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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:05:07 -0700 |
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+\newcommand{\Partref}[1]{\hyperref[part:#1]{Part~#1}} +\newcommand{\Sectionref}[1]{\hyperref[chapter:#1]{Section~#1}} +\newcommand{\Srefno}[1]{\hyperref[chapter:#1]{#1}} +\newcommand{\Appendixref}[1]{\hyperref[appendix:#1]{Appendix~#1}} + +\newcommand{\ie}{\textit{i.e.}} +\newcommand{\eg}{\textit{e.g.}} +\newcommand{\NB}{\textit{N.B.}} +\newcommand{\Item}[1]{\textit{#1}} + +\newcommand{\itema}{(\Item{a})} +\newcommand{\itemb}{(\Item{b})} +\newcommand{\itemc}{(\Item{c})} + +\newcommand{\Z}{\phantom{0}} + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} +\pagenumbering{Alph} +\pagestyle{empty} +\BookMark{-1}{Front Matter} +%%%% PG BOILERPLATE %%%% +\BookMark{0}{PG Boilerplate} +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +The Project Gutenberg EBook of Relativity: The Special and the General +Theory, by Albert Einstein + +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: Relativity: The Special and the General Theory + A Popular Exposition, 3rd ed. + +Author: Albert Einstein + +Translator: Robert W. Lawson + +Release Date: May 15, 2011 [EBook #36114] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK RELATIVITY *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang. (This ebook was produced using +OCR text generously provided by the University of Toronto +Robarts Library through the Internet Archive.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\frontmatter +\pagestyle{empty} +\begin{center} +\bfseries \Huge RELATIVITY \\ +\medskip +\normalsize THE SPECIAL \textit{\&} THE GENERAL THEORY \\ +\medskip +\small A POPULAR EXPOSITION +\vfill + +\footnotesize BY \\ +\Large ALBERT EINSTEIN, Ph.D. \\ +\smallskip\normalfont\scriptsize +PROFESSOR OF PHYSICS IN THE UNIVERSITY OF BERLIN +\vfill + +\footnotesize AUTHORISED TRANSLATION BY \\ +\normalsize \textbf{ROBERT W. LAWSON, D.Sc.} \\ +\smallskip\scriptsize UNIVERSITY OF SHEFFIELD +\vfill + +\footnotesize WITH FIVE DIAGRAMS \\ +AND A PORTRAIT OF THE AUTHOR +\vfill\vfill + +THIRD EDITION +\vfill\vfill + + +\normalsize\bfseries METHUEN \& CO. LTD. \\ +36 ESSEX STREET W.C. \\ +LONDON +\end{center} +\PageSep{iv} +\begin{PubInfo} +\PubRow{This Translation was first Published}{August 19th 1920} +\PubRow{Second Edition}{September 1920} +\PubRow{Third Edition}{1920} +\end{PubInfo} +\PageSep{v} + + +\Preface + +\First{The} present book is intended, as far as possible, +to give an exact insight into the theory of Relativity +to those readers who, from a general +scientific and philosophical point of view, are interested +in the theory, but who are not conversant with the +mathematical apparatus\footnote + {The mathematical fundaments of the special theory of + relativity are to be found in the original papers of H.~A. Lorentz, + A.~Einstein, H.~Minkowski, published under the title \textit{Das + Relativitätsprinzip} (The Principle of Relativity) in B.~G. + Teubner's collection of monographs \textit{Fortschritte der mathematischen + Wissenschaften} (Advances in the Mathematical + Sciences), also in M.~Laue's exhaustive book \textit{Das Relativitätsprinzip}---published + by Friedr.\ Vieweg \&~Son, Braunschweig. + The general theory of relativity, together with the necessary + parts of the theory of invariants, is dealt with in the author's + book \textit{Die Grundlagen der allgemeinen Relativitätstheorie} (The + Foundations of the General Theory of Relativity) Joh.\ Ambr.\ + Barth,~1916; this book assumes some familiarity with the special + theory of relativity.} +of theoretical physics. The +work presumes a standard of education corresponding +to that of a university matriculation examination, +and, despite the shortness of the book, a fair amount +of patience and force of will on the part of the reader. +The author has spared himself no pains in his endeavour +\PageSep{vi} +to present the main ideas in the simplest and most intelligible +form, and on the whole, in the sequence and connection +in which they actually originated. In the interest +of clearness, it appeared to me inevitable that I should +repeat myself frequently, without paying the slightest +attention to the elegance of the presentation. I adhered +scrupulously to the precept of that brilliant theoretical +physicist L.~Boltzmann, according to whom matters of +elegance ought to be left to the tailor and to the cobbler. +I make no pretence of having withheld from the reader +difficulties which are inherent to the subject. On the +other hand, I have purposely treated the empirical +physical foundations of the theory in a ``step-motherly'' +fashion, so that readers unfamiliar with physics may +not feel like the wanderer who was unable to see the +forest for trees. May the book bring some one a few +happy hours of suggestive thought! + +\Signature[\textit{December}, 1916]{A. EINSTEIN} + + +\SectTitle{Note to the Third Edition} + +\First{In} the present year (1918) an excellent and detailed +manual on the general theory of relativity, written +by H.~Weyl, was published by the firm Julius +Springer (Berlin). This book, entitled \textit{Raum---Zeit---Materie} +(Space---Time---Matter), may be warmly recommended +to mathematicians and physicists. +\PageSep{vii} + + +\Section{Biographical Note} + +\First{Albert Einstein} is the son of German-Jewish +parents. He was born in~1879 in the +town of Ulm, Würtemberg, Germany. His +schooldays were spent in Munich, where he attended +the \textit{Gymnasium} until his sixteenth year. After leaving +school at Munich, he accompanied his parents to Milan, +whence he proceeded to Switzerland six months later +to continue his studies. + +From 1896 to 1900 Albert Einstein studied mathematics +and physics at the Technical High School in +Zurich, as he intended becoming a secondary school +(\textit{Gymnasium}) teacher. For some time afterwards he +was a private tutor, and having meanwhile become +naturalised, he obtained a post as engineer in the Swiss +Patent Office in~1902 which position he occupied till +1909. The main ideas involved in the most important +of Einstein's theories date back to this period. Amongst +these may be mentioned: \textit{The Special Theory of Relativity}, +\textit{Inertia of Energy}, \textit{Theory of the Brownian Movement}, +and the \textit{Quantum-Law of the Emission and Absorption of Light}~(1905). +These were followed some years +\PageSep{viii} +later by the \textit{Theory of the Specific Heat of Solid Bodies}, +and the fundamental idea of the \textit{General Theory of +Relativity}. + +During the interval 1909~to~1911 he occupied the post +of Professor \textit{Extraordinarius} at the University of Zurich, +afterwards being appointed to the University of Prague, +Bohemia, where he remained as Professor \textit{Ordinarius} +until~1912. In the latter year Professor Einstein +accepted a similar chair at the \textit{Polytechnikum}, Zurich, +and continued his activities there until~1914, when he +received a call to the Prussian Academy of Science, +Berlin, as successor to Van't~Hoff. Professor Einstein +is able to devote himself freely to his studies at the +Berlin Academy, and it was here that he succeeded in +completing his work on the \textit{General Theory of Relativity} +(1915--17). Professor Einstein also lectures on various +special branches of physics at the University of Berlin, +and, in addition, he is Director of the Institute for +Physical Research of the \textit{Kaiser Wilhelm Gesellschaft}. + +Professor Einstein has been twice married. His first +wife, whom he married at Berne in~1903, was a fellow-student +from Serbia. There were two sons of this +marriage, both of whom are living in Zurich, the elder +being sixteen years of age. Recently Professor Einstein +married a widowed cousin, with whom he is now living +in Berlin. + +\Signature{R. W. L.} +\PageSep{ix} + +\Section{Translator's Note} + +\First{In} presenting this translation to the English-reading +public, it is hardly necessary for me to +enlarge on the Author's prefatory remarks, except +to draw attention to those additions to the book which +do not appear in the original. + +At my request, Professor Einstein kindly supplied +me with a portrait of himself, by one of Germany's +most celebrated artists. \Appendixref{III}, on ``The +Experimental Confirmation of the General Theory of +Relativity,'' has been written specially for this translation. +Apart from these valuable additions to the book, +I have included a biographical note on the Author, +and, at the end of the book, an Index and a list of +English references to the subject. This list, which is more +suggestive than exhaustive, is intended as a guide to those +readers who wish to pursue the subject farther. + +I desire to tender my best thanks to my colleagues +Professor S.~R. Milner,~D.Sc., and Mr.~W.~E. Curtis, +A.R.C.Sc.,~F.R.A.S., also to my friend Dr.~Arthur +Holmes, A.R.C.Sc.,~F.G.S., of the Imperial College, +for their kindness in reading through the manuscript, +\PageSep{x} +for helpful criticism, and for numerous suggestions. I +owe an expression of thanks also to Messrs.\ Methuen +for their ready counsel and advice, and for the care +they have bestowed on the work during the course of +its publication. + +\Signature{ROBERT W. LAWSON} + +\noindent\textsc{The Physics Laboratory} \\ +\hspace*{\parindent}\textsc{The University of Sheffield} \\ +\hspace*{3\parindent}\textit{June} 12, 1920 +\PageSep{xi} +\TableofContents % [** TN: Auto-generate the table of contents] +\iffalse %%%% Start of table of contents text %%%% +CONTENTS + +PART I + +THE SPECIAL THEORY OF RELATIVITY + +PAGE + + I. Physical Meaning of Geometrical Propositions . 1 + II. The System of Co-ordinates . 5 +III. Space and Time in Classical Mechanics . . 9 + IV. The Galileian System of Co-ordinates . .11 + V. The Principle of Relativity (in the Restricted + Sense) . . . . . .12 + VI. The Theorem of the Addition of Velocities employed + in Classical Mechanics . . 16 +VII. The Apparent Incompatibility of the Law of + Propagation of Light with the Principle of + Relativity . . . . 17 + +VIII. On the Idea of Time in Physics . . .21 + IX. The Relativity of Simultaneity . . .25 + X. On the Relativity of the Conception of Distance 28 + XI. The Lorentz Transformation . . .30 + XII. The Behaviour of Measuring-Rods and Clocks + in Motion . . . . 35 +\PageSep{xii} +XIII. Theorem of the Addition of Velocities. The + Experiment of Fizeau . . 3 %[** TN: Edge of page cut off] + XIV. The Heuristic Value of the Theory of Relativity 4 + XV. General Results of the Theory . . .4, + XVI. Experience and the Special Theory of Relativity 4 +XVII. Minkowski's Four-dimensional Space . . 5; + +PART II +THE GENERAL THEORY OF RELATIVITY + +XVIII. Special and General Principle of Relativity . 5 + XIX. The Gravitational Field . . . .6 + XX. The Equality of Inertial and Gravitational Mass + as an Argument for the General Postulate + of Relativity ..... + XXI. In what Respects are the Foundations of Classical + Mechanics and of the Special Theory + of Relativity unsatisfactory? . + XXII. A Few Inferences from the General Principle of + Relativity ..... +XXIII. Behaviour of Clocks and Measuring-Rods on a + Rotating Body of Reference . + XXIV. Euclidean and Non-Euclidean Continuum + XXV. Gaussian Co-ordinates .... + XXVI. The Space-time Continuum of the Special + Theory of Relativity considered as a + Euclidean Continuum +\PageSep{xiii} +PAGE + + XXVII. The Space-time Continuum of the General + Theory of Relativity is not a Euclidean + Continuum . . . . 93 +XXVIII. Exact Formulation of the General Principle of + Relativity . . . . 97 + XXIX. The Solution of the Problem of Gravitation on + the Basis of the General Principle of + Relativity ..... 100 + +PART III + +CONSIDERATIONS ON THE UNIVERSE +AS A WHOLE + + XXX. Cosmological Difficulties of Newton's Theory 105 + XXXI. The Possibility of a ``Finite'' and yet ``Unbounded'' + Universe. . . . 108 + XXXII. The Structure of Space according to the + General Theory of Relativity . . 113 + +APPENDICES + + I. Simple Derivation of the Lorentz Transformation . 115 + II. Minkowski's Four-dimensional Space (``World'') + [Supplementary to Section XVII.] . . 121 +III. The Experimental Confirmation of the General + Theory of Relativity . . . .123 +(a) Motion of the Perihelion of Mercury . 124 +(b) Deflection of Light by a Gravitational Field 126 +(c) Displacement of Spectral Lines towards the + Red . . . . . 129 + +BIBLIOGRAPHY . . . . . . 133 + +INDEX . . . . . . .135 +\fi %%%% End of table of contents text %%%% +\PageSep{xiv} +\FlushRunningHeads +\begin{CenterPage} + \bfseries\LARGE RELATIVITY \\[8pt] + \normalsize THE SPECIAL AND THE GENERAL THEORY +\end{CenterPage} +\PageSep{1} +\index{Manifold|see{Continuum}}% + + +\Part{I}{The Special Theory of Relativity}{Special Theory of Relativity} +\index{Special theory of relativity|(}% + +\Chapter[Geometrical Propositions] +{I}{Physical Meaning of Geometrical +Propositions} + +\First{In} your schooldays most of you who read this +\index{Euclidean geometry}% +book made acquaintance with the noble building of +Euclid's geometry, and you remember---perhaps +with more respect than love---the magnificent structure, +on the lofty staircase of which you were chased about +for uncounted hours by conscientious teachers. By +reason of your past experience, you would certainly +regard everyone with disdain who should pronounce even +the most out-of-the-way proposition of this science to +be untrue. But perhaps this feeling of proud certainty +would leave you immediately if some one were to ask +you: ``What, then, do you mean by the assertion that +these propositions are true?'' Let us proceed to give +this question a little consideration. + +Geometry sets out from certain conceptions such as +\index{Geometrical ideas!truth of|(}% +``plane,'' ``point,'' and ``straight line,'' with which +\index{Plane}% +\index{Point}% +\index{Straight line|(}% +\PageSep{2} +we are able to associate more or less definite ideas, and +from certain simple propositions (axioms) which, +\index{Axioms}% +\index{Axioms!truth of}% +\index{Geometrical ideas!propositions}% +in virtue of these ideas, we are inclined to accept as +``true.'' Then, on the basis of a logical process, the +justification of which we feel ourselves compelled to +admit, all remaining propositions are shown to follow +from those axioms, \ie\ they are proven. A proposition +is then correct (``true'') when it has been derived in the +recognised manner from the axioms. The question +of the ``truth'' of the individual geometrical propositions +\index{Truth@{``Truth''}}% +is thus reduced to one of the ``truth'' of the +axioms. Now it has long been known that the last +question is not only unanswerable by the methods of +geometry, but that it is in itself entirely without meaning. +We cannot ask whether it is true that only one +straight line goes through two points. We can only +say that Euclidean geometry deals with things called +\index{Euclidean geometry}% +``straight lines,'' to each of which is ascribed the property +of being uniquely determined by two points +situated on it. The concept ``true'' does not tally with +the assertions of pure geometry, because by the word +``true'' we are eventually in the habit of designating +always the correspondence with a ``real'' object; +geometry, however, is not concerned with the relation +of the ideas involved in it to objects of experience, but +only with the logical connection of these ideas among +themselves. + +It is not difficult to understand why, in spite of this, +we feel constrained to call the propositions of geometry +``true.'' Geometrical ideas correspond to more or less +\index{Geometrical ideas}% +exact objects in nature, and these last are undoubtedly +the exclusive cause of the genesis of those ideas. Geometry +ought to refrain from such a course, in order to +\PageSep{3} +give to its structure the largest possible logical unity. +The practice, for example, of seeing in a ``distance'' +two marked positions on a practically rigid body is +something which is lodged deeply in our habit of thought. +We are accustomed further to regard three points as +being situated on a straight line, if their apparent +positions can be made to coincide for observation with +one eye, under suitable choice of our place of observation. + +If, in pursuance of our habit of thought, we now +supplement the propositions of Euclidean geometry by +\index{Euclidean geometry!propositions of}% +the single proposition that two points on a practically +rigid body always correspond to the same distance +\index{Distance (line-interval)}% +(line-interval), independently of any changes in position +to which we may subject the body, the propositions of +Euclidean geometry then resolve themselves into propositions +on the possible relative position of practically +\index{Relative!position}% +rigid bodies.\footnote + {It follows that a natural object is associated also with a + straight line. Three points $A$,~$B$ and~$C$ on a rigid body thus + lie in a straight line when, the points $A$~and~$C$ being given, $B$ + is chosen such that the sum of the distances $AB$~and~$BC$ is as + short as possible. This incomplete suggestion will suffice for + our present purpose.} +Geometry which has been supplemented +in this way is then to be treated as a branch of physics. +We can now legitimately ask as to the ``truth'' of +geometrical propositions interpreted in this way, since +we are justified in asking whether these propositions +are satisfied for those real things we have associated +with the geometrical ideas. In less exact terms we can +\index{Geometrical ideas}% +express this by saying that by the ``truth'' of a geometrical +proposition in this sense we understand its +validity for a construction with ruler and compasses. +\index{Straight line|)}% +\PageSep{4} + +Of course the conviction of the ``truth'' of geometrical +propositions in this sense is founded exclusively +on rather incomplete experience. For the present we +shall assume the ``truth'' of the geometrical propositions, +then at a later stage (in the general theory of +relativity) we shall see that this ``truth'' is limited, +and we shall consider the extent of its limitation. +\index{Geometrical ideas!truth of|)}% +\PageSep{5} + + +\Chapter{II}{The System of Co-ordinates} +\index{System of co-ordinates}% + +\First{On} the basis of the physical interpretation of distance +\index{Distance (line-interval)}% +\index{Distance (line-interval)!physical interpretation of}% +\index{Measuring-rod}% +\index{Reference-body}% +which has been indicated, we are also +in a position to establish the distance between +two points on a rigid body by means of measurements. +For this purpose we require a ``distance'' (rod~$S$) +which is to be used once and for all, and which we +employ as a standard measure. If, now, $A$~and~$B$ are +two points on a rigid body, we can construct the +line joining them according to the rules of geometry; +then, starting from~$A$, we can mark off the distance~$S$ +time after time until we reach~$B$. The number of +these operations required is the numerical measure +of the distance~$AB$. This is the basis of all measurement +of length.\footnote + {Here we have assumed that there is nothing left over, \ie\ + that the measurement gives a whole number. This difficulty + is got over by the use of divided measuring-rods, the introduction + of which does not demand any fundamentally new method.} + +Every description of the scene of an event or of the +position of an object in space is based on the specification +of the point on a rigid body (body of reference) +with which that event or object coincides. This applies +not only to scientific description, but also to everyday +life. If I analyse the place specification ``Trafalgar +\index{Place specification}% +\PageSep{6} +Square, London,''\footnote + {I have chosen this as being more familiar to the English + reader than the ``Potsdamer Platz, Berlin,'' which is referred to + in the original. (R.~W.~L.)} +I arrive at the following result. +The earth is the rigid body to which the specification +of place refers; ``Trafalgar Square, London,'' is a +well-defined point, to which a name has been assigned, +and with which the event coincides in space.\footnote + {It is not necessary here to investigate further the significance + of the expression ``coincidence in space.'' This conception is + sufficiently obvious to ensure that differences of opinion are + scarcely likely to arise as to its applicability in practice.} + +This primitive method of place specification deals +\index{Place specification}% +only with places on the surface of rigid bodies, and is +dependent on the existence of points on this surface +which are distinguishable from each other. But we +can free ourselves from both of these limitations without +altering the nature of our specification of position. +\index{Conception of mass!position}% +If, for instance, a cloud is hovering over Trafalgar +Square, then we can determine its position relative to +the surface of the earth by erecting a pole perpendicularly +on the Square, so that it reaches the cloud. The +length of the pole measured with the standard measuring-rod, +\index{Measuring-rod}% +combined with the specification of the position of +the foot of the pole, supplies us with a complete place +specification. On the basis of this illustration, we are +able to see the manner in which a refinement of the conception +of position has been developed. + +\itema~We imagine the rigid body, to which the place +specification is referred, supplemented in such a manner +that the object whose position we require is reached by +the completed rigid body. + +\itemb~In locating the position of the object, we make +use of a number (here the length of the pole measured +\PageSep{7} +with the measuring-rod) instead of designated points of +reference. + +\itemc~We speak of the height of the cloud even when the +pole which reaches the cloud has not been erected. +By means of optical observations of the cloud from +different positions on the ground, and taking into account +the properties of the propagation of light, we determine +the length of the pole we should have required in order +to reach the cloud. + +From this consideration we see that it will be advantageous +\index{Physics}% +if, in the description of position, it should be +possible by means of numerical measures to make ourselves +independent of the existence of marked positions +(possessing names) on the rigid body of reference. In +\index{Reference-body}% +the physics of measurement this is attained by the +\index{Physics!of measurement}% +application of the Cartesian system of co-ordinates. +\index{Cartesian system of co-ordinates}% + +This consists of three plane surfaces perpendicular +to each other and rigidly attached to a rigid body. +Referred to a system of co-ordinates, the scene of any +event will be determined (for the main part) by the +specification of the lengths of the three perpendiculars +or co-ordinates $(x, y, z)$ which can be dropped from the +scene of the event to those three plane surfaces. The +lengths of these three perpendiculars can be determined +by a series of manipulations with rigid measuring-rods +performed according to the rules and methods laid +down by Euclidean geometry. + +In practice, the rigid surfaces which constitute the +system of co-ordinates are generally not available; +furthermore, the magnitudes of the co-ordinates are not +actually determined by constructions with rigid rods, but +by indirect means. If the results of physics and astronomy +\index{Astronomy}% +are to maintain their clearness, the physical meaning +\PageSep{8} +of specifications of position must always be sought +in accordance with the above considerations.\footnote + {A refinement and modification of these views does not become + necessary until we come to deal with the general theory of + relativity, treated in the second part of this book.} + +We thus obtain the following result: Every description +of events in space involves the use of a rigid body +to which such events have to be referred. The resulting +relationship takes for granted that the laws of Euclidean +\index{Distance (line-interval)}% +\index{Euclidean geometry!propositions of}% +geometry hold for ``distances,'' the ``distance'' being +represented physically by means of the convention of +two marks on a rigid body. +\PageSep{9} + + +\Chapter{III}{Space and Time in Classical Mechanics} +\index{Classical mechanics}% +\index{Space}% + +\Change{}{``}\First{The} purpose of mechanics is to describe how +bodies change their position in space with +\index{Position}% +time.'' I should load my conscience with grave +sins against the sacred spirit of lucidity were I to +formulate the aims of mechanics in this way, without +serious reflection and detailed explanations. Let us +proceed to disclose these sins. + +It is not clear what is to be understood here by +\index{Reference-body|(}% +``position'' and ``space.'' I stand at the window of a +railway carriage which is travelling uniformly, and drop +a stone on the embankment, without throwing it. Then, +disregarding the influence of the air resistance, I see the +stone descend in a straight line. A pedestrian who +\index{Straight line}% +observes the misdeed from the footpath notices that the +stone falls to earth in a parabolic curve. I now ask: +Do the ``positions'' traversed by the stone lie ``in +reality'' on a straight line or on a parabola? Moreover, +\index{Parabola}% +what is meant here by motion ``in space''? From the +considerations of the previous section the answer is +self-evident. In the first place, we entirely shun the +vague word ``space,'' of which, we must honestly +acknowledge, we cannot form the slightest conception, +and we replace it by ``motion relative to a +practically rigid body of reference.'' The positions +relative to the body of reference (railway carriage or +embankment) have already been defined in detail in the +\PageSep{10} +preceding section. If instead of ``body of reference'' +we insert ``system of co-ordinates,'' which is a useful +\index{System of co-ordinates}% +idea for mathematical description, we are in a position +to say: The stone traverses a straight line relative to a +\index{Straight line}% +system of co-ordinates rigidly attached to the carriage, +but relative to a system of co-ordinates rigidly attached +to the ground (embankment) it describes a parabola. +\index{Parabola}% +With the aid of this example it is clearly seen that there +is no such thing as an independently existing trajectory +\index{Trajectory}% +(lit. ``path-curve''\footnotemark), but only a trajectory relative to a +\index{Path-curve}% +particular body of reference. +\footnotetext{That is, a curve along which the body moves.