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Relativity

Albert
Einstein
Relativity
The Special and the General Theory
Authorised translation by Robert W. Lawson
London and New York

First published in 1916
English edition ﬁrst published in the United Kingdom 1920 by Methuen & Co. Ltd Fifteenth, enlarged edition January 1954 First published by Routledge 1993
First published in Routledge Classics 2001 by Routledge 11 New Fetter Lane, London EC4P 4EE 29 West 35th Street, New York, NY 10001
Routledge is an imprint of the Taylor & Francis Group
This edition published in the Taylor & Francis e-Library, 2002.
All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers.
British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library
ISBN 0–415–25538–4 (hbk) ISBN 0–415–25384–5 (pbk) ISBN 0-203-19871-9 Master e-book ISBN ISBN 0-203-19874-3 (Glassbook Format)

CONTENTS

Preface

ix

PART I The Special Theory of Relativity

1 Physical Meaning of Geometrical Propositions

3

2 The System of Co-ordinates

6

3 Space and Time in Classical Mechanics

10

4 The Galileian System of Co-ordinates

12

5 The Principle of Relativity (in the Restricted

Sense)

14

6 The Theorem of the Addition of Velocities

Employed in Classical Mechanics

18

7 The Apparent Incompatibility of the Law of

Propagation of Light with the Principle of

Relativity

20

8 On the Idea of Time in Physics

23

9 The Relativity of Simultaneity

27

10 On the Relativity of the Conception of Distance 30

11 The Lorentz Transformation

32

vi contents

12 The Behaviour of Measuring-Rods and

Clocks in Motion

37

13 Theorem of the Addition of Velocities. The

Experiment of Fizeau

40

14 The Heuristic Value of the Theory of Relativity

44

15 General Results of the Theory

46

16 Experience and the Special Theory

of Relativity

51

17 Minkowski’s Four-dimensional Space

56

PART II The General Theory of Relativity

18 Special and General Principle of Relativity

61

19 The Gravitational Field

65

20 The Equality of Inertial and Gravitational

Mass as an Argument for the General

Postulate of Relativity

68

21 In what Respects are the Foundations of

Classical Mechanics and of the Special

Theory of Relativity Unsatisfactory?

72

22 A Few Inferences from the General Principle

of Relativity

75

23 Behaviour of Clocks and Measuring-Rods on

a Rotating Body of Reference

79

24 Euclidean and non-Euclidean Continuum

83

25 Gaussian Co-ordinates

87

26 The Space-Time Continuum of the Special

Theory of Relativity Considered as a

Euclidean Continuum

91

27 The Space-Time Continuum of the General

Theory of Relativity is not a Euclidean

Continuum

94

28 Exact Formulation of the General Principle of

Relativity

97

contents vii

29 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

30 Cosmological Difﬁculties of Newton’s Theory

107

31 The Possibility of a “Finite” and yet

“Unbounded” Universe

110

32 The Structure of Space according to the

General Theory of Relativity

115

APPENDICES

1 Simple Derivation of the Lorentz

Transformation [Supplementary to Section 11] 117

2 Minkowski’s Four-dimensional Space

(“World”) [Supplementary to Section 17]

124

3 The Experimental Conﬁrmation of the

General Theory of Relativity

126

(a) Motion of the Perihelion of Mercury

127

(b) Deﬂection of Light by a Gravitational Field 129

(c) Displacement of Spectral Lines towards

the Red

132

4 The Structure of Space according to the

General Theory of Relativity [Supplementary

to Section 32]

136

5 Relativity and the Problem of Space

139

Bibliography

159

Index

161

PREFACE
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 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 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

x preface
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!

December, 1916

A. EINSTEIN

NOTE TO THE FIFTEENTH EDITION
In this edition I have added, as a ﬁfth appendix, a presentation of my views on the problem of space in general and on the gradual modiﬁcations of our ideas on space resulting from the inﬂuence of the relativistic view-point. I wished to show that space-time is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept “empty space” loses its meaning.

