Relativity: The Specialand General Theory

Albert EinsteinTranslated by Robert W. Lawson

This public-domain (U.S.) text was originallypublished by Henry Holt and Company, 1920.The present edition was created from mate-rial archived at bartleby.com. The text wasconverted to LATEX by Ron Burkey, and mod-ified (both for formatting reasons, and to cor-rect obvious errors). All mathematical expres-sions which had been represented as scannedgraphics have been translated to pure LATEX.All scanned figures have been completely re-drawn, except for Fig. 5 (in which only thetext has been redrawn). Revision D differsfrom revision C in Appendix I, where the def-inition of the symbol b and the left-hand sideof equation 14 have been corrected thanks toa report by Charles Cameron. D1 has the fur-ther change that GIF images are changed toJPEG for greater portability of the LATEX. Re-port other problems to info@sandroid.org.

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Contents

Preface v

Biographical Note ix

Translator’s Note xi

Part I: The Special Theoryof Relativity 3Physical Meaning of Geometrical

Propositions 3

The System of Co-ordinates 7

Space and Time in Classical Mechanics 11

The Galileian System of Co-ordinates 15

The Principle of Relativity (In theRestricted Sense) 17

The Theorem of the Addition ofVelocities Employed in ClassicalMechanics 23

The Apparent Incompatibility of theLaw of Propagation of Light with

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the Principle of Relativity 25

On the Idea of Time in Physics 29

The Relativity of Simultaneity 35

On the Relativity of the Conceptionof Distance 39

The Lorentz Transformation 41

The Behaviour of Measuring-Rodsand Clocks in Motion 47

Theorem of the Addition of Velocities.The Experiment of Fizeau 51

The Heuristic Value of the Theory ofRelativity 57

General Results of the Theory 59

Experience and the Special Theory ofRelativity 65

Minkowski’s Four-Dimensional Space 71

Part II: The General Theoryof Relativity 77Special and General Principle of

Relativity 77

The Gravitational Field 83

The Equality of Inertial and GravitationalMass as an Argument for the GeneralPostulate of Relativity 87

iii

In What Respects Are the Foundations ofClassical Mechanics and of the SpecialTheory of Relativity Unsatisfactory? 93

A Few Inferences from the GeneralTheory of Relativity 97

Behaviour of Clocks and Measuring Rodson a Rotating Body of Reference 103

Euclidean and Non-EuclideanContinuum 109

Gaussian Co-ordinates 115

The Space-Time Continuum of the SpecialTheory of Relativity Considered as aEuclidean Continuum 121

The Space-Time Continuum of theGeneral Theory of Relativity is nota Euclidean Continuum 125

Exact Formulation of theGeneral Principle of Relativity 129

The Solution of the Problem ofGravitation on the Basis of the GeneralPrinciple of Relativity 133

Part III: Considerations on theUniverse as a Whole 141Cosmological Difficulties of Newton’s

Theory 141

The Possibility of a “Finite” and Yet“Unbounded” Universe 145

iv

The Structure of Space According to theGeneral Theory of Relativity 151

Appendices 157Simple Derivation of the Lorentz

Transformation 157

Minkowski’s Four-Dimensional Space(“World”) 165

The Experimental Confirmation of theGeneral Theory of Relativity 167

Bibliography 179

Preface

The present book is intended, as far as pos-sible, to give an exact insight into the the-ory of Relativity to those readers who, froma general scientific and philosophical point ofview, are interested in the theory, but who arenot conversant with the mathematical appa-ratus1 of theoretical physics. The work pre-sumes a standard of education correspondingto that of a university matriculation examina-tion, and, despite the shortness of the book,a fair amount of patience and force of will onthe part of the reader. The author has spared

1The mathematical fundaments of the special theoryof relativity are to be found in the original papers of H.A. Lorentz, A. Einstein, H. Minkowski published un-der the title Das Relativitäts-prinzip (The Principle ofRelativity) in B. G. Teubner’s collection of monographsFortschritte der mathematischen Wissenschaften (Ad-vances in the Mathematical Sciences), also in M. Laue’sexhaustive book Das Relativitäts prinzip—publishedby Friedr. Vieweg & Son, Braunschweig. The generaltheory of relativity, together with the necessary partsof the theory of invariants, is dealt with in the author’sbook Die Grundlagen der allgemeinen Relativitätstheo-rie (The Foundations of the General Theory of Relativ-ity)—Joh. Ambr. Barth, 1916; this book assumes somefamiliarity with the special theory of relativity.

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himself no pains in his endeavour to presentthe main ideas in the simplest and most in-telligible form, and on the whole, in the se-quence and connection in which they actuallyoriginated. In the interest of clearness, it ap-peared to me inevitable that I should repeatmyself frequently, without paying the slight-est attention to the elegance of the presenta-tion. I adhered scrupulously to the precept ofthat brilliant theoretical physicist, L. Boltz-mann, according to whom matters of eleganceought to be left to the tailor and to the cobbler.I make no pretence of having withheld fromthe reader difficulties which are inherent tothe subject. On the other hand, I have pur-posely treated the empirical physical founda-tions of the theory in a “step-motherly” fash-ion, so that readers unfamiliar with physicsmay not feel like the wanderer who was un-able to see the forest for trees. May the bookbring some one a few happy hours of sugges-tive thought!

A. EINSTEINDECEMBER, 1916

NOTE TO THE THIRD EDI-TIONIn the present year (1918) an excellent anddetailed manual on the general theory of rel-ativity, written by H. Weyl, was publishedby the firm Julius Springer (Berlin). Thisbook, entitled Raum—Zeit—Materie (Space—

Preface vii

Time—Matter), may be warmly recommendedto mathematicians and physicists.

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Biographical Note

Albert Einstein is the son of German-Jewishparents. He was born in 1879 in the town ofUlm, Würtemberg, Germany. His schooldayswere spent in Munich, where he attendedthe Gymnasium until his sixteenth year. Af-ter leaving school at Munich, he accompaniedhis parents to Milan, whence he proceeded toSwitzerland six months later to continue hisstudies.

From 1896 to 1900 Albert Einstein stud-ied mathematics and physics at the TechnicalHigh School in Zurich, as he intended becom-ing a secondary school (Gymnasium) teacher.For some time afterwards he was a privatetutor, and having meanwhile become natu-ralised, he obtained a post as engineer inthe Swiss Patent Office in 1902, which posi-tion he occupied till 1909. The main ideasinvolved in the most important of Einstein’stheories date back to this period. Amongstthese may be mentioned: The Special The-ory of Relativity, Inertia of Energy, Theory ofthe Brownian Movement, and the Quantum-Law of the Emission and Absorption of Light(1905). These were followed some years later

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by the Theory of the Specific Heat of SolidBodies, and the fundamental idea of the Gen-eral Theory of Relativity.

During the intervel 1909 to 1911 he oc-cupied the post of Professor Extraordinariusat the University of Prague, Bohemia, wherehe remained as Professor Ordinarius until1912. In the latter year Professor Einstein ac-cepted a similar chair at the Polytechnikum,Zurich, and continued his activities there un-til 1914, when he received a call to the Prus-sian Academy of Science, Berlin, as successorto Van’t Hoff. Professor Einstein is able to de-vote himself freely to his studies at the BerlinAcademy, and it was here that he succeeded incompleting his work on the General Theory ofRelativity (1915-17). Professor Einstein alsolectures on various special branches of physicsat the University of Berlin, and, in addition,he is Director of the Institute for Physical Re-search of the Kaiser Wilhelm Gesellschaft.

Professor Einstein has been twice married.His first wife, whom he married at Berne in1903, was a fellow-student from Serbia. Therewere two sons of this marriage, both of whomare living in Zurich, the elder being sixteenyears of age. Recently Professor Einstein mar-ried a widowed cousin, with whom he is nowliving in Berlin.