} + +In order to have a \emph{complete} description of the motion, +we must specify how the body alters its position \emph{with +time}; \ie\ for every point on the trajectory it must be +stated at what time the body is situated there. These +data must be supplemented by such a definition of +time that, in virtue of this definition, these time-values +can be regarded essentially as magnitudes (results of +measurements) capable of observation. If we take our +stand on the ground of classical mechanics, we can +satisfy this requirement for our illustration in the +following manner. We imagine two clocks of identical +\index{Clocks}% +construction; the man at the railway-carriage window +is holding one of them, and the man on the footpath +the other. Each of the observers determines +the position on his own reference-body occupied by the +stone at each tick of the clock he is holding in his +hand. In this connection we have not taken account +of the inaccuracy involved by the finiteness of the +velocity of propagation of light. With this and with a +\index{Velocity of light}% +second difficulty prevailing here we shall have to deal +in detail later. +\PageSep{11} + + +\Chapter{IV}{The Galileian System of Co-ordinates} +\index{Galileian system of co-ordinates}% +\index{System of co-ordinates}% + +\First{As} is well known, the fundamental law of the +mechanics of Galilei-Newton, which is known +\index{Galilei}% +\index{Newton}% +as the \emph{law of inertia}, can be stated thus: +\index{Law of inertia}% +A body removed sufficiently far from other bodies +continues in a state of rest or of uniform motion +in a straight line. This law not only says something +about the motion of the bodies, but it also +indicates the reference-bodies or systems of co-ordinates, +permissible in mechanics, which can be used +in mechanical description. The visible fixed stars are +\index{Fixed stars}% +bodies for which the law of inertia certainly holds to a +high degree of approximation. Now if we use a system +of co-ordinates which is rigidly attached to the earth, +then, relative to this system, every fixed star describes +a circle of immense radius in the course of an astronomical +day, a result which is opposed to the statement +\index{Astronomical day}% +of the law of inertia. So that if we adhere to this law +we must refer these motions only to systems of co-ordinates +relative to which the fixed stars do not move +in a circle. A system of co-ordinates of which the state +of motion is such that the law of inertia holds relative to +it is called a ``Galileian system of co-ordinates.'' The +laws of the mechanics of Galilei-Newton can be regarded +as valid only for a Galileian system of co-ordinates. +\index{Reference-body|)}% +\PageSep{12} + + +\Chapter{V}{The Principle of Relativity (In the +Restricted Sense)} + +\First{In} order to attain the greatest possible clearness, +let us return to our example of the railway carriage +supposed to be travelling uniformly. We call its +motion a uniform translation (``uniform'' because +\index{Uniform translation}% +it is of constant velocity and direction, ``translation'' +because although the carriage changes its position +relative to the embankment yet it does not rotate +in so doing). Let us imagine a raven flying through +the air in such a manner that its motion, as observed +from the embankment, is uniform and in a straight +line. If we were to observe the flying raven from +the moving railway carriage, we should find that the +motion of the raven would be one of different velocity +and direction, but that it would still be uniform +and in a straight line. Expressed in an abstract +manner we may say: If a mass~$m$ is moving uniformly +in a straight line with respect to a co-ordinate +system~$K$, then it will also be moving uniformly and in a +straight line relative to a second co-ordinate system~$K'$, +provided that the latter is executing a uniform +translatory motion with respect to~$K$. In accordance +with the discussion contained in the preceding section, +it follows that: +\PageSep{13} + +If $K$~is a Galileian co-ordinate system, then every other +\index{Galileian system of co-ordinates}% +co-ordinate system~$K'$ is a Galileian one, when, in relation +to~$K$, it is in a condition of uniform motion of translation. +\index{Motion!of heavenly bodies}% +Relative to~$K'$ the mechanical laws of Galilei-Newton +\index{Laws of Galilei-Newton}% +hold good exactly as they do with respect to~$K$. + +We advance a step farther in our generalisation when +we express the tenet thus: If, relative to~$K$, $K'$~is a +uniformly moving co-ordinate system devoid of rotation, +then natural phenomena run their course with respect to~$K'$ +according to exactly the same general laws as with +respect to~$K$. This statement is called the \emph{principle +of relativity} (in the restricted sense). + +As long as one was convinced that all natural phenomena +were capable of representation with the help of +classical mechanics, there was no need to doubt the +\index{Classical mechanics}% +\index{Classical mechanics!truth of}% +validity of this principle of relativity. But in view of +\index{Principle of relativity|(}% +the more recent development of electrodynamics and +\index{Electrodynamics}% +optics it became more and more evident that classical +\index{Optics}% +mechanics affords an insufficient foundation for the +physical description of all natural phenomena. At this +juncture the question of the validity of the principle of +relativity became ripe for discussion, and it did not +appear impossible that the answer to this question +might be in the negative. + +Nevertheless, there are two general facts which at the +outset speak very much in favour of the validity of the +principle of relativity. Even though classical mechanics +does not supply us with a sufficiently broad basis for the +theoretical presentation of all physical phenomena, +still we must grant it a considerable measure of ``truth,'' +since it supplies us with the actual motions of the +heavenly bodies with a delicacy of detail little short of +wonderful. The principle of relativity must therefore +\PageSep{14} +apply with great accuracy in the domain of \emph{mechanics}. +\index{Classical mechanics}% +But that a principle of such broad generality should +hold with such exactness in one domain of phenomena, +and yet should be invalid for another, is \textit{a~priori} not +very probable. + +We now proceed to the second argument, to which, +moreover, we shall return later. If the principle of relativity +(in the restricted sense) does not hold, then the +Galileian co-ordinate systems $K$,~$K'$, $K''$,~etc., which are +\index{Galileian system of co-ordinates}% +moving uniformly relative to each other, will not be +\emph{equivalent} for the description of natural phenomena. +\index{Equivalent}% +In this case we should be constrained to believe that +natural laws are capable of being formulated in a particularly +simple manner, and of course only on condition +that, from amongst all possible Galileian co-ordinate +systems, we should have chosen \emph{one}~($K_{0}$) of a particular +state of motion as our body of reference. We should +\index{Motion}% +then be justified (because of its merits for the description +of natural phenomena) in calling this system ``absolutely +at rest,'' and all other Galileian systems~$K$ ``in motion.'' +\index{Rest}% +If, for instance, our embankment were the system~$K_{0}$, +then our railway carriage would be a system~$K$, +relative to which less simple laws would hold than with +respect to~$K_{0}$. This diminished simplicity would be +due to the fact that the carriage~$K$ would be in motion +(\ie\ ``really'') with respect to~$K_{0}$. In the general laws +of nature which have been formulated with reference +to~$K$, the magnitude and direction of the velocity +of the carriage would necessarily play a part. We should +expect, for instance, that the note emitted by an organ-pipe +\index{Organ-pipe, note of}% +placed with its axis parallel to the direction of +travel would be different from that emitted if the axis +of the pipe were placed perpendicular to this direction. +\PageSep{15} +Now in virtue of its motion in an orbit round the sun, +\index{Motion!of heavenly bodies}% +our earth is comparable with a railway carriage travelling +with a velocity of about $30$~kilometres per~second. +If the principle of relativity were not valid we should +therefore expect that the direction of motion of the +earth at any moment would enter into the laws of nature, +and also that physical systems in their behaviour would +be dependent on the orientation in space with respect +to the earth. For owing to the alteration in direction +of the velocity of revolution of the earth in the course +of a year, the earth cannot be at rest relative to the +hypothetical system~$K_{0}$ throughout the whole year. +However, the most careful observations have never +revealed such anisotropic properties in terrestrial physical +\index{Terrestrial space}% +space, \ie\ a physical non-equivalence of different +directions. This is very powerful argument in favour +of the principle of relativity. +\index{Principle of relativity|)}% +\PageSep{16} + + +\Chapter{VI}{The Theorem of the Addition of Velocities +employed in Classical Mechanics} +\index{Addition of velocities}% +\index{Classical mechanics}% + +\First{Let} us suppose our old friend the railway carriage +to be travelling along the rails with a constant +velocity~$v$, and that a man traverses the length of +the carriage in the direction of travel with a velocity~$w$. +How quickly or, in other words, with what velocity~$W$ +does the man advance relative to the embankment +during the process? The only possible answer seems to +result from the following consideration: If the man were +to stand still for a second, he would advance relative to +the embankment through a distance~$v$ equal numerically +to the velocity of the carriage. As a consequence of +his walking, however, he traverses an additional distance~$w$ +relative to the carriage, and hence also relative to the +embankment, in this second, the distance~$w$ being +numerically equal to the velocity with which he is +walking. Thus in total he covers the distance $W = v + w$ +relative to the embankment in the second considered. +We shall see later that this result, which expresses +the theorem of the addition of velocities employed in +classical mechanics, cannot be maintained; in other +words, the law that we have just written down does not +hold in reality. For the time being, however, we shall +assume its correctness. +\PageSep{17} + + +\Chapter{VII}{The Apparent Incompatibility of the +Law of Propagation of Light with +the Principle of Relativity} +\index{Propagation of light}% + +\First{There} is hardly a simpler law in physics than +that according to which light is propagated in +empty space. Every child at school knows, or +believes he knows, that this propagation takes place +in straight lines with a velocity $c = 300,000$~km./sec. +At all events we know with great exactness that this +velocity is the same for all colours, because if this were +not the case, the minimum of emission would not be +observed simultaneously for different colours during +the eclipse of a fixed star by its dark neighbour. By +\index{DeSitter@{De Sitter}}% +\index{Eclipse of star}% +means of similar considerations based on observations +of double stars, the Dutch astronomer De~Sitter +\index{Double stars}% +was also able to show that the velocity of propagation +of light cannot depend on the velocity of motion +of the body emitting the light. The assumption that +this velocity of propagation is dependent on the direction +``in space'' is in itself improbable. + +In short, let us assume that the simple law of the +constancy of the velocity of light~$c$ (in vacuum) is +\index{Velocity of light}% +justifiably believed by the child at school. Who would +imagine that this simple law has plunged the conscientiously +thoughtful physicist into the greatest +\PageSep{18} +intellectual difficulties? Let us consider how these +difficulties arise. + +Of course we must refer the process of the propagation +of light (and indeed every other process) to a rigid +reference-body (co-ordinate system). As such a system +\index{Reference-body}% +let us again choose our embankment. We shall imagine +the air above it to have been removed. If a ray of +light be sent along the embankment, we see from the +above that the tip of the ray will be transmitted with +the velocity~$c$ relative to the embankment. Now let +us suppose that our railway carriage is again travelling +along the railway lines with the velocity~$v$, and that +its direction is the same as that of the ray of light, but +its velocity of course much less. Let us inquire about +the velocity of propagation of the ray of light relative +to the carriage. It is obvious that we can here apply the +consideration of the previous section, since the ray of +light plays the part of the man walking along relatively +to the carriage. The velocity~$W$ of the man relative +to the embankment is here replaced by the velocity +of light relative to the embankment. $w$~is the required +velocity of light with respect to the carriage, and we +\index{Velocity of light}% +have +\[ +w = c - v. +\] +The velocity of propagation of a ray of light relative to +the carriage thus comes out smaller than~$c$. + +But this result comes into conflict with the principle +of relativity set forth in \Sectionref{V}. For, like every +other general law of nature, the law of the transmission +of light \textit{in~vacuo} must, according to the principle of +relativity, be the same for the railway carriage as +reference-body as when the rails are the body of reference. +\PageSep{19} +But, from our above consideration, this would +appear to be impossible. If every ray of light is propagated +relative to the embankment with the velocity~$c$, +then for this reason it would appear that another law +of propagation of light must necessarily hold with respect +\index{Propagation of light}% +to the carriage---a result contradictory to the principle +of relativity. + +In view of this dilemma there appears to be nothing +else for it than to abandon either the principle of relativity +\index{Principle of relativity}% +or the simple law of the propagation of light \textit{in~vacuo}. +Those of you who have carefully followed the +preceding discussion are almost sure to expect that +we should retain the principle of relativity, which +appeals so convincingly to the intellect because it is so +natural and simple. The law of the propagation of +light \textit{in~vacuo} would then have to be replaced by a +more complicated law conformable to the principle of +relativity. The development of theoretical physics +shows, however, that we cannot pursue this course. +The epoch-making theoretical investigations of H.~A. +Lorentz on the electrodynamical and optical phenomena +\index{Electrodynamics}% +\index{Optics}% +\index{Lorentz, H. A.}% +connected with moving bodies show that experience +in this domain leads conclusively to a theory of electromagnetic +phenomena, of which the law of the constancy +of the velocity of light \textit{in~vacuo} is a necessary consequence. +Prominent theoretical physicists were therefore +more inclined to reject the principle of relativity, +in spite of the fact that no empirical data had been +found which were contradictory to this principle. + +At this juncture the theory of relativity entered the +arena. As a result of an analysis of the physical conceptions +of time and space, it became evident that \emph{in +\index{Space!conception of}% +\index{Time!conception of}% +reality there is not the least incompatibility between the +\PageSep{20} +principle of relativity and the law of propagation of light}, +\index{Principle of relativity}% +\index{Propagation of light}% +and that by systematically holding fast to both these +laws a logically rigid theory could be arrived at. This +theory has been called the \emph{special theory of relativity} +\index{Special theory of relativity}% +to distinguish it from the extended theory, with which +we shall deal later. In the following pages we shall +present the fundamental ideas of the special theory of +relativity. +\PageSep{21} + + +\Chapter{VIII}{On the Idea of Time in Physics} +\index{Time!in Physics}% + +\First{Lightning} has struck the rails on our railway +embankment at two places $A$~and~$B$ far distant +from each other. I make the additional assertion +that these two lightning flashes occurred simultaneously. +If I ask you whether there is sense in this statement, +you will answer my question with a decided +``Yes.'' But if I now approach you with the request +to explain to me the sense of the statement more +precisely, you find after some consideration that the +answer to this question is not so easy as it appears at +first sight. + +After some time perhaps the following answer would +occur to you: ``The significance of the statement is +clear in itself and needs no further explanation; of +course it would require some consideration if I were to +be commissioned to determine by observations whether +in the actual case the two events took place simultaneously +or not.'' I cannot be satisfied with this answer +for the following reason. Supposing that as a result +of ingenious considerations an able meteorologist were +to discover that the lightning must always strike the +places $A$~and~$B$ simultaneously, then we should be faced +with the task of testing whether or not this theoretical +result is in accordance with the reality. We encounter +\PageSep{22} +the same difficulty with all physical statements in which +the conception ``simultaneous'' plays a part. The +concept does not exist for the physicist until he has the +possibility of discovering whether or not it is fulfilled +in an actual case. We thus require a definition of +simultaneity such that this definition supplies us with +\index{Simultaneity}% +the method by means of which, in the present case, he +can decide by experiment whether or not both the +lightning strokes occurred simultaneously. As long +as this requirement is not satisfied, I allow myself to be +deceived as a physicist (and of course the same applies +if I am not a physicist), when I imagine that I am able +to attach a meaning to the statement of simultaneity. +(I would ask the reader not to proceed farther until he +is fully convinced on this point.) + +After thinking the matter over for some time you +then offer the following suggestion with which to test +simultaneity. By measuring along the rails, the +connecting line~$AB$ should be measured up and an +observer placed at the mid-point~$M$ of the distance~$AB$. +This observer should be supplied with an arrangement +(\eg\ two mirrors inclined at~$90°$) which allows him +visually to observe both places $A$~and~$B$ at the same +time. If the observer perceives the two flashes of +lightning at the same time, then they are simultaneous. + +I am very pleased with this suggestion, but for all +that I cannot regard the matter as quite settled, because +I feel constrained to raise the following objection: +``Your definition would certainly be right, if I only +knew that the light by means of which the observer +at~$M$ perceives the lightning flashes travels along the +length $A\longrightarrow M$ with the same velocity as along the +length $B\longrightarrow M$. But an examination of this supposition +\PageSep{23} +would only be possible if we already had at our +disposal the means of measuring time. It would thus +appear as though we were moving here in a logical circle.'' + +After further consideration you cast a somewhat +disdainful glance at me---and rightly so---and you +declare: ``I maintain my previous definition nevertheless, +because in reality it assumes absolutely nothing +about light. There is only \emph{one} demand to be made of +the definition of simultaneity, namely, that in every +real case it must supply us with an empirical decision +as to whether or not the conception that has to +be defined is fulfilled. That my definition satisfies +this demand is indisputable. That light requires the +same time to traverse the path $A\longrightarrow M$ as for the path +$B\longrightarrow M$ is in reality neither a \emph{supposition nor a hypothesis} +about the physical nature of light, but a \emph{stipulation} +which I can make of my own \Change{freewill}{free will} in order to arrive +at a definition of simultaneity.'' + +It is clear that this definition can be used to give an +exact meaning not only to \emph{two} events, but to as many +events as we care to choose, and independently of the +positions of the scenes of the events with respect to the +\index{Reference-body}% +body of reference\footnote + {We suppose further, that, when three events $A$,~$B$ and~$C$ + occur in different places in such a manner that $A$~is simultaneous + with~$,$ and $B$~is simultaneous with~$C$ (simultaneous + in the sense of the above definition), then the criterion for the + simultaneity of the pair of events $A$,~$C$ is also satisfied. This + assumption is a physical hypothesis about the law of propagation + of light; it must certainly be fulfilled if we are to maintain the + law of the constancy of the velocity of light \textit{in~vacuo}.} +(here the railway embankment). +We are thus led also to a definition of ``time'' in physics. +For this purpose we suppose that clocks of identical +\index{Clocks}% +construction are placed at the points $A$,~$B$ and~$C$ of +\PageSep{24} +\index{Simultaneity|(}% +the railway line (co-ordinate system), and that they +are set in such a manner that the positions of their +pointers are simultaneously (in the above sense) the +same. Under these conditions we understand by the +``time'' of an event the reading (position of the hands) +\index{Time!of an event}% +of that one of these clocks which is in the immediate +vicinity (in space) of the event. In this manner a +time-value is associated with every event which is +essentially capable of observation. + +This stipulation contains a further physical hypothesis, +the validity of which will hardly be doubted without +empirical evidence to the contrary. It has been assumed +that all these clocks go \emph{at the same rate} if they are of +identical construction. Stated more exactly: When +two clocks arranged at rest in different places of a +reference-body are set in such a manner that a \emph{particular} +position of the pointers of the one clock is \emph{simultaneous} +(in the above sense) with the \emph{same} position of the +pointers of the other clock, then identical ``settings'' +are always simultaneous (in the sense of the above +definition). +\PageSep{25} + + +\Chapter{IX}{The Relativity of Simultaneity} + +\First{Up} to now our considerations have been referred +\index{Reference-body}% +to a particular body of reference, which we +have styled a ``railway embankment.'' We +suppose a very long train travelling along the rails +with the constant velocity~$v$ and in the direction indicated +in \Figref{1}. People travelling in this train will +with advantage use the train as a rigid reference-body +(co-ordinate system); they regard all events in +%[Illustration: Fig. 1.] +\Figure{025} +reference to the train. Then every event which takes +place along the line also takes place at a particular +point of the train. Also the definition of simultaneity +can be given relative to the train in exactly the same +way as with respect to the embankment. As a natural +consequence, however, the following question arises: + +Are two events (\eg\ the two strokes of lightning $A$ +and~$B$) which are simultaneous \emph{with reference to the +railway embankment} also simultaneous \emph{relatively to the +train}? We shall show directly that the answer must +be in the negative. + +When we say that the lightning strokes $A$~and~$B$ are +\PageSep{26} +simultaneous with respect to the embankment, we +mean: the rays of light emitted at the places $A$~and~$B$, +where the lightning occurs, meet each other at the +mid-point~$M$ of the length $A\longrightarrow B$ of the embankment. +But the events $A$~and~$B$ also correspond to positions $A$~and~$B$ +\index{Time!of an event}% +on the train. Let $M'$~be the mid-point of the +distance $A\longrightarrow B$ on the travelling train. Just when +the flashes\footnote + {As judged from the embankment.} +of lightning occur, this point~$M'$ naturally +coincides with the point~$M$, but it moves towards the +right in the diagram with the velocity~$v$ of the train. If +an observer sitting in the position~$M'$ in the train did +not possess this velocity, then he would remain permanently +at~$M$, and the light rays emitted by the +flashes of lightning $A$~and~$B$ would reach him simultaneously, +\ie\ they would meet just where he is situated. +Now in reality (considered with reference to the railway +embankment) he is hastening towards the beam of light +coming from~$B$, whilst he is riding on ahead of the beam +of light coming from~$A$. Hence the observer will see +the beam of light emitted from~$B$ earlier than he will +see that emitted from~$A$. Observers who take the railway +train as their reference-body must therefore come +\index{Reference-body}% +to the conclusion that the lightning flash~$B$ took place +earlier than the lightning flash~$A$. We thus arrive at +the important result: + +Events which are simultaneous with reference to the +embankment are not simultaneous with respect to the +train, and \textit{vice versa} (relativity of simultaneity). Every +\index{Simultaneity|)}% +\index{Simultaneity!relativity of}% +reference-body (co-ordinate system) has its own particular +time; unless we are told the reference-body to which +the statement of time refers, there is no meaning in a +statement of the time of an event. +\PageSep{27} + +Now before the advent of the theory of relativity +it had always tacitly been assumed in physics that the +statement of time had an absolute significance, \ie\ +that it is independent of the state of motion of the body +of reference. But we have just seen that this assumption +is incompatible with the most natural definition +of simultaneity; if we discard this assumption, then +the conflict between the law of the propagation of +light \textit{in~vacuo} and the principle of relativity (developed +in \Sectionref{VII}) disappears. + +We were led to that conflict by the considerations +of \Sectionref{VI}, which are now no longer tenable. In +that section we concluded that the man in the carriage, +who traverses the distance~$w$ \emph{per~second} relative to the +carriage, traverses the same distance also with respect to +the embankment \emph{in each second} of time. But, according +to the foregoing considerations, the time required by a +particular occurrence with respect to the carriage must +not be considered equal to the duration of the same +occurrence as judged from the embankment (as reference-body). +Hence it cannot be contended that the +man in walking travels the distance~$w$ relative to the +railway line in a time which is equal to one second as +judged from the embankment. + +Moreover, the considerations of \Sectionref{VI} are based +on yet a second assumption, which, in the light of a +strict consideration, appears to be arbitrary, although +it was always tacitly made even before the introduction +of the theory of relativity. +\PageSep{28} + + +\Chapter{X}{On the Relativity of the Conception +of Distance} +\index{Distance (line-interval)}% +\index{Distance (line-interval)!relativity of}% + +\First{Let} us consider two particular points on the train\footnote + {\eg\ the middle of the first and of the hundredth carriage.