June 9th 1952

A. EINSTEIN

Part I
The Special Theory of Relativity

1
PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS
In your schooldays most of you who read this 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 “plane,” “point,” and “straight line,” with which we are able to associate more or less deﬁnite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as “true.” Then, on the basis of a logical process, the

4 special theory of relativity
justiﬁcation of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. 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 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 “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 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 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 the single proposition that two points on a practically rigid body always

geometrical propositions 5
correspond to the same distance (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 rigid bodies.1 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 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.
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.
1 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.

2
THE SYSTEM OF CO-ORDINATES
On the basis of the physical interpretation of distance 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.1
Every description of the scene of an event or of the position of
1 Here we have assumed that there is nothing left over, i.e. 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.

the system of co-ordinates 7
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 Square, London,”1 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.2
This primitive method of place speciﬁcation deals 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. 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, 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.
(a) 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.
(b) In locating the position of the object, we make use of a
1 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.) 2 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.

8 special theory of relativity
number (here the length of the pole measured with the measuring-rod) instead of designated points of reference.
(c) 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 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 the physics of measurement this is attained by the application of the Cartesian system of co-ordinates.
This consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body. Referred to as 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 are to maintain their clearness, the physical meaning of speciﬁcations of position must always be sought in accordance with the above considerations.1
1 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.

the system of co-ordinates 9
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 geometry hold for “distances,” the “distance” being represented physically by means of the convention of two marks on a rigid body.

3
SPACE AND TIME IN CLASSICAL MECHANICS
The purpose of mechanics is to describe how bodies change their position in space with “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 “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 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, 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

space and time in classical mechanics 11
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 preceding section. If instead of “body of reference” we insert “system of co-ordinates,” which is a useful idea for mathematical description, we are in a position to say: The stone traverses a straight line relative to a 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. With the aid of this example it is clearly seen that there is no such thing as an independently existing trajectory (lit. “pathcurve”1), but only a trajectory relative to a particular body of reference.
In order to have a complete description of the motion, we must specify how the body alters its position with time; i.e. 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 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 second diﬃculty prevailing here we shall have to deal in detail later.
1 That is, a curve along which the body moves.

4
THE GALILEIAN SYSTEM OF CO-ORDINATES
As is well known, the fundamental law of the mechanics of Galilei-Newton, which is known as the law of inertia, can be stated thus: 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 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 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

the galileian system of co-ordinates 13
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.

5
THE PRINCIPLE OF RELATIVITY (IN THE RESTRICTED SENSE)
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 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

the principle of relativity 15
accordance with the discussion contained in the preceding section, it follows that:
If K is a Galileian co-ordinate system, then every other coordinate system K′ is a Galileian one, when, in relation to K, it is in a condition of uniform motion of translation. Relative to K′ the mechanical 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 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 validity of this principle of relativity. But in view of the more recent development of electrodynamics and optics it became more and more evident that classical 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 apply with great accuracy in the domain of mechanics. But that a principle of such broad generality should hold with such exactness in one

16 special theory of relativity
domain of phenomena, and yet should be invalid for another, is 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 moving uniformly relative to each other, will not be equivalent for the description of natural phenomena. 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 one (K0) of a particular state of motion as our body of reference. We should 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.” If, for instance, our embankment were the system K0, then our railway carriage would be a system K, relative to which less simple laws would hold than with respect to K0. This diminished simplicity would be due to the fact that the carriage K would be in motion (i.e. “really”) with respect to K0. 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 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. Now in virtue of its motion in an orbit round the sun, 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 principle of relativity 17
the velocity of revolution of the earth in the course of a year, the earth cannot be at rest relative to the hypothetical system K0 throughout the whole year. However, the most careful observations have never revealed such anisotropic properties in terrestrial physical space, i.e. a physical non-equivalence of diﬀerent directions. This is a very powerful argument in favour of the principle of relativity.

6
THE THEOREM OF THE ADDITION OF VELOCITIES EMPLOYED IN CLASSICAL
MECHANICS
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

classical mechanics 19
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.

