point we can obtain a one-vector which we may call +the four-dimensional gradient of~$F$ at the point in question by simple +multiplication; we have +\[ +\grad F= \Qop F + = \vc{k}_1\, \frac{\partial F}{\partial x_1} + + \vc{k}_2\, \frac{\partial F}{\partial x_2} + + \vc{k}_3\, \frac{\partial F}{\partial x_3} + - \vc{k}_4\, \frac{\partial F}{\partial x_4}. +\Tag{280} +\] +If we have a one-vector field, with a vector~$\vc{f}$ whose value varies +from point to point we may obtain by inner multiplication a scalar +quantity which we may call the four-dimensional divergence of~$\vc{f}$\DPtypo{ we}{. We} +have +\[ +\divg\vc{f} = \Qop · \vc{f} + = \frac{\partial f_1}{\partial x_1} + + \frac{\partial f_2}{\partial x_2} + + \frac{\partial f_3}{\partial x_3} + + \frac{\partial f_4}{\partial x_4}. +\Tag{280} +\] +Taking the outer product with quad we may obtain a two-vector, the +%% -----File: 214.png---Folio 200------- +four-dimensional curl of~$\vc{f}$, +\[ +\begin{aligned}%[** TN: Re-aligning] +\curl \vc{f} = \Qop × \vc{f} + &= \left(\frac{\partial f_2}{\partial x_1} + - \frac{\partial f_1}{\partial x_2}\right) \vc{k}_{12} + + \left(\frac{\partial f_3}{\partial x_1} + - \frac{\partial f_1}{\partial x_3}\right) \vc{k}_{13} \\ + &+ \left(\frac{\partial f_4}{\partial x_1} + + \frac{\partial f_1}{\partial x_4}\right) \vc{k}_{14} + + \left(\frac{\partial f_3}{\partial x_2} + - \frac{\partial f_2}{\partial x_3}\right) \vc{k}_{23} \\ + &+ \left(\frac{\partial f_4}{\partial x_2} + + \frac{\partial f_2}{\partial x_4}\right) \vc{k}_{24} + + \left(\frac{\partial f_4}{\partial x_3} + + \frac{\partial f_3}{\partial x_4}\right) \vc{k}_{34}. +\end{aligned} +\Tag{282} +\] +By similar methods we could apply quad to a two-vector function~$\vc{F}$ +and obtain the one-vector function $\Qop · \vc{F}$ and the three-vector function +$\Qop × \vc{F}$. + +\Paragraph{186.} Still regarding $\Qop$ as a one-vector we may obtain a number of +important expressions containing~$\Qop$ more than once; we have: +\begin{align*} +\Qop × (\Qop F) &= 0, \quad(283) & +\Qop × (\Qop × \vc{f}) &= 0,\quad (286) \\ +% +\Qop · (\Qop · \vc{F}) &= 0, \quad (284) & +\Qop × (\Qop × \vc{F}) &= 0, \quad (287) \\ +% +\Qop · (\Qop · \frakF) &= 0, \quad (285) && +\end{align*} +\begin{align*} +\Qop · (\Qop × \vc{f}) + &= \Qop (\Qop · \vc{f}) - (\Qop · \Qop)\vc{f}, +\Tag{288} \\ +\Qop · (\Qop × \vc{F}) + &= \Qop × (\Qop · \vc{F}) + (\Qop · \Qop)\vc{F}, +\Tag{289}\\ +\Qop · (\Qop × \frakF) + &= \Qop × (\Qop · \frakF) - (\Qop · \Qop)\frakF. +\Tag{290} +\end{align*} + +The operator $\Qop · \Qop$ or~$\Qop^2$ has long been known under the name +of the D'Alembertian, +\[ +\Qop^2 = \frac{\partial^2}{\partial {x_1}^2} + + \frac{\partial^2}{\partial {x_2}^2} + + \frac{\partial^2}{\partial {x_3}^2} + - \frac{\partial^2}{\partial {x_4}^2} + = \Delta^2 - \frac{\partial^2}{c^2\, \partial t^2}. +\Tag{291} +\] + +From the definition of the complement of a vector given in the +previous section it may be shown by carrying out the proper expansions +that +\[ +(\Qop × \phi)^* = \Qop · \phi^*, +\Tag{292} +\] +where $\phi$~is a vector of any number of dimensions. + +\Subsubsection{187}{Tensors.} In analogy to three-dimensional tensors we may +define a four-dimensional tensor as a quantity with sixteen components +as given in the following table: +\[ +T = \left\{ +\begin{matrix} +T_{11} & T_{12} & T_{13} &T_{14}, \\ +T_{21} & T_{22} & T_{23} &T_{24}, \\ +T_{31} & T_{32} & T_{33} &T_{34}, \\ +T_{41} & T_{42} & T_{43} &T_{44}, +\end{matrix} +\right. +\Tag{293} +\] +%% -----File: 215.png---Folio 201------- +with the additional requirement that the divergence of the tensor, +defined as follows, shall itself be a one-vector. +\[ +\settowidth{\TmpLen}{$\ds\frac{\partial T_{12}}{\partial x_2} + +\frac{\partial T_{13}}{\partial x_3} + +\frac{\partial T_{14}}{\partial x_4}\,$}% +\begin{aligned} +\divg T &= \left\{ + \frac{\partial T_{11}}{\partial x_1} + + \frac{\partial T_{12}}{\partial x_2} + + \frac{\partial T_{13}}{\partial x_3} + + \frac{\partial T_{14}}{\partial x_4}\right\}\vc{k}_1 \\ + &+ \left\{\frac{\partial T_{21}}{\partial x_1} + + \makebox[\TmpLen][l]{$\cdots$} \right\}\vc{k}_2 \\ + &+ \left\{\frac{\partial T_{31}}{\partial x_1} + + \makebox[\TmpLen][l]{$\cdots$} \right\}\vc{k}_3 \\ + &+ \left\{\frac{\partial T_{41}}{\partial x_1} + + \makebox[\TmpLen][l]{$\cdots$} \right\}\vc{k}_4 \\ +\end{aligned} +\Tag{294} +\] + +\Subsubsection{188}{The Rotation of Axes.} Before proceeding to the application +of our four-dimensional analysis to the actual problems of relativity +theory we may finally consider the changes in the components of a +vector which would be produced by a rotation of the axes. We have +already pointed out that the quantity $({x_1}^2 + {x_2}^2 + {x_3}^2 - {x_4}^2)$ is an +invariant in our space for any set of rectangular coördinates having +the same origin since it is the square of the magnitude of a radius +vector, and have noted that in this way we have obtained for the +quantity $(x^2 + y^2 + z^2 - c^2t^2)$ the desired invariance which is characteristic +of the Lorentz transformation. In fact we may look upon +the Lorentz transformation as a rotation from a given set of axes to a +new set, with a corresponding re-expression of quantities in terms of +the new components. The particular form of Lorentz transformation, +familiar in preceding chapters, in which the new set of spatial axes +has a velocity component relative to the original set, in the $X$\DPchg{-}{~}direction +alone, will be found to correspond to a rotation of the axes in which +only the directions of the $X_1$~and~$X_4$ axes are changed, the $X_2$~and~$X_3$ +axes remaining unchanged in direction. + +Let us consider a one-vector +\[ +\vc{a} + = (a_1\vc{k}_1 + a_2\vc{k}_2 + a_3\vc{k}_3 + a_4\vc{k}_4) + = ({a_1}'\vc{k_1}' + {a_2}'\vc{k_2}' + {a_3}'\vc{k_3}' + {a_4}'\vc{k_4}'), +\] +where $a_1$,~$a_2$,~$a_3$ and~$a_4$ are the component magnitudes, using a set of +axes which have $\vc{k}_1$,~$\vc{k}_2$,~$\vc{k}_3$ and~$\vc{k}_4$ as unit vectors and ${a_1}'$,~${a_2}'$,~${a_3}'$ and~${a_4}'$ +the corresponding