R. W. L.

Translator’s Note

In presenting this translation to the English-reading public, it is hardly necessary for meto enlarge on the Author’s prefatory remarks,except to draw attention to those additions tothe book which do not appear in the original.

At my request, Professor Einstein kindlysupplied me with a portrait of himself, byone of Germany’s most celebrated artists. Ap-pendix III, on “The Experimental Confirma-tion of the General Theory of Relativity,” hasbeen written specially for this translation.Apart from these valuable additions to thebook, I have included a biographical note onthe Author, and, at the end of the book, an In-dex and a list of English references to the sub-ject. This list, which is more suggestive thanexhaustive, is intended as a guide to thosereaders who wish to pursue the subject far-ther.

I desire to tender my best thanks to my col-leagues Professor S. R. Milner, D.Sc., and Mr.W. E. Curtis, A.R.C.Sc., F.R.A.S., also to myfriend Dr. Arthur Holmes, A.R.C.SC., F.G.S.,of the Imperial College, for their kindness inreading through the manuscript, for helpful

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criticism, and for numerous suggestions. Iowe an expression of thanks also to Messrs.Methuen for their ready counsel and advice,and for the care they have bestowed on thework during the course of its publication.

ROBERT W. LAWSONTHE PHYSICS LABORATORY

THE UNIVERSITY OF SHEFFIELDJUNE 12, 1920

Part I: The SpecialTheory ofRelativity

1

I. PhysicalMeaning ofGeometricalPropositions

In your schooldays most of you who read thisbook made acquaintance with the noble build-ing of Euclid’s geometry, and you remember—perhaps with more respect than love—themagnificent structure, on the lofty staircaseof which you were chased about for uncountedhours by conscientious teachers. By reason ofyour past experience, you would certainly re-gard every one with disdain who should pro-nounce even the most out-of-the-way proposi-tion of this science to be untrue. But perhapsthis feeling of proud certainty would leaveyou immediately if some one were to ask you:“What, then, do you mean by the assertionthat these propositions are true?” Let us pro-ceed to give this question a little considera-tion.

Geometry sets out from certain concep-tions such as “plane,” “point,” and “straight

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line,” with which we are able to associate moreor less definite ideas, and from certain sim-ple propositions (axioms) which, in virtue ofthese ideas, we are inclined to accept as “true.”Then, on the basis of a logical process, thejustification of which we feel ourselves com-pelled to admit, all remaining propositionsare shown to follow from those axioms, i.e.they are proven. A proposition is then correct(“true”) when it has been derived in the recog-nised manner from the axioms. The ques-tion of the “truth” of the individual geomet-rical propositions is thus reduced to one of the“truth” of the axioms. Now it has long beenknown that the last question is not only unan-swerable by the methods of geometry, but thatit is in itself entirely without meaning. Wecannot ask whether it is true that only onestraight line goes through two points. We canonly say that Euclidean geometry deals withthings called “straight line,” to each of whichis ascribed the property of being uniquely de-termined by two points situated on it. Theconcept “true” does not tally with the asser-tions of pure geometry, because by the word“true” we are eventually in the habit of des-ignating always the correspondence with a“real” object; geometry, however, is not con-cerned with the relation of the ideas involvedin it to objects of experience, but only with thelogical connection of these ideas among them-selves.

It is not difficult to understand why, inspite of this, we feel constrained to call thepropositions of geometry “true.” Geometrical

Physical Meaning of Geometry 5

ideas correspond to more or less exact objectsin nature, and these last are undoubtedly theexclusive cause of the genesis of those ideas.Geometry ought to refrain from such a course,in order to give to its structure the largest pos-sible logical unity. The practice, for example,of seeing in a “distance” two marked positionson a practically rigid body is something whichis lodged deeply in our habit of thought. Weare accustomed further to regard three pointsas being situated on a straight line, if theirapparent positions can be made to coincidefor observation with one eye, under suitablechoice of our place of observation.

If, in pursuance of our habit of thought,we now supplement the propositions of Eu-clidean geometry by the single propositionthat two points on a practically rigid body al-ways correspond to the same distance (line-interval), independently of any changes in po-sition to which we may subject the body, thepropositions of Euclidean geometry then re-solve themselves into propositions on the pos-sible relative position of practically rigid bod-ies.2 Geometry which has been supplementedin this way is then to be treated as a branchof physics. We can now legitimately ask asto the “truth” of geometrical propositions in-terpreted in this way, since we are justified

2It follows that a natural object is associated alsowith a straight line. Three points A, B and C on a rigidbody thus lie in a straight when, the points A and Cbeing given, B is chosen such that the sum of the dis-tances AB and BC is as short as possible. This incom-plete suggestion will suffice for our present purpose.

6 Relativity

in asking whether these propositions are sat-isfied for those real things we have associ-ated with the geometrical ideas. In less ex-act terms we can express this by saying thatby the “truth” of a geometrical proposition inthis sense we understand its validity for a con-struction with ruler and compasses.

Of course the conviction of the “truth”of geometrical propositions in this sense isfounded exclusively on rather incomplete ex-perience. For the present we shall assume the“truth” of the geometrical propositions, thenat a later stage (in the general theory of rela-tivity) we shall see that this “truth” is limited,and we shall consider the extent of its limita-tion.

II. The System ofCo-ordinates

On the basis of the physical interpretation ofdistance which has been indicated, we are alsoin a position to establish the distance betweentwo points on a rigid body by means of mea-surements. For this purpose we require a “dis-tance” (rod S) which is to be used once and forall, and which we employ as a standard mea-sure. If, now, A and B are two points on arigid body, we can construct the line joiningthem according to the rules of geometry; then,starting from A, we can mark off the distanceS time after time until we reach B. The num-ber of these operations required is the numer-ical measure of the distance AB. This is thebasis of all measurement of length.3

Every description of the scene of an eventor of the position of an object in space is basedon the specification of the point on a rigid body

3Here we have assumed that there is nothing leftover, i.e. that the measurement gives a whole num-ber. This difficulty is got over by the use of dividedmeasuring-rods, the introduction of which does not de-mand any fundamentally new method.

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(body of reference) with which that event orobject coincides. This applies not only to sci-entific description, but also to everyday life.If I analyse the place specification “Trafal-gar Square, London,”4 I arrive at the follow-ing result. The earth is the rigid body towhich the specification of place refers; “Trafal-gar Square, London” is a well-defined point,to which a name has been assigned, and withwhich the event coincides in space.5

This primitive method of place specifica-tion deals only with places on the surface ofrigid bodies, and is dependent on the exis-tence of points on this surface which are dis-tinguishable from each other. But we can freeourselves from both of these limitations with-out altering the nature of our specification ofposition. If, for instance, a cloud is hover-ing over Trafalgar Square, then we can deter-mine its position relative to the surface of theearth by erecting a pole perpendicularly onthe Square, so that it reaches the cloud. Thelength of the pole measured with the standardmeasuring-rod, combined with the specifica-tion of the position of the foot of the pole, sup-plies us with a complete place specification.On the basis of this illustration, we are able

4I have chosen this as being more familiar to the En-glish reader than the “Potsdamer Platz, Berlin,” whichis referred to in the original. (R. W. L.)

5It is not necessary here to investigate further thesignificance of the expression “coincidence in space.”This conception is sufficiently obvious to ensure thatdifferences of opinion are scarcely likely to arise as toits applicability in practice.

The System of Co-ordinates 9

to see the manner in which a refinement ofthe conception of position has been developed.

(a) We imagine the rigid body, to which theplace specification is referred, supplementedin such a manner that the object whose posi-tion we require is reached by the completedrigid body.

(b) In locating the position of the object,we make use of a number (here the lengthof the pole measured with the measuring-rod)instead of designated points of reference.

(c) We speak of the height of the cloud evenwhen the pole which reaches the cloud hasnot been erected. By means of optical obser-vations of the cloud from different positionson the ground, and taking into account theproperties of the propagation of light, we de-termine the length of the pole we should haverequired in order to reach the cloud.