7
THE APPARENT INCOMPATIBILITY OF THE LAW OF PROPAGATION OF LIGHT WITH THE PRINCIPLE
OF RELATIVITY
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 means of similar considerations based on observations of double stars, the Dutch astronomer De Sitter 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

the propagation of light 21
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 justiﬁably believed by the child at school. Who would imagine that this simple law has plunged the conscientiously thoughtful physicist into the greatest 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 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 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 Section 5. For, like every other general law of nature, the law of the transmission of light 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

22 special theory of relativity
reference. 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 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 or the simple law of the propagation of light 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 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 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 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 in reality there is not the least incompatibility between the principle of relativity and the law of 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 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.

8
ON THE IDEA OF TIME IN PHYSICS
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,

24 special theory of relativity
then we should be faced with the task of testing whether or not this theoretical result is in accordance with the reality. We encounter 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 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 (e.g. 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 only I knew that the light by means of which the observer at M perceives the lightning ﬂashes travels along the length A → M with the same velocity as along the length B → M. But an examination of this supposition 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.”

idea of time in physics 25
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 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 → M as for the path B → M is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own freewill 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 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 body of reference1 (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 construction are placed at the points A, B and C of 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) of that of 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
1 We suppose further, that, when three events A, B and C occur in diﬀerent places in such a manner that A is simultaneous with B, 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 in vacuo.

26 special theory of relativity
validity of which will hardly be doubted without empirical evidence to the contrary. It has been assumed that all these clocks go 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 particular position of the pointers of the one clock is simultaneous (in the above sense) with the same position of the pointers of the other clock, then identical “settings” are always simultaneous (in the sense of the above deﬁnition).

9
THE RELATIVITY OF SIMULTANEITY
Up to now our considerations have been referred 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 Fig. I. People travelling in this train will with advantage use the train
Figure 1
as a rigid reference-body (co-ordinate system); they regard all events in 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

28 special theory of relativity
embankment. As a natural consequence, however, the following question arises:
Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous 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 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 → B of the embankment. But the events A and B also correspond to positions A and B on the train. Let M′ be the mid-point of the distance A → B on the travelling train. Just when the ﬂashes1 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, i.e. 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 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 vice versa (relativity of simultaneity). Every reference-body (coordinate system) has its own particular time; unless we are told
1 As judged from the embankment.

the relativity of simultaneity 29
the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.
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, i.e. 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 in vacuo and the principle of relativity (developed in Section 7) disappears.
We were led to that conﬂict by the considerations of Section 6, which are now no longer tenable. In that section we concluded that the man in the carriage, who traverses the distance w per second relative to the carriage, traverses the same distance also with respect to the embankment 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 Section 6 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.

10
ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE
Let us consider two particular points on the train1 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 (coordinate system). An observer in the train measures the interval by marking oﬀ his measuring-rod in a straight line (e.g. 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
1 e.g. the middle of the ﬁrst and of the twentieth carriage.

the relativity of distance 31
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 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 Section 8. The distance between these points A and B is then measured by repeated application of the measuring-rod along the embankment.
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 Section 6. Namely, if the man in the carriage covers the distance w in a unit of time— measured from the train,—then this distance—as measured from the embankment—is not necessarily also equal to w.

11
THE LORENTZ TRANSFORMATION
The results of the last three sections show that the apparent incompatibility of the law of propagation of light with the principle of relativity (Section 7) has been derived by means of a consideration which borrowed two unjustiﬁable hypotheses from classical mechanics; these are as follows:
(1) The time-interval (time) between two events is independent of the condition of motion of the body of reference.
(2) The space-interval (distance) between two points of a rigid body is independent of the condition of motion of the body of reference.
If we drop these hypotheses, then the dilemma of Section 7 disappears, because the theorem of the addition of velocities derived in Section 6 becomes invalid. The possibility presents itself that the law of the propagation of light in vacuo may be compatible with the principle of relativity, and the question

the lorentz transformation 33
arises: How have we to modify the considerations of Section 6 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 Section 6 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 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 referencebodies, 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 Section 2 we can imagine this referencebody 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 to the impenetrability of solid bodies. In every such framework we imagine three surfaces perpendicular to each other marked out,