magnitudes using another set of mutually perpendicular +axes with the unit vectors $\vc{k_1}'$,~$\vc{k_2}'$,~$\vc{k_3}'$ and~$\vc{k_4}'$. Our problem, +%% -----File: 216.png---Folio 202------- +now, is to find relations between the magnitudes $a_1$,~$a_2$,~$a_3$ and~$a_4$ and +${a_1}'$,~${a_2}'$,~${a_3}'$ and~${a_4}'$. + +We have already seen\DPtypo{}{,} \DPchg{sections (\Secnumref{179})~and~(\Secnumref{183})}{Sections \Secnumref{179}~and~\Secnumref{183}}, that we may obtain +any desired component magnitude of a vector by taking its inner +product with a unit vector in the desired direction, reversing the +sign if the subscript~$4$ is involved. We may obtain in this way an +expression for~$a_1$ in terms of ${a_1}'$,~${a_2}'$,~${a_3}'$ and~${a_4}'$. We have +\begin{align*} +a_1 = \vc{a}·\vc{k}_1 + &= ({a_1}'{\vc{k}_1}' + {a_2}'{\vc{k}_2}' + + {a_3}'{\vc{k}_3}' + {a_4}'{\vc{k}_4}') · {\vc{k}_1} \\ + &= {a_1}'{\vc{k}_1}' · \vc{k}_1 + {a_2}'{\vc{k}_2}' · \vc{k}_1 + + {a_3}'{\vc{k}_3}' · \vc{k}_1 + {a_4}'{\vc{k}_4}' · \vc{k}_1. +\Tag{295} +\end{align*} +By similar multiplications with $\vc{k_2}$,~$\vc{k_3}$ and~$\vc{k_4}$ we may obtain expressions +for $a_2$,~$a_3$ and~$-a_4$. The results can be tabulated in the convenient +form +\[ +\begin{array}{c|*{4}{l|}} + & \Neg{a_1}' & \Neg{a_2}' & \Neg{a_3}' & \Neg{a_4}' \\ +\hline +a_1 & \Neg{\vc{k}_1}' · \vc{k}_1 & \Neg{\vc{k}_2}' · \vc{k}_1 + & \Neg{\vc{k}_3}' · \vc{k}_1 & \Neg{\vc{k}_4}' · \vc{k}_1 \\ +\hline +a_2 & \Neg{\vc{k}_1}' · \vc{k}_2 & \Neg{\vc{k}_2}' · \vc{k}_2 + & \Neg{\vc{k}_3}' · \vc{k}_2 & \Neg{\vc{k}_4}' · \vc{k}_2 \\ +\hline +a_3 & \Neg{\vc{k}_1}' · \vc{k}_3 & \Neg{\vc{k}_2}' · \vc{k}_3 + & \Neg{\vc{k}_3}' · \vc{k}_3 & \Neg{\vc{k}_4}' · \vc{k}_3 \\ +\hline +a_4 & -{\vc{k}_1}' · \vc{k}_4 & -{\vc{k}_2}' · \vc{k}_4 + & -{\vc{k}_3}' · \vc{k}_4 & -{\vc{k}_4}' · \vc{k}_4 \\ +\hline +\end{array} +\Tag{296} +\] + +Since the square of the magnitude of the vector, $({a_1}^2 + {a_2}^2 + {a_3}^2 +- {a_4}^2)$, is a quantity which is to be independent of the choice of axes, +we shall have certain relations holding between the quantities ${\vc{k}_1}'· \vc{k}_1$, +${\vc{k}_1}' · \vc{k}_2$, etc. These relations, which are analogous to the familiar +%% -----File: 217.png---Folio 203------- +conditions of orthogonality in Euclidean space, can easily be shown +to be +\[ +\begin{aligned} +({\vc{k}_1}'· \vc{k}_1)^2 + ({\vc{k}_1}'· \vc{k}_2)^2 + ({\vc{k}_1}'· \vc{k}_3)^2 - ({\vc{k}_1}'· \vc{k}_4)^2 &= 1,\\ +({\vc{k}_2}'· \vc{k}_1)^2 + ({\vc{k}_2}'· \vc{k}_2)^2 + ({\vc{k}_2}'· \vc{k}_3)^2 - ({\vc{k}_2}'· \vc{k}_4)^2 &= 1,\\ +({\vc{k}_3}'· \vc{k}_1)^2 + ({\vc{k}_3}'· \vc{k}_2)^2 + ({\vc{k}_3}'· \vc{k}_3)^2 - ({\vc{k}_3}'· \vc{k}_4)^2 &= 1,\\ +({\vc{k}_4}'· \vc{k}_1)^2 + ({\vc{k}_4}'· \vc{k}_2)^2 + ({\vc{k}_4}'· \vc{k}_3)^2 - ({\vc{k}_4}'· \vc{k}_4)^2 &= - 1, +\end{aligned} +\Tag{297} +\] +and +\begin{align*}%[** TN: Re-breaking] +({\vc{k}_1}'· \vc{k}_1)({\vc{k}_2}' · \vc{k}_1) + &+ ({\vc{k}_1}' · \vc{k}_2)({\vc{k}_2}' · \vc{k}_2) \\ + &+ ({\vc{k}_1}' · \vc{k}_3)({\vc{k}_2}' · \vc{k}_3) + - ({\vc{k}_1}' · \vc{k}_4)({\vc{k}_2}' · \vc{k}_4) = 0, +\end{align*} +etc., for each of the six pairs of vertical columns in table~(296). + +Since we shall often be interested in a simple rotation in which +the directions of the $X_2$~and~$X_3$ axes are not changed, we shall be able +to simplify this table for that particular case by writing +\[ +{\vc{k}_2}' = \vc{k}_2,\qquad +{\vc{k}_3}' = \vc{k}_3, +\] +and noting the simplifications thus introduced in the products of the +unit vectors, we shall obtain +\[ +\begin{array}{*{5}{c|}} + & \Neg {a_1}' & {a_2}' & {a_3}' & \Neg {a_4}' \\ +\hline +a_1 & \Neg{\vc{k}_1}' · \vc{k}_1 & 0 & 0 & \Neg{\vc{k}_4}' · \vc{k}_1 \\ +\hline +a_2 & \Neg 0 & 1 & 0 & \Neg 0 \\ +\hline +a_3 & \Neg 0 & 0 & 1 & \Neg 0 \\ +\hline +a_4 & -{\vc{k}_1}' · \vc{k}_4 & 0 & 0 & -{\vc{k}_4}' · \vc{k}_4 \\ +\hline +\end{array} +\Tag{298} +\] +%% -----File: 218.png---Folio 204------- + +If now we call~$\phi$ the angle of rotation between the two time axes +${OX_4}'$~and~$OX_4$, we may write, in accordance with equation~(262), +\[ +-{\vc{k}_4}' · \vc{k}_4 = \cosh \phi. +\] + +Since we must preserve the orthogonal relations~(297) and may +also make use of the well-known expression of hyperbolic trigonometry +\[ +\cosh^2 \phi - \sinh^2 \phi = 1, +\] +we may now rewrite our transformation table in the form +\[ +\begin{array}{*{5}{c|}} + & {a_1}' & {a_2}' & {a_3}' & {a_4}' \\ +\hline +a_1 & \cosh\phi & 0 & 0 & \sinh \phi \\ +\hline +a_2 & 0 & 1 & 0 & 0 \\ +\hline +a_3 & 0 & 0 & 1 & 0 \\ +\hline +a_4 & \sinh \phi & 0 & 0 & \cosh \phi \\ +\hline +\end{array} +\Tag{299} +\] + +By a similar process we may obtain transformation tables for the +components of a two-vector~$\vc{A}$. Expressing~$\vc{A}$ in terms of the unit +vectors ${\vc{k}_{12}}'$,~${\vc{k}_{13}}'$, ${\vc{k}_{14}}'$,~etc., and taking successive inner products with +the unit vectors $\vc{k}_{12}$,~$\vc{k}_{13}$, $\vc{k}_{14}$,~etc., we may obtain transformation +equations which can be expressed by the \hyperref[table:300]{tabulation~(300)} shown on +the following page.\DPnote{[** TN: No need for varioref]} +%% -----File: 219.png---Folio 205------- +\begin{sidewaystable}[p] +\phantomsection\label{table:300}% +\renewcommand{\arraystretch}{3} +\[ +\begin{array}{c|*{6}{r|}} + & \multicolumn{1}{c|}{{A_{12}}'} & \multicolumn{1}{c|}{{A_{13}}'} + & \multicolumn{1}{c|}{{A_{14}}'} & \multicolumn{1}{c|}{{A_{23}}'} + & \multicolumn{1}{c|}{{A_{24}}'} & \multicolumn{1}{c|}{{A_{34}}'} \\ +\hline +A_{12} & {\vc{k}_{12}}' · \vc{k}_{12} & {\vc{k}_{13}}' · \vc{k}_{12} + & {\vc{k}_{14}}' · \vc{k}_{12} & {\vc{k}_{23}}' · \vc{k}_{12} + & {\vc{k}_{24}}' · \vc{k}_{12} & {\vc{k}_{34}}' · \vc{k}_{12} \\ +\hline +A_{13} & {\vc{k}_{12}}' · \vc{k}_{13} & {\vc{k}_{13}}' · \vc{k}_{13} + & {\vc{k}_{14}}' · \vc{k}_{13} & {\vc{k}_{23}}' · \vc{k}_{13} + & {\vc{k}_{24}}' · \vc{k}_{13} & {\vc{k}_{34}}' · \vc{k}_{13} \\ +\hline +A_{14} &-{\vc{k}_{12}}' · \vc{k}_{14} & -{\vc{k}_{13}}' · \vc{k}_{14} + & -{\vc{k}_{14}}' · \vc{k}_{14} & -{\vc{k}_{23}}' · \vc{k}_{14} + & -{\vc{k}_{24}}' · \vc{k}_{14} & -{\vc{k}_{34}}' · \vc{k}_{14} \\ +\hline +A_{23} & {\vc{k}_{12}}' · \vc{k}_{23} & {\vc{k}_{13}}' · \vc{k}_{23} + & {\vc{k}_{14}}' · \vc{k}_{23} & {\vc{k}_{23}}' · \vc{k}_{23} + & {\vc{k}_{24}}' · \vc{k}_{23} & {\vc{k}_{34}}' · \vc{k}_{23} \\ +\hline +A_{24} & -{\vc{k}_{12}}' · \vc{k}_{24} & -{\vc{k}_{13}}' · \vc{k}_{24} + & -{\vc{k}_{14}}' · \vc{k}_{24} & -{\vc{k}_{23}}' · \vc{k}_{24} + & -{\vc{k}_{24}}' · \vc{k}_{24} & -{\vc{k}_{34}}' · \vc{k}_{24} \\ +\hline +A_{34} & -{\vc{k}_{12}}' · \vc{k}_{34} & -{\vc{k}_{13}}' · \vc{k}_{34} + & -{\vc{k}_{14}}' · \vc{k}_{34} & -{\vc{k}_{23}}' · \vc{k}_{34} + & -{\vc{k}_{24}}' · \vc{k}_{34} & -{\vc{k}_{34}}' · \vc{k}_{34} \\ +\hline +\end{array} +\Tag{300} +\] +\end{sidewaystable} + +For the particular case of a rotation in which the direction of the +$X_2$~and~$X_3$ axes are not changed we shall have +\[ +{\vc{k}_2}' = \vc{k}_2,\qquad +{\vc{k}_3}' = \vc{k}_3, +\] +and very considerable simplification will be introduced. We shall +have, for example, +\begin{alignat*}{4} +&{\vc{k}_{12}}'· \vc{k}_{12} + &&= ({\vc{k}_1}' × {\vc{k}_2}') · (\vc{k}_1 × \vc{k}_2) + &&= ({\vc{k}_1}' × \vc{k}_2) · (\vc{k}_1 × \vc{k}_2) + &&= {\vc{k}_1}' · \vc{k}_1, \\ +&{\vc{k}_{13}}' · \vc{k}_{12} + &&= ({\vc{k}_1}' × {\vc{k}_3}') · (\vc{k}_1 × \vc{k}_2) + &&= ({\vc{k}_1}' × \vc{k}_3 ) · (\vc{k}_1 × \vc{k}_2) + &&= 0, \\ +&\text{etc.} +\end{alignat*} +Making these and similar substitutions and introducing, as before, +%% -----File: 220.png---Folio 206------- +the relation $-\DPtypo{{\vc{k}'}_4}{{\vc{k}_4}'} · \vc{k}_4 = \cosh \phi$ where $\phi$~is the non-Euclidean angle +between the two time axes, we may write our transformation table +in the form +\[ +\begin{array}{*{7}{c|}} + & \Neg{A_{12}}' & \Neg{A_{13}}' & {A_{14}}' & {A_{23}}' & {A_{24}}' &{A_{34}}' \\ +\hline +A_{12} & \Neg\cosh\phi & 0 & 0 & 0 & \sinh\phi & 0 \\ +\hline +A_{13} & \Neg0 & \Neg\cosh\phi & 0 & 0 & 0 & \sinh\phi \\ +\hline +A_{14} & \Neg0 & \Neg0 & 1 & 0 & 0 & 0 \\ +\hline +A_{23} & \Neg0 & \Neg0 & 0 & 1 & 0 & 0 \\ +\hline +A_{24} & -\sinh\phi & 0 & 0 & 0 & \cosh\phi & 0 \\ +\hline +A_{34} & \Neg0 & -\sinh\phi & 0 & 0 & 0 & \cosh\phi \\ +\hline +\end{array} +\Tag{301} +\] + +\Subsubsection{189}{Interpretation of the Lorentz Transformation as a Rotation +of Axes.} We may now show that the Lorentz transformation may +be looked upon as a change from a given set of axes to a rotated set. + +Since the angle~$\phi$ which occurs in our transformation tables is +that between the $\vc{k}_4$~axis and the new ${\vc{k}_4}'$~axis, we may write, in accordance +with equations (265)~and~(266), +\[ +\cosh \phi = \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}, \qquad +\sinh \phi = \frac{\smfrac{V}{c}}{\sqrt{1 - \smfrac{V^2}{c^2}}}, +\] +where $V$~is the velocity between the two sets of space axes which +correspond to the original and the rotated set of four-dimensional +axes. This will permit us to rewrite our transformation table for the +%% -----File: 221.png---Folio 207------- +components of a one-vector in the forms +\begin{gather*} +\phantomsection\label{table:302}% +\renewcommand{\arraystretch}{2} +\begin{array}{*{5}{c|}} + & {a_1}' & {a_2}' & {a_3}' & {a_4}' \\ +\hline +a_1 & \dfrac{1} {\sqrt{1 - \smfrac{V^2}{c^2}}}& 0 & 0 + & \dfrac{V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\ +\hline +a_2 & 0 & 1 & 0 & 0 \\ +\hline +a_3 & 0 & 0 & 1 & 0 \\ +\hline +a_4 & \dfrac{V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} & 0 & 0 + & \dfrac{1} {\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\ +\hline +\end{array} \\ +\Tag{302} \\ +\renewcommand{\arraystretch}{2} +\begin{array}{*{5}{c|}} + & a_1 & a_2 & a_3 & a_4 \\ +\hline +{a_1}' & \dfrac{1} {\sqrt{1 - \smfrac{V^2}{c^2}}} & 0 & 0 + & \dfrac{-V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\ +\hline +{a_2}' & 0 & 1 & 0 & 0 \\ +\hline +{a_3}' & 0 & 0 & 1 & 0 \\ +\hline +{a_4}' & \dfrac{-V/c}{\sqrt{1 - \smfrac{V^2}{c^2}}} & 0 & 0 + & \dfrac{1} {\sqrt{1 - \smfrac{V^2}{c^2}}} \rule[-32pt]{0pt}{48pt} \\ +\hline +\end{array} +\end{gather*} + +Consider now any point $P(x_1, x_2, x_3, x_4)$. The radius vector from +the origin to this point will be $\vc{r} = (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4)$, or, +making use of the relations between $x_1$,~$x_2$, $x_3$,~$x_4$ and $x$,~$y$, $z$,~$t$ given +by equations~(252), we may write +\[ +\vc{r} = (x\vc{k}_1 + y\vc{k}_2 + z\vc{k}_3 + ct\vc{k}_4). +\] +Applying our transformation table to the components of this one-vector, +we obtain the familiar equations for the Lorentz transformation +\begin{align*} +x' &= \frac{x - Vt}{\sqrt{1 - \smfrac{V^2}{c^2}}}, \\ +%% -----File: 222.png---Folio 208------- +y' &= y, \\ +z' &= z, \\ +t' &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\left(t - \frac{V}{c^2}\, x\right). +\end{align*} + +We thus see that the Lorentz transformation is to be interpreted +in our four-dimensional analysis as a rotation of axes. + +\Subsubsection{190}{Graphical Representation.} Although we have purposely restricted +ourselves in the foregoing treatment to methods of attack +which are almost purely analytical rather than geometrical in nature, +the importance of a graphical representation of our four-dimensional +manifold should not be neglected. The difficulty of representing all +four axes on a single piece of two-dimensional paper is not essentially +different from that encountered in the graphical representation of the +facts of ordinary three-dimensional solid geometry, and these difficulties +can often be solved by considering only one pair of axes at a +time, say $OX_1$~and~$OX_4$, and plotting the occurrences in the $X_1OX_4$ +plane. The fact that the geometry of this plane is a non-Euclidean +one presents a more serious complication since the figures that we +draw on our sheet of paper will obviously be Euclidean in nature, +but this difficulty also can be met if we make certain conventions as +to the significance of the lines we draw, conventions which are fundamentally +not so very unlike the conventions by which we interpret as +solid, a figure drawn in ordinary perspective. + +Consider for example the diagram shown in \Figref{18}, where we +have drawn a pair of perpendicular axes, $OX_1$,~and~$OX_4$ and the +two unit hyperbolæ given by the equations +\[ +\begin{aligned} +{x_1}^2 - {x_4}^2 &= 1, \\ +{x_1}^2 - {x_4}^2 &= -1, +\end{aligned} +\Tag{303} +\] +together with their asymptotes, $OA$~and~$OB$, given by the equation +\[ +{x_1}^2 - {x_4}^2 = 0. +\Tag{304} +\] +This purely Euclidean figure permits, as a matter of fact, a fairly +satisfactory representation of the non-Euclidean properties of the +manifold with which we have been dealing. +%% -----File: 223.png---Folio 209------- + +$OX_1$~and~$OX_4$ may be considered as perpendicular axes in the +non-Euclidean $X_1OX_4$~plane. Radius vectors lying in the quadrant~$AOB$\DPtypo{,}{} +will have a greater component along the~$X_4$ than along the $X_1$~axis +and hence will be $\delta$-vectors with the magnitude $s = \sqrt{{x_4}^2 - {x_1}^2}$, +where $x_1$~and~$x_4$ are the coördinates of the terminal of the vector. +\begin{figure}[hbt] + \begin{center} + \Fig{18} + \Input[4in]{223} + \end{center} +\end{figure} +$\gamma$-radius-vectors will lie in the quadrant~$BOC$ and will have the magnitude +$s = \sqrt{{x_1}^2 - {x_4}^2}$. Radius vectors lying along the asymptotes +$OA$~and~$OB$ will have zero magnitudes ($s = \sqrt{{x_1}^2 - {x_4}^2} = 0$) and +hence will be singular vectors. + +Since the two hyperbolæ have the equations ${x_1}^2 - {x_4}^2 = 1$ and +${x_1}^2 - {x_4}^2 = -1$, rays such as $Oa$,~${Oa}'$, $Ob$,~etc., starting from the +origin and terminating on the hyperbolæ, will all have unit magnitude. +Hence we may consider the hyperbolæ as representing unit pseudo-circles +in our non-Euclidean plane and consider the rays as representing +the radii of these pseudo-circles. + +A non-Euclidean rotation of axes will then be represented by +changing from the axes $OX_1$~and~$OX_4$ to ${OX_1}'$~and~${OX_4}'$, and taking +${Oa}'$~and~${Ob}'$ as unit distances along the axes instead of $Oa$~and~$Ob$. +%% -----File: 224.png---Folio 210------- + +It is easy to show, as a matter of fact, that such a change of axes +and units does correspond to the Lorentz transformation. Let $x_1$~and~$x_4$ +be the coördinates of any point with respect to the original +axes $OX_1$~and~$OX_4$, and ${x_1}''$~and~${x_4}''$ the coördinates of the same point +referred to the oblique axes ${OX_1}'$~and~${OX_4}'$, no change having yet +been made in the actual lengths of the units of measurement. Then, +by familiar equations of analytical geometry, we shall have +\[ +\begin{aligned} +x_1 &= {x_1}'' \cos\theta + {x_4}'' \sin\theta, \\ +x_4 &= {x_1}'' \sin\theta + {x_4}'' \cos\theta, +\end{aligned} +\Tag{305} +\] +where $\theta$ is the angle~$X_1O{X_1}'$. + +We have, moreover, from the properties of the hyperbola, +\[ +\frac{{Oa}'}{Oa} = \frac{{Ob}'}{Ob} + = \frac{1}{\sqrt{\cos^2\theta - \sin^2\theta}}, +\] +and hence if we represent by ${x_1}'$~and~${x_4}'$ the coördinates of the point +with respect to the oblique axes and use $O{a}'$~and~$O{b}'$ as unit distances +instead of $Oa$~and~$Ob$, we shall obtain +\begin{align*} +x_1 &= {x_1}'\, \frac{\cos\theta}{\sqrt{\cos^2\theta - \sin^2\theta}} + + {x_4}'\, \frac{\sin\theta}{\sqrt{\cos^2\theta - \sin^2\theta}}, \\ +x_4 &= {x_1}'\, \frac{\sin\theta}{\sqrt{\cos^2\theta - \sin^2\theta}} + + {x_4}'\, \frac{\cos\theta}{\sqrt{\cos^2\theta - \sin^2\theta}}. +\end{align*} + +It is evident, however, that we may write +\[ +\frac{\sin\theta}{\cos\theta} = \tan\theta = \frac{dx_1}{ dx_4} = \frac{V}{c}, +\] +where $V$ may be regarded as the relative velocity of our two sets of +space axes. Introducing this into the above equations and also +writing $x_1 = x$, $x_4 = ct$, ${x_1}' = x'$, ${x_4}' = ct'$, we may obtain the familiar +equations +\begin{align*} +x &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}}\, (x' + Vt'), \\ +t &= \frac{1}{\sqrt{1 - \smfrac{V^2}{c^2}}} \left(t' + \frac{V}{c^2}\, x'\right). +\end{align*} +%% -----File: 225.png---Folio 211------- +We thus see that our diagrammatic representation of non-Euclidean +rotation in the ${X_1}OX_4$~plane does as a matter of fact correspond to +the Lorentz transformation. + +Diagrams of this kind can now be used to study various kinematical +events. $\delta$-curves can be drawn in the quadrant~$AOB$ to represent +the space-time trajectories of particles, their form can be investigated +using different sets of rotated axes, and the equations for +the transformation of velocities and accelerations thus studied. +$\gamma$-lines perpendicular to the particular time axis used can be drawn to +correspond to the instantaneous positions of actual lines in ordinary +space and studies made of the Lorentz shortening. Singular vectors +along the asymptote~$OB$ can be used to represent the trajectory of a +ray of light and it can be shown that our rotation of axes is so devised +as to leave unaltered, the angle between such singular vectors and the +$OX_4$~axis, corresponding to the fact that the velocity of light must +appear the same to all observers. Further development of the possibilities +of graphical representation of the properties of our non-Euclidean +space may be left to the reader. + + +\Section[II]{Applications of the Four-Dimensional Analysis.} + +\Paragraph{191.} We may now apply our four-dimensional methods to a +number of problems in the fields of kinematics, mechanics and electromagnetics. +Our general plan will be to express the laws of the particular +field in question in four-dimensional language, making use of +four-dimensional vector quantities of a kinematical, mechanical, or +electromagnetic nature. Since the components of these vectors +along the three spatial axes and the temporal axis will be closely +related to the ordinary quantities familiar in kinematical, mechanical, +and electrical discussions, there will always be an easy transition from +our four-dimensional language to that ordinarily used in such discussions, +and necessarily used when actual numerical computations +are to be made. We shall find, however, that our four-dimensional +language introduces an extraordinary brevity into the statement of a +number of important laws of physics. + + +%[** TN: Heading set like a \Section in original] +\Subsection{Kinematics.