} +travelling along the embankment with the +velocity~$v$, and inquire as to their distance apart. +We already know that it is necessary to have a body of +reference for the measurement of a distance, with respect +to which body the distance can be measured up. It is +the simplest plan to use the train itself as reference-body +(co-ordinate system). An observer in the train +measures the interval by marking off his measuring-rod +\index{Measuring-rod}% +in a straight line (\eg\ along the floor of the carriage) +as many times as is necessary to take him from the one +marked point to the other. Then the number which +tells us how often the rod has to be laid down is the +required distance. + +It is a different matter when the distance has to be +judged from the railway line. Here the following +method suggests itself. If we call $A'$~and~$B'$ the two +points on the train whose distance apart is required, +then both of these points are moving with the velocity~$v$ +along the embankment. In the first place we require to +determine the points $A$~and~$B$ of the embankment which +are just being passed by the two points $A'$~and~$B'$ at a +\PageSep{29} +particular time~$t$---judged from the embankment. +These points $A$~and~$B$ of the embankment can be determined +by applying the definition of time given in +\Sectionref{VIII}. The distance between these points $A$~and~$B$ +\index{Distance (line-interval)}% +is then measured by repeated application of the +measuring-rod along the embankment. + +\textit{A~priori} it is by no means certain that this last +measurement will supply us with the same result as +the first. Thus the length of the train as measured +from the embankment may be different from that +obtained by measuring in the train itself. This +circumstance leads us to a second objection which must +be raised against the apparently obvious consideration +of \Sectionref{VI}. Namely, if the man in the carriage +covers the distance~$w$ in a unit of time---\emph{measured from +the train},---then this distance---\emph{as measured from the +embankment}---is not necessarily also equal to~$w$. +\PageSep{30} + + +\Chapter{XI}{The Lorentz Transformation} + +\First{The} results of the last three sections show +that the apparent incompatibility of the law +of propagation of light with the principle of +relativity (\Sectionref{VII}) has been derived by means of +a consideration which borrowed two unjustifiable +hypotheses from classical mechanics; these are as +\index{Classical mechanics}% +follows: +\begin{itemize} +\item[(1)] The time-interval (time) between two events is +\index{Time-interval}% + independent of the condition of motion of the + body of reference. + +\item[(2)] The space-interval (distance) between two points +\index{Space!interval@{-interval}}% + of a rigid body is independent of the condition + of motion of the body of reference. +\end{itemize} + +If we drop these hypotheses, then the dilemma of +\Sectionref{VII} disappears, because the theorem of the addition +of velocities derived in \Sectionref{VI} becomes invalid. +The possibility presents itself that the law of the propagation +of light \textit{in~vacuo} may be compatible with the +principle of relativity, and the question arises: How +have we to modify the considerations of \Sectionref{VI} +in order to remove the apparent disagreement between +these two fundamental results of experience? This +question leads to a general one. In the discussion of +\PageSep{31} +\Sectionref{VI} we have to do with places and times relative +both to the train and to the embankment. How are +we to find the place and time of an event in relation to +the train, when we know the place and time of the +event with respect to the railway embankment? Is +there a thinkable answer to this question of such a +nature that the law of transmission of light \textit{in~vacuo} +does not contradict the principle of relativity? In +other words: Can we conceive of a relation between +place and time of the individual events relative to both +reference-bodies, such that every ray of light possesses +the velocity of transmission~$c$ relative to the embankment +and relative to the train? This question leads to +a quite definite positive answer, and to a perfectly definite +transformation law for the space-time magnitudes of +an event when changing over from one body of reference +to another. + +Before we deal with this, we shall introduce the +following incidental consideration. Up to the present +we have only considered events taking place along the +embankment, which had mathematically to assume the +function of a straight line. In the manner indicated +in \Sectionref{II} we can imagine this reference-body supplemented +laterally and in a vertical direction by means of +a framework of rods, so that an event which takes place +anywhere can be localised with reference to this framework. +Similarly, we can imagine the train travelling +with the velocity~$v$ to be continued across the whole of +space, so that every event, no matter how far off it +may be, could also be localised with respect to the second +framework. Without committing any fundamental error, +we can disregard the fact that in reality these frameworks +would continually interfere with each other, owing +\PageSep{32} +\index{Propagation of light}% +to the impenetrability of solid bodies. In every such +framework we imagine three surfaces perpendicular to +each other marked out, and designated as ``co-ordinate +\index{Coordinate@{Co-ordinate}!planes}% +planes'' (``co-ordinate system''). A co-ordinate +system~$K$ then corresponds to the embankment, and a +co-ordinate system~$K'$ to the train. An event, wherever +it may have taken place, would be fixed in space with +respect to~$K$ by the three perpendiculars $x$,~$y$,~$z$ on the +co-ordinate planes, and with regard to time by a time-value~$t$. +Relative to~$K'$, \emph{the +same event} would be fixed +in respect of space and time +by corresponding values $x'$,~$y'$, +$z'$,~$t'$, which of course are +not identical with $x$,~$y$, $z$,~$t$. +It has already been set +forth in detail how these +magnitudes are to be regarded +as results of physical measurements. +%[Illustration: Fig. 2.] +\Figure[2in]{032} + +Obviously our problem can be exactly formulated in +the following manner. What are the values $x'$,~$y'$, $z'$,~$t'$, +of an event with respect to~$K'$, when the magnitudes +$x$,~$y$, $z$,~$t$, of the same event with respect to~$K$ are given? +The relations must be so chosen that the law of the +transmission of light \textit{in~vacuo} is satisfied for one and the +same ray of light (and of course for every ray) with +respect to $K$ and~$K'$. For the relative orientation in +space of the co-ordinate systems indicated in the diagram +(\Figref{2}), this problem is solved by means of the +equations: +\begin{align*} +x' &= \frac{x - vt}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}\Add{,}\displaybreak[1] \\ +\PageSep{33} +y' &= y\Add{,} \\ +z' &= z\Add{,} \\ +t' &= \frac{t - \dfrac{v}{c^{2}}·x}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}\Change{}{.} +\end{align*} +This system of equations is known as the ``Lorentz +\index{Lorentz, H. A.!transformation}% +transformation.''\footnote + {A simple derivation of the Lorentz transformation is given + in \Appendixref{I}.} + +If in place of the law of transmission of light we had +taken as our basis the tacit assumptions of the older +mechanics as to the absolute character of times and +lengths, then instead of the above we should have +obtained the following equations: +\begin{align*} +x' &= x - vt\Add{,} \\ +y' &= y\Add{,} \\ +z' &= z\Add{,} \\ +t' &= t. +\end{align*} +This system of equations is often termed the ``Galilei +\index{Galilei!transformation}% +transformation.'' The Galilei transformation can be +obtained from the Lorentz transformation by substituting +an infinitely large value for the velocity of +light~$c$ in the latter transformation. + +Aided by the following illustration, we can readily +see that, in accordance with the Lorentz transformation, +the law of the transmission of light \textit{in~vacuo} +is satisfied both for the reference-body~$K$ and for the +reference-body~$K'$. A light-signal is sent along the +\index{Light-signal}% +positive $x$-axis, and this light-stimulus advances in +\index{Light-stimulus}% +accordance with the equation +\[ +x = ct, +\] +\PageSep{34} +\ie\ with the velocity~$c$. According to the equations of +the Lorentz transformation, this simple relation between +$x$~and~$t$ involves a relation between $x'$~and~$t'$. In point +of fact, if we substitute for~$x$ the value~$ct$ in the first +and fourth equations of the Lorentz transformation, +we obtain: +\begin{align*} +x' &= \frac{(c - v)t}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}\Add{,} \\ +t' &= \frac{\left(1 - \dfrac{v}{c}\right)t}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}, +\end{align*} +from which, by division, the expression +\[ +x' = ct' +\] +immediately follows. If referred to the system~$K'$, the +propagation of light takes place according to this +equation. We thus see that the velocity of transmission +relative to the reference-body~$K'$ is also equal to~$c$. The +same result is obtained for rays of light advancing in +any other direction whatsoever. Of course this is not +surprising, since the equations of the Lorentz transformation +were derived conformably to this point of +view. +\PageSep{35} + + +\Chapter{XII}{The Behaviour of Measuring-Rods and +Clocks in Motion} + +\First{I place} a metre-rod in the $x'$-axis of~$K'$ in such a +manner that one end (the beginning) coincides with +the point $x' = 0$, whilst the other end (the end of the +rod) coincides with the point $x' = 1$. What is the length +of the metre-rod relatively to the system~$K$? In order +to learn this, we need only ask where the beginning of the +rod and the end of the rod lie with respect to~$K$ at a +particular time~$t$ of the system~$K$. By means of the first +equation of the Lorentz transformation the values of +these two points at the time $t = 0$ can be shown to be +\begin{align*} +x_{\text{(beginning of rod)}} + &= 0·\sqrt{1 - \frac{v^{2}}{c^{2}}}\Add{,} \\ +x_{\text{(end of rod)}} + &= 1·\sqrt{1 - \frac{v^{2}}{c^{2}}}, +\end{align*} +the distance between the points being~$\sqrt{1 - \dfrac{v^{2}}{c^{2}}}$. But +the metre-rod is moving with the velocity~$v$ relative to~$K$. +It therefore follows that the length of a rigid metre-rod +moving in the direction of its length with a velocity~$v$ +is $\sqrt{1 - v^{2}/c^{2}}$~of a metre. The rigid rod is thus +shorter when in motion than when at rest, and the +more quickly it is moving, the shorter is the rod. For +the velocity $v = c$ we should have $\sqrt{1 - v^{2}/c^{2}} = 0$, and +for still greater velocities the square-root becomes +\PageSep{36} +imaginary. From this we conclude that in the theory +of relativity the velocity~$c$ plays the part of a limiting +\index{Limiting velocity ($c$)}% +velocity, which can neither be reached nor exceeded +by any real body. + +Of course this feature of the velocity~$c$ as a limiting +velocity also clearly follows from the equations of the +Lorentz transformation, for these become meaningless +if we choose values of~$v$ greater than~$c$. + +If, on the contrary, we had considered a metre-rod +at rest in the $x$-axis with respect to~$K$, then we should +have found that the length of the rod as judged from~$K'$ +would have been~$\sqrt{1 - v^{2}/c^{2}}$; this is quite in accordance +with the principle of relativity which forms the +basis of our considerations. + +\textit{A~priori} it is quite clear that we must be able to +learn something about the physical behaviour of measuring-rods +and clocks from the equations of transformation, +for the magnitudes $x$,~$y$, $z$,~$t$, are nothing more nor +less than the results of measurements obtainable by +means of measuring-rods and clocks. If we had based +our considerations on the Galilei transformation we +\index{Galilei!transformation}% +should not have obtained a contraction of the rod as a +consequence of its motion. + +Let us now consider a seconds-clock which is permanently +\index{Seconds-clock}% +situated at the origin ($x' = 0$) of~$K'$. $t' = 0$ +and $t' = 1$ are two successive ticks of this clock. The +first and fourth equations of the Lorentz transformation +give for these two ticks: +\begin{align*} +t &= 0 \\ +\intertext{and} +t &= \frac{1}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}. +\end{align*} +\PageSep{37} + +As judged from~$K$, the clock is moving with the +velocity~$v$; as judged from this reference-body, the +\index{Reference-body}% +time which elapses between two strokes of the clock +is not one second, but $\dfrac{1}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}$~seconds, \ie\ a somewhat +larger time. As a consequence of its motion +the clock goes more slowly than when at rest. Here +also the velocity~$c$ plays the part of an unattainable +limiting velocity. +\index{Limiting velocity ($c$)}% +\PageSep{38} + + +\Chapter{XIII}{Theorem of the Addition of Velocities. +The Experiment of Fizeau} +\index{Addition of velocities}% + +\First{Now} in practice we can move clocks and +measuring-rods only with velocities that are +small compared with the velocity of light; hence +we shall hardly be able to compare the results of the +previous section directly with the reality. But, on the +other hand, these results must strike you as being very +singular, and for that reason I shall now draw another +conclusion from the theory, one which can easily be +derived from the foregoing considerations, and which +has been most elegantly confirmed by experiment. + +In \Sectionref{VI} we derived the theorem of the addition +of velocities in one direction in the form which also +results from the hypotheses of classical mechanics. This +theorem can also be deduced readily from the Galilei +\index{Galilei!transformation}% +transformation (\Sectionref{XI}). In place of the man +walking inside the carriage, we introduce a point moving +relatively to the co-ordinate system~$K'$ in accordance +with the equation +\[ +x' = wt'. +\] +By means of the first and fourth equations of the Galilei +transformation we can express $x'$~and~$t'$ in terms of $x$~and~$t$, +and we then obtain +\[ +x = (v + w)t. +\] +\PageSep{39} +This equation expresses nothing else than the law of +motion of the point with reference to the system~$K$ +(of the man with reference to the embankment). We +denote this velocity by the symbol~$W$, and we then +obtain, as in \Sectionref{VI}, +\[ +W = v + w. +\Tag{(A)} +\] + +But we can carry out this consideration just as well +on the basis of the theory of relativity. In the equation +\[ +x' = wt' +\] +we must then express $x'$~and~$t'$ in terms of $x$~and~$t$, making +use of the first and fourth equations of the \emph{Lorentz +\index{Lorentz, H. A.!transformation}% +transformation}. Instead of the equation~\Eqref{(A)} we then +obtain the equation +\[ +W = \frac{v + w}{1 + \dfrac{vw}{c^{2}}}, +\Tag{(B)} +\] +which corresponds to the theorem of addition for +velocities in one direction according to the theory of +relativity. The question now arises as to which of these +two theorems is the better in accord with experience. On +this point we are enlightened by a most important experiment +which the brilliant physicist Fizeau performed more +\index{Fizeau}% +\index{Fizeau!experiment of}% +than half a century ago, and which has been repeated +since then by some of the best experimental physicists, +so that there can be no doubt about its result. The +experiment is concerned with the following question. +Light travels in a motionless liquid with a particular +velocity~$w$. How quickly does it travel in the direction +of the arrow in the tube~$T$ (see the accompanying diagram, +\Figref{3}) when the liquid above mentioned is flowing +through the tube with a velocity~$v$? +\PageSep{40} + +In accordance with the principle of relativity we shall +\index{Propagation of light!in liquid}% +certainly have to take for granted that the propagation +of light always takes place with the same velocity~$w$ +\emph{with respect to the liquid}, whether the latter is in motion +with reference to other bodies or not. The velocity +of light relative to the liquid and the velocity of the +latter relative to the tube are thus known, and we +require the velocity of light relative to the tube. + +It is clear that we have the problem of \Sectionref{VI} +again before us. The tube plays the part of the railway +embankment or of the co-ordinate system~$K$, the liquid +plays the part of the carriage or of the co-ordinate +system~$K'$, and finally, the light plays the part of the +%[Illustration: Fig. 3.] +\Figure[2in]{040} +man walking along the carriage, or of the moving point +in the present section. If we denote the velocity of the +light relative to the tube by~$W$, then this is given +by the equation \Eqref{(A)}~or~\Eqref{(B)}, according as the Galilei +transformation or the Lorentz transformation corresponds +to the facts. Experiment\footnote + {Fizeau found $W = w + v\left(1 - \dfrac{1}{n^{2}}\right)$, where $n = \dfrac{c}{w}$ is the index of + refraction of the liquid. On the other hand, owing to the smallness + of~$\dfrac{vw}{c^{2}}$ as compared with~$1$, we can replace~\Eqref{(B)} in the first + place by $W = (w + v) \left(1 - \dfrac{vw}{c^{2}}\right)$, or to the same order of approximation + by $w + v \left(1 - \dfrac{1}{n^{2}}\right)$, which agrees with Fizeau's result.} +decides in favour +of equation~\Eqref{(B)} derived from the theory of relativity, and +the agreement is, indeed, very exact. According to +\PageSep{41} +recent and most excellent measurements by Zeeman, the +\index{Zeeman}% +influence of the velocity of flow~$v$ on the propagation of +light is represented by formula~\Eqref{(B)} to within one per +cent. %[** TN: [sic] two words] + +Nevertheless we must now draw attention to the fact +that a theory of this phenomenon was given by H.~A. +Lorentz long before the statement of the theory of +\index{Lorentz, H. A.}% +relativity. This theory was of a purely electrodynamical +nature, and was obtained by the use of particular +hypotheses as to the electromagnetic structure of matter. +This circumstance, however, does not in the least +diminish the conclusiveness of the experiment as a +crucial test in favour of the theory of relativity, for the +electrodynamics of Maxwell-Lorentz, on which the +\index{Electrodynamics}% +\index{Maxwell}% +original theory was based, in no way opposes the theory +of relativity. Rather has the latter been developed +from electrodynamics as an astoundingly simple combination +and generalisation of the hypotheses, formerly +independent of each other, on which electrodynamics +was built. +\PageSep{42} + + +\Chapter{XIV}{The Heuristic Value of the Theory of +Relativity} +\index{Heuristic value of relativity}% + +\First{Our} train of thought in the foregoing pages can be +epitomised in the following manner. Experience +has led to the conviction that, on the one hand, +the principle of relativity holds true, and that on the +other hand the velocity of transmission of light \textit{in~vacuo} +has to be considered equal to a constant~$c$. By uniting +these two postulates we obtained the law of transformation +for the rectangular co-ordinates $x$,~$y$,~$z$ and the time~$t$ +of the events which constitute the processes of nature. +\index{Processes of Nature}% +In this connection we did not obtain the Galilei transformation, +\index{Galilei!transformation}% +but, differing from classical mechanics, +the \emph{Lorentz transformation}. +\index{Lorentz, H. A.!transformation}% + +The law of transmission of light, the acceptance of +which is justified by our actual knowledge, played an +important part in this process of thought. Once in +possession of the Lorentz transformation, however, +we can combine this with the principle of relativity, +and sum up the theory thus: + +Every general law of nature must be so constituted +that it is transformed into a law of exactly the same +form when, instead of the space-time variables $x$,~$y$, $z$,~$t$ +of the original co-ordinate system~$K$, we introduce new +space-time variables $x'$,~$y'$, $z'$,~$t'$ of a co-ordinate system~$K'$. +\PageSep{43} +In this connection the relation between the +ordinary and the accented magnitudes is given by the +Lorentz transformation. Or, in brief: General laws +of nature are co-variant with respect to Lorentz transformations. +\index{Covariant@{Co-variant}}% + +This is a definite mathematical condition that the +theory of relativity demands of a natural law, and in +virtue of this, the theory becomes a valuable heuristic aid +in the search for general laws of nature. If a general +law of nature were to be found which did not satisfy +this condition, then at least one of the two fundamental +assumptions of the theory would have been disproved. +Let us now examine what general results the latter +theory has hitherto evinced. +\PageSep{44} + + +\Chapter{XV}{General Results of the Theory} + +\First{It} is clear from our previous considerations that the +(special) theory of relativity has grown out of electrodynamics +\index{Electrodynamics}% +and optics. In these fields it has not +\index{Optics}% +appreciably altered the predictions of theory, but it +has considerably simplified the theoretical structure, +\ie\ the derivation of laws, and---what is incomparably +\index{Derivation of laws}% +more important---it has considerably reduced the +number of independent hypotheses forming the basis of +\index{Basis of theory}% +theory. The special theory of relativity has rendered +the Maxwell-Lorentz theory so plausible, that the latter +\index{Lorentz, H. A.}% +\index{Maxwell}% +would have been generally accepted by physicists +even if experiment had decided less unequivocally in its +favour. + +Classical mechanics required to be modified before it +\index{Classical mechanics}% +could come into line with the demands of the special +theory of relativity. For the main part, however, +this modification affects only the laws for rapid motions, +in which the velocities of matter~$v$ are not very small as +compared with the velocity of light. We have experience +of such rapid motions only in the case of electrons +\index{Electron}% +and ions; for other motions the variations from the laws +\index{Ions}% +of classical mechanics are too small to make themselves +evident in practice. We shall not consider the motion +\index{Motion!of heavenly bodies}% +of stars until we come to speak of the general theory of +relativity. In accordance with the theory of relativity +\PageSep{45} +the kinetic energy of a material point of mass~$m$ is no +\index{Kinetic energy}% +longer given by the well-known expression +\[ +m\frac{v^{2}}{2}\Change{.}{,} +\] +but by the expression +\[ +\frac{mc^{2}}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}. +\] +This expression approaches infinity as the velocity~$v$ +approaches the velocity of light~$c$. The velocity must +therefore always remain less than~$c$, however great may +be the energies used to produce the acceleration. If +we develop the expression for the kinetic energy in the +form of a series, we obtain +\[ +mc^{2} + m\frac{v^{2}}{2} + \frac{3}{8}m\frac{v^4}{c^{2}} + \dots. +\] + +When $\dfrac{v^{2}}{c^{2}}$ is small compared with unity, the third +of these terms is always small in comparison with the +second, which last is alone considered in classical +mechanics. The first term~$mc^{2}$ does not contain +the velocity, and requires no consideration if we are only +dealing with the question as to how the energy of a +point-mass depends on the velocity. We shall speak +\index{Point-mass, energy of}% +of its essential significance later. + +The most important result of a general character to +\index{Conservation of energy}% +\index{Conservation of energy!mass}% +which the special theory of relativity has led is concerned +with the conception of mass. Before the advent of +\index{Conception of mass}% +relativity, physics recognised two conservation laws of +fundamental importance, namely, the law of the conservation +of energy and the law of the conservation of +mass; these two fundamental laws appeared to be quite +\PageSep{46} +independent of each other. By means of the theory of +relativity they have been united into one law. We shall +now briefly consider how this unification came about, +and what meaning is to be attached to it. + +The principle of relativity requires that the law of the +conservation of energy should hold not only with reference +to a co-ordinate system~$K$, but also with respect +to every co-ordinate system~$K'$ which is in a state of +uniform motion of translation relative to~$K$, or, briefly, +relative to every ``Galileian'' system of co-ordinates. +\index{Galileian system of co-ordinates}% +In contrast to classical mechanics, the Lorentz transformation +is the deciding factor in the transition from +one such system to another. + +By means of comparatively simple considerations +we are led to draw the following conclusion from +these premises, in conjunction with the fundamental +equations of the electrodynamics of Maxwell: A body +\index{Maxwell!fundamental equations}% +\index{Absorption of energy}% +moving with the velocity~$v$, which absorbs\footnote + {$E_{0}$~is the energy taken up, as judged from a co-ordinate + system moving with the body.} +an amount +of energy~$E_{0}$ in the form of radiation without suffering +\index{Radiation}% +an alteration in velocity in the process, has, as a consequence, +its energy increased by an amount +\[ +\frac{E_{0}}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}. +\] + +In consideration of the expression given above for the +kinetic energy of the body, the required energy of the +body comes out to be +\[ +\frac{\left(m + \dfrac{E_{0}}{c^{2}}\right)c^{2}} + {\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}. +\] +\PageSep{47} + +Thus the body has the same energy as a body of mass +$\left(m + \dfrac{E_{0}}{c^{2}}\right)$ moving with the velocity~$v$. Hence we can +say: If a body takes up an amount of energy~$E_{0}$, then +its inertial mass increases by an amount~$\dfrac{E_{0}}{c^{2}}$; the +\index{Inertial mass}% +inertial mass of a body is not a constant, but varies +according to the change in the energy of the body. +The inertial mass of a system of bodies can even be +regarded as a measure of its energy. The law of the +conservation of the mass of a system becomes identical +with the law of the conservation of energy, and is only +\index{Conservation of energy!mass}% +valid provided that the system neither takes up nor sends +out energy. Writing the expression for the energy in +the form +\[ +\frac{mc^{2} + E_{0}}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}, +\] +we see that the term~$mc^{2}$, which has hitherto attracted +our attention, is nothing else than the energy possessed +by the body\footnote + {As judged from a co-ordinate system moving with the body.} +before it absorbed the energy~$E_{0}$. + +A direct comparison of this relation with experiment +is not possible at the present time, owing to the fact that +the changes in energy~$E_{0}$ to which we can subject a +system are not large enough to make themselves +perceptible as a change in the inertial mass of the +system. $\dfrac{E_{0}}{c^{2}}$~is too small in comparison with the mass~$m$, +which was present before the alteration of the energy. +It is owing to this circumstance that classical mechanics +was able to establish successfully the conservation of +mass as a law of independent validity. +\PageSep{48} + +Let me add a final remark of a fundamental nature. +The success of the Faraday-Maxwell interpretation of +\index{Faraday}% +\index{Maxwell|(}% +electromagnetic action at a distance resulted in physicists +\index{Action at a distance}% +becoming convinced that there are no such things as +instantaneous actions at a distance (not involving an +intermediary medium) of the type of Newton's law of +\index{Newton's!law of gravitation}% +gravitation. According to the theory of relativity, +action at a distance with the velocity of light always +takes the place of instantaneous action at a distance or +of action at a distance with an infinite velocity of transmission. +This is connected with the fact that the +velocity~$c$ plays a fundamental rôle in this theory. In +\Partref{II} we shall see in what way this result becomes +modified in the general theory of relativity. +\PageSep{49} + + +\Chapter{XVI}{Experience and the Special Theory of +Relativity} +\index{Experience}% + +\First{To} what extent is the special theory of relativity +supported by experience? This question is not +easily answered for the reason already mentioned +in connection with the fundamental experiment of Fizeau. +\index{Fizeau}% +The special theory of relativity has crystallised out +from the Maxwell-Lorentz theory of electromagnetic +\index{Lorentz, H. A.