34 special theory of relativity
and designated as “co-ordinate 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′, 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.
Figure 2
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 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 (Fig. 2), this problem is solved by means of the equations:
x − vt x′ =
Ί v2 1− c2

the lorentz transformation 35
y′ = y z′ = z
t − v .x c2
t′ =
Ί v2 1− c2
This system of equations is known as the “Lorentz transformation.”1
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:
x′ = x − vt y′ = y z′ = z t′ = t.
This system of equations is often termed the “Galilei 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 in vacuo is satisﬁed both for the referencebody K and for the reference-body K′. A light-signal is sent along the positive x-axis, and this light-stimulus advances in accordance with the equation
x = ct,
1 A simple derivation of the Lorentz transformation is given in Appendix 1.

36 special theory of relativity
i.e. 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:

Ί x′

=

(c − v)t

1

−

v2 c2

΂1 − v΃t
t′ = c ,
Ί v2 1− c2

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.

12
THE BEHAVIOUR OF MEASURING-RODS AND
CLOCKS IN MOTION

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 relative 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

Ί v2
x(beginning of rod) = 0 1 − c2

Ί x(end of rod) = 1.

1 − v2, c2

38 special theory of relativity
Ί v2
the distance between the points being 1 − . But the metrec2
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 Ί1 − v2/c2 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 Ί1 − v2/c2 = 0 and for still greater velocities the square-root becomes imaginary. From this we conclude that in the theory of relativity the velocity c plays the part of a limiting 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 Ί1 − v2/c2; this is quite in accordance with the principle of relativity which forms the basis of our considerations.
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 Galileian transformation we 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 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:

rods and clocks in motion 39

t=0

and

Ί t =

1

1

−

v2 c2

As judged from K, the clock is moving with the velocity v; as

judged from this reference-body, the time which elapses

between two strokes of the clock is not one second, but

1

Ί1

−

v2 c2

seconds,

i.e.

a

somewhat

larger

time.

As

a

con-

sequence 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.

13
THEOREM OF THE ADDITION OF VELOCITIES. THE
EXPERIMENT OF FIZEAU
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 Section 6 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 transformation (Section 11). 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

the experiment of fizeau 41

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.

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 Section 6,

W=v+w . . .

(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 Lorentz transformation. Instead
of the equation (A) we then obtain the equation

W= v+w . . .

(B),

vw

1 + c2

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 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

42 special theory of relativity
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, Fig. 3) when the liquid above mentioned is ﬂowing through the tube with a velocity v?

Figure 3

In accordance with the principle of relativity we shall certainly have to take for granted that the propagation of light always takes place with the same velocity w 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 Section 6 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 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 (A) or (B), according as the Galilei transformation or the Lorentz transformation corresponds to the facts. Experiment1 decides in

΂ ΃1

c

1 Fizeau found W = w + v 1 − , where n = is the index of refraction of the

n2

w

vw liquid. On the other hand, owing to the smallness of c2 as compared with 1,

΂ ΃ we

can

replace

(B)

in

the

ﬁrst

place

by

W

= (w + v)

1

−

vw c2

,

or

to

the

same

΂ ΃1
order of approximation by w + v 1 − . which agrees with Fizeau’s result. n2

the experiment of fizeau 43
favour of equation (B) derived from the theory of relativity, and the agreement is, indeed, very exact. According to recent and most excellent measurements by Zeeman, the inﬂuence of the velocity of ﬂow v on the propagation of light is represented by formula (B) to within one per cent.
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 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 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.

14
THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY
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 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. In this connection we did not obtain the Galilei transformation, but, diﬀering from classical mechanics, the Lorentz 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

heuristic value of relativity 45
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′. 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.
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.