} + +\Subsubsection{192}{Extended Position.} The position of a particle and the particular +instant at which it occupies that position can both be indicated +%% -----File: 226.png---Folio 212------- +by a point in our four-dimensional space. We can call this +the extended position of the particle and determine it by stating the +value of a four-dimensional radius vector +\[ +\vc{r} = (x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4). +\Tag{306} +\] + +\Subsubsection{193}{Extended Velocity.} Since the velocity of a real particle can +never exceed that of light, its changing position in space and time +will be represented by a $\delta$-curve. + +The equation for a unit vector tangent to this $\delta$-curve will be +\[ +\vc{w} = \frac{d\vc{r}}{ds} + = \left(\frac{dx_1}{ds}\, \vc{k}_1 + \frac{dx_2}{ds}\, \vc{k}_2 + + \frac{dx_3}{ds}\, \vc{k}_3 + \frac{dx_4}{ds}\, \vc{k}_4\right), +\Tag{307} +\] +where $ds$~indicates interval along the $\delta$-curve; and this important +vector~$\vc{w}$ may be called the extended velocity of the particle. + +Remembering that for a $\delta$-curve +\[ +ds = \sqrt{d{x_4}^2 - d{x_1}^2 - d{x_2}^2 - d{x_3}^2} + = c\, dt \sqrt{1 - \frac{u^2}{c^2}}, +\Tag{308} +\] +we may rewrite our expression for extended velocity in the form +\[ +\vc{w} = \frac{1}{\sqrt{1 - \smfrac{u^2}{c^2}}} + \left\{\frac{\vc{u}}{c} + \vc{k}_4\right\}, +\Tag{309} +\] +where $\vc{u}$ is evidently the ordinary three-dimensional velocity of the +particle. + +Since $\vc{w}$ is a four-dimensional vector in our imaginary space, we +may use our tables for transforming the components of~$\vc{w}$ from one +set of axes to another. We shall find that we may thus obtain transformation +equations for velocity identical with those already familiar +in \Chapref{IV}. + +The four components of $\vc{w}$ are +\[ +\frac{\smfrac{u_x}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{k}_1, \qquad +\frac{\smfrac{u_y}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{k}_2, \qquad +\frac{\smfrac{u_z}{c}}{\sqrt{1 - \smfrac{u^2}{c^2}}}\, \vc{k}_3, \qquad +\frac{\vc{k}_4}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +and with the help of \hyperref[table:302]{table~(302)} we may easily obtain, by making +simple algebraic substitutions, the following familiar transformation +%% -----File: 227.png---Folio 213------- +equations: +\begin{gather*}%[** TN: Re-breaking] + {u_x}' = \frac{u_x - V}{1 - \smfrac{u_xV}{c^2}},\qquad + {u_y}' = \frac{u_y\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}},\qquad + {u_z}' = \frac{u_z\sqrt{1 - \smfrac{V^2}{c^2}}}{1 - \smfrac{u_xV}{c^2}},\\ +\frac{1}{\sqrt{1 - \smfrac{{u'}^2}{c^2}}} + = \frac{1 - \smfrac{u_xV}{c^2}} + {\sqrt{1 - \smfrac{u^2}{c^2}}\, \sqrt{1 - \smfrac{V^2}{c^2}}}. +\end{gather*} + +This is a good example of the ease with which we can derive our +familiar transformation equations with the help of the four-dimensional +method. + +\Subsubsection{194}{Extended Acceleration.} We may define the extended acceleration +of a particle as the rate of curvature of the $\delta$-line which determines +its four-dimensional position. We have +\[ +c = \frac{d^2\vc{r}}{ds^2} = \frac{d\vc{w}}{ds} + = \frac{d}{ds}\left[ + \frac{\smfrac{\vc{u}}{c} + \vc{k}_4} + {\sqrt{1 - \smfrac{u^2}{c^2}}}\right]. +\Tag{310} +\] +Or, introducing as before the relation $ds = c\, dt \sqrt{1 - \dfrac{u^2}{c^2}}$, we may write +\begin{multline*} +c = \frac{1}{c^2} \Biggl\{ + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)}\, \frac{d\vc{u}}{dt} + + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, \frac{u}{c^2}\, + \frac{du}{dt}\, \vc{u} \\ + + \frac{1}{\left(1 - \smfrac{u^2}{c^2}\right)^2}\, \frac{u}{c}\, + \frac{du}{dt}\, \vc{k}_4\Biggr\}, +\Tag{311} +\end{multline*} +%% -----File: 228.png---Folio 214------- +where $\vc{u}$ is evidently the ordinary three-dimensional velocity, and $\dfrac{d\vc{u}}{dt}$ +the three-dimensional acceleration; and we might now use our transformation +table to determine the transformation equations for acceleration +which we originally obtained in \Chapref{IV}. + +\Subsubsection{195}{The Velocity of Light.} As an interesting illustration of the +application to kinematics of our four-dimensional methods, we may +point out that the trajectory of a ray of light will be represented by a +singular line. Since the magnitude of all singular vectors is zero by +definition, we have for any singular line +\[ +{dx_1}^2 + {dx_2}^2 + {dx_3}^2 = {dx_4}^2, +\] +or, since the magnitude will be independent of any particular choice +of axes, we may also write +\[ +{{dx_1}'}^2 + {{dx_2}'}^2 + {{dx_3}'}^2 = {{dx_4}'}^2. +\] +Transforming the first of these equations we may write +\[ +\frac{{dx_1}^2 + {dx_2}^2 + {dx_3}^2 }{{dx_4}^2} + = \frac{dx^2 + dy^2 + dz^2 }{c^2\, dt^2} = 1 +\] +or +\[ +\frac{dl}{dt} = c. +\] +Similarly we could obtain from the second equation +\[ +\frac{dl'}{dt'} = c. +\] +We thus see that a singular line does as a matter of fact correspond +to the four-dimensional trajectory of a ray of light having the velocity~$c$, +and that our four-dimensional analysis corresponds to the requirements +of the second postulate of relativity that a ray of light shall +have the same velocity for all reference systems. + + +%[** TN: Heading set like a \Section in original] +\Subsection{The Dynamics of a Particle.} + +\Subsubsection{196}{Extended Momentum.