From this consideration we see that it willbe advantageous if, in the description of posi-tion, it should be possible by means of numer-ical measures to make ourselves independentof the existence of marked positions (possess-ing names) on the rigid body of reference. Inthe physics of measurement this is attainedby the application of the Cartesian system ofco-ordinates.

This consists of three plane surfaces per-pendicular to each other and rigidly attachedto a rigid body. Referred to a system of co-ordinates, the scene of any event will be de-termined (for the main part) by the specifica-tion of the lengths of the three perpendicularsor co-ordinates (x, y, z) which can be dropped

10 Relativity

from the scene of the event to those threeplane surfaces. The lengths of these three per-pendiculars can be determined by a series ofmanipulations with rigid measuring-rods per-formed according to the rules and methodslaid down by Euclidean geometry.

In practice, the rigid surfaces which consti-tute the system of co-ordinates are generallynot available; furthermore, the magnitudes ofthe co-ordinates are not actually determinedby constructions with rigid rods, but by indi-rect means. If the results of physics and as-tronomy are to maintain their clearness, thephysical meaning of specifications of positionmust always be sought in accordance with theabove considerations.6

We thus obtain the following result: Everydescription of events in space involves the useof a rigid body to which such events have tobe referred. The resulting relationship takesfor granted that the laws of Euclidean geom-etry hold for “distances,” the “distance” beingrepresented physically by means of the con-vention of two marks on a rigid body.

6A refinement and modification of these views doesnot become necessary until we come to deal with thegeneral theory of relativity, treated in the second partof this book.

III. Space and Timein ClassicalMechanics

“The purpose of mechanics is to describe howbodies change their position in space withtime.” I should load my conscience with gravesins against the sacred spirit of lucidity wereI to formulate the aims of mechanics in thisway, without serious reflection and detailedexplanations. Let us proceed to disclose thesesins.

It is not clear what is to be understoodhere by “position” and “space.” I stand at thewindow of a railway carriage which is travel-ling uniformly, and drop a stone on the em-bankment, without throwing it. Then, dis-regarding the influence of the air resistance,I see the stone descend in a straight line.A pedestrian who observes the misdeed fromthe footpath notices that the stone falls toearth in a parabolic curve. I now ask: Dothe “positions” traversed by the stone lie “inreality” on a straight line or on a parabola?Moreover, what is meant here by motion “in

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space”? From the considerations of the previ-ous section the answer is self-evident. In thefirst place, we entirely shun the vague word“space,” of which, we must honestly acknowl-edge, we cannot form the slightest conception,and we replace it by “motion relative to a prac-tically rigid body of reference.” The positionsrelative to the body of reference (railway car-riage or embankment) have already been de-fined in detail in the preceding section. Ifinstead of “body of reference” we insert “sys-tem of co-ordinates,” which is a useful idea formathematical description, we are in a posi-tion to say: The stone traverses a straight linerelative to a system of co-ordinates rigidly at-tached to the carriage, but relative to a systemof co-ordinates rigidly attached to the ground(embankment) it describes a parabola. Withthe aid of this example it is clearly seen thatthere is no such thing as an independently ex-isting trajectory (lit. “path-curve”7), but only atrajectory relative to a particular body of ref-erence.

In order to have a complete description ofthe motion, we must specify how the bodyalters its position with time; i.e. for everypoint on the trajectory it must be stated atwhat time the body is situated there. Thesedata must be supplemented by such a defi-nition of time that, in virtue of this defini-tion, these time-values can be regarded es-sentially as magnitudes (results of measure-ments) capable of observation. If we take our

7That is, a curve along which the body moves.

Space and Time in Classical Mechanics 13

stand on the ground of classical mechanics,we can satisfy this requirement for our illus-tration in the following manner. We imag-ine two clocks of identical construction; theman at the railway-carriage window is hold-ing one of them, and the man on the footpaththe other. Each of the observers determinesthe position on his own reference-body occu-pied by the stone at each tick of the clock heis holding in his hand. In this connection wehave not taken account of the inaccuracy in-volved by the finiteness of the velocity of prop-agation of light. With this and with a seconddifficulty prevailing here we shall have to dealin detail later.

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IV. The GalileianSystem ofCo-ordinates

As is well known, the fundamental law of themechanics of Galilei-Newton, which is knownas the law of inertia, can be stated thus: Abody removed sufficiently far from other bod-ies continues in a state of rest or of uniformmotion in a straight line. This law not onlysays something about the motion of the bod-ies, but it also indicates the reference-bodiesor systems of co-ordinates, permissible in me-chanics, which can be used in mechanical de-scription. The visible fixed stars are bodiesfor which the law of inertia certainly holds toa high degree of approximation. Now if weuse a system of co-ordinates which is rigidlyattached to the earth, then, relative to thissystem, every fixed star describes a circle ofimmense radius in the course of an astronom-ical day, a result which is opposed to the state-ment of the law of inertia. So that if we adhereto this law we must refer these motions only tosystems of co-ordinates relative to which the

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fixed stars do not move in a circle. A systemof co-ordinates of which the state of motion issuch that the law of inertia holds relative to itis called a “Galileian system of co-ordinates.”The laws of the mechanics of Galilei-Newtoncan be regarded as valid only for a Galileiansystem of co-ordinates.

V. The Principle ofRelativity (In theRestricted Sense)

In order to attain the greatest possible clear-ness, let us return to our example of the rail-way carriage supposed to be travelling uni-formly. We call its motion a uniform trans-lation (“uniform” because it is of constant ve-locity and direction, “translation” because al-though the carriage changes its position rel-ative to the embankment yet it does not ro-tate in so doing). Let us imagine a raven fly-ing through the air in such a manner that itsmotion, as observed from the embankment, isuniform and in a straight line. If we were toobserve the flying raven from the moving rail-way carriage, we should find that the motionof the raven would be one of different velocityand direction, but that it would still be uni-form and in a straight line. Expressed in anabstract manner we may say: If a mass m ismoving uniformly in a straight line with re-spect to a co-ordinate system K, then it willalso be moving uniformly and in a straight

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line relative to a second co-ordinate systemK ′, provided that the latter is executing a uni-form translatory motion with respect to K. Inaccordance with the discussion contained inthe preceding section, it follows that:

If K is a Galileian co-ordinate system, thenevery other co-ordinate system K ′ is a Galile-ian one, when, in relation to K, it is in a condi-tion of uniform motion of translation. Relativeto K ′ the mechanical laws of Galilei-Newtonhold good exactly as they do with respect toK.

We advance a step farther in our general-isation when we express the tenet thus: If,relative to K, K ′ is a uniformly moving co-ordinate system devoid of rotation, then natu-ral phenomena run their course with respectto K ′ according to exactly the same generallaws as with respect to K. This statementis called the principle of relativity (in the re-stricted sense).

As long as one was convinced that all nat-ural phenomena were capable of represen-tation with the help of classical mechanics,there was no need to doubt the validity ofthis principle of relativity. But in view of themore recent development of electrodynamicsand optics it became more and more evidentthat classical mechanics affords an insuffi-cient foundation for the physical descriptionof all natural phenomena. At this juncture thequestion of the validity of the principle of rel-ativity became ripe for discussion, and it didnot appear impossible that the answer to thisquestion might be in the negative.

The Restricted Principle of Relativity 19

Nevertheless, there are two general factswhich at the outset speak very much in favourof the validity of the principle of relativ-ity. Even though classical mechanics doesnot supply us with a sufficiently broad basisfor the theoretical presentation of all physicalphenomena, still we must grant it a consid-erable measure of “truth,” since it supplies uswith the actual motions of the heavenly bodieswith a delicacy of detail little short of wonder-ful. The principle of relativity must thereforeapply with great accuracy in the domain ofmechanics. But that a principle of such broadgenerality should hold with such exactness inone domain of phenomena, and yet should beinvalid for another, is a priori not very proba-ble.