15
GENERAL RESULTS OF THE THEORY
It is clear from our previous considerations that the (special) theory of relativity has grown out of electrodynamics and optics. In these ﬁelds it has not appreciably altered the predictions of theory, but it has considerably simpliﬁed the theoretical structure, i.e. the derivation of laws, and—what is incomparably more important—it has considerably reduced the number of independent hypotheses forming the basis of theory. The special theory of relativity has rendered the Maxwell-Lorentz theory so plausible, that the latter 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 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

general results of the theory 47
and ions; for other motions the variations from the laws of classical mechanics are too small to make themselves evident in practice. We shall not consider the motion of stars until we come to speak of the general theory of relativity. In accordance with the theory of relativity the kinetic energy of a material point of mass m is no longer given by the well-known expression
v2 m= ,
2
but by the expression
mc2 .
Ί v2 1 − c2
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
v2 3 v4 mc2 + m + m + . . . .
2 8 c2
v2 When c2 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 mc2 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 of its essential signiﬁcance later.

48 special theory of relativity
The most important result of a general character to which the special theory of relativity has led is concerned with the conception of mass. Before the advent of 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 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 coordinate 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. 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 moving with the velocity v, which absorbs1 an amount of energy E0 in the form of radiation without suﬀering an alteration in velocity in the process, has, as a consequence, its energy increased by an amount
E0 .
Ί v2 1 − c2
In consideration of the expression given above for the kinetic
1 E0 is the energy taken up, as judged from a co-ordinate system moving with the body.

general results of the theory 49

energy of the body, the required energy of the body comes out to be

΂ ΃ m + E0 c2 c2 .
Ί v2 1− c2

΂ ΃ Thus the body has the same energy as a body of mass

m

+

E0 c2

moving with the velocity v. Hence we can say: If a body takes

up an amount of energy E0 then its inertial mass increases by an amount E0; the inertial mass of a body is not a constant, but
c2

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 valid provided that the system neither takes

up nor sends out energy. Writing the expression for the energy

in the form

mc2 + E0,

Ί1

−v2 c2

we see that the term mc2, which has hitherto attracted our attention, is nothing else than the energy possessed by the body1 before it absorbed the energy E0.
A direct comparison of this relation with experiment is not possible at the present time (1920; see Note, p. 50), owing to the

1 As judged from a co-ordinate system moving with the body.

50 special theory of relativity
fact that the changes in energy E0 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. E0 is too small in
c2 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.
Let me add a ﬁnal remark of a fundamental nature. The success of the Faraday–Maxwell interpretation of electromagnetic action at a distance resulted in physicists 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 gravitation. According to the theory of relativity, action at a distance with the velocity of light always takes the place of instantaneous action at a distance or of action at a distance with an inﬁnite velocity of transmission. This is connected with the fact that the velocity c plays a fundamental rôle in this theory. In Part II we shall see in what way this result becomes modiﬁed in the general theory of relativity.
N.—With the advent of nuclear transformation processes, which result from the bombardment of elements by α-particles, protons, deuterons, neutrons or γ-rays, the equivalence of mass and energy expressed by the relation E = mc2 has been amply conﬁrmed. The sum of the reacting masses, together with the mass equivalent of the kinetic energy of the bombarding particle (or photon), is always greater than the sum of the resulting masses. The diﬀerence is the equivalent mass of the kinetic energy of the particles generated, or of the released electromagnetic energy (γ-photons). In the same way, the mass of a spontaneously disintegrating radioactive atom is always greater than the sum of the masses of the resulting atoms by the mass equivalent of the kinetic energy of the particles generated (or of the photonic energy). Measurements of the energy of the rays emitted in nuclear reactions, in combination with the equations of such reactions, render it possible to evaluate atomic weights to a high degree of accuracy.
R. W. L.

16
EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY
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. The special theory of relativity has crystallised out from the Maxwell–Lorentz theory of electromagnetic phenomena. Thus all facts of experience which support the electromagnetic theory also support the theory of 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 inﬂuence of the radial components of the relative motions of the ﬁxed stars with respect to the earth on

52 special theory of relativity
the colour of the light reaching us from them. The 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 arguments in favour of the Maxwell-Lorentz theory, 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 auxiliary hypothesis, which in itself—i.e. without making use of the theory of relativity—appears extraneous.
It is known that cathode rays and the so-called β-rays emitted by radioactive substances consist of 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 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 hitherto remained obscure to us.1 If we now assume that the relative distances between the electrical masses constituting the
1 The general theory of relativity renders it likely that the electrical masses of an electron are held together by gravitational forces.