} We may define the extended momentum +of a material particle as equal to the product~$m_0\vc{w}$ of its mass~$m_0$, +measured when at rest, and its extended velocity~$\vc{w}$. In accordance +%% -----File: 229.png---Folio 215------- +with equation~(309) for extended velocity, we may write then, for +the extended momentum, +\[ +m_0\vc{w} = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} + \left(\frac{\vc{u}}{c} + \vc{k}_4\right). +\Tag{312} +\] +Or, if in accordance with our considerations of \Chapref{VI} we put +for the mass of the particle at the velocity~$u$ +\[ +m = \frac{m_0}{\sqrt{1 - \smfrac{u^2}{c^2}}}, +\] +we may write +\[ +m_0\vc{w} = m\, \frac{\vc{u}}{c} + m\vc{k}_4. +\Tag{313} +\] +We note that the space component of this vector is ordinary momentum +and the time component has the magnitude of mass, and by +applying our \hyperref[table:302]{transformation table~(302)} we can derive very simply +the transformation equations for mass and momentum already +obtained in \Chapref{VI}. + +\Subsubsection{197}{The Conservation Laws.} We may now express the laws for +the dynamics of a system of particles in a very simple form by stating +the principle that the extended momentum of a system of particles is a +quantity which remains constant in all interactions of the particles, +we have then +\[ +\Sum m_0\vc{w} + = \Sum\left(\frac{m\vc{u}}{c} + m\vc{k}_4 \right) + = \text{ a constant}, +\Tag{314} +\] +where the summation $\Sum$ extends over all the particles of the system. + +It is evident that this one principle really includes the three +principles of the conservation of momentum, mass, and energy. +This is true because in order for the vector~$\Sum m_0\vc{w}$ to be a constant +quantity, its components along each of the four axes must be constant, +and as will be seen from the above equation this necessitates +the constancy of the momentum~$\Sum m\vc{u}$, of the total mass~$\Sum m$, and of +the total energy~$\Sum \dfrac{m}{c^2}$. +%% -----File: 230.png---Folio 216------- + + +%[** TN: Heading set like a \Section in original] +\Subsection{The Dynamics of an Elastic Body.} + +Our four-dimensional methods may also be used to present the +results of our theory of elasticity in a very compact form. + +\Subsubsection{198}{The Tensor of Extended Stress.} In order to do this we shall +first need to define an expression which may be called the four-dimensional +stress in the elastic medium. For this purpose we may take the +symmetrical tensor~$T_m$ defined by the following table: +\[ +T_m = \left\{ +\begin{matrix} +p_{xx} & p_{xy} & p_{xz} & cg_x, \\ +p_{yx} & p_{yy} & p_{yz} & cg_y, \\ +p_{zx} & p_{zy} & p_{zz} & cg_z, \\ +\dfrac{s_x}{c} & \dfrac{s_y}{c} & \dfrac{s_z}{c} & w, +\end{matrix} +\right. +\Tag{315} +\] +where the spatial components of~$T_m$ are equal to the components of +the symmetrical tensor~$\vc{p}$ which we have already defined in \Chapref{X} +and the time components are related to the density of momentum~$\vc{g}$, +density of energy flow~$\vc{s}$ and energy density~$w$, as shown in the tabulation. + +From the symmetry of this tensor we may infer at once the simple +relation between density of momentum and density of energy flow: +\[ +\vc{g} = \frac{\vc{s}}{c^2}, +\Tag{316} +\] +with which we have already become familiar in \Secref{132}. + +\Subsubsection{199}{The Equation of Motion.} We may, moreover, express the +equation of motion for an elastic medium unacted on by external +forces in the very simple form +\[ +\divg T_m = 0. +\Tag{317} +\] + +It will be seen from our definition of the divergence of a four-dimensional +tensor, \Secref{187}, that this one equation is in reality +equivalent to the two equations +\begin{align*} +\divg\vc{p} + \frac{\partial\vc{g}}{\partial t} &= 0 +\Tag{318} \\ +\intertext{and} +\divg\vc{s} + \frac{\partial w}{\partial t} &= 0. +\end{align*} +%% -----File: 231.png---Folio 217------- +The first of these equations is identical with~(184) of Chapter~X, %[** TN: Not a useful cross-reference] +which we found to be the equation for the motion of an elastic medium +in the absence of external forces, and the second of these equations +expresses the principle of the conservation of energy. + +The elegance and simplicity of this four-dimensional method of +expressing the results of our laborious calculations in \Chapref{X} cannot +fail to be appreciated. + + +%[** TN: Heading set like a \Section in original] +\Subsection{Electromagnetics.} + +We also find it possible to express the laws of the electromagnetic +field very simply in our four-dimensional language. + +\Subsubsection{200}{Extended Current.} We may first define the extended current, +a simple but important one-vector, whose value at any point will depend +on the density and velocity of charge at that point. We shall +take as the equation of definition +\[ +\vc{q} = \rho_0\vc{w} + = \rho \left\{\frac{\vc{u}}{c} + \vc{k}_4\right\}, +\Tag{319} +\] +where +\[ +\rho = \frac{\rho_0}{\sqrt{1 - \smfrac{u^2}{c^2}}} +\] +is the density of charge at the point in question. + +\Subsubsection{201}{The Electromagnetic Vector $\vc{M}$.} We may further define a +two-vector~$\vc{M}$ which will be directly related to the familiar vectors +strength of electric field~$\vc{e}$ and strength of magnetic field~$\vc{h}$ by the +equation of definition +\begin{align*} +\vc{M} &= (h_1\vc{k}_{23} + h_2\vc{k}_{31} + h_3\vc{k}_{12} + - e_1\vc{k}_{14} - e_2\vc{k}_{24} - e_3\vc{k}_{34}) \\ +%[** TN: Hack to get equation number vertically centered] +\intertext{or\hfill(320)} +\vc{M^*} &= (e_1\vc{k}_{23} + e_2\vc{k}_{31} + e_3\vc{k}_{12} + + h_1\vc{k}_{14} + h_2\vc{k}_{24} + h_3\vc{k}_{34}), +\end{align*} +where $e_1$,~$e_2$,~$e_3$, and $h_1$,~$h_2$,~$h_3$ are the components of $\vc{e}$~and~$\vc{h}$. + +\Subsubsection{202}{The Field Equations.} We may now state the laws of the +electromagnetic field in the extremely simple form +\begin{align*} +\Qop · \vc{M} &= \vc{q}, \Tag{321} \\ +\Qop × \vc{M} &= 0. \Tag{322} +\end{align*} +%% -----File: 232.png---Folio 218------- + +These two simple equations are, as a matter of fact, completely +equivalent to the four field equations which we made fundamental +for our treatment of electromagnetic theory in \Chapref{XII}. Indeed +if we treat~$\Qop$ formally as a one-vector +\[ +\left(\vc{k}_1\, \frac{\partial}{\partial x_1} + + \vc{k}_2\, \frac{\partial}{\partial x_2} + + \vc{k}_3\, \frac{\partial}{\partial x_3} + - \vc{k}_4\, \frac{\partial}{\partial x_4}\right) +\] +and apply it to the electromagnetic vector~$\vc{M}$ expressed in the extended +form given in the equation of definition~(320) we shall obtain from~(321) +the two equations +\begin{align*} +\curl \vc{h} - \frac{1}{c}\, \frac{\partial\vc{e}}{\partial t} + &= \rho\, \frac{\vc{u}}{c}, \\ +\divg\vc{e} &= \rho, \\ +\intertext{and from (322)} +\divg \vc{h} &= 0,\\ +\curl \vc{e} + \frac{1}{c}\, \frac{\partial\vc{h}}{\partial t} &= 0, +\end{align*} +where we have made the substitution $x_4 = ct$. These are of course +the familiar field equations for the Maxwell-Lorentz theory of electromagnetism. + +\Subsubsection{203}{The Conservation of Electricity.