}% +phenomena. Thus all facts of experience which support +the electromagnetic theory also support the theory of +\index{Electromagnetic theory}% +relativity. As being of particular importance, I mention +here the fact that the theory of relativity enables us to +predict the effects produced on the light reaching us +from the fixed stars. These results are obtained in an +exceedingly simple manner, and the effects indicated, +which are due to the relative motion of the earth with +reference to those fixed stars, are found to be in accord +with experience. We refer to the yearly movement of +the apparent position of the fixed stars resulting from the +motion of the earth round the sun (aberration), and to the +\index{Aberration}% +influence of the radial components of the relative +motions of the fixed stars with respect to the earth on +the colour of the light reaching us from them. The +\PageSep{50} +latter effect manifests itself in a slight displacement +of the spectral lines of the light transmitted to us from +a fixed star, as compared with the position of the same +spectral lines when they are produced by a terrestrial +source of light (Doppler principle). The experimental +\index{Doppler principle}% +arguments in favour of the Maxwell-Lorentz theory, +\index{Lorentz, H. A.|(}% +which are at the same time arguments in favour of the +theory of relativity, are too numerous to be set forth +here. In reality they limit the theoretical possibilities +to such an extent, that no other theory than that of +Maxwell and Lorentz has been able to hold its own when +tested by experience. + +But there are two classes of experimental facts +hitherto obtained which can be represented in the +Maxwell-Lorentz theory only by the introduction of an +\index{Maxwell|)}% +auxiliary hypothesis, which in itself---\ie\ without +making use of the theory of relativity---appears extraneous. + +It is known that cathode rays and the so-called +\index{beta-rays@{$\beta$-rays}}% +\index{Cathode rays}% +$\beta$-rays emitted by radioactive substances consist of +\index{Radioactive substances}% +negatively electrified particles (electrons) of very small +inertia and large velocity. By examining the deflection +of these rays under the influence of electric and magnetic +fields, we can study the law of motion of these particles +very exactly. + +In the theoretical treatment of these electrons, we are +faced with the difficulty that electrodynamic theory of +itself is unable to give an account of their nature. For +since electrical masses of one sign repel each other, the +negative electrical masses constituting the electron would +\index{Electron}% +necessarily be scattered under the influence of their +mutual repulsions, unless there are forces of another +kind operating between them, the nature of which has +\PageSep{51} +hitherto remained obscure to us.\footnote + {The general theory of relativity renders it likely that the + electrical masses of an electron are held together by gravitational +\index{Electron!electrical masses of}% + forces.} +If we now assume +that the relative distances between the electrical masses +constituting the electron remain unchanged during the +motion of the electron (rigid connection in the sense of +classical mechanics), we arrive at a law of motion of the +electron which does not agree with experience. Guided +by purely formal points of view, H.~A.~Lorentz was the +first to introduce the hypothesis that the particles +constituting the electron experience a contraction +in the direction of motion in consequence of that motion, +the amount of this contraction being proportional to +the expression~$\sqrt{1 - \dfrac{v^{2}}{c^{2}}}$. This hypothesis, which is +not justifiable by any electrodynamical facts, supplies us +then with that particular law of motion which has +been confirmed with great precision in recent years. + +The theory of relativity leads to the same law of +motion, without requiring any special hypothesis whatsoever +as to the structure and the behaviour of the +electron. We arrived at a similar conclusion in \Sectionref{XIII} +in connection with the experiment of Fizeau, the +\index{Fizeau}% +result of which is foretold by the theory of relativity +without the necessity of drawing on hypotheses as to +the physical nature of the liquid. + +The second class of facts to which we have alluded +has reference to the question whether or not the motion +of the earth in space can be made perceptible in terrestrial +experiments. We have already remarked in \Sectionref{V} +that all attempts of this nature led to a negative result. +Before the theory of relativity was put forward, it was +\PageSep{52} +difficult to become reconciled to this negative result, +for reasons now to be discussed. The inherited +prejudices about time and space did not allow any +\index{Time!conception of}% +\index{Space}% +doubt to arise as to the prime importance of the +Galilei transformation for changing over from one +\index{Galilei!transformation}% +body of reference to another. Now assuming that the +Maxwell-Lorentz equations hold for a reference-body~$K$, +\index{Maxwell}% +we then find that they do not hold for a reference-body~$K'$ +moving uniformly with respect to~$K$, if we +assume that the relations of the Galileian transformation +exist between the co-ordinates of $K$~and~$K'$. It +thus appears that of all Galileian co-ordinate systems +one~($K$) corresponding to a particular state of motion +is physically unique. This result was interpreted +physically by regarding $K$ as at rest with respect to a +hypothetical æther of space. On the other hand, +all co-ordinate systems~$K'$ moving relatively to~$K$ were +to be regarded as in motion with respect to the æther. +\index{Aether}% +\index{Aether!-drift}% +To this motion of~$K'$ against the æther (``æther-drift'' +relative to~$K'$) were assigned the more complicated +laws which were supposed to hold relative to~$K'$. +Strictly speaking, such an æther-drift ought also to be +assumed relative to the earth, and for a long time the +efforts of physicists were devoted to attempts to detect +the existence of an æther-drift at the earth's surface. + +In one of the most notable of these attempts Michelson +\index{Michelson|(}% +devised a method which appears as though it must be +decisive. Imagine two mirrors so arranged on a rigid +body that the reflecting surfaces face each other. A +ray of light requires a perfectly definite time~$T$ to pass +from one mirror to the other and back again, if the whole +system be at rest with respect to the æther. It is found +by calculation, however, that a slightly different time~$T'$ +\PageSep{53} +is required for this process, if the body, together with +the mirrors, be moving relatively to the æther. And +\index{Aether!-drift}% +yet another point: it is shown by calculation that for +a given velocity~$v$ with reference to the æther, this +time~$T'$ is different when the body is moving perpendicularly +to the planes of the mirrors from that resulting +when the motion is parallel to these planes. Although +the estimated difference between these two times is +exceedingly small, Michelson and Morley performed an +\index{Morley}% +experiment involving interference in which this difference +should have been clearly detectable. But the experiment +gave a negative result---a fact very perplexing +to physicists. Lorentz and FitzGerald rescued the +\index{FitzGerald}% +\index{Lorentz, H. A.|)}% +theory from this difficulty by assuming that the motion +of the body relative to the æther produces a contraction +of the body in the direction of motion, the amount of contraction +being just sufficient to compensate for the difference +in time mentioned above. Comparison with the +discussion in \Sectionref{XII} shows that also from the standpoint +of the theory of relativity this solution of the +difficulty was the right one. But on the basis of the +theory of relativity the method of interpretation is +incomparably more satisfactory. According to this +theory there is no such thing as a ``specially favoured'' +(unique) co-ordinate system to occasion the introduction +of the æther-idea, and hence there can be no æther-drift, +nor any experiment with which to demonstrate it. +Here the contraction of moving bodies follows from +the two fundamental principles of the theory without +the introduction of particular hypotheses; and as the +prime factor involved in this contraction we find, not +the motion in itself, to which we cannot attach any +meaning, but the motion with respect to the body of +\PageSep{54} +reference chosen in the particular case in point. Thus +for a co-ordinate system moving with the earth the +mirror system of Michelson and Morley is not shortened, +\index{Michelson|)}% +\index{Morley}% +but it \emph{is} shortened for a co-ordinate system which is at +rest relatively to the sun. +\PageSep{55} + + +\Chapter{XVII}{Minkowski's Four-dimensional Space} +\index{Minkowski|(}% +\index{Space}% + +\First{The} non-mathematician is seized by a mysterious +shuddering when he hears of ``four-dimensional'' +things, by a feeling not unlike that awakened by +thoughts of the occult. And yet there is no more +common-place statement than that the world in which +\index{World}% +we live is a four-dimensional space-time continuum. +\index{Continuum}% + +Space is a three-dimensional continuum. By this +\index{Space co-ordinates}% +\index{Three-dimensional}% +\index{Time!coordinate@{co-ordinate}}% +we mean that it is possible to describe the position of a +point (at rest) by means of three numbers (co-ordinates) +$x$,~$y$,~$z$, and that there is an indefinite number of points +in the neighbourhood of this one, the position of which +can be described by co-ordinates such as $x_{1}$,~$y_{1}$,~$z_{1}$, which +may be as near as we choose to the respective values of +the co-ordinates $x$,~$y$,~$z$ of the first point. In virtue of the +latter property we speak of a ``continuum,'' and owing +to the fact that there are three co-ordinates we speak of +it as being ``three-dimensional.'' + +Similarly, the world of physical phenomena which was +briefly called ``world'' by Minkowski is naturally +four-dimensional in the space-time sense. For it is +composed of individual events, each of which is described +by four numbers, namely, three space +co-ordinates $x$,~$y$,~$z$ and a time co-ordinate, the time-value~$t$. +The ``world'' is in this sense also a continuum; +for to every event there are as many ``neighbouring'' +\PageSep{56} +events (realised or at least thinkable) as we care to +choose, the co-ordinates $x_{1}$,~$y_{1}$, $z_{1}$,~$t_{1}$ of which differ +by an indefinitely small amount from those of the +event $x$,~$y$, $z$,~$t$ originally considered. That we have not +been accustomed to regard the world in this sense as a +\index{World}% +four-dimensional continuum is due to the fact that in +physics, before the advent of the theory of relativity, +time played a different and more independent rôle, as +compared with the space co-ordinates. It is for this +reason that we have been in the habit of treating time +as an independent continuum. As a matter of fact, +according to classical mechanics, time is absolute, +\ie\ it is independent of the position and the condition +of motion of the system of co-ordinates. We see this +expressed in the last equation of the Galileian transformation +($t' = t$). + +The four-dimensional mode of consideration of the +``world'' is natural on the theory of relativity, since +according to this theory time is robbed of its independence. +This is shown by the fourth equation of the +Lorentz transformation: +\[ +t' = \frac{t - \dfrac{v}{c^{2}}x}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}. +\] +Moreover, according to this equation the time difference~$\Delta t'$ +\index{Space!interval@{-interval}}% +\index{Time-interval}% +of two events with respect to~$K'$ does not in general +vanish, even when the time difference~$\Delta t$ of the same +events with reference to~$K$ vanishes. Pure ``space-distance'' +of two events with respect to~$K$ results in +``time-distance'' of the same events with respect to~$K'$. +But the discovery of Minkowski, which was of importance +\PageSep{57} +for the formal development of the theory of relativity, +does not lie here. It is to be found rather in +the fact of his recognition that the four-dimensional +space-time continuum of the theory of relativity, in its +\index{Continuum!three-dimensional}% +most essential formal properties, shows a pronounced +relationship to the three-dimensional continuum of +Euclidean geometrical space.\footnote + {Cf.\ the somewhat more detailed discussion in \Appendixref{II}.} +In order to give due +prominence to this relationship, however, we must +replace the usual time co-ordinate~$t$ by an imaginary +magnitude~$\sqrt{-1}·ct$ proportional to it. Under these +conditions, the natural laws satisfying the demands of +the (special) theory of relativity assume mathematical +forms, in which the time co-ordinate plays exactly the +same rôle as the three space co-ordinates. Formally, +these four co-ordinates correspond exactly to the three +space co-ordinates in Euclidean geometry. It must be +\index{Euclidean geometry}% +\index{Euclidean space}% +clear even to the non-mathematician that, as a consequence +of this purely formal addition to our knowledge, +the theory perforce gained clearness in no mean +measure. + +These inadequate remarks can give the reader only a +vague notion of the important idea contributed by Minkowski. +Without it the general theory of relativity, of +which the fundamental ideas are developed in the following +pages, would perhaps have got no farther than its +long clothes. Minkowski's work is doubtless difficult of +\index{Minkowski|)}% +access to anyone inexperienced in mathematics, but +since it is not necessary to have a very exact grasp of +this work in order to understand the fundamental ideas +of either the special or the general theory of relativity, +I shall at present leave it here, and shall revert to it +only towards the end of \Partref{II}. +\index{Special theory of relativity|)}% +\PageSep{58} +% [Blank page] +\PageSep{59} + + +\Part{II}{The General Theory of Relativity}{General Theory of Relativity} +\index{General theory of relativity|(}% + +\Chapter{XVIII}{Special and General Principle of +Relativity} +\index{Laws of Galilei-Newton!of Nature}% + +\First{The} basal principle, which was the pivot of all +our previous considerations, was the \emph{special} +principle of relativity, \ie\ the principle of the +physical relativity of all \emph{uniform} motion. Let us once +\index{Uniform translation}% +more analyse its meaning carefully. + +It was at all times clear that, from the point of view +of the idea it conveys to us, every motion must only +be considered as a relative motion. Returning to the +illustration we have frequently used of the embankment +and the railway carriage, we can express the fact of the +motion here taking place in the following two forms, +both of which are equally justifiable: +\begin{itemize} +\item[\itema] The carriage is in motion relative to the embankment. + +\item[\itemb] The embankment is in motion relative to the + carriage. +\end{itemize} + +In \itema~the embankment, in \itemb~the carriage, serves as +the body of reference in our statement of the motion +taking place. If it is simply a question of detecting +\PageSep{60} +or of describing the motion involved, it is in principle +\index{Motion}% +immaterial to what reference-body we refer the motion. +\index{Reference-body}% +As already mentioned, this is self-evident, but it must +not be confused with the much more comprehensive +statement called ``the principle of relativity,'' which +\index{Principle of relativity}% +we have taken as the basis of our investigations. + +The principle we have made use of not only maintains +that we may equally well choose the carriage or the +embankment as our reference-body for the description +of any event (for this, too, is self-evident). Our principle +rather asserts what follows: If we formulate the general +laws of nature as they are obtained from experience, +\index{Experience}% +by making use of +\begin{itemize} +\item[\itema] the embankment as reference-body, +\item[\itemb] the railway carriage as reference-body, +\end{itemize} +then these general laws of nature (\eg\ the laws of +mechanics or the law of the propagation of light \textit{in~vacuo}) +have exactly the same form in both cases. This can +also be expressed as follows: For the \emph{physical} description +of natural processes, neither of the reference-bodies +$K$,~$K'$ is unique (lit.\ ``specially marked out'') as +compared with the other. Unlike the first, this latter +statement need not of necessity hold \textit{a~priori}; it is +not contained in the conceptions of ``motion'' and +``reference-body'' and derivable from them; only +\emph{experience} can decide as to its correctness or incorrectness. + +Up to the present, however, we have by no means +maintained the equivalence of \emph{all} bodies of reference~$K$ +in connection with the formulation of natural laws. +Our course was more on the following lines. In the +first place, we started out from the assumption that +there exists a reference-body~$K$, whose condition of +\PageSep{61} +\index{Law of inertia}% +motion is such that the Galileian law holds with respect +to it: A particle left to itself and sufficiently far removed +from all other particles moves uniformly in a straight +line. With reference to~$K$ (Galileian reference-body) the +laws of nature were to be as simple as possible. But +in addition to~$K$, all bodies of reference~$K'$ should be +given preference in this sense, and they should be exactly +equivalent to~$K$ for the formulation of natural laws, +provided that they are in a state of \emph{uniform rectilinear +and non-rotary motion} with respect to~$K$; all these +bodies of reference are to be regarded as Galileian +reference-bodies. The validity of the principle of +relativity was assumed only for these reference-bodies, +but not for others (\eg\ those possessing motion of a +different kind). In this sense we speak of the \emph{special} +principle of relativity, or special theory of relativity. + +In contrast to this we wish to understand by the +``general principle of relativity'' the following statement: +All bodies of reference $K$,~$K'$,~etc., are equivalent +for the description of natural phenomena (formulation of +the general laws of nature), whatever may be their +state of motion. But before proceeding farther, it +ought to be pointed out that this formulation must be +replaced later by a more abstract one, for reasons which +will become evident at a later stage. + +Since the introduction of the special principle of +relativity has been justified, every intellect which +strives after generalisation must feel the temptation +to venture the step towards the general principle of +relativity. But a simple and apparently quite reliable +consideration seems to suggest that, for the present +at any rate, there is little hope of success in such an +attempt. Let us imagine ourselves transferred to our +\PageSep{62} +\index{Law of inertia}% +old friend the railway carriage, which is travelling at a +uniform rate. As long as it is moving uniformly, the +occupant of the carriage is not sensible of its motion, +and it is for this reason that he can without reluctance +interpret the facts of the case as indicating that the +carriage is at rest but the embankment in motion. +Moreover, according to the special principle of relativity, +this interpretation is quite justified also from a physical +point of view. + +If the motion of the carriage is now changed into a +non-uniform motion, as for instance by a powerful +\index{Non-uniform motion}% +application of the brakes, then the occupant of the +carriage experiences a correspondingly powerful jerk +forwards. The retarded motion is manifested in the +mechanical behaviour of bodies relative to the person +in the railway carriage. The mechanical behaviour is +different from that of the case previously considered, +and for this reason it would appear to be impossible +that the same mechanical laws hold relatively to the non-uniformly +moving carriage, as hold with reference to the +carriage when at rest or in uniform motion. At all +events it is clear that the Galileian law does not hold +with respect to the non-uniformly moving carriage. +Because of this, we feel compelled at the present juncture +to grant a kind of absolute physical reality to non-uniform +motion, in opposition to the general principle +of relativity. But in what follows we shall soon see +that this conclusion cannot be maintained. +\PageSep{63} + + +\Chapter{XIX}{The Gravitational Field} + +``\First{If} we pick up a stone and then let it go, why does it +fall to the ground?'' The usual answer to this +question is: ``Because it is attracted by the earth.'' +Modern physics formulates the answer rather differently +for the following reason. As a result of the more careful +study of electromagnetic phenomena, we have come +to regard action at a distance as a process impossible +without the intervention of some intermediary medium. +If, for instance, a magnet attracts a piece of iron, we +cannot be content to regard this as meaning that the +magnet acts directly on the iron through the intermediate +empty space, but we are constrained to imagine---after +the manner of Faraday---that the magnet +\index{Faraday}% +always calls into being something physically real in +the space around it, that something being what we call a +``magnetic field.'' In its turn this magnetic field +\index{Magnetic field}% +operates on the piece of iron, so that the latter strives +to move towards the magnet. We shall not discuss +here the justification for this incidental conception, +which is indeed a somewhat arbitrary one. We shall +only mention that with its aid electromagnetic phenomena +can be theoretically represented much more +satisfactorily than without it, and this applies particularly +\index{Electromagnetic theory!waves}% +to the transmission of electromagnetic waves. +\PageSep{64} +The effects of gravitation also are regarded in an analogous +\index{Gravitation}% +manner. + +The action of the earth on the stone takes place indirectly. +The earth produces in its surroundings a +gravitational field, which acts on the stone and produces +\index{Gravitational field}% +its motion of fall. As we know from experience, the +intensity of the action on a body diminishes according +to a quite definite law, as we proceed farther and farther +away from the earth. From our point of view this +means: The law governing the properties of the gravitational +field in space must be a perfectly definite one, in +order correctly to represent the diminution of gravitational +action with the distance from operative bodies. +It is something like this: The body (\eg\ the earth) produces +a field in its immediate neighbourhood directly; +the intensity and direction of the field at points farther +removed from the body are thence determined by +the law which governs the properties in space of the +gravitational fields themselves. + +In contrast to electric and magnetic fields, the gravitational +field exhibits a most remarkable property, which +is of fundamental importance for what follows. Bodies +which are moving under the sole influence of a gravitational +field receive an acceleration, \emph{which does not in the +\index{Acceleration}% +least depend either on the material or on the physical +state of the body}. For instance, a piece of lead and +a piece of wood fall in exactly the same manner in a +gravitational field (\textit{in~vacuo}), when they start off from +rest or with the same initial velocity. This law, which +holds most accurately, can be expressed in a different +form in the light of the following consideration. + +According to Newton's law of motion, we have +\index{Newton's!law of motion}% +\[ +(\text{Force}) = (\text{inertial mass}) × (\text{acceleration}), +\] +\PageSep{65} +where the ``inertial mass'' is a characteristic constant +\index{Inertial mass}% +of the accelerated body. If now gravitation is the +cause of the acceleration, we then have +%[** TN: Re-breaking next two displayed equations] +\begin{multline*} +(\text{Force}) + = (\text{gravitational mass}) \\ + × (\text{intensity of the gravitational field}), +\index{Gravitational mass}% +\end{multline*} +where the ``gravitational mass'' is likewise a characteristic +constant for the body. From these two relations +follows: +\begin{multline*} +(\text{acceleration}) + = \frac{(\text{gravitational mass})}{(\text{inertial mass})} \\ + × (\text{intensity of the gravitational field}). +\end{multline*} + +If now, as we find from experience, the acceleration is +to be independent of the nature and the condition of the +body and always the same for a given gravitational +field, then the ratio of the gravitational to the inertial +mass must likewise be the same for all bodies. By a +suitable choice of units we can thus make this ratio +equal to unity. We then have the following law: +The \emph{gravitational} mass of a body is equal to its \emph{inertial} +mass. + +It is true that this important law had hitherto been +recorded in mechanics, but it had not been \emph{interpreted}. +A satisfactory interpretation can be obtained only if we +recognise the following fact: \emph{The same} quality of a +body manifests itself according to circumstances as +``inertia'' or as ``weight'' (lit.