We now proceed to the second argument,to which, moreover, we shall return later. Ifthe principle of relativity (in the restrictedsense) does not hold, then the Galileian co-ordinate systems K, K ′, K ′′, etc., which aremoving uniformly relative to each other, willnot be equivalent for the description of nat-ural phenomena. In this case we should beconstrained to believe that natural laws arecapable of being formulated in a particularlysimple manner, and of course only on condi-tion that, from amongst all possible Galileianco-ordinate systems, we should have chosenone (K0) of a particular state of motion as ourbody of reference. We should then be justi-fied (because of its merits for the descriptionof natural phenomena) in calling this system“absolutely at rest,” and all other Galileian

20 Relativity

systems K “in motion.” If, for instance, ourembankment were the system K0, then ourrailway carriage would be a system K, rela-tive to which less simple laws would hold thanwith respect to K0. This diminished simplic-ity would be due to the fact that the carriageK would be in motion (i.e. “really”) with re-spect to K0. In the general laws of naturalphenomena which have been formulated withreference to K, the magnitude and directionof the velocity of the carriage would necessar-ily play a part. We should expect, for instance,that the note emitted by an organ-pipe placedwith its axis parallel to the direction of travelwould be different from that emitted if theaxis of the pipe were placed perpendicular tothis direction. Now in virtue of its motion inan orbit round the sun, our earth is compa-rable with a railway carriage travelling witha velocity of about 30 kilometres per second.If the principle of relativity were not validwe should therefore expect that the directionof motion of the earth at any moment wouldenter into the laws of nature, and also thatphysical systems in their behaviour would bedependent on the orientation in space withrespect to the earth. For owing to the al-teration in direction of the velocity of rota-tion of the earth in the course of a year, theearth cannot be at rest relative to the hypo-thetical system K0 throughout the whole year.However, the most careful observations havenever revealed such anisotropic properties interrestrial physical space, i.e. a physical non-equivalence of different directions. This is a

The Restricted Principle of Relativity 21

very powerful argument in favour of the prin-ciple of relativity.

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VI. The Theorem ofthe Addition ofVelocitiesEmployed inClassicalMechanics

Let us suppose our old friend the railway car-riage to be travelling along the rails with aconstant velocity v, and that a man traversesthe length of the carriage in the direction oftravel with a velocity w. How quickly, or, inother words, with what velocity W does theman advance relative to the embankment dur-ing the process? The only possible answerseems to result from the following consider-ation: If the man were to stand still for asecond, he would advance relative to the em-bankment through a distance v equal numer-ically to the velocity of the carriage. As aconsequence of his walking, however, he tra-verses an additional distance w relative to

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the carriage, and hence also relative to theembankment, in this second, the distance wbeing numerically equal to the velocity withwhich he is walking. Thus in total he coversthe distance W = v + w relative to the em-bankment in the second considered. We shallsee later that this result, which expresses thetheorem of the addition of velocities employedin classical mechanics, cannot be maintained;in other words, the law that we have just writ-ten down does not hold in reality. For the timebeing, however, we shall assume its correct-ness.

VII. The ApparentIncompatibility ofthe Law ofPropagation ofLight with thePrinciple ofRelativity

There is hardly a simpler law in physics thanthat according to which light is propagated inempty space. Every child at school knows,or believes he knows, that this propagationtakes place in straight lines with a velocity c =300000 km./sec. At all events we know withgreat exactness that this velocity is the samefor all colours, because if this were not thecase, the minimum of emission would not beobserved simultaneously for different coloursduring the eclipse of a fixed star by its darkneighbour. By means of similar considera-tions based on observations of double stars,

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the Dutch astronomer De Sitter was also ableto show that the velocity of propagation oflight cannot depend on the velocity of motionof the body emitting the light. The assump-tion that this velocity of propagation is depen-dent on the direction “in space” is in itself im-probable.

In short, let us assume that the simple lawof the constancy of the velocity of light c (invacuum) is justifiably believed by the child atschool. Who would imagine that this simplelaw has plunged the conscientiously thought-ful physicist into the greatest intellectual dif-ficulties? Let us consider how these difficul-ties arise.

Of course we must refer the process of thepropagation of light (and indeed every otherprocess) to a rigid reference-body (co-ordinatesystem). As such a system let us again chooseour embankment. We shall imagine the airabove it to have been removed. If a ray oflight be sent along the embankment, we seefrom the above that the tip of the ray will betransmitted with the velocity c relative to theembankment. Now let us suppose that ourrailway carriage is again travelling along therailway lines with the velocity v, and that itsdirection is the same as that of the ray of light,but its velocity of course much less. Let us in-quire about the velocity of propagation of theray of light relative to the carriage. It is obvi-ous that we can here apply the considerationof the previous section, since the ray of lightplays the part of the man walking along rel-atively to the carriage. The velocity W of the

The Apparent Incompatibility 27

man relative to the embankment is here re-placed by the velocity of light relative to theembankment. w is the required velocity oflight with respect to the carriage, and we have

w = c− v.

The velocity of propagation of a ray of lightrelative to the carriage thus comes out smallerthan c.

But this result comes into conflict with theprinciple of relativity set forth in Section V.For, like every other general law of nature, thelaw of the transmission of light in vacuo must,according to the principle of relativity, be thesame for the railway carriage as reference-body as when the rails are the body of refer-ence. But, from our above consideration, thiswould appear to be impossible. If every rayof light is propagated relative to the embank-ment with the velocity c, then for this reasonit would appear that another law of propaga-tion of light must necessarily hold with re-spect to the carriage—a result contradictoryto the principle of relativity.

In view of this dilemma there appears tobe nothing else for it than to abandon eitherthe principle of relativity or the simple lawof the propagation of light in vacuo. Thoseof you who have carefully followed the pre-ceding discussion are almost sure to expectthat we should retain the principle of relativ-ity, which appeals so convincingly to the intel-lect because it is so natural and simple. Thelaw of the propagation of light in vacuo would

28 Relativity

then have to be replaced by a more compli-cated law conformable to the principle of rela-tivity. The development of theoretical physicsshows, however, that we cannot pursue thiscourse. The epoch-making theoretical investi-gations of H. A. Lorentz on the electrodynam-ical and optical phenomena connected withmoving bodies show that experience in thisdomain leads conclusively to a theory of elec-tromagnetic phenomena, of which the law ofthe constancy of the velocity of light in vacuois a necessary consequence. Prominent theo-retical physicists were therefore more inclinedto reject the principle of relativity, in spite ofthe fact that no empirical data had been foundwhich were contradictory to this principle.

At this juncture the theory of relativity en-tered the arena. As a result of an analysisof the physical conceptions of time and space,it became evident that in reality there is notthe least incompatibility between the princi-ple of relativity and the law of propagationof light, and that by systematically holdingfast to both these laws a logically rigid the-ory could be arrived at. This theory has beencalled the special theory of relativity to distin-guish it from the extended theory, with whichwe shall deal later. In the following pageswe shall present the fundamental ideas of thespecial theory of relativity.

VIII. On the Idea ofTime in Physics

Lightning has struck the rails on our railwayembankment at two places A and B far dis-tant from each other. I make the additionalassertion that these two lightning flashes oc-curred simultaneously. If now I ask youwhether there is sense in this statement, youwill answer my question with a decided “Yes.”But if I now approach you with the requestto explain to me the sense of the statementmore precisely, you find after some considera-tion that the answer to this question is not soeasy as it appears at first sight.