} We may also obtain very +easily an equation for the conservation of electric charge. In accordance +with equation~(284) we may write as a necessary mathematical +identity +\[ +\Qop · (\Qop · \vc{M}) = 0. +\Tag{323} +\] +Noting that $\Qop · \vc{M} = \vc{q}$, this may be expanded to give us the equation +of continuity. +\[ +\divg \rho\vc{u} + \frac{\partial\rho}{\partial t} = 0. +\Tag{324} +\] + +\Subsubsection{204}{The Product $\vc{M}·\vc{q}$.} We have thus shown the form taken by +the four field equations when they are expressed in four dimensional +language. Let us now consider with the help of our four-dimensional +methods what can be said about the forces which determine the +motion of electricity under the action of the electromagnetic field. + +Consider the inner product of the electromagnetic vector and +%% -----File: 233.png---Folio 219------- +the extended current: +\begin{multline*} +\vc{M} · \vc{q} + = (h_1\vc{k}_{23} + h_2\vc{k}_{31} + h_3\vc{k}_{12} + - e_1\vc{k}_{14} - e_2\vc{k}_{24} - e_3\vc{k}_{34}) + · \rho \left\{\frac{\vc{u}}{c} + \vc{k}_4\right\} \\ + = \rho \left\{\vc{e} + \frac{[\vc{u} × \vc{h}]^*}{c}\right\} + + \rho\, \frac{\vc{e} · \vc{h}}{c}\vc{k}_4. +\Tag{325} +\end{multline*} +We see that the space component of this vector is equal to the expression +which we have already found in \Chapref{XII} as the force +acting on the charge contained in unit volume, and the time component +is proportional to the work done by this force on the moving +charge; hence we may write the equation +\[ +\vc{M} · \vc{q} = \left\{\vc{f} + \frac{\vc{f} · \vc{u}}{c}\, \vc{k}_4\right\}, +\Tag{326} +\] +an expression which contains the same information as that given by +the so-called fifth fundamental equation of electromagnetic theory, +$\vc{f}$~being the force exerted by the electromagnetic field per unit volume +of charged material. + +\Subsubsection{205}{The Extended Tensor of Electromagnetic Stress.} We may +now show the possibility of defining a four-dimensional tensor~$T_e$, such +that the important quantity $\vc{M} · \vc{q}$ shall be equal to~$-\divg T_e$. This +will be valuable since we shall then be able to express the equation +of motion for a combined mechanical and electrical system in a very +simple and beautiful form. + +Consider the symmetrical tensor +\[ +T_e = +\left\{ +\begin{matrix} +T_{11} & T_{12} & T_{13} & T_{14}, \\ +T_{21} & T_{22} & T_{23} & T_{24}, \\ +T_{31} & T_{32} & T_{33} & T_{34}, \\ +T_{41} & T_{42} & T_{43} & T_{44}, +\end{matrix} +\right. +\Tag{327} +\] +defined by the expression +\[ +\begin{aligned} +T_{jk} &= \tfrac{1}{2} + \{M_{j1}M_{k1} + M_{j2}M_{k2} + M_{j3}M_{k3} - M_{j4}M_{k4} \\ + &\qquad + + {M_{j1}}^*{M_{k1}}^* + {M_{j2}}^*{M_{k2}}^* + + {M_{j3}}^*{M_{k3}}^* - {M_{j4}}^*{M_{k4}}^*\}, +\end{aligned} +\Tag{328} +\] +where $j$, $k = 1$, $2$, $3$, $4$. +%% -----File: 234.png---Folio 220------- + +It can then readily be shown by expansion that +\[ +-\divg T_e = \vc{M} · (\Qop · \vc{M}) + \vc{M}^* · (\Qop · \vc{M}^*). +\] +But, in accordance with equations (321),~(326),~(292) and~(322), this +is equivalent to +\[ +-\divg T_e = \vc{M} · \vc{q} +%[** TN: Keeping () in numerator, cf. (326) above] + = \left\{\vc{f} + \frac{(\vc{f} · \vc{u})}{c}\, \vc{k}_4\right\}. +\Tag{329} +\] + +Since in free space the value of the force~$\vc{f}$ is zero, we may write +for free space the equation +\[ +\divg T_e = 0. +\Tag{330} +\] + +This one equation is equivalent, as a matter of fact, to two important +and well-known equations of electromagnetic theory. If we +develop the components $T_{11}$,~$T_{12}$,~etc., of our tensor in accordance +with equations (328)~and~(320) we find that we can write +\[ +T_e = +\left\{ +\renewcommand{\arraystretch}{2} +\begin{matrix} +\psi_{xx} & \psi_{xy} & \psi_{xz} & \dfrac{S_x}{c}, \\ +\psi_{yx} & \psi_{yy} & \psi_{yz} & \dfrac{S_y}{c}, \\ +\psi_{zx} & \psi_{zxy} & \psi_{zz} & \dfrac{S_z}{c}, \\ +\dfrac{s_x}{c}& \dfrac{s_x}{c} & \dfrac{s_x}{c} & w, +\end{matrix} +\right. +\Tag{331} +\] +where we shall have +\[ +\begin{aligned} +\psi_{xx} + &= -\tfrac{1}{2}({e_x}^2 - {e_y}^2 - {e_z}^2 + {h_x}^2 - {h_y}^2 - {h_z}^2), \\ +\psi_{xy} + &= -(e_xh_y + h_xh_y), \\ +\text{etc.}& \\ +s_x &= c(e_yh_z - e_zh_y), \\ +\text{etc.}& \\ +w &= \tfrac{1}{2}(e^2 + h^2), +\end{aligned} +\Tag{332} +\] +$\psi$ thus being equivalent to the well-known Maxwell three-dimensional +stress tensor, $s_x$,~$s_y$,~etc., being the components of the Poynting vector +$c\, [\vc{e} × \vc{h}]^*$, and $w$~being the familiar expression for density of electromagnetic +%% -----File: 235.png---Folio 221------- +energy $\dfrac{e^2 + h^2}{s}$. We thus see that equation~(330) is equivalent +to the two equations +\begin{align*} +\divg \psi + \frac{1}{c^2}\, \frac{\partial s}{\partial t} = 0, \\ +\divg \vc{s} + \frac{\partial w}{\partial t} = 0. +\end{align*} +The first of these is the so-called equation of electromagnetic momentum, +and the second, Poynting's equation for the flow of electromagnetic +energy. + +\Subsubsection{206}{Combined Electrical and Mechanical Systems.} For a point +not in free space where mechanical and electrical systems are both +involved, taking into account our previous considerations, we may +now write the equation of motion for a combined electrical and +mechanical system in the very simple form +\[ +\divg T_m + \divg T_e = 0. +\] +And we may point out in closing that we may reasonably expect all +forces to be of such a nature that our most general equation of motion +for any continuous system can be written in the form +\[ +\divg T_1 + \divg T_2 + \cdots = 0. +\] +%% -----File: 236.png---Folio 222------- + + +\Appendix{I}{Symbols for Quantities.