\ ``heaviness''). In the +\index{Inertia}% +\index{Weight (heaviness)}% +following section we shall show to what extent this is +actually the case, and how this question is connected +with the general postulate of relativity. +\PageSep{66} + + +\Chapter{XX}{The Equality of Inertial and Gravitational +Mass as an Argument for the +General Postulate of Relativity} + +\First{We} imagine a large portion of empty space, so far +removed from stars and other appreciable +masses, that we have before us approximately +the conditions required by the fundamental law of Galilei. +It is then possible to choose a Galileian reference-body for +this part of space (world), relative to which points at +rest remain at rest and points in motion continue permanently +in uniform rectilinear motion. As reference-body +let us imagine a spacious chest resembling a room +\index{Chest}% +with an observer inside who is equipped with apparatus. +Gravitation naturally does not exist for this observer. +He must fasten himself with strings to the floor, +otherwise the slightest impact against the floor will +cause him to rise slowly towards the ceiling of the +room. + +To the middle of the lid of the chest is fixed externally +a hook with rope attached, and now a ``being'' (what +\index{Being@{``Being''}}% +kind of a being is immaterial to us) begins pulling at +this with a constant force. The chest together with the +observer then begin to move ``upwards'' with a +uniformly accelerated motion. In course of time their +velocity will reach unheard-of values---provided that +\PageSep{67} +we are viewing all this from another reference-body +which is not being pulled with a rope. + +But how does the man in the chest regard the process? +The acceleration of the chest will be transmitted to him +\index{Acceleration}% +by the reaction of the floor of the chest. He must +therefore take up this pressure by means of his legs if +he does not wish to be laid out full length on the floor. +He is then standing in the chest in exactly the same way +as anyone stands in a room of a house on our earth. +If he release a body which he previously had in his +hand, the acceleration of the chest will no longer be +transmitted to this body, and for this reason the body +will approach the floor of the chest with an accelerated +relative motion. The observer will further convince +himself \emph{that the acceleration of the body towards the floor +of the chest is always of the same magnitude, whatever +kind of body he may happen to use for the experiment}. + +Relying on his knowledge of the gravitational field +\index{Gravitational field}% +(as it was discussed in the preceding section), the man +in the chest will thus come to the conclusion that he +and the chest are in a gravitational field which is constant +with regard to time. Of course he will be puzzled for +a moment as to why the chest does not fall, in this +gravitational field. Just then, however, he discovers +the hook in the middle of the lid of the chest and the +rope which is attached to it, and he consequently comes +to the conclusion that the chest is suspended at rest in +the gravitational field. + +Ought we to smile at the man and say that he errs +in his conclusion? I do not believe we ought to if we +wish to remain consistent; we must rather admit that +his mode of grasping the situation violates neither reason +nor known mechanical laws. Even though it is being +\PageSep{68} +accelerated with respect to the ``Galileian space'' +first considered, we can nevertheless regard the chest +as being at rest. We have thus good grounds for +extending the principle of relativity to include bodies +of reference which are accelerated with respect to each +other, and as a result we have gained a powerful argument +for a generalised postulate of relativity. + +We must note carefully that the possibility of this +mode of interpretation rests on the fundamental +property of the gravitational field of giving all bodies +\index{Gravitational mass}% +the same acceleration, or, what comes to the same thing, +on the law of the equality of inertial and gravitational +mass. If this natural law did not exist, the man in +the accelerated chest would not be able to interpret +the behaviour of the bodies around him on the supposition +of a gravitational field, and he would not be justified +on the grounds of experience in supposing his reference-body +to be ``at rest.'' + +Suppose that the man in the chest fixes a rope to the +inner side of the lid, and that he attaches a body to the +free end of the rope. The result of this will be to stretch +the rope so that it will hang ``vertically'' downwards. +If we ask for an opinion of the cause of tension in the +rope, the man in the chest will say: ``The suspended +body experiences a downward force in the gravitational +field, and this is neutralised by the tension of the rope; +what determines the magnitude of the tension of the +rope is the \emph{gravitational mass} of the suspended body.'' +On the other hand, an observer who is poised freely in +space will interpret the condition of things thus: ``The +rope must perforce take part in the accelerated motion +of the chest, and it transmits this motion to the body +attached to it. The tension of the rope is just large +\PageSep{69} +enough to effect the acceleration of the body. That +which determines the magnitude of the tension of the +rope is the \emph{inertial mass} of the body.'' Guided by +\index{Inertial mass}% +this example, we see that our extension of the principle +of relativity implies the \emph{necessity} of the law of the +equality of inertial and gravitational mass. Thus we +have obtained a physical interpretation of this law. + +From our consideration of the accelerated chest we +see that a general theory of relativity must yield important +results on the laws of gravitation. In point of +\index{Gravitation}% +fact, the systematic pursuit of the general idea of relativity +has supplied the laws satisfied by the gravitational +field. Before proceeding farther, however, I +must warn the reader against a misconception suggested +by these considerations. A gravitational field exists +for the man in the chest, despite the fact that there was +no such field for the co-ordinate system first chosen. +Now we might easily suppose that the existence of a +gravitational field is always only an \emph{apparent} one. We +might also think that, regardless of the kind of gravitational +field which may be present, we could always +choose another reference-body such that \emph{no} gravitational +field exists with reference to it. This is by no means +true for all gravitational fields, but only for those of +quite special form. It is, for instance, impossible to +choose a body of reference such that, as judged from it, +the gravitational field of the earth (in its entirety) +vanishes. + +We can now appreciate why that argument is not +convincing, which we brought forward against the +general principle of relativity at the end of \Sectionref{XVIII}. +It is certainly true that the observer in the railway +carriage experiences a jerk forwards as a result of the +\PageSep{70} +application of the brake, and that he recognises in this the +non-uniformity of motion (retardation) of the carriage. +But he is compelled by nobody to refer this jerk to a +``real'' acceleration (retardation) of the carriage. He +\index{Acceleration}% +might also interpret his experience thus: ``My body of +reference (the carriage) remains permanently at rest. +With reference to it, however, there exists (during the +period of application of the brakes) a gravitational +field which is directed forwards and which is variable +with respect to time. Under the influence of this field, +the embankment together with the earth moves non-uniformly +in such a manner that their original velocity +in the backwards direction is continuously reduced.'' +\PageSep{71} + + +\Chapter{XXI}{In what Respects are the Foundations +of Classical Mechanics and of the +Special Theory of Relativity unsatisfactory?} +\index{Classical mechanics}% +\index{Laws of Galilei-Newton!of Nature}% + +\First{We} have already stated several times that +classical mechanics starts out from the following +law: Material particles sufficiently far +removed from other material particles continue to +move uniformly in a straight line or continue in a +state of rest. We have also repeatedly emphasised +that this fundamental law can only be valid for +bodies of reference~$K$ which possess certain unique +states of motion, and which are in uniform translational +motion relative to each other. Relative to other reference-bodies~$K$ +the law is not valid. Both in classical +mechanics and in the special theory of relativity we +therefore differentiate between reference-bodies~$K$ +relative to which the recognised ``laws of nature'' can +be said to hold, and reference-bodies~$K$ relative to which +these laws do not hold. + +But no person whose mode of thought is logical can +rest satisfied with this condition of things. He asks: +``How does it come that certain reference-bodies (or +their states of motion) are given priority over other +reference-bodies (or their states of motion)? \emph{What is +\PageSep{72} +the reason for this preference?}\Change{}{''} In order to show clearly +what I mean by this question, I shall make use of a +comparison. + +I am standing in front of a gas range. Standing +alongside of each other on the range are two pans so +much alike that one may be mistaken for the other. +Both are half full of water. I notice that steam is being +emitted continuously from the one pan, but not from the +other. I am surprised at this, even if I have never seen +either a gas range or a pan before. But if I now notice +a luminous something of bluish colour under the first +pan but not under the other, I cease to be astonished, +even if I have never before seen a gas flame. For I +can only say that this bluish something will cause the +emission of the steam, or at least \emph{possibly} it may do so. +If, however, I notice the bluish something in neither +case, and if I observe that the one continuously emits +steam whilst the other does not, then I shall remain +astonished and dissatisfied until I have discovered +some circumstance to which I can attribute the different +behaviour of the two pans. + +Analogously, I seek in vain for a real something in +classical mechanics (or in the special theory of relativity) +to which I can attribute the different behaviour +of bodies considered with respect to the reference-systems +$K$~and~$K'$.\footnote + {The objection is of importance more especially when the state + of motion of the reference-body is of such a nature that it does + not require any external agency for its maintenance, \eg\ in + the case when the reference-body is rotating uniformly.} +Newton saw this objection and +\index{Newton}% +attempted to invalidate it, but without success. But +E.~Mach recognised it most clearly of all, and because +\index{Mach, E.}% +of this objection he claimed that mechanics must be +\PageSep{73} +placed on a new basis. It can only be got rid of by +means of a physics which is conformable to the general +principle of relativity, since the equations of such a +theory hold for every body of reference, whatever +may be its state of motion. +\PageSep{74} + + +\Chapter{XXII}{A Few Inferences from the General +Principle of Relativity} + +\First{The} considerations of \Sectionref{XX} show that the +general principle of relativity puts us in a position +to derive properties of the gravitational field in a +\index{Gravitational field}% +purely theoretical manner. Let us suppose, for instance, +that we know the space-time ``course'' for any natural +process whatsoever, as regards the manner in which it +takes place in the Galileian domain relative to a +Galileian body of reference~$K$. By means of purely +theoretical operations (\ie\ simply by calculation) we are +then able to find how this known natural process +appears, as seen from a reference-body~$K'$ which is +accelerated relatively to~$K$. But since a gravitational +field exists with respect to this new body of reference~$K'$, +our consideration also teaches us how the gravitational +field influences the process studied. + +For example, we learn that a body which is in a state +of uniform rectilinear motion with respect to~$K$ (in +accordance with the law of Galilei) is executing an +accelerated and in general curvilinear motion with +\index{Curvilinear motion}% +respect to the accelerated reference-body~$K'$ (chest). +This acceleration or curvature corresponds to the influence +on the moving body of the gravitational field +prevailing relatively to~$K'$. It is known that a gravitational +field influences the movement of bodies in this +\PageSep{75} +way, so that our consideration supplies us with nothing +essentially new. + +However, we obtain a new result of fundamental +\index{Propagation of light!in gravitational fields}% +importance when we carry out the analogous consideration +for a ray of light. With respect to the Galileian +reference-body~$K$, such a ray of light is transmitted +rectilinearly with the velocity~$c$. It can easily be shown +that the path of the same ray of light is no longer a +straight line when we consider it with reference to the +accelerated chest (reference-body~$K'$). From this we +conclude, \emph{that, in general, rays of light are propagated +curvilinearly in gravitational fields}. In two respects +this result is of great importance. + +In the first place, it can be compared with the reality. +Although a detailed examination of the question shows +that the curvature of light rays required by the general +theory of relativity is only exceedingly small for the +gravitational fields at our disposal in practice, its estimated +magnitude for light rays passing the sun at +grazing incidence is nevertheless $1.7$~seconds of arc. +This ought to manifest itself in the following way. +As seen from the earth, certain fixed stars appear to be +in the neighbourhood of the sun, and are thus capable +of observation during a total eclipse of the sun. At such +times, these stars ought to appear to be displaced +outwards from the sun by an amount indicated above, +as compared with their apparent position in the sky +when the sun is situated at another part of the heavens. +The examination of the correctness or otherwise of this +deduction is a problem of the greatest importance, the +early solution of which is to be expected of astronomers.\footnote + {By means of the star photographs of two expeditions equipped + by a Joint Committee of the Royal and Royal Astronomical + Societies, the existence of the deflection of light demanded by + theory was confirmed during the solar eclipse of 29th~May, 1919. +\index{Solar eclipse}% + (Cf.\ \Appendixref{III}.)} +\PageSep{76} + +In the second place our result shows that, according +to the general theory of relativity, the law of the constancy +of the velocity of light \textit{in~vacuo}, which constitutes +\index{Velocity of light}% +one of the two fundamental assumptions in the +special theory of relativity and to which we have +already frequently referred, cannot claim any unlimited +validity. A curvature of rays of light can only take +place when the velocity of propagation of light varies +with position. Now we might think that as a consequence +of this, the special theory of relativity and with +it the whole theory of relativity would be laid in the +dust. But in reality this is not the case. We can only +conclude that the special theory of relativity cannot +claim an unlimited domain of validity; its results +hold only so long as we are able to disregard the influences +of gravitational fields on the phenomena +(\eg\ of light). + +Since it has often been contended by opponents of +the theory of relativity that the special theory of +relativity is overthrown by the general theory of relativity, +it is perhaps advisable to make the facts of the +case clearer by means of an appropriate comparison. +Before the development of electrodynamics the laws +\index{Electrodynamics}% +of electrostatics were looked upon as the laws of +\index{Electrostatics}% +electricity. At the present time we know that +\index{Electricity}% +electric fields can be derived correctly from electrostatic +considerations only for the case, which is never +strictly realised, in which the electrical masses are quite +at rest relatively to each other, and to the co-ordinate +system. Should we be justified in saying that for this +\PageSep{77} +reason electrostatics is overthrown by the field-equations +of Maxwell in electrodynamics? Not in the least. +\index{Maxwell!fundamental equations}% +Electrostatics is contained in electrodynamics as a +limiting case; the laws of the latter lead directly to +those of the former for the case in which the fields are +invariable with regard to time. No fairer destiny +could be allotted to any physical theory, than that it +should of itself point out the way to the introduction +of a more comprehensive theory, in which it lives on +as a limiting case. + +In the example of the transmission of light just dealt +with, we have seen that the general theory of relativity +enables us to derive theoretically the influence of a +gravitational field on the course of natural processes, +\index{Gravitational field}% +the laws of which are already known when a gravitational +field is absent. But the most attractive problem, +to the solution of which the general theory of relativity +supplies the key, concerns the investigation of the laws +satisfied by the gravitational field itself. Let us consider +this for a moment. + +We are acquainted with space-time domains which +behave (approximately) in a ``Galileian'' fashion under +suitable choice of reference-body, \ie\ domains in which +gravitational fields are absent. If we now refer such +a domain to a reference-body~$K'$ possessing any kind +of motion, then relative to~$K'$ there exists a gravitational +field which is variable with respect to space and +time.\footnote + {This follows from a generalisation of the discussion in \Sectionref{XX}.} +The character of this field will of course depend +on the motion chosen for~$K'$. According to the general +theory of relativity, the general law oi the gravitational +field must be satisfied for all gravitational fields obtainable +\PageSep{78} +in this way. Even though by no means all gravitational +fields can be produced in this way, yet we may +entertain the hope that the general law of gravitation +\index{Gravitation}% +will be derivable from such gravitational fields of a +special kind. This hope has been realised in the most +beautiful manner. But between the clear vision of +this goal and its actual realisation it was necessary to +surmount a serious difficulty, and as this lies deep at +the root of things, I dare not withhold it from the reader. +We require to extend our ideas of the space-time continuum +\index{Continuum!space-time}% +still farther. +\PageSep{79} + + +\Chapter{XXIII}{Behaviour of Clocks and Measuring-Rods +on a Rotating Body of Reference} + +\First{Hitherto} I have purposely refrained from +speaking about the physical interpretation of +space- and time-data in the case of the general +theory of relativity. As a consequence, I am guilty of a +certain slovenliness of treatment, which, as we know +from the special theory of relativity, is far from being +unimportant and pardonable. It is now high time that +we remedy this defect; but I would mention at the +outset, that this matter lays no small claims on the +patience and on the power of abstraction of the reader. + +We start off again from quite special cases, which we +\index{Galileian system of co-ordinates}% +have frequently used before. Let us consider a space-time +domain in which no gravitational field exists +relative to a reference-body~$K$ whose state of motion +\index{Reference-body!rotating}% +has been suitably chosen. $K$~is then a Galileian reference-body +as regards the domain considered, and the +results of the special theory of relativity hold relative +to~$K$. Let us suppose the same domain referred to a +second body of reference~$K'$, which is rotating uniformly +with respect to~$K$. In order to fix our ideas, we shall +imagine~$K'$ to be in the form of a plane circular disc, +which rotates uniformly in its own plane about its +centre. An observer who is sitting eccentrically on the +\PageSep{80} +disc~$K'$ is sensible of a force which acts outwards in a +radial direction, and which would be interpreted as an +effect of inertia (centrifugal force) by an observer who +\index{Centrifugal force}% +was at rest with respect to the original reference-body~$K$. +But the observer on the disc may regard his disc as a +reference-body which is ``at rest''; on the basis of the +general principle of relativity he is justified in doing this. +The force acting on himself, and in fact on all other +bodies which are at rest relative to the disc, he regards +as the effect of a gravitational field. Nevertheless, +the space-distribution of this gravitational field is of a +kind that would not be possible on Newton's theory of +\index{Newton's!law of gravitation}% +gravitation.\footnote + {The field disappears at the centre of the disc and increases + proportionally to the distance from the centre as we proceed + outwards.} +But since the observer believes in the +general theory of relativity, this does not disturb him; +he is quite in the right when he believes that a general +law of gravitation can be formulated---a law which not +only explains the motion of the stars correctly, but +also the field of force experienced by himself. + +The observer performs experiments on his circular +disc with clocks and measuring-rods. In doing so, it +\index{Clocks}% +\index{Measuring-rod}% +is his intention to arrive at exact definitions for the +signification of time- and space-data with reference +to the circular disc~$K'$, these definitions being based on +his observations. What will be his experience in this +enterprise? + +To start with, he places one of two identically constructed +clocks at the centre of the circular disc, and the +other on the edge of the disc, so that they are at rest +relative to it. We now ask ourselves whether both +clocks go at the same rate from the standpoint of the +\PageSep{81} +non-rotating Galileian reference-body~$K$. As judged +from this body, the clock at the centre of the disc has +no velocity, whereas the clock at the edge of the disc +is in motion relative to~$K$ in consequence of the rotation. +\index{Rotation}% +According to a result obtained in \Sectionref{XII}, it follows +that the latter clock goes at a rate permanently slower +than that of the clock at the centre of the circular disc, +\ie\ as observed from~$K$. It is obvious that the same effect +would be noted by an observer whom we will imagine +sitting alongside his clock at the centre of the circular +disc. Thus on our circular disc, or, to make the case +more general, in every gravitational field, a clock will +go more quickly or less quickly, according to the position +in which the clock is situated (at rest). For this reason +it is not possible to obtain a reasonable definition of time +with the aid of clocks which are arranged at rest with +\index{Clocks}% +respect to the body of reference. A similar difficulty +presents itself when we attempt to apply our earlier +definition of simultaneity in such a case, but I do not +\index{Simultaneity}% +wish to go any farther into this question. + +Moreover, at this stage the definition of the space +\index{Space co-ordinates}% +co-ordinates also presents insurmountable difficulties. +If the observer applies his standard measuring-rod +\index{Measuring-rod}% +(a rod which is short as compared with the radius of +the disc) tangentially to the edge of the disc, then, as +judged from the Galileian system, the length of this rod +will be less than~$1$, since, according to \Sectionref{XII}, moving +bodies suffer a shortening in the direction of the motion. +On the other hand, the measuring-rod will not experience +a shortening in length, as judged from~$K$, if it is applied +to the disc in the direction of the radius. If, then, the +observer first measures the circumference of the disc +with his measuring-rod and then the diameter of the +\PageSep{82} +disc, on dividing the one by the other, he will not obtain +as quotient the familiar number $\pi = 3.14\dots$, but +a larger number,\footnote + {Throughout this consideration we have to use the Galileian + (non-rotating) system~$K$ as reference-body, since we may only + assume the validity of the results of the special theory of relativity + relative to~$K$ (relative to~$K'$ a gravitational field prevails).} +whereas of course, for a disc which is +at rest with respect to~$K$, this operation would yield~$\pi$ +\index{Value of $\pi$}% +exactly. This proves that the propositions of Euclidean +\index{Euclidean geometry}% +geometry cannot hold exactly on the rotating disc, nor +in general in a gravitational field, at least if we attribute +the length~$1$ to the rod in all positions and in every +orientation. Hence the idea of a straight line also loses +\index{Straight line}% +its meaning. We are therefore not in a position to +define exactly the co-ordinates $x$,~$y$,~$z$ relative to the +disc by means of the method used in discussing the +special theory, and as long as the co-ordinates and times +of events have not been defined, we cannot assign an +exact meaning to the natural laws in which these occur. + +Thus all our previous conclusions based on general +relativity would appear to be called in question. In +reality we must make a subtle detour in order to be +able to apply the postulate of general relativity exactly. +I shall prepare the reader for this in the +following paragraphs. +\PageSep{83} + + +\Chapter{XXIV}{Euclidean and Non-Euclidean Continuum} +\index{Continuum}% + +\First{The} surface of a marble table is spread out in front +of me. I can get from any one point on this +table to any other point by passing continuously +from one point to a ``neighbouring'' one, and repeating +this process a (large) number of times, or, in other words, +by going from point to point without executing ``jumps.'' +I am sure the reader will appreciate with sufficient +clearness what I mean here by ``neighbouring'' and by +``jumps'' (if he is not too pedantic). We express this +property of the surface by describing the latter as a +continuum. + +Let us now imagine that a large number of little rods +of equal length have been made, their lengths being +small compared with the dimensions of the marble +slab. When I say they are of equal length, I mean that +one can be laid on any other without the ends overlapping. +We next lay four of these little rods on the +marble slab so that they constitute a quadrilateral +figure (a square), the diagonals of which are equally +long. To ensure the equality of the diagonals, we make +use of a little testing-rod. To this square we add +similar ones, each of which has one rod in common +with the first. We proceed in like manner with each of +these squares until finally the whole marble slab is +\PageSep{84} +laid out with squares. The arrangement is such, that +each side of a square belongs to two squares and each +corner to four squares. + +It is a veritable wonder that we can carry out this +business without getting into the greatest difficulties. +We only need to think of the following. If at any +moment three squares meet at a corner, then two sides +of the fourth square are already laid, and, as a consequence, +the arrangement of the remaining two sides of +the square is already completely determined. But I +am now no longer able to adjust the quadrilateral so +that its diagonals may be equal. If they are equal +of their own accord, then this is an especial favour +of the marble slab and of the little rods, about which I +can only be thankfully surprised. We must needs +experience many such surprises if the construction is to +be successful. + +If everything has really gone smoothly, then I say +that the points of the marble slab constitute a Euclidean +\index{Distance (line-interval)}% +\index{Continuum!Euclidean}% +continuum with respect to the little rod, which has been +used as a ``distance'' (line-interval). By choosing +one corner of a square as ``origin,'' I can characterise +every other corner of a square with reference to this +origin by means of two numbers. I only need state +how many rods I must pass over when, starting from the +origin, I proceed towards the ``right'' and then ``upwards,'' +in order to arrive at the corner of the square +under consideration. These two numbers are then the +``Cartesian co-ordinates'' of this corner with reference +\index{Cartesian system of co-ordinates}% +to the ``Cartesian co-ordinate system'' which is determined +by the arrangement of little rods. + +By making use of the following modification of this +abstract experiment, we recognise that there must also +\PageSep{85} +\index{Measurement of length}% +be cases in which the experiment would be unsuccessful. +We shall suppose that the rods ``expand'' by an amount +proportional to the increase of temperature. We heat +the central part of the marble slab, but not the periphery, +in which case two of our little rods can still be +brought into coincidence at every position on the table. +But our construction of squares must necessarily come +into disorder during the heating, because the little rods +on the central region of the table expand, whereas +those on the outer part do not. + +With reference to our little rods---defined as unit +lengths---the marble slab is no longer a Euclidean continuum, +and we are also no longer in the position of defining +Cartesian co-ordinates directly with their aid, +since the above construction can no longer be carried +out. But since there are other things which are not +influenced in a similar manner to the little rods (or +perhaps not at all) by the temperature of the table, it is +possible quite naturally to maintain the point of view +that the marble slab is a ``Euclidean continuum.'' +This can be done in a satisfactory manner by making a +more subtle stipulation about the measurement or the +comparison of lengths. + +But if rods of every kind (\ie\ of every material) were +to behave \emph{in the same way} as regards the influence of +temperature when they are on the variably heated +marble slab, and if we had no other means of detecting +the effect of temperature than the geometrical behaviour +of our rods in experiments analogous to the one +described above, then our best plan would be to assign +the distance \emph{one} to two points on the slab, provided that +the ends of one of our rods could be made to coincide +with these two points; for how else should we define +\PageSep{86} +the distance without our proceeding being in the highest +measure grossly arbitrary? The method of Cartesian +co-ordinates must then be discarded, and replaced by +another which does not assume the validity of Euclidean +\index{Continuum!Euclidean}% +\index{Continuum!non-Euclidean}% +\index{Euclidean geometry}% +\index{Euclidean space}% +geometry for rigid bodies.