After some time perhaps the following an-swer would occur to you: “The significanceof the statement is clear in itself and needsno further explanation; of course it wouldrequire some consideration if I were to becommissioned to determine by observationswhether in the actual case the two eventstook place simultaneously or not.” I cannotbe satisfied with this answer for the follow-ing reason. Supposing that as a result of in-genious considerations an able meteorologist

29

30 Relativity

were to discover that the lightning must al-ways strike the places A and B simultane-ously, then we should be faced with the taskof testing whether or not this theoretical re-sult is in accordance with the reality. We en-counter the same difficulty with all physicalstatements in which the conception “simulta-neous” plays a part. The concept does not ex-ist for the physicist until he has the possi-bility of discovering whether or not it is ful-filled in an actual case. We thus require adefinition of simultaneity such that this def-inition supplies us with the method by meansof which, in the present case, he can decide byexperiment whether or not both the lightningstrokes occurred simultaneously. As long asthis requirement is not satisfied, I allow my-self to be deceived as a physicist (and of coursethe same applies if I am not a physicist), whenI imagine that I am able to attach a meaningto the statement of simultaneity. (I would askthe reader not to proceed farther until he isfully convinced on this point.)

After thinking the matter over for sometime you then offer the following suggestionwith which to test simultaneity. By mea-suring along the rails, the connecting lineAB should be measured up and an observerplaced at the mid-point M of the distanceAB. This observer should be supplied withan arrangement (e.g. two mirrors inclined at90◦) which allows him visually to observe bothplaces A and B at the same time. If the ob-server perceives the two flashes of lightningat the same time, then they are simultaneous.

On the Idea of Time in Physics 31

I am very pleased with this suggestion,but for all that I cannot regard the matteras quite settled, because I feel constrained toraise the following objection: “Your definitionwould certainly be right, if I only knew thatthe light by means of which the observer at Mperceives the lightning flashes travels alongthe length A → M with the same velocity asalong the length B → M . But an examinationof this supposition would only be possible if wealready had at our disposal the means of mea-suring time. It would thus appear as thoughwe were moving here in a logical circle.”

After further consideration you cast asomewhat disdainful glance at me—andrightly so—and you declare: “I maintain myprevious definition nevertheless, because inreality it assumes absolutely nothing aboutlight. There is only one demand to be madeof the definition of simultaneity, namely, thatin every real case it must supply us with anempirical decision as to whether or not theconception that has to be defined is fulfilled.That my definition satisfies this demand is in-disputable. That light requires the same timeto traverse the path A → M as for the pathB → M is in reality neither a suppositionnor a hypothesis about the physical nature oflight, but a stipulation which I can make ofmy own freewill in order to arrive at a defini-tion of simultaneity.”

It is clear that this definition can be usedto give an exact meaning not only to twoevents, but to as many events as we care tochoose, and independently of the positions of

32 Relativity

the scenes of the events with respect to thebody of reference8 (here the railway embank-ment). We are thus led also to a definition of“time” in physics. For this purpose we sup-pose that clocks of identical construction areplaced at the points A, B and C of the rail-way line (co-ordinate system), and that theyare set in such a manner that the positionsof their pointers are simultaneously (in theabove sense) the same. Under these condi-tions we understand by the “time” of an eventthe reading (position of the hands) of thatone of these clocks which is in the immedi-ate vicinity (in space) of the event. In thismanner a time-value is associated with everyevent which is essentially capable of observa-tion.

This stipulation contains a further phys-ical hypothesis, the validity of which willhardly be doubted without empirical evidenceto the contrary. It has been assumed that allthese clocks go at the same rate if they are ofidentical construction. Stated more exactly:When two clocks arranged at rest in differentplaces of a reference-body are set in such amanner that a particular position of the point-ers of the one clock is simultaneous (in the

8We suppose further that, when three events A, Band C take place in different places in such a mannerthat, if A is simultaneous with B, and B is simultane-ous with C (simultaneous in the sense of the above def-inition), then the criterion for the simultaneity of thepair of events A, C is also satisfied. This assumption isa physical hypothesis about the law of propagation oflight; it must certainly be fulfilled if we are to maintainthe law of the constancy of the velocity of light in vacuo.

On the Idea of Time in Physics 33

above sense) with the same position of thepointers of the other clock, then identical “set-tings” are always simultaneous (in the senseof the above definition).

34 Relativity

IX. The Relativityof Simultaneity

Up to now our considerations have been re-ferred to a particular body of reference, whichwe have styled a “railway embankment.” Wesuppose a very long train travelling along therails with the constant velocity v and in thedirection indicated in Fig. 1. People travel-ling in this train will with advantage use thetrain as a rigid reference-body (co-ordinatesystem); they regard all events in reference tothe train. Then every event which takes placealong the line also takes place at a particularpoint of the train. Also the definition of si-multaneity can be given relative to the trainin exactly the same way as with respect to theembankment. As a natural consequence, how-ever, the following question arises:

Are two events (e.g. the two strokes oflightning A and B) which are simultaneouswith reference to the railway embankmentalso simultaneous relatively to the train? Weshall show directly that the answer must bein the negative.

35

36 Relativity

FIG. 1.

When we say that the lightning strokes Aand B are simultaneous with respect to theembankment, we mean: the rays of light emit-ted at the places A and B, where the lightningoccurs, meet each other at the mid-point M ofthe length A → B of the embankment. Butthe events A and B also correspond to posi-tions A and B on the train. Let M ′ be themid-point of the distance A → B on the trav-elling train. Just when the flashes9 of light-ning occur, this point M ′ naturally coincideswith the point M , but it moves towards theright in the diagram with the velocity v of thetrain. If an observer sitting in the position M ′

in the train did not possess this velocity, thenhe would remain permanently at M , and thelight rays emitted by the flashes of lightningA and B would reach him simultaneously, i.e.they would meet just where he is situated.Now in reality (considered with reference tothe railway embankment) he is hastening to-wards the beam of light coming from B, whilsthe is riding on ahead of the beam of light com-ing from A. Hence the observer will see thebeam of light emitted from B earlier than hewill see that emitted from A. Observers whotake the railway train as their reference-body

9As judged from the embankment.

The Relativity of Simultaneity 37

must therefore come to the conclusion that thelightning flash B took place earlier than thelightning flash A. We thus arrive at the im-portant result:

Events which are simultaneous with refer-ence to the embankment are not simultane-ous with respect to the train, and vice versa(relativity of simultaneity). Every reference-body (co-ordinate system) has its own particu-lar time; unless we are told the reference-bodyto which the statement of time refers, there isno meaning in a statement of the time of anevent.

Now before the advent of the theory of rel-ativity it had always tacitly been assumed inphysics that the statement of time had an ab-solute significance, i.e. that it is independentof the state of motion of the body of reference.But we have just seen that this assumption isincompatible with the most natural definitionof simultaneity; if we discard this assump-tion, then the conflict between the law of thepropagation of light in vacuo and the principleof relativity (developed in Section VII) disap-pears.

We were led to that conflict by the con-siderations of Section VI, which are now nolonger tenable. In that section we concludedthat the man in the carriage, who traversesthe distance w per second relative to the car-riage, traverses the same distance also withrespect to the embankment in each secondof time. But, according to the foregoing con-siderations, the time required by a particularoccurrence with respect to the carriage must

38 Relativity

not be considered equal to the duration of thesame occurrence as judged from the embank-ment (as reference-body). Hence it cannot becontended that the man in walking travels thedistance w relative to the railway line in atime which is equal to one second as judgedfrom the embankment.

Moreover, the considerations of Section VIare based on yet a second assumption, which,in the light of a strict consideration, appearsto be arbitrary, although it was always tacitlymade even before the introduction of the the-ory of relativity.

X. On the Relativityof the Conceptionof Distance

Let us consider two particular points on thetrain10 travelling along the embankment withthe velocity v, and inquire as to their distanceapart. We already know that it is necessaryto have a body of reference for the measure-ment of a distance, with respect to which bodythe distance can be measured up. It is thesimplest plan to use the train itself as thereference-body (co-ordinate system). An ob-server in the train measures the interval bymarking off his measuring-rod in a straightline (e.g. along the floor of the carriage) asmany times as is necessary to take him fromthe one marked point to the other. Then thenumber which tells us how often the rod hasto be laid down is the required distance.