experience and relativity 53
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 form of the electron experiences a contraction in the direction of motion in consequence of that motion, the contracted length being pro-
Ί v2
portional to the expression 1 - . This hypothesis, which is c2
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 Section 13 in connection with the experiment of Fizeau, the 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 Section 5 that all attempts of this nature led to a negative result. Before the theory of relativity was put forward, it was diﬁcult to become reconciled to this negative result, for reasons now to be discussed. The inherited prejudices about time and space did not allow any doubt to arise as to the prime importance of the Galileian transformation for changing over from one body of reference to another. Now assuming that the Maxwell–Lorentz equations hold for a reference-body K, 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,

54 special theory of relativity
one (K) corresponding to a particular state of motion is physically unique. This result was interpreted physically by regarding K as at rest with respect to a hypothetical æther of space. On the other hand, all co-ordinate systems K′ moving relatively to K were to be regarded as in motion with respect to the æther. To this motion of K′ against the æther (“æther-drift” relative to K′) were attributed the more complicated laws which were supposed to hold relative to K′. Strictly speaking, such an æther-drift ought also to be assumed relative to the earth, and for a long time the eﬀorts of physicists were devoted to attempts to detect the existence of an æther-drift at the earth’s surface.
In one of the most notable of these attempts Michelson devised a method which appears as though it must be decisive. Imagine two mirrors so arranged on a rigid body that the reﬂecting surfaces face each other. A ray of light requires a perfectly deﬁnite time T to pass from one mirror to the other and back again, if the whole system be at rest with respect to the æther. It is found by calculation, however, that a slightly diﬀerent time T′ is required for this process, if the body, together with the mirrors, be moving relatively to the æther. And yet another point: it is shown by calculation that for a given velocity v with reference to the æther, this time T′ is diﬀerent when the body is moving perpendicularly to the planes of the mirrors from that resulting when the motion is parallel to these planes. Although the estimated diﬀerence between these two times is exceedingly small, Michelson and Morley performed an 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 theory from this diﬃculty by assuming that the motion of the body relative to the æther produces a contraction of the body in the direction of motion, the amount of contraction being just sufﬁcient to compensate for the diﬀerence in time mentioned above. Comparison with the discussion in Section 12 shows that

experience and relativity 55
also from the standpoint of the theory of relativity this solution of the diﬃculty was the right one. But on the basis of the theory of relativity the method of interpretation is incomparably more satisfactory. According to this theory there is no such thing as a “specially favoured” (unique) co-ordinate system to occasion the introduction of the æther-idea, and hence there can be no æther-drift, nor any experiment with which to demonstrate it. Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses; and as the prime factor involved in this contraction we ﬁnd, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. Thus for a coordinate system moving with the earth the mirror system of Michelson and Morley is not shortened, but it is shortened for a co-ordinate system which is at rest relatively to the sun.

17
MINKOWSKI’S FOURDIMENSIONAL SPACE
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 we live is a four-dimensional space-time continuum.
Space is a three-dimensional continuum. By this 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 x1, y1, z1, 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

four-dimensional space 57

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” events (realised or at least thinkable) as we care to choose, the co-ordinates x1, y1, z1, t1 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 fourdimensional continuum is due to the fact that in physics, before the advent of the theory of relativity, time played a diﬀerent and more independent rôle, as compared with the space coordinates. 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, i.e. 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:

Ί t′ =

t

−

v c2

x

.

1

−

v2 c2

Moreover, according to this equation the time diﬀerence ∆t′ of two events with respect to K′ does not in general vanish, even when the time diﬀerence ∆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 for

58 special theory of relativity
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 most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space.1 In order to give due prominence to this relationship, however, we must replace the usual time coordinate t by an imaginary magnitude Ί− 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 role 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 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 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 leave it there at present, and revert to it only towards the end of Part II.
1 Cf. the somewhat more detailed discussion in Appendix 2.