\footnote + {Mathematicians have been confronted with our problem in the + following form. If we are given a surface (\eg\ an ellipsoid) in + Euclidean three-dimensional space, then there exists for this + surface a two-dimensional geometry, just as much as for a plane + surface. Gauss undertook the task of treating this two-dimensional +\index{Gauss}% + geometry from first principles, without making use of the + fact that the surface belongs to a Euclidean continuum of + three dimensions. If we imagine constructions to be made with + rigid rods \emph{in the surface} (similar to that above with the marble + slab), we should find that different laws hold for these from those + resulting on the basis of Euclidean plane geometry. The surface + is not a Euclidean continuum with respect to the rods, and we + cannot define Cartesian co-ordinates \emph{in the surface}. Gauss + indicated the principles according to which we can treat the + geometrical relationships in the surface, and thus pointed out + the way to the method of Riemann of treating multi-dimensional, +\index{Riemann}% + non-Euclidean \textit{continua}. Thus it is that mathematicians + long ago solved the formal problems to which we are led by the + general postulate of relativity.} +The reader will notice that +the situation depicted here corresponds to the one +brought about by the general postulate of relativity +(\Sectionref{XXIII}). +\PageSep{87} + + +\Chapter{XXV}{Gaussian Co-ordinates} + +\First{According} to Gauss, this combined analytical +\index{Gauss}% +and geometrical mode of handling the problem +can be arrived at in the following way. We +imagine a system of arbitrary curves (see \Figref{4}) +drawn on the surface of the table. These we designate +as $u$-curves, and we indicate each of them by +means of a number. The curves $u = 1$, $u = 2$ and +$u = 3$ are drawn in the diagram. Between the curves +$u = 1$ and $u = 2$ we must imagine an infinitely large +number to be drawn, all of which correspond +%[Illustration: Fig. 4.] +\WFigure{2in}{087} +to real +numbers lying between $1$~and~$2$. We have then +a system of $u$-curves, and +this ``infinitely dense'' system +covers the whole surface +of the table. These +$u$-curves must not intersect +each other, and through each +point of the surface one and +only one curve must pass. +Thus a perfectly definite +value of~$u$ belongs to every point on the surface of the +marble slab. In like manner we imagine a system of +$v$-curves drawn on the surface. These satisfy the same +conditions as the $u$-curves, they are provided with numbers +\PageSep{88} +in a corresponding manner, and they may likewise +be of arbitrary shape. It follows that a value of~$u$ and +a value of~$v$ belong to every point on the surface of the +table. We call these two numbers the co-ordinates +of the surface of the table (Gaussian co-ordinates). +\index{Gaussian co-ordinates|(}% +For example, the point~$P$ in the diagram has the Gaussian +co-ordinates $u = 3$, $v = 1$. Two neighbouring points $P$ +and~$P'$ on the surface then correspond to the co-ordinates +\begin{align*} +&P: &&u, v \\ +&P': &&u + du, v + dv, +\end{align*} +where $du$~and~$dv$ signify very small numbers. In a +similar manner we may indicate the distance (line-interval) +\index{Distance (line-interval)}% +between $P$~and~$P'$, as measured with a little +rod, by means of the very small number~$ds$. Then +according to Gauss we have +\[ +ds^{2} = g_{11}\, du^{2} + 2g_{12}\, du\, dv + g_{22}\, dv^{2}, +\] +where $g_{11}$,~$g_{12}$,~$g_{22}$, are magnitudes which depend in a +perfectly definite way on $u$~and~$v$. The magnitudes $g_{11}$,~$g_{12}$ +and~$g_{22}$ determine the behaviour of the rods relative +to the $u$-curves and $v$-curves, and thus also relative +to the surface of the table. For the case in which the +points of the surface considered form a Euclidean continuum +\index{Continuum!Euclidean}% +with reference to the measuring-rods, but +only in this case, it is possible to draw the $u$-curves +and $v$-curves and to attach numbers to them, in such a +manner, that we simply have: +\[ +ds^{2} = du^{2} + dv^{2}. +\] +Under these conditions, the $u$-curves and $v$-curves are +straight lines in the sense of Euclidean geometry, and +\index{Euclidean geometry}% +\index{Straight line}% +they are perpendicular to each other. Here the Gaussian +co-ordinates are simply Cartesian ones. It is clear +\PageSep{89} +that Gauss co-ordinates are nothing more than an +association of two sets of numbers with the points of +the surface considered, of such a nature that numerical +values differing very slightly from each other are +associated with neighbouring points ``in space.'' + +So far, these considerations hold for a continuum +\index{Continuum!four-dimensional}% +of two dimensions. But the Gaussian method can be +applied also to a continuum of three, four or more +dimensions. If, for instance, a continuum of four +dimensions be supposed available, we may represent +it in the following way. With every point of the +continuum we associate arbitrarily four numbers, $x_{1}$,~$x_{2}$, +$x_{3}$,~$x_{4}$, which are known as ``co-ordinates.'' Adjacent +points correspond to adjacent values of the co-ordinates. +If a distance~$ds$ is associated with the adjacent points +\index{Adjacent points}% +$P$~and~$P'$, this distance being measurable and well-defined +from a physical point of view, then the following +formula holds: +\[ +ds^{2} = g_{11}\, {dx_{1}}^{2} + + 2g_{12}\, dx_{1}\, dx_{2} \Add{+} \dots + + g_{44}\, {dx_{4}}^{2}, +\] +where the magnitudes $g_{11}$,~etc., have values which vary +with the position in the continuum. Only when the +continuum is a Euclidean one is it possible to associate +the co-ordinates $x_{1}$\Add{,}\ldots\Add{,}~$x_{4}$ with the points of the +continuum so that we have simply +\[ +ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + {dx_{4}}^{2}. +\] +In this case relations hold in the four-dimensional +continuum which are analogous to those holding in our +three-dimensional measurements. + +However, the Gauss treatment for~$ds^{2}$ which we have +given above is not always possible. It is only possible +when sufficiently small regions of the continuum under +consideration may be regarded as Euclidean continua. +\PageSep{90} +For example, this obviously holds in the case of the +marble slab of the table and local variation of temperature. +The temperature is practically constant for a small +part of the slab, and thus the geometrical behaviour of +the rods is \emph{almost} as it ought to be according to the +rules of Euclidean geometry. Hence the imperfections +\index{Continuum!non-Euclidean}% +of the construction of squares in the previous section +do not show themselves clearly until this construction +is extended over a considerable portion of the surface +of the table. + +We can sum this up as follows: Gauss invented a +\index{Gauss}% +method for the mathematical treatment of continua in +general, in which ``size-relations'' (``distances'' between +\index{Size-relations}% +neighbouring points) are defined. To every point of a +continuum are assigned as many numbers (Gaussian co-ordinates) +as the continuum has dimensions. This is +done in such a way, that only one meaning can be attached +to the assignment, and that numbers (Gaussian co-ordinates) +\index{Gaussian co-ordinates|)}% +which differ by an indefinitely small amount +are assigned to adjacent points. The Gaussian co-ordinate +system is a logical generalisation of the Cartesian +co-ordinate system. It is also applicable to non-Euclidean +continua, but only when, with respect to the defined +``size'' or ``distance,'' small parts of the continuum +under consideration behave more nearly like a Euclidean +system, the smaller the part of the continuum under +our notice. +\PageSep{91} + + +\Chapter{XXVI}{The Space-Time Continuum of the Special +Theory of Relativity considered as +a Euclidean Continuum} +\index{Continuum!four-dimensional}% +\index{Continuum!space-time|(}% + +\First{We} are now in a position to formulate more +exactly the idea of Minkowski, which was +\index{Minkowski}% +only vaguely indicated in \Sectionref{XVII}. +In accordance with the special theory of relativity, +certain co-ordinate systems are given preference +for the description of the four-dimensional, space-time +continuum. We called these ``Galileian co-ordinate +\index{Galileian system of co-ordinates}% +systems.'' For these systems, the four co-ordinates +$x$,~$y$, $z$,~$t$, which determine an event or---in other +words---a point of the four-dimensional continuum, are +defined physically in a simple manner, as set forth in +detail in the first part of this book. For the transition +from one Galileian system to another, which is moving +uniformly with reference to the first, the equations of +the Lorentz transformation are valid. These last +\index{Lorentz, H. A.!transformation}% +form the basis for the derivation of deductions from the +special theory of relativity, and in themselves they are +nothing more than the expression of the universal +validity of the law of transmission of light for all Galileian +\index{Propagation of light}% +systems of reference. + +Minkowski found that the Lorentz transformations +satisfy the following simple conditions. Let us consider +\PageSep{92} +two neighbouring events, the relative position of which +in the four-dimensional continuum is given with respect +\index{Continuum!four-dimensional}% +to a Galileian reference-body~$K$ by the space co-ordinate +\index{Coordinate@{Co-ordinate}!differences}% +\index{Coordinate@{Co-ordinate}!differentials}% +differences $dx$,~$dy$,~$dz$ and the time-difference~$dt$. With +reference to a second Galileian system we shall suppose +that the corresponding differences for these two events +are $dx'$,~$dy'$, $dz'$,~$dt'$. Then these magnitudes always +fulfil the condition\footnote + {Cf.\ Appendices I~and~II\@. The relations which are derived + there for the co-ordinates themselves are valid also for co-ordinate + \emph{differences}, and thus also for co-ordinate differentials + (indefinitely small differences).} +\[ +dx^{2} + dy^{2} + dz^{2} - c^{2}\, dt^{2} + = dx'^{2} + dy'^{2} + dz'^{2} - c^{2}\, dt'^{2}. +\] + +The validity of the Lorentz transformation follows +from this condition. We can express this as follows: +The magnitude +\[ +ds^{2} = dx^{2} + dy^{2} + dz^{2} - c^{2}\, dt^{2}, +\] +which belongs to two adjacent points of the four-dimensional +space-time continuum, has the same value +for all selected (Galileian) reference-bodies. If we replace +$x$,~$y$, $z$,~$\sqrt{-1}\,ct$, by $x_{1}$,~$x_{2}$, $x_{3}$,~$x_{4}$, we also obtain the +result that +\[ +ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + {dx_{4}}^{2} +\] +is independent of the choice of the body of reference. +We call the magnitude~$ds$ the ``distance'' apart of the +two events or four-dimensional points. + +Thus, if we choose as time-variable the imaginary +variable~$\sqrt{-1}\,ct$ instead of the real quantity~$t$, we can +regard the space-time continuum---in accordance with +the special theory of relativity---as a ``Euclidean'' +\index{Continuum!Euclidean}% +four-dimensional continuum, a result which follows +from the considerations of the preceding section. +\PageSep{93} + + +\Chapter{XXVII}{The Space-Time Continuum of the +General Theory of Relativity is +not a Euclidean Continuum} + +\First{In} the first part of this book we were able to make use +of space-time co-ordinates which allowed of a simple +and direct physical interpretation, and which, according +to \Sectionref{XXVI}, can be regarded as four-dimensional +Cartesian co-ordinates. This was possible on the basis +of the law of the constancy of the velocity of light. But +according to \Sectionref{XXI}, the general theory of relativity +cannot retain this law. On the contrary, we arrived at +the result that according to this latter theory the +velocity of light must always depend on the co-ordinates +when a gravitational field is present. In connection +\index{Gravitational field}% +with a specific illustration in \Sectionref{XXIII}, we found +that the presence of a gravitational field invalidates the +definition of the co-ordinates and the time, which led us +to our objective in the special theory of relativity. + +In view of the results of these considerations we are +led to the conviction that, according to the general +principle of relativity, the space-time continuum cannot +be regarded as a Euclidean one, but that here we have +the general case, corresponding to the marble slab with +local variations of temperature, and with which we +made acquaintance as an example of a two-dimensional +\PageSep{94} +continuum. Just as it was there impossible to construct +\index{Continuum!two-dimensional}% +\index{Continuum!four-dimensional}% +a Cartesian co-ordinate system from equal rods, so +here it is impossible to build up a system (reference-body) +from rigid bodies and clocks, which shall be of +\index{Clocks}% +such a nature that measuring-rods and clocks, arranged +\index{Measuring-rod}% +rigidly with respect to one another, shall indicate position +and time directly. Such was the essence of the +difficulty with which we were confronted in \Sectionref{XXIII}. + +But the considerations of Sections \Srefno{XXV}~and~\Srefno{XXVI} +show us the way to surmount this difficulty. We refer the +four-dimensional space-time continuum in an arbitrary +manner to Gauss co-ordinates. We assign to every +\index{Gaussian co-ordinates}% +point of the continuum (event) four numbers, $x_{1}$,~$x_{2}$, +$x_{3}$,~$x_{4}$ (co-ordinates), which have not the least direct +physical significance, but only serve the purpose of +numbering the points of the continuum in a definite +but arbitrary manner. This arrangement does not even +need to be of such a kind that we must regard $x_{1}$,~$x_{2}$,~$x_{3}$ as +``space'' co-ordinates and $x_{4}$~as a ``time'' co-ordinate. + +The reader may think that such a description of the +world would be quite inadequate. What does it mean +to assign to an event the particular co-ordinates $x_{1}$,~$x_{2}$, +$x_{3}$,~$x_{4}$, if in themselves these co-ordinates have no +significance? More careful consideration shows, however, +that this anxiety is unfounded. Let us consider, +for instance, a material point with any kind of motion. +If this point had only a momentary existence without +duration, then it would be described in space-time by a +single system of values $x_{1}$,~$x_{2}$, $x_{3}$,~$x_{4}$. Thus its permanent +existence must be characterised by an infinitely large +number of such systems of values, the co-ordinate values +of which are so close together as to give continuity; +\PageSep{95} +corresponding to the material point, we thus have a +(uni-dimensional) line in the four-dimensional continuum. +\index{Continuity}% +In the same way, any such lines in our continuum +correspond to many points in motion. The only statements +having regard to these points which can claim +a physical existence are in reality the statements about +their encounters. In our mathematical treatment, +such an encounter is expressed in the fact that the two +lines which represent the motions of the points in +question have a particular system of co-ordinate values, +$x_{1}$,~$x_{2}$, $x_{3}$,~$x_{4}$, in common. After mature consideration +the reader will doubtless admit that in reality such +encounters constitute the only actual evidence of a +time-space nature with which we meet in physical +statements. + +When we were describing the motion of a material +\index{Encounter (space-time coincidence)}% +point relative to a body of reference, we stated +nothing more than the encounters of this point with +particular points of the reference-body. We can also +determine the corresponding values of the time by the +observation of encounters of the body with clocks, in +\index{Clocks}% +conjunction with the observation of the encounter of the +hands of clocks with particular points on the dials. +It is just the same in the case of space-measurements by +means of measuring-rods, as a little consideration will +show. + +The following statements hold generally: Every +physical description resolves itself into a number of +statements, each of which refers to the space-time +coincidence of two events $A$~and~$B$. In terms of +Gaussian co-ordinates, every such statement is expressed +by the agreement of their four co-ordinates $x_{1}$,~$x_{2}$, $x_{3}$,~$x_{4}$. +Thus in reality, the description of the time-space +\PageSep{96} +continuum by means of Gauss co-ordinates completely +\index{Gaussian co-ordinates|(}% +replaces the description with the aid of a body of reference, +without suffering from the defects of the latter +mode of description; it is not tied down to the Euclidean +character of the continuum which has to be represented. +\index{Continuum!space-time|)}% +\PageSep{97} + + +\Chapter{XXVIII}{Exact Formulation of the General +Principle of Relativity} +\index{General theory of relativity}% + +\First{We} are now in a position to replace the provisional +formulation of the general principle +of relativity given in \Sectionref{XVIII} by +an exact formulation. The form there used, ``All +bodies of reference $K$,~$K'$,~etc., are equivalent for +the description of natural phenomena (formulation of +the general laws of nature), whatever may be their +state of motion,'' cannot be maintained, because the +use of rigid reference-bodies, in the sense of the method +followed in the special theory of relativity, is in general +not possible in space-time description. The Gauss +co-ordinate system has to take the place of the body of +reference. The following statement corresponds to the +fundamental idea of the general principle of relativity: +``\emph{All Gaussian co-ordinate systems are essentially equivalent +for the formulation of the general laws of nature.}'' + +We can state this general principle of relativity in still +another form, which renders it yet more clearly intelligible +than it is when in the form of the natural +extension of the special principle of relativity. According +to the special theory of relativity, the equations +which express the general laws of nature pass over into +equations of the same form when, by making use of the +Lorentz transformation, we replace the space-time +\index{Lorentz, H. A.!transformation}% +\PageSep{98} +variables $x$,~$y$, $z$,~$t$, of a (Galileian) reference-body~$K$ +by the space-time variables $x'$,~$y'$, $z'$,~$t'$, of a new reference-body~$K'$. +According to the general theory +of relativity, on the other hand, by application of +\emph{arbitrary substitutions} of the Gauss variables $x_{1}$,~$x_{2}$, $x_{3}$,~$x_{4}$, +\index{Arbitrary substitutions}% +the equations must pass over into equations of the same +form; for every transformation (not only the Lorentz +\index{Lorentz, H. A.!transformation}% +transformation) corresponds to the transition of one +Gauss co-ordinate system into another. + +If we desire to adhere to our ``old-time'' three-dimensional +\index{Law of inertia}% +view of things, then we can characterise +the development which is being undergone by the +fundamental idea of the general theory of relativity +as follows: The special theory of relativity has reference +to Galileian domains, \ie\ to those in which no gravitational +field exists. In this connection a Galileian reference-body +\index{Galileian system of co-ordinates}% +serves as body of reference, \ie\ a rigid +body the state of motion of which is so chosen that the +Galileian law of the uniform rectilinear motion of +``isolated'' material points holds relatively to it. + +Certain considerations suggest that we should refer +the same Galileian domains to \emph{non-Galileian} reference-bodies +\index{Non-Galileian reference-bodies}% +also. A gravitational field of a special kind is +\index{Gravitational field}% +then present with respect to these bodies (cf.\ Sections \Srefno{XX} +and~\Srefno{XXIII}). + +In gravitational fields there are no such things as rigid +\index{Time!in Physics}% +bodies with Euclidean properties; thus the fictitious rigid +body of reference is of no avail in the general theory of +relativity. The motion of clocks is also influenced by +\index{Clocks|(}% +gravitational fields, and in such a way that a physical +definition of time which is made directly with the aid of +clocks has by no means the same degree of plausibility +as in the special theory of relativity. +\PageSep{99} +\index{Laws of Galilei-Newton!of Nature}% +\index{Time!coordinate@{co-ordinate}}% + +For this reason non-rigid reference-bodies are used, +which are as a whole not only moving in any way +whatsoever, but which also suffer alterations in form +\textit{ad~lib.}\ during their motion. Clocks, for which the law of +motion is of any kind, however irregular, serve for the +definition of time. We have to imagine each of these +clocks fixed at a point on the non-rigid reference-body. +\index{Reference-mollusk|(}% +These clocks satisfy only the one condition, that the +``readings'' which are observed simultaneously on +adjacent clocks (in space) differ from each other by an +\index{Space!point@{-point}}% +indefinitely small amount. This non-rigid reference-body, +which might appropriately be termed a ``reference-mollusk,'' +is in the main equivalent to a Gaussian four-dimensional +co-ordinate system chosen arbitrarily. +That which gives the ``mollusk'' a certain comprehensibleness +as compared with the Gauss co-ordinate +system is the (really unjustified) formal retention of +the separate existence of the space co-ordinates as +\index{Space co-ordinates}% +opposed to the time co-ordinate. Every point on the +mollusk is treated as a space-point, and every material +point which is at rest relatively to it as at rest, so long as +the mollusk is considered as reference-body. The +general principle of relativity requires that all these +mollusks can be used as reference-bodies with equal +right and equal success in the formulation of the general +laws of nature; the laws themselves must be quite +independent of the choice of mollusk. + +The great power possessed by the general principle +of relativity lies in the comprehensive limitation which +is imposed on the laws of nature in consequence of what +we have seen above. +\PageSep{100} + + +\Chapter{XXIX}{The Solution of the Problem of Gravitation +on the Basis of the General +Principle of Relativity} + +\First{If} the reader has followed all our previous considerations, +he will have no further difficulty in +understanding the methods leading to the solution +of the problem of gravitation. + +We start off from a consideration of a Galileian +domain, \ie\ a domain in which there is no gravitational +field relative to the Galileian reference-body~$K$. The +\index{Galileian system of co-ordinates}% +behaviour of measuring-rods and clocks with reference +\index{Measuring-rod}% +to~$K$ is known from the special theory of relativity, +likewise the behaviour of ``isolated'' material points; +the latter move uniformly and in straight lines. + +Now let us refer this domain to a random Gauss co-ordinate +system or to a ``mollusk'' as reference-body~$K'$. +Then with respect to~$K'$ there is a gravitational +field~$G$ (of a particular kind). We learn the behaviour +of measuring-rods and clocks and also of freely-moving +material points with reference to~$K'$ simply by mathematical +transformation. We interpret this behaviour +as the behaviour of measuring-rods, clocks and material +\index{Clocks|)}% +points under the influence of the gravitational field~$G$. +\index{Gravitational field}% +Hereupon we introduce a hypothesis: that the influence +of the gravitational field on measuring-rods, +\index{Gaussian co-ordinates|)}% +\PageSep{101} +clocks and freely-moving material points continues to +take place according to the same laws, even in the case +when the prevailing gravitational field is \emph{not} derivable +\index{Gravitational field}% +from the Galileian special case, simply by means of a +transformation of co-ordinates. + +The next step is to investigate the space-time +behaviour of the gravitational field~$G$, which was derived +from the Galileian special case simply by transformation +of the co-ordinates. This behaviour is formulated +in a law, which is always valid, no matter how the +\index{Matter}% +reference-body (mollusk) used in the description may +\index{Reference-mollusk|)}% +be chosen. + +This law is not yet the \emph{general} law of the gravitational +field, since the gravitational field under consideration is +of a special kind. In order to find out the general +law-of-field of gravitation we still require to obtain a +generalisation of the law as found above. This can be +obtained without caprice, however, by taking into +consideration the following demands: +\begin{itemize} +\item[\itema] The required generalisation must likewise satisfy + the general postulate of relativity. + +\item[\itemb] If there is any matter in the domain under consideration, + only its inertial mass, and thus +\index{Inertial mass}% + according to \Sectionref{XV} only its energy is of + importance for its effect in exciting a field. + +\item[\itemc] Gravitational field and matter together must + satisfy the law of the conservation of energy +\index{Conservation of energy}% +\index{Conservation of energy!impulse}% +\index{Kinetic energy}% + (and of impulse). +\end{itemize} + +Finally, the general principle of relativity permits +us to determine the influence of the gravitational field +on the course of all those processes which take place +according to known laws when a gravitational field is +\PageSep{102} +absent, \ie\ which have already been fitted into the +frame of the special theory of relativity. In this connection +we proceed in principle according to the method +which has already been explained for measuring-rods, +\index{Measuring-rod}% +clocks and freely-moving material points. +\index{Clocks}% + +The theory of gravitation derived in this way from +\index{Gravitation}% +the general postulate of relativity excels not only in +its beauty; nor in removing the defect attaching to +classical mechanics which was brought to light in \Sectionref{XXI}; +\index{Classical mechanics}% +nor in interpreting the empirical law of the equality +of inertial and gravitational mass; but it has also +\index{Gravitational mass}% +\index{Inertial mass}% +already explained a result of observation in astronomy, +\index{Astronomy}% +against which classical mechanics is powerless. + +If we confine the application of the theory to the +case where the gravitational fields can be regarded as +being weak, and in which all masses move with respect +to the co-ordinate system with velocities which are +small compared with the velocity of light, we then obtain +as a first approximation the Newtonian theory. Thus +the latter theory is obtained here without any particular +assumption, whereas Newton had to introduce the +\index{Newton}% +hypothesis that the force of attraction between mutually +attracting material points is inversely proportional to +the square of the distance between them. If we increase +the accuracy of the calculation, deviations from +the theory of Newton make their appearance, practically +all of which must nevertheless escape the test of +observation owing to their smallness. + +We must draw attention here to one of these deviations. +According to Newton's theory, a planet moves +round the sun in an ellipse, which would permanently +maintain its position with respect to the fixed stars, +if we could disregard the motion of the fixed stars +\index{Motion!of heavenly bodies}% +\PageSep{103} +themselves and the action of the other planets under +consideration. Thus, if we correct the observed motion +of the planets for these two influences, and if Newton's +theory be strictly correct, we ought to obtain for the +orbit of the planet an ellipse, which is fixed with reference +to the fixed stars. This deduction, which can +be tested with great accuracy, has been confirmed +for all the planets save one, with the precision that is +capable of being obtained by the delicacy of observation +attainable at the present time. The sole exception +is Mercury, the planet which lies nearest the sun. Since +\index{Mercury}% +\index{Mercury!orbit of}% +the time of Leverrier, it has been known that the ellipse +\index{Leverrier}% +corresponding to the orbit of Mercury, after it has been +corrected for the influences mentioned above, is not +stationary with respect to the fixed stars, but that it +rotates exceedingly slowly in the plane of the orbit +and in the sense of the orbital motion. The value +obtained for this rotary movement of the orbital ellipse +was $43$~seconds of arc per~century, an amount ensured +to be correct to within a few seconds of arc. This +effect can be explained by means of classical mechanics +\index{Classical mechanics}% +only on the assumption of hypotheses which have +little probability, and which were devised solely for +this purpose. + +On the basis of the general theory of relativity, it +is found that the ellipse of every planet round the sun +must necessarily rotate in the manner indicated above; +that for all the planets, with the exception of Mercury, +this rotation is too small to be detected with the delicacy +of observation possible at the present time; but that in +the case of Mercury it must amount to $43$~seconds of +arc per century, a result which is strictly in agreement +with observation. +\PageSep{104} + +Apart from this one, it has hitherto been possible to +make only two deductions from the theory which admit +of being tested by observation, to wit, the curvature +\index{Curvature of light-rays}% +of light rays by the gravitational field of the sun,\footnote + {Observed by Eddington and others in~1919. (Cf.