It is a different matter when the distancehas to be judged from the railway line. Herethe following method suggests itself. If we call

10E.g. the middle of the first and of the hundredthcarriage.

39

40 Relativity

A′ and B′ the two points on the train whosedistance apart is required, then both of thesepoints are moving with the velocity v alongthe embankment. In the first place we requireto determine the points A and B of the em-bankment which are just being passed by thetwo points A′ and B′ at a particular time t—judged from the embankment. These points Aand B of the embankment can be determinedby applying the definition of time given in Sec-tion VIII. The distance between these points Aand B is then measured by repeated applica-tion of the measuring-rod along the embank-ment.

A priori it is by no means certain thatthis last measurement will supply us withthe same result as the first. Thus the lengthof the train as measured from the embank-ment may be different from that obtained bymeasuring in the train itself. This circum-stance leads us to a second objection whichmust be raised against the apparently obvi-ous consideration of Section VI. Namely, if theman in the carriage covers the distance w ina unit of time—measured from the train,—then this distance—as measured from theembankment—is not necessarily also equal tow.

XI. The LorentzTransformation

The results of the last three sections showthat the apparent incompatibility of the law ofpropagation of light with the principle of rela-tivity (Section VII) has been derived by meansof a consideration which borrowed two unjus-tifiable hypotheses from classical mechanics;these are as follows: The time-interval (time)between two events is independent of the con-dition of motion of the body of reference. Thespace-interval (distance) between two pointsof a rigid body is independent of the conditionof motion of the body of reference.

If we drop these hypotheses, then thedilemma of Section VII disappears, becausethe theorem of the addition of velocities de-rived in Section VI becomes invalid. The pos-sibility presents itself that the law of the prop-agation of light in vacuo may be compatiblewith the principle of relativity, and the ques-tion arises: How have we to modify the con-siderations of Section VI in order to removethe apparent disagreement between these twofundamental results of experience? This ques-

41

42 Relativity

tion leads to a general one. In the discussionof Section VI we have to do with places andtimes relative both to the train and to the em-bankment. How are we to find the place andtime of an event in relation to the train, whenwe know the place and time of the event withrespect to the railway embankment? Is therea thinkable answer to this question of such anature that the law of transmission of lightin vacuo does not contradict the principle ofrelativity? In other words: Can we conceiveof a relation between place and time of theindividual events relative to both reference-bodies, such that every ray of light possessesthe velocity of transmission c relative to theembankment and relative to the train? Thisquestion leads to a quite definite positive an-swer, and to a perfectly definite transforma-tion law for the space-time magnitudes of anevent when changing over from one body ofreference to another.

Before we deal with this, we shall in-troduce the following incidental considera-tion. Up to the present we have only consid-ered events taking place along the embank-ment, which had mathematically to assumethe function of a straight line. In the man-ner indicated in Section II we can imagine thisreference-body supplemented laterally and ina vertical direction by means of a frameworkof rods, so that an event which takes placeanywhere can be localised with reference tothis framework. Similarly, we can imaginethe train travelling with the velocity v to becontinued across the whole of space, so that

The Lorentz Transformation 43

every event, no matter how far off it may be,could also be localised with respect to the sec-ond framework. Without committing any fun-damental error, we can disregard the fact thatin reality these frameworks would continuallyinterfere with each other, owing to the impen-etrability of solid bodies. In every such frame-work we imagine three surfaces perpendicu-lar to each other marked out, and designatedas “co-ordinate planes” (“co-ordinate system”).A co-ordinate system K then corresponds tothe embankment, and a co-ordinate system K ′

to the train. An event, wherever it may havetaken place, would be fixed in space with re-spect to K by the three perpendiculars x, y,z on the co-ordinate planes, and with regardto time by a time-value t. Relative to K ′, thesame event would be fixed in respect of spaceand 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 detailhow these magnitudes are to be regarded asresults of physical measurements.

Obviously our problem can be exactly for-mulated in the following manner. What arethe values x′, y′, z′, t′ of an event with respectto K ′, when the magnitudes x, y, z, t, of thesame event with respect to K are given? Therelations must be so chosen that the law of thetransmission of light in vacuo is satisfied forone and the same ray of light (and of course forevery ray) with respect to K and K ′. For therelative orientation in space of the co-ordinatesystems indicated in the diagram (Fig. 2), this

44 Relativity

problem is solved by means of the equations:

x′ =x− vt√1− v2

c2

y′ = y

z′ = z

t′ =t− v

c2x√

1− v2

c2

.

This system of equations is known as the“Lorentz transformation.”11

FIG. 2.

If in place of the law of transmission oflight we had taken as our basis the tacit as-sumptions of the older mechanics as to the ab-solute character of times and lengths, then in-stead of the above we should have obtained

11A simple derivation of the Lorentz transformationis given in Appendix I.

The Lorentz Transformation 45

the following equations:

x′ = x− vt

y′ = y

z′ = z

t′ = t.

This system of equations is often termedthe “Galilei transformation.” The Galileitransformation can be obtained from theLorentz transformation by substituting an in-finitely large value for the velocity of light c inthe latter transformation.

Aided by the following illustration, wecan readily see that, in accordance with theLorentz transformation, the law of the trans-mission of light in vacuo is satisfied both forthe reference-body K and for the reference-body K ′. A light-signal is sent along the pos-itive x-axis, and this light-stimulus advancesin accordance with the equation

x = ct;

i.e. with the velocity c. According to the equa-tions of the Lorentz transformation, this sim-ple relation between x and t involves a rela-tion between x′ and t′. In point of fact, if wesubstitute for x the value ct in the first andfourth equations of the Lorentz transforma-tion, we obtain

x′ =(c− v)t√

1− v2

c2

t′ =(1− v

c)t√

1− v2

c2

,

46 Relativity

from which, by division, the expression

x′ = ct′

immediately follows. If referred to the sys-tem K ′, the propagation of light takes placeaccording to this equation. We thus see thatthe velocity of transmission relative to thereference-body K ′ is also equal to c. The sameresult is obtained for rays of light advancingin any other direction whatsoever. Of coursethis is not surprising, since the equations ofthe Lorentz transformation were derived con-formably to this point of view.

XII. The Behaviourof Measuring-Rodsand Clocks inMotion

I place a metre-rod in the x′-axis of K ′ in sucha manner that one end (the beginning) coin-cides with the point x′ = 0, whilst the otherend (the end of the rod) coincides with thepoint x′ = 1. What is the length of the metre-rod relatively to the system K? In order tolearn this, we need only ask where the be-ginning of the rod and the end of the rod liewith respect to K at a particular time t ofthe system K. By means of the first equa-tion of the Lorentz transformation the valuesof these two points at the time t = 0 can beshown to be

x(beginning of rod) = 0 ·√

1− v2

c2

x(end of rod) = 1 ·√

1− v2

c2;

the distance between the points being

47

48 Relativity

√1− v2

c2.

But the metre-rod is moving with the ve-locity v relative to K. It therefore follows thatthe length of a rigid metre-rod moving in thedirection of its length with a velocity v is√

1− v2/c2

of a metre. The rigid rod is thus shorter whenin motion than when at rest, and the morequickly it is moving, the shorter is the rod. Forthe velocity v = c we should have√

1− v2/c2 = 0,

and for still greater velocities the square-rootbecomes imaginary. From this we concludethat in the theory of relativity the velocity cplays the part of a limiting velocity, which canneither be reached nor exceeded by any realbody.

Of course this feature of the velocity c as alimiting velocity also clearly follows from theequations of the Lorentz transformation, forthese become meaningless if we choose valuesof v greater than c.