Part II
The General Theory of Relativity

18
SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY
The basal principle, which was the pivot of all our previous considerations, was the special principle of relativity, i.e. the principle of the physical relativity of all uniform motion. Let us once 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 be considered only 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:
(a) The carriage is in motion relative to the embankment. (b) The embankment is in motion relative to the carriage.
In (a) the embankment, in (b) the carriage, serves as the body of reference in our statement of the motion taking place. If it is simply a question of detecting or of describing the motion

62 the general theory of relativity
involved, it is in principle immaterial to what reference-body we refer the motion. 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 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, by making use of
(a) the embankment as reference-body, (b) the railway carriage as reference-body,
then these general laws of nature (e.g. the laws of mechanics or the law of the propagation of light in vacuo) have exactly the same form in both cases. This can also be expressed as follows: For the physical description of natural processes, neither of the referencebodies K, K′ is unique (lit. “specially marked out”) as compared with the other. Unlike the ﬁrst, this latter statement need not of necessity hold a priori; it is not contained in the conceptions of “motion” and “reference-body” and derivable from them; only experience can decide as to its correctness or incorrectness.
Up to the present, however, we have by no means maintained the equivalence of 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 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

special and general principle 63
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 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 (e.g. those possessing motion of a diﬀerent kind). In this sense we speak of the 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 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 nonuniform motion, as for instance by a powerful application of the brakes, then the occupant of the carriage experiences a

64 the general theory of relativity
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.

19
THE GRAVITATIONAL FIELD
“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 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 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

66 the general theory of relativity
theoretically represented much more satisfactorily than without it, and this applies particularly to the transmission of electromagnetic waves. The eﬀects of gravitation also are regarded in an analogous 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 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 (e.g. 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, which does not in the 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 (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
(Force) = (inertial mass) × (acceleration),
where the “inertial mass” is a characteristic constant of the

the gravitational field 67
accelerated body. If now gravitation is the cause of the acceleration, we then have
(Force) = (gravitational mass) × (intensity of the gravitational ﬁeld),
where the “gravitational mass” is likewise a characteristic constant for the body. From these two relations follows:
(acceleration) = (gravitational mass) × (intensity of the (inertial mass) gravitational ﬁeld).
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 gravitational mass of a body is equal to its inertial mass.
It is true that this important law had hitherto been recorded in mechanics, but it had not been interpreted. A satisfactory interpretation can be obtained only if we recognise the following fact: The same quality of a body manifests itself according to circumstances as “inertia” or as “weight” (lit. “heaviness”). In the 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.

20
THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS
AS AN ARGUMENT FOR THE GENERAL POSTULATE
OF RELATIVITY
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 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.

inertial and gravitational mass 69
To the middle of the lid of the chest is ﬁxed externally a hook with rope attached, and now a “being” (what 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 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 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 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 (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.

70 the general theory of relativity
Even though it is being 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 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 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 enough to eﬀect the acceleration of the body. That which determines the magnitude of the tension of the rope is the inertial mass of the body.” Guided by this example, we see that our extension of the principle of relativity implies the necessity of the law of the equality of inertial and gravitational mass. Thus we have obtained a physical interpretation of this law.

inertial and gravitational mass 71
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 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 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 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 Section 18. It is certainly true that the observer in the railway carriage experiences a jerk forwards as a result of the 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 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.”

21
IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE SPECIAL THEORY
OF RELATIVITY UNSATISFACTORY?
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 referencebodies K relative to which the recognised “laws of nature” can be

mechanics and relativity 73
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)? What is the reason for this preference? 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 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′.1 Newton saw this objection and attempted to invalidate it, but without success. But E. Mach recognised it most clearly of all, and because of this objection he claimed that mechanics must be placed on a new
1 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, e.g. in the case when the reference-body is rotating uniformly.

74 the general theory of relativity
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.