\ \Appendixref{III}.)} +\index{Eddington}% +and a displacement of the spectral lines of light reaching +\index{Displacement of spectral lines}% +us from large stars, as compared with the corresponding +lines for light produced in an analogous manner terrestrially +(\ie\ by the same kind of molecule). I do not +doubt that these deductions from the theory will be +confirmed also. +\index{General theory of relativity|)}% +\PageSep{105} + + +\Part{III}{Considerations on the Universe as +a Whole}{Considerations on the Universe} + +\Chapter{XXX}{Cosmological Difficulties of Newton's +Theory} +\index{Newton}% + +\First{Apart} from the difficulty discussed in \Sectionref{XXI}, +there is a second fundamental difficulty +attending classical celestial mechanics, which, +\index{Celestial mechanics}% +to the best of my knowledge, was first discussed in +detail by the astronomer Seeliger. If we ponder over +\index{Seeliger}% +the question as to how the universe, considered as a +whole, is to be regarded, the first answer that suggests +itself to us is surely this: As regards space (and time) +\index{Space}% +\index{Time!conception of}% +the universe is infinite. There are stars everywhere, +so that the density of matter, although very variable +in detail, is nevertheless on the average everywhere the +same. In other words: However far we might travel +through space, we should find everywhere an attenuated +swarm of fixed stars of approximately the same kind +and density. + +This view is not in harmony with the theory of +Newton. The latter theory rather requires that the +universe should have a kind of centre in which the +\PageSep{106} +density of the stars is a maximum, and that as we +proceed outwards from this centre the group-density +\index{Group-density of stars}% +of the stars should diminish, until finally, at great +distances, it is succeeded by an infinite region of emptiness. +The stellar universe ought to be a finite island in +\index{Stellar universe}% +the infinite ocean of space.\footnote + {\textit{Proof}---According to the theory of Newton, the number of + ``lines of force'' which come from infinity and terminate in a +\index{Lines of force}% + mass~$m$ is proportional to the mass~$m$. If, on the average, the + mass-density~$\rho_{0}$ is constant throughout the universe, then a + sphere of volume~$V$ will enclose the average mass~$\rho_{0}V$. Thus + the number of lines of force passing through the surface~$F$ of the + sphere into its interior is proportional to~$\rho_{0}V$. For unit area + of the surface of the sphere the number of lines of force which + enters the sphere is thus proportional to~$\rho_{0}\dfrac{V}{F}$ or to~$\rho_{0}R$. Hence + the intensity of the field at the surface would ultimately become + infinite with increasing radius~$R$ of the sphere, which is impossible.} + +This conception is in itself not very satisfactory. +It is still less satisfactory because it leads to the result +that the light emitted by the stars and also individual +stars of the stellar system are perpetually passing out +into infinite space, never to return, and without ever +again coming into interaction with other objects of +nature. Such a finite material universe would be +destined to become gradually but systematically impoverished. + +In order to escape this dilemma, Seeliger suggested a +\index{Intensity of gravitational field}% +\index{Seeliger}% +modification of Newton's law, in which he assumes that +\index{Newton's!law of gravitation}% +for great distances the force of attraction between two +masses diminishes more rapidly than would result from +the inverse square law. In this way it is possible for the +mean density of matter to be constant everywhere, +even to infinity, without infinitely large gravitational +fields being produced. We thus free ourselves from the +\PageSep{107} +distasteful conception that the material universe ought +to possess something of the nature of a centre. Of +course we purchase our emancipation from the fundamental +difficulties mentioned, at the cost of a modification +and complication of Newton's law which has +neither empirical nor theoretical foundation. We can +imagine innumerable laws which would serve the same +purpose, without our being able to state a reason why +one of them is to be preferred to the others; for any +one of these laws would be founded just as little on +more general theoretical principles as is the law of +Newton. +\PageSep{108} + + +\Chapter{XXXI}{The Possibility of a ``Finite'' and yet +``Unbounded'' Universe} +\index{Universe (World) structure of}% + +\First{But} speculations on the structure of the universe +also move in quite another direction. The +development of non-Euclidean geometry led to +\index{Euclidean geometry}% +\index{Non-Euclidean geometry}% +the recognition of the fact, that we can cast doubt on the +\emph{infiniteness} of our space without coming into conflict +with the laws of thought or with experience (Riemann, +\index{Riemann}% +Helmholtz). These questions have already been treated +\index{Helmholtz}% +in detail and with unsurpassable lucidity by Helmholtz +and Poincaré, whereas I can only touch on them +\index{Poincare@{Poincaré}}% +briefly here. + +In the first place, we imagine an existence in two-dimensional +\index{Being@{``Being''}}% +\index{Space!two-dimensional}% +space. Flat beings with flat implements, +and in particular flat rigid measuring-rods, are free to +move in a \emph{plane}. For them nothing exists outside of +\index{Plane}% +this plane: that which they observe to happen to +themselves and to their flat ``things'' is the all-inclusive +reality of their plane. In particular, the constructions +of plane Euclidean geometry can be carried out by +means of the rods, \eg\ the lattice construction, considered +\index{Lattice}% +in \Sectionref{XXIV}. In contrast to ours, the universe of +these beings is two-dimensional; but, like ours, it extends +to infinity. In their universe there is room for an +infinite number of identical squares made up of rods, +\PageSep{109} +\ie\ its volume (surface) is infinite. If these beings say +their universe is ``plane,'' there is sense in the statement, +\index{Plane}% +\index{Universe!Euclidean}% +because they mean that they can perform the constructions +of plane Euclidean geometry with their rods. +\index{Euclidean geometry}% +In this connection the individual rods always represent +\index{Distance (line-interval)}% +the same distance, independently of their position. + +Let us consider now a second two-dimensional existence, +but this time on a spherical surface instead of on +\index{Spherical!surface}% +a plane. The flat beings with their measuring-rods +and other objects fit exactly on this surface and they +are unable to leave it. Their whole universe of observation +extends exclusively over the surface of the sphere. +Are these beings able to regard the geometry of their +universe as being plane geometry and their rods withal +as the realisation of ``distance''? They cannot do +this. For if they attempt to realise a straight line, they +\index{Straight line}% +will obtain a curve, which we ``three-dimensional +beings'' designate as a great circle, \ie\ a self-contained +line of definite finite length, which can be measured +up by means of a measuring-rod. Similarly, this +universe has a finite area that can be compared with the +area of a square constructed with rods. The great +charm resulting from this consideration lies in the +recognition of the fact that \emph{the universe of these beings is +finite and yet has no limits}. + +But the spherical-surface beings do not need to go +on a world-tour in order to perceive that they are not +\index{World}% +living in a Euclidean universe. They can convince +themselves of this on every part of their ``world,'' +provided they do not use too small a piece of it. Starting +from a point, they draw ``straight lines'' (arcs of circles +as judged in three-dimensional space) of equal length +in all directions. They will call the line joining the +\PageSep{110} +free ends of these lines a ``circle.'' For a plane surface, +the ratio of the circumference of a circle to its diameter, +both lengths being measured with the same rod, is, +according to Euclidean geometry of the plane, equal to +a constant value~$\pi$, which is independent of the diameter +\index{Value of $\pi$}% +of the circle. On their spherical surface our flat beings +would find for this ratio the value +\[ +\pi = \frac{\sin\left(\dfrac{r}{R}\right)}{\left(\dfrac{r}{R}\right)}, +\] +\ie\ a smaller value than~$\pi$, the difference being the +more considerable, the greater is the radius of the +circle in comparison with the radius~$R$ of the ``world-sphere.'' +\index{World!sphere@{-sphere}}% +By means of this relation the spherical beings +can determine the radius of their universe (``world''), +even when only a relatively small part of their world-sphere +is available for their measurements. But if this +part is very small indeed, they will no longer be able to +demonstrate that they are on a spherical ``world'' and +not on a Euclidean plane, for a small part of a spherical +surface differs only slightly from a piece of a plane of +the same size. + +Thus if the spherical-surface beings are living on a +planet of which the solar system occupies only a negligibly +small part of the spherical universe, they have no means +of determining whether they are living in a finite or in +an infinite universe, because the ``piece of universe'' +to which they have access is in both cases practically +plane, or Euclidean. It follows directly from this +discussion, that for our sphere-beings the circumference +of a circle first increases with the radius until the ``circumference +\PageSep{111} +\index{Universe (World) structure of!circumference of}% +of the universe'' is reached, and that it +\index{Universe!Euclidean}% +\index{Universe!spherical}% +thenceforward gradually decreases to zero for still +further increasing values of the radius. During this +process the area of the circle continues to increase +more and more, until finally it becomes equal to the +total area of the whole ``world-sphere.'' +\index{World!sphere@{-sphere}}% + +Perhaps the reader will wonder why we have placed +our ``beings'' on a sphere rather than on another closed +surface. But this choice has its justification in the fact +that, of all closed surfaces, the sphere is unique in possessing +the property that all points on it are equivalent. I +admit that the ratio of the circumference~$c$ of a circle +to its radius~$r$ depends on~$r$, but for a given value of~$r$ +it is the same for all points of the ``world-sphere''; +in other words, the ``world-sphere'' is a ``surface of +constant curvature.'' + +To this two-dimensional sphere-universe there is a +three-dimensional analogy, namely, the three-dimensional +spherical space which was discovered by Riemann. Its +\index{Riemann}% +points are likewise all equivalent. It possesses a finite +volume, which is determined by its ``radius'' ($2\pi^{2}R^{3}$). +Is it possible to imagine a spherical space? To imagine +a space means nothing else than that we imagine an +epitome of our ``space'' experience, \ie\ of experience +that we can have in the movement of ``rigid'' bodies. +In this sense we \emph{can} imagine a spherical space. + +Suppose we draw lines or stretch strings in all directions +from a point, and mark off from each of these +the distance~$r$ with a measuring-rod. All the free end-points +\index{Measuring-rod}% +of these lengths lie on a spherical surface. We +\index{Spherical!space}% +can specially measure up the area~($F$) of this surface +by means of a square made up of measuring-rods. If +the universe is Euclidean, then $F = 4\pi r^{2}$; if it is spherical, +\PageSep{112} +then $F$~is always less than~$4\pi r^{2}$. With increasing +values of~$r$, $F$~increases from zero up to a maximum +value which is determined by the ``world-radius,'' but +\index{World!radius@{-radius}}% +for still further increasing values of~$r$, the area gradually +diminishes to zero. At first, the straight lines which +radiate from the starting point diverge farther and +farther from one another, but later they approach +each other, and finally they run together again at a +``counter-point'' to the starting point. Under such +\index{Counter-Point}% +conditions they have traversed the whole spherical +space. It is easily seen that the three-dimensional +spherical space is quite analogous to the two-dimensional +spherical surface. It is finite (\ie\ of finite volume), and +\index{Spherical!space}% +has no bounds. + +It may be mentioned that there is yet another kind +of curved space: ``elliptical space.'' It can be regarded +\index{Elliptical space}% +as a curved space in which the two ``counter-points'' +are identical (indistinguishable from each other). An +elliptical universe can thus be considered to some +\index{Universe!elliptical}% +extent as a curved universe possessing central symmetry. + +It follows from what has been said, that closed spaces +without limits are conceivable. From amongst these, +the spherical space (and the elliptical) excels in its +simplicity, since all points on it are equivalent. As a +result of this discussion, a most interesting question +arises for astronomers and physicists, and that is +whether the universe in which we live is infinite, or +whether it is finite in the manner of the spherical universe. +Our experience is far from being sufficient to +enable us to answer this question. But the general +theory of relativity permits of our answering it with a +moderate degree of certainty, and in this connection the +difficulty mentioned in \Sectionref{XXX} finds its solution. +\PageSep{113} + + +\Chapter{XXXII}{The Structure of Space according to +the General Theory of Relativity} +\index{Motion!of heavenly bodies}% +\index{Universe (World) structure of}% + +\First{According} to the general theory of relativity, +the geometrical properties of space are not independent, +but they are determined by matter. +Thus we can draw conclusions about the geometrical +structure of the universe only if we base our considerations +on the state of the matter as being something +that is known. We know from experience that, for a +suitably chosen co-ordinate system, the velocities of +the stars are small as compared with the velocity of +transmission of light. We can thus as a rough approximation +arrive at a conclusion as to the nature of +the universe as a whole, if we treat the matter as being +at rest. + +We already know from our previous discussion that the +behaviour of measuring-rods and clocks is influenced by +\index{Clocks}% +\index{Measuring-rod}% +gravitational fields, \ie\ by the distribution of matter. +\index{Gravitational field}% +This in itself is sufficient to exclude the possibility of +the exact validity of Euclidean geometry in our universe. +\index{Euclidean geometry}% +But it is conceivable that our universe differs +only slightly from a Euclidean one, and this notion +seems all the more probable, since calculations show +that the metrics of surrounding space is influenced only +to an exceedingly small extent by masses even of the +\PageSep{114} +magnitude of our sun. We might imagine that, as +regards geometry, our universe behaves analogously +\index{Universe!elliptical}% +\index{Universe!space expanse (radius) of}% +\index{Universe!spherical}% +to a surface which is irregularly curved in its individual +parts, but which nowhere departs appreciably from a +plane: something like the rippled surface of a lake. +Such a universe might fittingly be called a quasi-Euclidean +universe. As regards its space it would be +infinite. But calculation shows that in a quasi-Euclidean +universe the average density of matter +would necessarily be \emph{nil}. Thus such a universe could +not be inhabited by matter everywhere; it would +present to us that unsatisfactory picture which we +portrayed in \Sectionref{XXX}. + +If we are to have in the universe an average density +of matter which differs from zero, however small may +be that difference, then the universe cannot be quasi-Euclidean. +\index{Quasi-Euclidean universe}% +On the contrary, the results of calculation +indicate that if matter be distributed uniformly, the +universe would necessarily be spherical (or elliptical). +Since in reality the detailed distribution of matter is +not uniform, the real universe will deviate in individual +parts from the spherical, \ie\ the universe will be quasi-spherical. +\index{Quasi-spherical universe}% +But it will be necessarily finite. In fact, the +theory supplies us with a simple connection\footnote + {For the ``radius''~$R$ of the universe we obtain the equation + \[ + R^{2} = \frac{2}{\kappa \rho}. + \] + The use of the C.G.S. system in this equation gives $\dfrac{2}{\kappa} = 1.08 × 10^{27}$; +is the average density of the matter.} +between +the space-expanse of the universe and the average +density of matter in it. +\PageSep{115} + + +\Appendix{I}{Simple Derivation of the Lorentz +Transformation}{[Supplementary to \Sectionref{XI}]} +\index{Lorentz, H. A.!transformation}% + +\First{For} the relative orientation of the co-ordinate +systems indicated in \Figref{2}, the $x$-axes of both +systems permanently coincide. In the present +case we can divide the problem into parts by considering +first only events which are localised on the $x$-axis. Any +such event is represented with respect to the co-ordinate +system~$K$ by the abscissa~$x$ and the time~$t$, and with +respect to the system~$K'$ by the abscissa~$x'$ and the +time~$t'$. We require to find $x'$~and~$t'$ when $x$~and~$t$ are +given. + +A light-signal, which is proceeding along the positive +\index{Light-signal}% +axis of~$x$, is transmitted according to the equation +\[ +x = ct +\] +or +\[ +x - ct = 0. +\Tag{(1)} +\] +Since the same light-signal has to be transmitted relative +to~$K'$ with the velocity~$c$, the propagation relative to +the system~$K'$ will be represented by the analogous +formula +\[ +x' - ct' = 0. +\Tag{(2)} +\] +Those space-time points (events) which satisfy~\Eqref{(1)} must +\PageSep{116} +also satisfy~\Eqref{(2)}. Obviously this will be the case when +the relation +\[ +(x' - ct') = \lambda(x - ct)\Change{.}{} +\Tag{(3)} +\] +is fulfilled in general, where $\lambda$~indicates a constant; for, +according to~\Eqref{(3)}, the disappearance of~$(x - ct)$ involves +the disappearance of~$(x' - ct')$. + +If we apply quite similar considerations to light rays +which are being transmitted along the negative $x$-axis, +we obtain the condition +\[ +(x' + ct') = \mu(x + ct). +\Tag{(4)} +\] + +By adding (or subtracting) equations \Eqref{(3)}~and~\Eqref{(4)}, and +introducing for convenience the constants $a$~and~$b$ in +place of the constants $\lambda$~and~$\mu$, where +\begin{align*} +a &= \frac{\lambda + \mu}{2} +\intertext{and} +b &= \frac{\lambda - \mu}{2}, +\end{align*} +we obtain the equations +\[ +\left. +\begin{aligned} +x' &= ax - bct\Add{,} \\ +ct' &= act - bx. +\end{aligned} +\right\} +\Tag{(5)} +\] + +We should thus have the solution of our problem, +if the constants $a$~and~$b$ were known. These result +from the following discussion. + +For the origin of~$K'$ we have permanently $x' = 0$, and +hence according to the first of the equations~\Eqref{(5)} +\[ +x = \frac{bc}{a} t. +\] + +If we call~$v$ the velocity with which the origin of~$K'$ is +moving relative to~$K$, we then have +\[ +v = \frac{bc}{a}. +\Tag{(6)} +\] +\PageSep{117} + +The same value~$v$ can be obtained from equation~\Eqref{(5)}, +if we calculate the velocity of another point of~$K'$ +relative to~$K$, or the velocity (directed towards the +\index{Relative!velocity}% +negative $x$-axis) of a point of~$K$ with respect to~$K'$. In +short, we can designate~$v$ as the relative velocity of the +two systems. + +Furthermore, the principle of relativity teaches us +that, as judged from~$K$, the length of a unit measuring-rod +\index{Measuring-rod}% +which is at rest with reference to~$K'$ must be exactly +the same as the length, as judged from~$K'$, of a unit +measuring-rod which is at rest relative to~$K$. In order +to see how the points of the $x'$-axis appear as viewed +from~$K$, we only require to take a ``snapshot'' of~$K'$ +\index{Instantaneous photograph (snapshot)}% +from~$K$; this means that we have to insert a particular +value of~$t$ (time of~$K$), \eg\ $t = 0$. For this value of~$t$ +we then obtain from the first of the equations~\Eqref{(5)} +\[ +x' = ax. +\] + +Two points of the $x'$-axis which are separated by the +distance $\Delta x' = 1$ when measured in the $K'$~system are +thus separated in our instantaneous photograph by the +distance +\[ +\Delta x = \frac{1}{a}. +\Tag{(7)} +\] + +But if the snapshot be taken from~$K'$\Change{}{ }($t' = 0$), and if +we eliminate~$t$ from the equations~\Eqref{(5)}, taking into +account the expression~\Eqref{(6)}, we obtain +\[ +x' = a\left(1 - \frac{v^{2}}{c^{2}}\right)x. +\] + +From this we conclude that two points on the $x$-axis +and separated by the distance~$1$ (relative to~$K$) will +be represented on our snapshot by the distance +\[ +\Delta x' = a\left(1 - \frac{v^{2}}{c^{2}}\right). +\Tag{(7a)} +\] +\PageSep{118} + +But from what has been said, the two snapshots +must be identical; hence $\Delta x$~in~\Eqref{(7)} must be equal to +$\Delta x'$~in~\Eqref{(7a)}, so that we obtain +\[ +a^{2} = \frac{1}{1 - \dfrac{v^{2}}{c^{2}}}. +\Tag{(7b)} +\] + +The equations \Eqref{(6)}~and~\Eqref{(7b)} determine the constants $a$~and~$b$. +By inserting the values of these constants in~\Eqref{(5)}, +we obtain the first and the fourth of the equations +given in \Sectionref{XI}. +\[ +\left. +\begin{aligned} +x' &= \frac{x - vt}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}\Add{,} \\ +t' &= \frac{t - \dfrac{v}{c^{2}}x}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}. +\end{aligned} +\right\} +\Tag{(8)} +\] + +Thus we have obtained the Lorentz transformation +\index{Lorentz, H. A.!transformation}% +for events on the $x$-axis. It satisfies the condition +\[ +x'^{2} - c^{2} t'^{2} = x^{2} - c^{2} t^{2}. +\Tag{(8a)} +\] + +The extension of this result, to include events which +take place outside the $x$-axis, is obtained by retaining +equations~\Eqref{(8)} and supplementing them by the relations +\[ +\left. +\begin{aligned} +y' &= y\Add{,} \\ +z' &= z. +\end{aligned} +\right\} +\Tag{(9)} +\] +In this way we satisfy the postulate of the constancy of +the velocity of light \textit{in~vacuo} for rays of light of arbitrary +\index{Velocity of light}% +direction, both for the system~$K$ and for the system~$K'$. +This may be shown in the following manner. + +We suppose a light-signal sent out from the origin +\index{Light-signal}% +of~$K$ at the time $t = 0$. It will be propagated according +to the equation +\[ +r = \sqrt{x^{2} + y^{2} + z^{2}} = ct, +\] +\PageSep{119} +or, if we square this equation, according to the equation +\[ +x^{2} + y^{2} + z^{2} - c^{2} t^{2} = 0. +\Tag{(10)} +\] + +It is required by the law of propagation of light, in +\index{Propagation of light}% +conjunction with the postulate of relativity, that the +transmission of the signal in question should take place---as +judged from~$K'$---in accordance with the corresponding +formula +\[ +r' = ct', +\] +or, +\[ +x'^{2} + y'^{2} + z'^{2} - c^{2} t'^{2} = 0. +\Tag{(10a)} +\] +In order that equation~\Eqref{(10a)} may be a consequence of +equation~\Eqref{(10)}, we must have +\[ +x'^{2} + y'^{2} + z'^{2} - c^{2} t'^{2} + = \sigma(x^{2} + y^{2} + z^{2} - c^{2} t^{2}). +\Tag{(11)} +\] + +Since equation~\Eqref{(8a)} must hold for points on the +$x$-axis, we thus have $\sigma = 1$. It is easily seen that the +Lorentz transformation really satisfies equation~\Eqref{(11)} +\index{Lorentz, H. A.!transformation}% +for $\sigma = 1$; for \Eqref{(11)}~is a consequence of \Eqref{(8a)}~and~\Eqref{(9)}, +and hence also of \Eqref{(8)}~and~\Eqref{(9)}. We have thus derived +the Lorentz transformation. + +The Lorentz transformation represented by \Eqref{(8)}~and~\Eqref{(9)} +still requires to be generalised. Obviously it is +immaterial whether the axes of~$K'$ be chosen so that +they are spatially parallel to those of~$K$. It is also not +essential that the velocity of translation of~$K'$ with +respect to~$K$ should be in the direction of the $x$-axis. +A simple consideration shows that we are able to +construct the Lorentz transformation in this general +sense from two kinds of transformations, viz.\ from +Lorentz transformations in the special sense and from +purely spatial transformations, which corresponds to +the replacement of the rectangular co-ordinate system +\PageSep{120} +by a new system with its axes pointing in other +directions. + +Mathematically, we can characterise the generalised +Lorentz transformation thus: +\index{Lorentz, H. A.!transformation!