If, on the contrary, we had considered ametre-rod at rest in the x-axis with respect toK, then we should have found that the lengthof the rod as judged from K ′ would have been√

1− v2/c2;

Moving Measuring-Rods and Clocks 49

this is quite in accordance with the principleof relativity which forms the basis of our con-siderations.

A priori it is quite clear that we must beable to learn something about the physicalbehaviour of measuring-rods and clocks fromthe equations of transformation, for the mag-nitudes x, y, z, t, are nothing more nor lessthan the results of measurements obtainableby means of measuring-rods and clocks. If wehad based our considerations on the Galileitransformation we should not have obtaineda contraction of the rod as a consequence ofits motion.

Let us now consider a seconds-clock whichis permanently situated at the origin (x′ = 0)of K ′. t′ = 0 and t′ = 1 are two successive ticksof this clock. The first and fourth equations ofthe Lorentz transformation give for these twoticks:

t = 0

and

t =1√

1− v2

c2

.

As judged from K, the clock is moving withthe velocity v; as judged from this reference-body, the time which elapses between twostrokes of the clock is not one second, but

1√1− v2

c2

seconds, i.e. a somewhat larger time. As aconsequence of its motion the clock goes more

50 Relativity

slowly than when at rest. Here also the veloc-ity c plays the part of an unattainable limitingvelocity.

XIII. Theorem ofthe Addition ofVelocities. TheExperiment ofFizeau

Now in practice we can move clocks andmeasuring-rods only with velocities that aresmall compared with the velocity of light;hence we shall hardly be able to compare theresults of the previous section directly withthe reality. But, on the other hand, these re-sults must strike you as being very singular,and for that reason I shall now draw anotherconclusion from the theory, one which can eas-ily be derived from the foregoing considera-tions, and which has been most elegantly con-firmed by experiment.

In Section VI we derived the theorem ofthe addition of velocities in one direction inthe form which also results from the hypothe-ses of classical mechanics. This theorem canalso be deduced readily from the Galilei trans-

51

52 Relativity

formation (Section XI). In place of the manwalking inside the carriage, we introduce apoint moving relatively to the co-ordinate sys-tem K ′ in accordance with the equation

x′ = wt′.

By means of the first and fourth equationsof 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 thanthe law of motion of the point with referenceto the system K (of the man with referenceto the embankment). We denote this velocityby the symbol W , and we then obtain, as inSection VI,

W = v + w. (1)

But we can carry out this considerationjust as well on the basis of the theory of rel-ativity. In the equation

x′ = wt′,

we must then express x′ and t′ in terms of xand t, making use of the first and fourth equa-tions of the Lorentz transformation. Insteadof the equation (1) we then obtain the equa-tion

W =v + w

1 + vwc2

, (2)

which corresponds to the theorem of additionfor velocities in one direction according to the

Theorem of the Addition of Velocities 53

theory of relativity. The question now arisesas to which of these two theorems is the bet-ter in accord with experience. On this pointwe are enlightened by a most important ex-periment which the brilliant physicist Fizeauperformed more than half a century ago, andwhich has been repeated since then by someof the best experimental physicists, so thatthere can be no doubt about its result. Theexperiment is concerned with the followingquestion. Light travels in a motionless liquidwith a particular velocity w. How quickly doesit travel in the direction of the arrow in thetube T (see the accompanying diagram, Fig.3) when the liquid above mentioned is flowingthrough the tube with a velocity v?

FIG. 3.

In accordance with the principle of relativ-ity we shall certainly have to take for grantedthat the propagation of light always takesplace with the same velocity w with respectto the liquid, whether the latter is in motionwith reference to other bodies or not. The ve-locity of light relative to the liquid and the ve-locity of the latter relative to the tube are thusknown, and we require the velocity of lightrelative to the tube.

54 Relativity

It is clear that we have the problem of Sec-tion VI again before us. The tube plays thepart of the railway embankment or of the co-ordinate system K, the liquid plays the partof the carriage or of the co-ordinate systemK ′, and finally, the light plays the part of theman walking along the carriage, or of the mov-ing point in the present section. If we denotethe velocity of the light relative to the tube byW , then this is given by the equation (1) or(2), according as the Galilei transformation orthe Lorentz transformation corresponds to thefacts. Experiment12 decides in favour of equa-tion (2) derived from the theory of relativity,

12Fizeau found

W = w + v(1− 1n2

),

where

n =c

w

is the index of refraction of the liquid. On the otherhand, owing to the smallness of

vw

c2

as compared with 1, we can replace (2) in the first placeby

W = (w + v)(1− vw

c2),

or to the same order of approximation by

w + v(1− 1n2

),

which agrees with Fizeau’s result.

Theorem of the Addition of Velocities 55

and the agreement is, indeed, very exact. Ac-cording to recent and most excellent measure-ments by Zeeman, the influence of the velocityof flow v on the propagation of light is repre-sented by formula (2) to within one per cent.

Nevertheless we must now draw attentionto the fact that a theory of this phenomenonwas given by H. A. Lorentz long before thestatement of the theory of relativity. This the-ory was of a purely electrodynamical nature,and was obtained by the use of particular hy-potheses as to the electromagnetic structureof matter. This circumstance, however, doesnot in the least diminish the conclusivenessof the experiment as a crucial test in favourof the theory of relativity, for the electrody-namics of Maxwell-Lorentz, on which the orig-inal theory was based, in no way opposes thetheory of relativity. Rather has the latterbeen developed from electrodynamics as anastoundingly simple combination and gener-alisation of the hypotheses, formerly indepen-dent of each other, on which electrodynamicswas built.

56 Relativity

XIV. The HeuristicValue of the Theoryof Relativity

Our train of thought in the foregoing pagescan be epitomised in the following manner.Experience has led to the conviction that, onthe one hand, the principle of relativity holdstrue, and that on the other hand the velocityof transmission of light in vacuo has to be con-sidered equal to a constant c. By uniting thesetwo postulates we obtained the law of trans-formation for the rectangular co-ordinates x,y, z and the time t of the events which consti-tute the processes of nature. In this connec-tion we did not obtain the Galilei transforma-tion, but, differing from classical mechanics,the Lorentz transformation.

The law of transmission of light, the accep-tance of which is justified by our actual knowl-edge, played an important part in this processof thought. Once in possession of the Lorentztransformation, however, we can combine thiswith the principle of relativity, and sum up thetheory thus:

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58 Relativity

Every general law of nature must be soconstituted that it is transformed into a lawof exactly the same form when, instead ofthe space-time variables x, y, z, t of theoriginal co-ordinate system K, we introducenew space-time variables x′, y′, z′, t′ of a co-ordinate system K ′. In this connection the re-lation between the ordinary and the accentedmagnitudes is given by the Lorentz transfor-mation. Or, in brief: General laws of natureare co-variant with respect to Lorentz trans-formations.

This is a definite mathematical conditionthat the theory of relativity demands of a nat-ural law, and in virtue of this, the theory be-comes a valuable heuristic aid in the searchfor general laws of nature. If a general lawof nature were to be found which did not sat-isfy this condition, then at least one of thetwo fundamental assumptions of the theorywould have been disproved. Let us now ex-amine what general results the latter theoryhas hitherto evinced.

XV. General Resultsof the Theory

It is clear from our previous considerationsthat the (special) theory of relativity hasgrown out of electrodynamics and optics. Inthese fields it has not appreciably altered thepredictions of theory, but it has considerablysimplified the theoretical structure, i.e. thederivation of laws, and—what is incompara-bly more important—it has considerably re-duced the number of independent hypothesesforming the basis of theory. The special the-ory of relativity has rendered the Maxwell-Lorentz theory so plausible, that the latterwould have been generally accepted by physi-cists even if experiment had decided less un-equivocally in its favour.