} + +\AppSection{Scalar Quantities}{Scalar Quantities. \(Indicated by Italic type.\)} + +\begin{longtable}{rl} +$c$& speed of light.\\ +$e$& electric charge.\\ +$E$& energy.\\ +$H$& kinetic potential.\\ +$K$& kinetic energy.\\ +$l$, $m$, $n$& direction cosines.\\ +$L$& Lagrangian function.\\ +$p$& pressure.\\ +$Q$& quantity of electricity.\\ +$S$& entropy.\\ +$t$& time.\\ +$T$& temperature, function $\ds\Sum m_0c^2 \left(1-\sqrt{1-\frac{u^2}{c^2}}\;\right)$.\\ +$U$& potential energy.\\ +$v$& volume.\\ +$V$& relative speed of coördinate systems, volume.\\ +$w$& energy density.\\ +$W$& work.\\ +$\epsilon$&dielectric constant.\\ +$\kappa$ &$\dfrac{1}{\sqrt{1-\smfrac{V^2}{c^2}}}$.\\ +$\mu$ &index of refraction, magnetic permeability.\\ +$\nu $ &frequency.\\ +$\rho$ &density of charge.\\ +$\sigma$ &electrical conductivity.\\ +$\phi$ &non-Euclidean angle between time axes.\\ +$\phi_1\phi_2\phi_3 \cdots $& generalized coördinates.\\ +$\psi$ &scalar potential.\\ +$\psi_1\psi_2\psi_3\cdots$ & generalized momenta. +\end{longtable} +%% -----File: 237.png---Folio 223------- + + +\AppSection{Vector Quantities}{Vector Quantities. \(Indicated by Clarendon type.\)} + +\begin{longtable}{r l} +$\vc{B}$& magnetic induction.\\ +$\vc{c}$& extended acceleration.\\ +$\vc{D}$& dielectric displacement.\\ +$\vc{e}$& electric field strength in free space.\\ +$\vc{E}$& electric field strength in a medium.\\ +$\vc{f}$& force per unit volume.\\ +$\vc{F}$& force acting on a particle.\\ +$\vc{g}$& density of momentum.\\ +$\vc{h}$& magnetic field strength in free space.\\ +$\vc{H}$& magnetic field strength in a medium.\\ +$\vc{i}$& density of electric current.\\ +$\vc{M}$& angular momentum, electromagnetic vector.\\ +$\vc{p}$& symmetrical elastic stress tensor.\\ +$\vc{q}$& extended current.\\ +$\vc{r}$& radius vector\DPtypo{}{.}\\ +$\vc{s}$& density of energy flow.\\ +$\vc{t}$& unsymmetrical elastic stress tensor.\\ +$\vc{u}$& velocity.\\ +$\vc{w}$& extended velocity.\\ +$\vc{\phi}$& vector potential. +\end{longtable} +%% -----File: 238.png---Folio 224------- + + +\Appendix{II}{Vector Notation.} + +\AppSection{Three Dimensional Space}{Three Dimensional Space.} + +%[** TN: No periods after items in this section.] +Unit Vectors, $\vc{i}\ \vc{j}\ \vc{k}$ + +Radius Vector, $\vc{r} = x\vc{i} + y\vc{j} + z\vc{k}$ + +Velocity, +\begin{align*} +\vc{u} = \frac{d\vc{r}}{dt} + &= \dot{x}\vc{i} + \dot{y}\vc{j} + \dot{z}\vc{k} \\ + &= u_x\vc{i} + u_y\vc{j} + u_z\vc{k} \\ +\intertext{\indent Acceleration,} +\dot{\vc{u}} = \frac{d^2\vc{r}}{dt^2} + &= \ddot{x}\vc{i} + \ddot{y}\vc{j} + \ddot{z}\vc{k} \\ + &= \dot{u}_x\vc{i} + \dot{u}_y\vc{j} + \dot{u}_z\vc{k} +\end{align*} + +Inner Product, +\[ +\vc{a}·\vc{b} = a_xb_x + a_yb_y + a_zb_z +\] + +Outer Product, +\[ +\vc{a} × \vc{b} + = (a_xb_y - a_yb_x)\vc{ij} + + (a_yb_z - a_zb_y)\vc{jk} + + (a_zb_x - a_xb_z)\vc{ki} +\] + +Complement of Outer Product, +\[ +[\vc{a} × \vc{b}]^* + = (a_yb_z - a_zb_y)\vc{i} + + (a_zb_x - a_xb_z)\vc{j} + + (a_xb_y - a_yb_x)\vc{k} +\] + +The Vector Operator Del or~$\nabla$, +\[ +\nabla + = \vc{i}\, \frac{\partial}{\partial x} + + \vc{j}\, \frac{\partial}{\partial y} + + \vc{k}\, \frac{\partial}{\partial z} +\] +\begin{align*} +\grad A &= \nabla A + = \vc{i}\, \frac{\partial A}{\partial x} + + \vc{j}\, \frac{\partial A}{\partial y} + + \vc{k}\, \frac{\partial A}{\partial z} \\ +\divg\vc{a} &= \nabla · \vc{a} + = \frac{\partial a_x}{\partial x} + + \frac{\partial a_y}{\partial y} + + \frac{\partial a_z}{\partial z} \\ +\curl\vc{a} &= [\nabla × \vc{a}]^* \\ + &= \left(\frac{\partial a_z}{\partial y} + - \frac{\partial a_y}{\partial z}\right) \vc{i} + + \left(\frac{\partial a_x}{\partial z} + - \frac{\partial a_z}{\partial x}\right) \vc{j} + + \left(\frac{\partial a_y}{\partial x} + - \frac{\partial a_x}{\partial y}\right) \vc{k} +\end{align*} +%% -----File: 239.png---Folio 225------- + +\AppSection{Non-Euclidean Four Dimensional Space.}{Non-Euclidean Four Dimensional Space.} + +Unit Vectors, $\vc{k}_1$ $\vc{k}_2$ $\vc{k}_3$ $\vc{k}_4$ + +Radius Vector, +\begin{align*} +\vc{r} &= x_1\vc{k}_1 + x_2\vc{k}_2 + x_3\vc{k}_3 + x_4\vc{k}_4 \\ + &= x\vc{i} + y\vc{j} + z\vc{k} + ct\vc{k}_4 +\end{align*} + +One Vector, +\[ +\vc{a} = a_1\vc{k}_1 + a_2\vc{k}_2 + a_3\vc{k}_3 + a_4\vc{k}_4 +\] + +Two Vector, +\[ +\vc{A} = A_{12}\vc{k}_{12} + A_{13}\vc{k}_{13} + + A_{14}\vc{k}_{14} + A_{23}\vc{k}_{23} + + A_{24}\vc{k}_{24} + A_{34}\vc{k}_{34} +\] + +Three Vector, +\[ +\Alpha = \frakA_{123}\vc{k}_{123} + \frakA_{124}\vc{k}_{124} + + \frakA_{134}\vc{k}_{134} + \frakA_{234}\vc{k}_{234} +\] + +Pseudo Scalar, +\[ +\vc{\alpha} = \alpha\vc{k}_{1234} +\] + +Transposition of Subscripts, +\[ +\vc{k}_{abc\cdots} = -\vc{k}_{bac\cdots} = \vc{k}_{bca\cdots} +\] + +Inner Product of One Vectors, + +(\textit{See \Secref{183}}). + +Outer Product of One Vectors, +\[ +\vc{k}_{ab\cdots} × \vc{k}_{nm\cdots} = \vc{k}_{ab\cdots nm\cdots} +\] + +Complement of a Vector, +\[ +\vc{\phi}^* = \phi·\vc{k}_{1234} +\] + +The Vector Operator Quad or~$\Qop$, +\[ +\Qop = \vc{k}_1\frac{\partial}{\partial x_1} + + \vc{k}_2\frac{\partial}{\partial x_2} + + \vc{k}_3\frac{\partial}{\partial x_3} + + \vc{k}_4\frac{\partial}{\partial x_4} +\] + +\cleardoublepage +\backmatter + +%%%% LICENSE %%%% +\pagenumbering{Alph} +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} +\pdfbookmark[0]{Project Gutenberg License}{License} +\fancyhf{} +\fancyhead[C]{\CtrHeading{Project Gutenberg License}} + +\begin{PGtext} +End of the Project Gutenberg EBook of The Theory of the Relativity of Motion, by +Richard Chace Tolman + +*** END OF THIS PROJECT GUTENBERG EBOOK THE THEORY OF THE RELATIVITY *** + +***** This file should be named 32857-pdf.pdf or 32857-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/2/8/5/32857/ + +Produced by Andrew D. 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Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..3737c18 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #32857 (https://www.gutenberg.org/ebooks/32857) |