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-rw-r--r--old/36114-pdf 2011-05-11.pdf13884
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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
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+% 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 %
+% %
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+%%%%%%%%%%%%%%%%%%%%%%%% 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
+ Relativittsprinzip} (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 Relativittsprinzip}---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 Relativittstheorie} (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, Wrtemberg, 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 rle 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 ther 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 ther.
+\index{Aether}%
+\index{Aether!-drift}%
+To this motion of~$K'$ against the ther (``ther-drift''
+relative to~$K'$) were assigned the more complicated
+laws which were supposed to hold relative to~$K'$.
+Strictly speaking, such an ther-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 ther-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 ther. 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 ther. And
+\index{Aether!-drift}%
+yet another point: it is shown by calculation that for
+a given velocity~$v$ with reference to the ther, 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 ther 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 ther-idea, and hence there can be no ther-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 rle, 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 rle 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 Poincar, whereas I can only touch on them
+\index{Poincare@{Poincar}}%
+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 Prfung der
+allgemeinen Relativitts-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@{Poincar}#Poincar 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]
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generated by cgit v1.2.3 (git 2.25.1) at 2025-12-25 22:01:27 +0000