(generalised)}% + +It expresses $x'$,~$y'$, $z'$,~$t'$, in terms of linear homogeneous +functions of $x$,~$y$, $z$,~$t$, of such a kind that the relation +\[ +x'^{2} + y'^{2} + z'^{2} - c^{2} t'^{2} + = x^{2} + y^{2} + z^{2} - c^{2} t^{2} +\Tag{(11a)} +\] +is satisfied identically. That is to say: If we substitute +their expressions in $x$,~$y$, $z$,~$t$, in place of $x'$,~$y'$, +$z'$,~$t'$, on the left-hand side, then the left-hand side of~\Eqref{(11a)} +agrees with the right-hand side. +\PageSep{121} + + +\Appendix{II}{Minkowski's Four-dimensional Space +(``World'')}{[Supplementary to \Sectionref{XVII}]} + +\First{We} can characterise the Lorentz transformation +\index{Lorentz, H. A.!transformation}% +still more simply if we introduce the imaginary~$\sqrt{-1}·ct$ +in place of~$t$, as time-variable. If, in +accordance with this, we insert +\begin{align*} +x_{1} &= x\Add{,} \\ +x_{2} &= y\Add{,} \\ +x_{3} &= z\Add{,} \\ +x_{4} &= \sqrt{-1}·ct, +\end{align*} +and similarly for the accented system~$K'$, then the +condition which is identically satisfied by the transformation +can be expressed thus: +\[ +x_{1}'^{2} + x_{2}'^{2} + x_{3}'^{2} + x_{4}'^{2} + = {x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2} + {x_{4}}^{2}. +\Tag{(12)} +\] + +That is, by the afore-mentioned choice of ``co-ordinates,'' +\Eqref{(11a)}~is transformed into this equation. + +We see from~\Eqref{(12)} that the imaginary time co-ordinate~$x_{4}$ +\index{Cartesian system of co-ordinates}% +\index{Euclidean geometry}% +\index{Euclidean space}% +\index{Space!three-dimensional}% +\index{Time!in Physics}% +enters into the condition of transformation in exactly +the same way as the space co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$. It +is due to this fact that, according to the theory of +\PageSep{122} +relativity, the ``time''~$x_{4}$ enters into natural laws in the +same form as the space co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$. + +A four-dimensional continuum described by the +\index{Continuum!four-dimensional}% +``co-or\-di\-nates'' $x_{1}$,~$x_{2}$, $x_{3}$,~$x_{4}$, was called ``world'' by +\index{World}% +\index{World!point@{-point}}% +Minkowski, who also termed a point-event a ``world-point.'' +\index{Minkowski}% +From a ``happening'' in three-dimensional +space, physics becomes, as it were, an ``existence'' in +the four-dimensional ``world.'' + +This four-dimensional ``world'' bears a close similarity +to the three-dimensional ``space'' of (Euclidean) +analytical geometry. If we introduce into the latter a +new Cartesian co-ordinate system $(x_{1}', x_{2}', x_{3}')$ with +the same origin, then $x_{1}'$,~$x_{2}'$,~$x_{3}'$, are linear homogeneous +functions of $x_{1}$,~$x_{2}$,~$x_{3}$, which identically satisfy the +equation +\[ +x_{1}'^{2} + x_{2}'^{2} + x_{3}'^{2} + = {x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2}. +\] +The analogy with~\Eqref{(12)} is a complete one. We can +regard Minkowski's ``world'' in a formal manner as a +four-dimensional Euclidean space (with imaginary +time co-ordinate); the Lorentz transformation corresponds +to a ``rotation'' of the co-ordinate system in the +\index{Rotation}% +four-dimensional ``world.'' +\PageSep{123} + + +\Appendix{III}{The Experimental Confirmation of the +General Theory of Relativity}{} +\index{Theory}% + +\First{From} a systematic theoretical point of view, we +may imagine the process of evolution of an empirical +science to be a continuous process of induction. +\index{Induction}% +Theories are evolved and are expressed in +short compass as statements of a large number of individual +observations in the form of empirical laws, +\index{Empirical laws}% +from which the general laws can be ascertained by comparison. +Regarded in this way, the development of a +science bears some resemblance to the compilation of a +classified catalogue. It is, as it were, a purely empirical +enterprise. + +But this point of view by no means embraces the whole +of the actual process; for it slurs over the important +part played by intuition and deductive thought in the +\index{Deductive thought}% +\index{Intuition}% +development of an exact science. As soon as a science +has emerged from its initial stages, theoretical advances +are no longer achieved merely by a process of arrangement. +Guided by empirical data, the investigator +rather develops a system of thought which, in general, +is built up logically from a small number of fundamental +assumptions, the so-called axioms. We call such a +\index{Axioms}% +system of thought a \emph{theory}. The theory finds the +\PageSep{124} +\index{Classical mechanics}% +\index{Darwinian theory}% +justification for its existence in the fact that it correlates +a large number of single observations, and it is just here +that the ``truth'' of the theory lies. +\index{Theory!truth of}% + +Corresponding to the same complex of empirical data, +there may be several theories, which differ from one +another to a considerable extent. But as regards the +deductions from the theories which are capable of +being tested, the agreement between the theories may +be so complete, that it becomes difficult to find such +deductions in which the two theories differ from each +other. As an example, a case of general interest is +available in the province of biology, in the Darwinian +\index{Biology}% +theory of the development of species by selection in +the struggle for existence, and in the theory of development +which is based on the hypothesis of the hereditary +transmission of acquired characters. + +We have another instance of far-reaching agreement +between the deductions from two theories in Newtonian +mechanics on the one hand, and the general theory of +relativity on the other. This agreement goes so far, +that up to the present we have been able to find only +a few deductions from the general theory of relativity +which are capable of investigation, and to which the +physics of pre-relativity days does not also lead, and +this despite the profound difference in the fundamental +assumptions of the two theories. In what follows, we +shall again consider these important deductions, and we +shall also discuss the empirical evidence appertaining to +them which has hitherto been obtained. + + +\Subsection{a}{Motion of the Perihelion of Mercury} +\index{Perihelion of Mercury|(}% + +According to Newtonian mechanics and Newton's +\index{Newton's!law of gravitation}% +law of gravitation, a planet which is revolving round the +\PageSep{125} +sun would describe an ellipse round the latter, or, more +correctly, round the common centre of gravity of the +sun and the planet. In such a system, the sun, or the +common centre of gravity, lies in one of the foci of the +orbital ellipse in such a manner that, in the course of a +planet-year, the distance sun-planet grows from a +minimum to a maximum, and then decreases again to +a minimum. If instead of Newton's law we insert a +\index{Newton}% +somewhat different law of attraction into the calculation, +we find that, according to this new law, the motion +would still take place in such a manner that the distance +sun-planet exhibits periodic variations; but in this +case the angle described by the line joining sun and +planet during such a period (from perihelion---closest +proximity to the sun---to perihelion) would differ from~$360°$. +The line of the orbit would not then be a closed +one, but in the course of time it would fill up an annular +part of the orbital plane, viz.\ between the circle of +least and the circle of greatest distance of the planet from +the sun. + +According also to the general theory of relativity, +which differs of course from the theory of Newton, a +small variation from the Newton-Kepler motion of a +\index{Kepler}% +planet in its orbit should take place, and in such a way, +that the angle described by the radius sun-planet +between one perihelion and the next should exceed that +corresponding to one complete revolution by an amount +given by +\[ ++\frac{24\pi^{3} a^{2}}{T^{2} c^{2} (1-e^{2})}. +\] + +(\NB---One complete revolution corresponds to the +angle~$2\pi$ in the absolute angular measure customary in +physics, and the above expression gives the amount by +\PageSep{126} +which the radius sun-planet exceeds this angle during +the interval between one perihelion and the next.) +In this expression $a$~represents the major semi-axis of +the ellipse, $e$~its eccentricity, $c$~the velocity of light, and +$T$~the period of revolution of the planet. Our result +may also be stated as follows: According to the general +theory of relativity, the major axis of the ellipse rotates +round the sun in the same sense as the orbital motion +of the planet. Theory requires that this rotation should +amount to $43$~seconds of arc per~century for the planet +Mercury, but for the other planets of our solar system its +\index{Mercury}% +\index{Mercury!orbit of}% +magnitude should be so small that it would necessarily +escape detection.\footnote + {Especially since the next planet Venus has an orbit that is +\index{Venus}% + almost an exact circle, which makes it more difficult to locate + the perihelion with precision.} + +In point of fact, astronomers have found that the +theory of Newton does not suffice to calculate the +observed motion of Mercury with an exactness corresponding +to that of the delicacy of observation attainable +at the present time. After taking account of all +the disturbing influences exerted on Mercury by the +remaining planets, it was found (Leverrier---1859---and +\index{Leverrier}% +Newcomb---1895) that an unexplained perihelial +\index{Newcomb}% +movement of the orbit of Mercury remained over, the +amount of which does not differ sensibly from the above-mentioned +$+43$~seconds of arc per~century. The uncertainty +of the empirical result amounts to a few +seconds only. +\index{Perihelion of Mercury|)}% + + +\Subsection{b}{Deflection of Light by a Gravitational +Field} + +In \Sectionref{XXII} it has been already mentioned that, +\PageSep{127} +according to the general theory of relativity, a ray of +light will experience a curvature of its path when passing +\index{Curvature of light-rays}% +\index{Curvature of light-rays!space}% +through a gravitational field, this curvature being similar +to that experienced by the path of a body which is +projected through a gravitational field. As a result of +this theory, we should expect that a ray of light which +is passing close to a heavenly body would be deviated +towards the latter. For a ray of light which passes the +sun at a distance of $\Delta$~sun-radii from its centre, the +angle of deflection~($\alpha$) should amount to +\[ +\alpha = \frac{\text{$1.7$~seconds of arc}}{\Delta}. +\] +It may be added that, according to the theory, half of +this deflection is produced by the +Newtonian field of attraction of the +sun, and the other half by the geometrical +modification (``curvature'') +of space caused by the sun. + +%[Illustration: Fig. 5.] +\WFigure{1in}{127} +This result admits of an experimental +\index{Solar eclipse}% +test by means of the photographic +registration of stars during +a total eclipse of the sun. The only +reason why we must wait for a total +eclipse is because at every other +time the atmosphere is so strongly +illuminated by the light from the +sun that the stars situated near the +sun's disc are invisible. The predicted effect can be +seen clearly from the accompanying diagram. If the +sun~($S$) were not present, a star which is practically +infinitely distant would be seen in the direction~$D_{1}$, as +observed from the earth. But as a consequence of the +\PageSep{128} +deflection of light from the star by the sun, the star +will be seen in the direction~$D_{2}$, \ie\ at a somewhat +greater distance from the centre of the sun than corresponds +to its real position. + +In practice, the question is tested in the following +way. The stars in the neighbourhood of the sun are +photographed during a solar eclipse. In addition, a +\index{Solar eclipse}% +\index{Stellar universe!photographs}% +second photograph of the same stars is taken when the +sun is situated at another position in the sky, \ie\ a few +months earlier or later. As compared with the standard +photograph, the positions of the stars on the eclipse-photograph +ought to appear displaced radially outwards +(away from the centre of the sun) by an amount +corresponding to the angle~$\alpha$. + +We are indebted to the Royal Society and to the +Royal Astronomical Society for the investigation of +this important deduction. Undaunted by the war and +by difficulties of both a material and a psychological +nature aroused by the war, these societies equipped +two expeditions---to Sobral (Brazil), and to the island of +Principe (West Africa)---and sent several of Britain's +most celebrated astronomers (Eddington, Cottingham, +\index{Cottingham}% +\index{Eddington}% +Crommelin, Davidson), in order to obtain photographs +\index{Crommelin}% +\index{Davidson}% +of the solar eclipse of 29th~May, 1919. The relative +discrepancies to be expected between the stellar photographs +obtained during the eclipse and the comparison +photographs amounted to a few hundredths of a millimetre +only. Thus great accuracy was necessary in +making the adjustments required for the taking of the +photographs, and in their subsequent measurement. + +The results of the measurements confirmed the theory +in a thoroughly satisfactory manner. The rectangular +components of the observed and of the calculated +\PageSep{129} +deviations of the stars (in seconds of arc) are set forth +in the following table of results: +\[ +\begin{array}{@{}c*{2}{>{\quad}cc}@{}} +%[** TN: Re-break first column heading to improve overall width] +\ColHead{1}{Number of}{Number of\\ the Star.} & +\ColHead{2}{Observed. Calculated.}{First Co-ordinate. \\[2pt] +$\overbrace{\text{Observed. Calculated.}}$} & +\ColHead{2}{Observed. Calculated.}{Second Co-ordinate. \\[2pt] +$\overbrace{\text{Observed. Calculated.}}$} \\ +11 & -0.19 & -0.22 & +0.16 & +0.02 \\ +\Z5 & +0.29 & +0.31 & -0.46 & -0.43 \\ +\Z4 & +0.11 & +0.10 & +0.83 & +0.74 \\ +\Z3 & +0.20 & +0.12 & +1.00 & +0.87 \\ +\Z6 & +0.10 & +0.04 & +0.57 & +0.40 \\ +10 & -0.08 & +0.09 & +0.35 & +0.32 \\ +\Z2 & +0.95 & +0.85 & -0.27 & -0.09 +\end{array} +\] + +\Subsection{c}{Displacement of Spectral Lines towards +the Red} +\index{Displacement of spectral lines}% + +In \Sectionref{XXIII} it has been shown that in a system~$K'$ +which is in rotation with regard to a Galileian system~$K$, +clocks of identical construction, and which are considered +\index{Clocks}% +\index{Clocks!rate of}% +at rest with respect to the rotating reference-body, +go at rates which are dependent on the positions +of the clocks. We shall now examine this dependence +quantitatively. A clock, which is situated at a distance~$r$ +from the centre of the disc, has a velocity relative to~$K$ +which is given by +\[ +v = \omega r, +\] +where $\omega$~represents the angular velocity of rotation of the +disc~$K'$ with respect to~$K$. If $\nu_{0}$~represents the number +of ticks of the clock per unit time (``rate'' of the clock) +relative to~$K$ when the clock is at rest, then the ``rate'' +of the clock~($\nu$) when it is moving relative to~$K$ with +a velocity~$v$, but at rest with respect to the disc, will, +in accordance with \Sectionref{XII}, be given by +\[ +\nu = \nu_{0} \sqrt{1 - \frac{v^{2}}{c^{2}}}, +\] +\PageSep{130} +or with sufficient accuracy by +\[ +\nu = \nu_{0} \left(1 - \tfrac{1}{2}\, \frac{v^{2}}{c^{2}}\right). +\] +This expression may also be stated in the following +form: +\[ +\nu = \nu_{0} \left(1 - \frac{1}{c^{2}}\, \frac{\omega^{2} r^{2}}{2}\right). +\] +If we represent the difference of potential of the centrifugal +force between the position of the clock and the +centre of the disc by~$\phi$, \ie\ the work, considered negatively, +which must be performed on the unit of mass +against the centrifugal force in order to transport it +\index{Centrifugal force}% +from the position of the clock on the rotating disc to +the centre of the disc, then we have +\[ +\phi = -\frac{\omega^{2} r^{2}}{2}. +\] +From this it follows that +\[ +\nu = \nu_{0} \left(1 + \frac{\phi}{c^{2}}\right). +\] +In the first place, we see from this expression that two +clocks of identical construction will go at different rates +when situated at different distances from the centre of +the disc. This result is also valid from the standpoint +of an observer who is rotating with the disc. + +Now, as judged from the disc, the latter is in a gravitational +\index{Gravitational field!potential of}% +field of potential~$\phi$, hence the result we have +obtained will hold quite generally for gravitational +fields. Furthermore, we can regard an atom which is +emitting spectral lines as a clock, so that the following +statement will hold: + +\emph{An atom absorbs or emits light of a frequency which is +\PageSep{131} +dependent on the potential of the gravitational field in +\index{Gravitational field!potential of}% +which it is situated.} + +The frequency of an atom situated on the surface of a +\index{Frequency of atom}% +heavenly body will be somewhat less than the frequency +of an atom of the same element which is situated in free +space (or on the surface of a smaller celestial body). +Now $\phi = -K\dfrac{M}{r}$, where $K$~is Newton's constant of +\index{Newton's!constant of gravitation}% +gravitation, and $M$~is the mass of the heavenly body. +Thus a displacement towards the red ought to take place +for spectral lines produced at the surface of stars as +compared with the spectral lines of the same element +produced at the surface of the earth, the amount of this +displacement being +\[ +\frac{\nu_{0} - \nu}{\nu_{0}} = \frac{K}{c^{2}}\, \frac{M}{r}. +\] + +For the sun, the displacement towards the red predicted +by theory amounts to about two millionths of +the wave-length. A trustworthy calculation is not +possible in the case of the stars, because in general +neither the mass~$M$ nor the radius~$r$ is known. + +It is an open question whether or not this effect +exists, and at the present time astronomers are working +with great zeal towards the solution. Owing to the +smallness of the effect in the case of the sun, it is difficult +to form an opinion as to its existence. Whereas +Grebe and Bachem (Bonn), as a result of their own +\index{Bachem}% +\index{Grebe}% +measurements and those of Evershed and Schwarzschild +\index{Evershed}% +\index{Schwarzschild}% +on the cyanogen bands, have placed the existence of +\index{Cyanogen bands}% +the effect almost beyond doubt, other investigators, +particularly St.~John, have been led to the opposite +\index{St. John@{St.\ John}}% +opinion in consequence of their measurements. +\PageSep{132} + +Mean displacements of lines towards the less refrangible +end of the spectrum are certainly revealed by +statistical investigations of the fixed stars; but up +to the present the examination of the available data +does not allow of any definite decision being arrived at, +as to whether or not these displacements are to be +referred in reality to the effect of gravitation. The +results of observation have been collected together, +and discussed in detail from the standpoint of the +question which has been engaging our attention here, +in a paper by E.~Freundlich entitled ``Zur Prüfung der +allgemeinen Relativitäts-Theorie'' (\textit{Die Naturwissenschaften}, +1919, No.~35, p.~520: Julius Springer, Berlin). + +At all events, a definite decision will be reached during +the next few years. If the displacement of spectral +lines towards the red by the gravitational potential +does not exist, then the general theory of relativity +will be untenable. On the other hand, if the cause of +the displacement of spectral lines be definitely traced +to the gravitational potential, then the study of this +displacement will furnish us with important information +\index{Mass of heavenly bodies}% +as to the mass of the heavenly bodies. +\PageSep{133} + + +\backmatter +\BookMark{-1}{Back Matter} +\Bibliography{WORKS IN ENGLISH ON EINSTEIN'S THEORY} + +\Bibsection{Introductory} + +\Bibitem{The Foundations of Einstein's Theory of Gravitation} +{Erwin Freundlich (translation by H.~L.~Brose). +Camb.\ Univ.\ Press, 1920.} + +\Bibitem{Space and Time in Contemporary Physics}{Moritz Schlick +(translation by H.~L.~Brose). Clarendon Press, +Oxford, 1920.} + + +\Bibsection{The Special Theory} + +\Bibitem{The Principle of Relativity}{E.~Cunningham. Camb.\ +Univ.\ Press.} + +\Bibitem{Relativity and the Electron Theory}{E.~Cunningham, Monographs +on Physics. Longmans, Green \&~Co.} + +\Bibitem{The Theory of Relativity}{L.~Silberstein. Macmillan \&~Co.} + +\Bibitem{The Space-Time Manifold of Relativity}{E.~B.~Wilson +and G.~N.~Lewis, \textit{Proc.\ Amer.\ Soc.\ Arts \&~Science}, +vol.~xlviii., No.~11, 1912.} + + +\Bibsection{The General Theory} + +\Bibitem{Report on the Relativity Theory of Gravitation}{A.~S. +Eddington. Fleetway Press Ltd., Fleet Street, +London.} +\PageSep{134} + +\Bibitem{On Einstein's Theory of Gravitation and its Astronomical +Consequences}{W.~de~Sitter, \textit{M.~N.~Roy.\ Astron.\ +Soc.},~lxxvi.\ p.~699, 1916; lxxvii.\ p.~155, 1916; lxxviii.\ +p.~3, 1917.} + +\Bibitem{On Einstein's Theory of Gravitation}{H.~A.~Lorentz, \textit{Proc.\ +Amsterdam Acad.}, vol.~xix. p.~1341, 1917.} + +\Bibitem{Space, Time and Gravitation}{W.~de~Sitter: \textit{The +Observatory}, No.~505, p.~412. Taylor \&~Francis, Fleet +Street, London.} + +\Bibitem{The Total Eclipse of 29th~May, 1919, and the Influence of +Gravitation on Light}{A.~S.~Eddington, \textit{ibid.}, +March~1919.} + +\Bibitem{Discussion on the Theory of Relativity}{\textit{M.~N.~Roy.\ Astron.\ +Soc.}, vol.~lxxx.\ No.~2, p.~96, December~1919.} + +\Bibitem{The Displacement of Spectrum Lines and the Equivalence +Hypothesis}{W.~G.~Duffield, \textit{M.~N.~Roy.\ Astron.\ Soc.}, +vol.~lxxx.\Change{;}{} No.~3, p.~262, 1920.} + +\Bibitem{Space, Time and Gravitation}{A.~S.~Eddington, Camb.\ Univ.\ +Press, 1920.} + + +\Bibsection{Also, Chapters in} + +\Bibitem{The Mathematical Theory of Electricity and Magnetism} +{J.~H. Jeans (4th~edition). Camb.\ Univ.\ Press, 1920.} + +\Bibitem{The Electron Theory of Matter}{O.~W.~Richardson. Camb.\ +Univ.\ Press.} +\PageSep{135} +\printindex % [** TN: Auto-generate the index] +\iffalse %%%% Start of index text %%%% +INDEX + +Aberration 49 + +Absorption of energy 46 + +Acceleration 64, 67, 70 + +Action at a distance 48 + +Addition of velocities 16, 38 + +Adjacent points 89 + +Aether 52 + drift@{-drift}#drift 52, 53 + +Arbitrary substitutions 98 + +Astronomy 7, 102 + +Astronomical day 11 + +Axioms 2, 123 + truth of 2 + +Bachem 131 + +Basis of theory 44 + +Being@{``Being''}#Being 66, 108 + +beta-rays@{$\beta$-rays}#rays 50 + +Biology 124 + +Cartesian system of co-ordinates 7, 84, 122 + +Cathode rays 50 + +Celestial mechanics 105 + +Centrifugal force 80, 130 + +Chest 66 + +Classical mechanics 9, 13, 14, 16, 30, 44, 71, 102, 103, 124 + truth of 13 + +Clocks 10, 23, 80, 81, 94, 95, 98-100, 102, 113, 129 + rate of 129 + +Conception of mass 45 + position 6 + +Conservation of energy 45, 101 + impulse 101 + mass 45, 47 + +Continuity 95 + +Continuum 55, 83 + two-dimensional 94 + three-dimensional 57 + four-dimensional 89, 91, 92, 94, 122 + space-time 78, 91-96 + Euclidean 84, 86, 88, 92 + non-Euclidean 86, 90 + +Coordinate@{Co-ordinate}#Co-ordinate + differences 92 + differentials 92 + planes 32 + +Cottingham 128 + +Counter-Point 112 + +Covariant@{Co-variant}#Co-variant 43 + +Crommelin 128 + +Curvature of light-rays 104, 127 + space 127 + +Curvilinear motion 74 + +Cyanogen bands 131 + +Darwinian theory 124 + +Davidson 128 + +Deductive thought 123 + +Derivation of laws 44 + +Desitter@{De Sitter}#De Sitter 17 + +Displacement of spectral lines 104, 129 + +Distance (line-interval) 3, 5, 8, 28, 29, 84, 88, 109 + physical interpretation of 5 + relativity of 28 + +Doppler principle 50 %. + +Double stars 17 + +Eclipse of star 17 + +Eddington 104, 128 +%\PageSep{136} + +Electricity 76 + +Electrodynamics 13, 19, 41, 44, 76 + +Electromagnetic theory 49 + waves 63 + +Electron 44, 50 %. + electrical masses of 51 + +Electrostatics 76 + +Elliptical space 112 + +Empirical laws 123 + +Encounter (space-time coincidence) 95 + +Equivalent 14 + +Euclidean geometry 1, 2, 57, 82, 86, 88, 108, 109, 113, 122 + propositions of 3, 8 + +%[** TN: Add explicit "Euclidean" heading] +Euclidean space 57, 86, 122 + +Evershed 131 + +Experience 49, 60 + +Faraday 48, 63 + +FitzGerald 53 + +Fixed stars 11 + +Fizeau 39, 49, 51 + experiment of 39 + +Frequency of atom 131 + +Galilei 11 + transformation 33, 36, 38, 42, 52 + +Galileian system of co-ordinates + 11, 13, 14, 46, 79, 91, 98, 100 + +Gauss 86, 87, 90 + +Gaussian co-ordinates 88-90, 94, 96-100 + +General theory of relativity 59-104, 97 + +Geometrical ideas 2, 3 + propositions 1 + truth of 2-4 + +Gravitation 64, 69, 78, 102 + +Gravitational field 64, 67, 74, 77, 93, 98, 100, 101, 113 + potential of 130, 131 + +%[** TN: Add explicit "Gravitational" heading] +Gravitational mass 65, 68, 102 + +Grebe 131 + +Group-density of stars 106 + +Helmholtz 108 + +Heuristic value of relativity#Heuristic 42 + +Induction 123 + +Inertia 65 + +Inertial mass 47, 65, 69, 101, 102 + +Instantaneous photograph (snapshot) 117 + +Intensity of gravitational field 106 + +Intuition 123 + +Ions 44 + +Kepler 125 + +Kinetic energy 45, 101 + +Lattice 108 + +Law of inertia 11, 61, 62, 98 + +Laws of Galilei-Newton 13 + of Nature 60, 71, 99 + +Leverrier 103, 126 + +Light-signal 33, 115, 118 + +Light-stimulus 33 + +Limiting velocity ($c$)#Limiting 36, 37 + +Lines of force 106 + +Lorentz, H. A.#Lorentz 19, 41, 44, 49, 50-53 + transformation 33, 39, 42, 91, 97, 98, 115, 118, 119, 121 + (generalised) 120 + +Mach, E.#Mach 72 + +Magnetic field 63 + +Manifold|see{Continuum} 0 + +Mass of heavenly bodies 132 + +Matter 101 + +Maxwell 41, 44, 48-50, 52 + fundamental equations 46, 77 + +Measurement of length 85 + +Measuring-rod 5, 6, 28, 80, 81, 94, 100, 102, 111, 113, 117 + +Mercury 103, 126 + orbit of 103, 126 + +Michelson 52-54 + +Minkowski 55-57, 91, 122 +%\PageSep{137} + +Morley 53, 54 + +Motion 14, 60 + of heavenly bodies 13, 15, 44, 102, 113 + +Newcomb 126 + +Newton 11, 72, 102, 105, 125 + +Newton's + constant of gravitation 131 + law of gravitation 48, 80, 106, 124 + law of motion 64 + +Non-Euclidean geometry 108 + +Non-Galileian reference-bodies 98 + +Non-uniform motion 62 + +Optics 13, 19, 44 + +Organ-pipe, note of 14 + +Parabola 9, 10 + +Path-curve 10 + +Perihelion of Mercury 124-126 + +Physics 7 + of measurement 7 + +Place specification 5, 6 + +Plane 1, 108, 109 + +Poincare@{Poincaré}#Poincaré 108 + +Point 1 + +Point-mass, energy of#Point-mass 45 + +Position 9 + +Principle of relativity 13-15, 19, 20, 60 + +Processes of Nature 42 + +Propagation of light 17, 19, 20, 32, 91, 119 + in liquid 40 + in gravitational fields 75 + +Quasi-Euclidean universe 114 + +Quasi-spherical universe 114 + +Radiation 46 + +Radioactive substances 50 + +Reference-body 5, 7, 9-11, 18, 23, 25, 26, 37, 60 + rotating 79 + +%[** TN: Add explicit "Reference-" heading] +Reference-mollusk 99-101 + +Relative + position 3 + velocity 117 + +Rest 14 + +Riemann 86, 108, 111 + +Rotation 81, 122 + +Schwarzschild 131 + +Seconds-clock 36 + +Seeliger 105, 106 + +Simultaneity 22, 24-26, 81 + relativity of 26 + +Size-relations 90 + +Solar eclipse 75, 127, 128 + +Space 9, 52, 55, 105 + conception of 19 + +Space co-ordinates 55, 81, 99 + +Space + interval@{-interval}#interval 30, 56 + point@{-point}#point 99 + two-dimensional 108 + three-dimensional 122 + +Special theory of relativity 1-57, 20 + +Spherical + surface 109 + space 111, 112 + +St. John@{St.\ John}#St.~John 131 + +Stellar universe 106 + photographs 128 + +Straight line 1-3, 9, 10, 82, 88, 109 + +System of co-ordinates 5, 10, 11 + +Terrestrial space 15 + +Theory 123 + truth of 124 + +Three-dimensional 55 + +Time + conception of 19, 52, 105 + coordinate@{co-ordinate}#co-ordinate 55, 99 + in Physics 21, 98, 122 + of an event 24, 26 + +Time-interval 30, 56 + +Trajectory 10 + +Truth@{``Truth''}#Truth 2 + +Uniform translation 12, 59 + +Universe (World) structure of 108, 113 + circumference of 111 +%\PageSep{138} + +Universe + elliptical 112, 114 + Euclidean 109, 111 + space expanse (radius) of 114 + spherical 111, 114 + +Value of $\pi$#$\pi$ 82, 110 + +Velocity of light 10, 17, 18, 76, 118 + +Venus 126 + +Weight (heaviness) 65 + +World 55, 56, 109, 122 + +World + point@{-point}#point 122 + radius@{-radius}#radius 112 + sphere@{-sphere}#sphere 110, 111 + +Zeeman 41 +\fi %%%% End of index text %%%% +\PageSep{139} +% [Blank page] +\PageSep{140} +\ifthenelse{\boolean{ForPrinting}}{\cleardoublepage\null}{} +\newpage +\begin{CenterPage} + \scriptsize + PRINTED BY \\[2pt] + MORRISON AND GIBB LIMITED \\[2pt] + EDINBURGH +\end{CenterPage} +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\cleardoublepage +\BookMark{0}{PG License} +\SetEvenHead{Licensing} +\SetOddHead{Licensing} +\pagenumbering{Roman} +\begin{PGtext} +End of the Project Gutenberg EBook of Relativity: The Special and the +General Theory, by Albert Einstein + +*** END OF THIS PROJECT GUTENBERG EBOOK RELATIVITY *** + +***** This file should be named 36114-pdf.pdf or 36114-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/6/1/1/36114/ + +Produced by Andrew D. 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