22
A FEW INFERENCES FROM THE GENERAL
PRINCIPLE OF RELATIVITY
The considerations of Section 20 show that the general principle of relativity puts us in a position to derive properties of the gravitational ﬁeld in a 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 (i.e. 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

76 the general theory of relativity
uniform rectilinear motion with respect to K (in accordance with the law of Galilei) is executing an accelerated and in general curvilinear motion with 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 way, so that our consideration supplies us with nothing essentially new.
However, we obtain a new result of fundamental 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, 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

inferences from relativity 77
importance, the early solution of which is to be expected of astronomers.1
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 in vacuo, which constitutes 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 (e.g. 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 of electrostatics were looked upon as the laws of electricity. At the present time we know that 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 reason electrostatics is overthrown by the ﬁeld-equations of Maxwell in electrodynamics? Not in the least. Electrostatics is contained
1 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 ﬁrst conﬁrmed during the solar eclipse of 29th May, 1919. (Cf. Appendix 3.)

78 the general theory of relativity
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, 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, i.e. 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.1 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 of the gravitational ﬁeld must be satisﬁed for all gravitational ﬁelds obtainable 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 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 still farther.
1 This follows from a generalisation of the discussion in Section 20.

23
BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY
OF REFERENCE
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 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 has been suitably chosen. K is then a Galileian reference-body as regards the domain considered, and

80 the general theory of relativity
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 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 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 gravitation.1 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 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
1 The ﬁeld disappears at the centre of the disc and increases proportionally to the distance from the centre as we proceed outwards.

behaviour of clocks and rods 81
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 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. According to a result obtained in Section 12, it follows that the latter clock goes at a rate permanently slower than that of the clock at the centre of the circular disc, i.e. 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 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 wish to go any farther into this question.
Moreover, at this stage the deﬁnition of the space co-ordinates also presents insurmountable diﬃculties. If the observer applies his standard 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 Section 12, 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 disc, on dividing the one by the other, he will not obtain as quotient the familiar number π = 3.14 . . ., but a

82 the general theory of relativity larger number,1 whereas of course, for a disc which is at rest with respect to K, this operation would yield exactly. This proves that the propositions of Euclidean 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 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 coordinates 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.
1 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).

24
EUCLIDEAN AND NONEUCLIDEAN CONTINUUM
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 over-lapping. 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

84 the general theory of relativity
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 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 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 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 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

euclidean and non-euclidean 85
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 (i.e. of every material) were to behave 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 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 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 geometry for rigid bodies.1 The reader will
1 Mathematicians have been confronted with our problem in the following form. If we are given a surface (e.g. 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 geometry from ﬁrst principles, without making use of the fact that the surface belongs to a Euclidean continuum of three dimensions. If

86 the general theory of relativity notice that the situation depicted here corresponds to the one brought about by the general postulate of relativity (Section 23).
we imagine constructions to be made with rigid rods 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 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, non-Euclidean continua. Thus it is that mathematicians long ago solved the formal problems to which we are led by the general postulate of relativity.

25
GAUSSIAN CO-ORDINATES
According to Gauss, this combined analytical and geometrical mode of handling the problem can be arrived at in the following way. We imagine a system of arbitrary curves (see Fig. 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 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.
Figure 4

88 the general theory of relativity
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 vcurves drawn on the surface. These satisfy the same conditions as the u-curves, they are provided with numbers 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). 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

P:

u, v

P′:

u + du, v + dv,

where du and dv signify very small numbers. In a similar manner we may indicate the 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

ds2 = g11du2 + 2g12dudv + g22dv2,
where g11, g12, g22, are magnitudes which depend in a perfectly deﬁnite way on u and v. The magnitudes g11, g12 and g22 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 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:

ds2 = du2 + dv2.

gaussian co-ordinates 89
Under these conditions, the u-curves and v-curves are straight lines in the sense of Euclidean geometry, and they are perpendicular to each other. Here the Gaussian co-ordinates are simply Cartesian ones. It is clear 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 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, x1, x2, x3, x4, 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 P and P′, this distance being measurable and well-deﬁned from a physical point of view, then the following formula holds:
ds2 = g11dx12 + 2g12dx1dx2 . . . . + g44dx42,
where the magnitudes g11, 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 x1 . . x4 with the points of the continuum so that we have simply
ds2 = dx12+dx22 + dx32 + dx42.
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 ds2 which we have given above is not always possible. It is only possible when suﬃciently