Classical mechanics required to be modi-fied before it could come into line with the de-mands of the special theory of relativity. Forthe main part, however, this modification af-fects only the laws for rapid motions, in whichthe velocities of matter v are not very small ascompared with the velocity of light. We haveexperience of such rapid motions only in the

59

60 Relativity

case of electrons and ions; for other motionsthe variations from the laws of classical me-chanics are too small to make themselves evi-dent in practice. We shall not consider the mo-tion of stars until we come to speak of the gen-eral theory of relativity. In accordance withthe theory of relativity the kinetic energy of amaterial point of mass m is no longer given bythe well-known expression

mv2

2,

but by the expression

mc2√1− v2

c2

.

This expression approaches infinity as thevelocity v approaches the velocity of light c.The velocity must therefore always remainless than c, however great may be the ener-gies used to produce the acceleration. If wedevelop the expression for the kinetic energyin the form of a series,

mc2 + mv2

2+

3

8m

v4

c2+ . . . .

Whenv2

c2

is small compared with unity, the third ofthese terms is always small in comparisonwith the second, which last is alone consid-ered in classical mechanics. The first term mc2

does not contain the velocity, and requires no

General Results of the Theory 61

consideration if we are only dealing with thequestion as to how the energy of a point-massdepends on the velocity. We shall speak of itsessential significance later.

The most important result of a generalcharacter to which the special theory of rel-ativity has led is concerned with the concep-tion of mass. Before the advent of relativity,physics recognised two conservation laws offundamental importance, namely, the law ofthe conservation of energy and the law of theconservation of mass; these two fundamentallaws appeared to be quite independent of eachother. By means of the theory of relativitythey have been united into one law. We shallnow briefly consider how this unification cameabout, and what meaning is to be attached toit.

The principle of relativity requires thatthe law of the conservation of energy shouldhold not only with reference to a co-ordinatesystem K, but also with respect to every co-ordinate system K ′ which is in a state of uni-form motion of translation relative to K, or,briefly, relative to every “Galileian” system ofco-ordinates. In contrast to classical mechan-ics, the Lorentz transformation is the decidingfactor in the transition from one such systemto another.

By means of comparatively simple consid-erations we are led to draw the following con-clusion from these premises, in conjunctionwith the fundamental equations of the elec-trodynamics of Maxwell: A body moving with

62 Relativity

the velocity v, which absorbs13 an amount ofenergy E0 in the form of radiation without suf-fering an alteration in velocity in the process,has, as a consequence, its energy increased byan amount

E0√1− v2

c2

.

In consideration of the expression givenabove for the kinetic energy of the body, therequired energy of the body comes out to be

(m + E0

c2)c2√

1− v2

c2

.

Thus the body has the same energy as a bodyof 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

c2,

the inertial mass of a body is not a constant,but varies according to the change in the en-ergy of the body. The inertial mass of a systemof bodies can even be regarded as a measure ofits energy. The law of the conservation of themass of a system becomes identical with thelaw of the conservation of energy, and is only

13E0 is the energy taken up, as judged from a co-ordinate system moving with the body.

General Results of the Theory 63

valid provided that the system neither takesup nor sends out energy. Writing the expres-sion for the energy in the form

mc2 + E0√1− v2

c2

,

we see that the term mc2, which has hithertoattracted our attention, is nothing else thanthe energy possessed by the body14 before itabsorbed the energy E0.

A direct comparison of this relation withexperiment is not possible at the present time,owing to the fact that the changes in energyE0 to which we can subject a system are notlarge enough to make themselves perceptibleas a change in the inertial mass of the system.

E0

c2

is too small in comparison with the mass m,which was present before the alteration of theenergy. It is owing to this circumstance thatclassical mechanics was able to establish suc-cessfully the conservation of mass as a law ofindependent validity.

Let me add a final remark of a funda-mental nature. The success of the Faraday-Maxwell interpretation of electromagnetic ac-tion at a distance resulted in physicists be-coming convinced that there are no suchthings as instantaneous actions at a distance(not involving an intermediary medium) of the

14As judged from a co-ordinate system moving withthe body.

64 Relativity

type of Newton’s law of gravitation. Accord-ing to the theory of relativity, action at a dis-tance with the velocity of light always takesthe place of instantaneous action at a distanceor of action at a distance with an infinite ve-locity of transmission. This is connected withthe fact that the velocity c plays a fundamen-tal rôle in this theory. In Part II we shall seein what way this result becomes modified inthe general theory of relativity.

XVI. Experienceand the SpecialTheory ofRelativity

To what extent is the special theory of rel-ativity supported by experience? This ques-tion is not easily answered for the reason al-ready mentioned in connection with the fun-damental experiment of Fizeau. The spe-cial theory of relativity has crystallised outfrom the Maxwell-Lorentz theory of electro-magnetic phenomena. Thus all facts of expe-rience which support the electromagnetic the-ory also support the theory of relativity. As be-ing of particular importance, I mention herethe fact that the theory of relativity enablesus to predict the effects produced on the lightreaching us from the fixed stars. These re-sults are obtained in an exceedingly simplemanner, and the effects indicated, which aredue to the relative motion of the earth withreference to those fixed stars, are found to bein accord with experience. We refer to the

65

66 Relativity

yearly movement of the apparent position ofthe fixed stars resulting from the motion ofthe earth round the sun (aberration), and tothe influence of the radial components of therelative motions of the fixed stars with respectto the earth on the colour of the light reach-ing us from them. The latter effect manifestsitself in a slight displacement of the spectrallines of the light transmitted to us from afixed star, as compared with the position of thesame spectral lines when they are producedby a terrestrial source of light (Doppler prin-ciple). The experimental arguments in favourof the Maxwell-Lorentz theory, which are atthe same time arguments in favour of the the-ory of relativity, are too numerous to be setforth here. In reality they limit the theoreticalpossibilities to such an extent, that no othertheory than that of Maxwell and Lorentz hasbeen able to hold its own when tested by ex-perience.

But there are two classes of experimen-tal facts hitherto obtained which can be rep-resented in the Maxwell-Lorentz theory onlyby the introduction of an auxiliary hypothe-sis, which in itself—i.e. without making use ofthe theory of relativity—appears extraneous.

It is known that cathode rays and theso-called β-rays emitted by radioactive sub-stances consist of negatively electrified parti-cles (electrons) of very small inertia and largevelocity. By examining the deflection of theserays under the influence of electric and mag-netic fields, we can study the law of motion ofthese particles very exactly.

Experience and Special Relativity 67

In the theoretical treatment of these elec-trons, we are faced with the difficulty thatelectrodynamic theory of itself is unable togive an account of their nature. For since elec-trical masses of one sign repel each other, thenegative electrical masses constituting theelectron would necessarily be scattered underthe influence of their mutual repulsions, un-less there are forces of another kind operat-ing between them, the nature of which hashitherto remained obscure to us.15 If we nowassume that the relative distances betweenthe electrical masses constituting the electronremain unchanged during the motion of theelectron (rigid connection in the sense of clas-sical mechanics), we arrive at a law of motionof the electron which does not agree with ex-perience. Guided by purely formal points ofview, H. A. Lorentz was the first to introducethe hypothesis that the particles constitutingthe electron experience a contraction in the di-rection of motion in consequence of that mo-tion, the amount of this contraction being pro-portional to the expression√

1− v2

c2

This hypothesis, which is not justifiable byany electrodynamical facts, supplies us thenwith that particular law of motion which hasbeen confirmed with great precision in recentyears.

15The general theory of relativity renders it likelythat the electrical masses of an electron are held to-gether by gravitational forces.

68 Relativity

The theory of relativity leads to the samelaw of motion, without requiring any specialhypothesis whatsoever as to the structure andthe behaviour of the electron. We arrived at asimilar conclusion in Section XIII in connec-tion with the experiment of Fizeau, the resultof which is fore-told by the theory of relativitywithout the necessity of drawing on hypothe-ses as to the physical nature of the liquid.

The second class of facts to which we havealluded has reference to the question whetheror not the motion of the earth in space can bemade perceptible in terrestrial experiments.We have already remarked in Section V thatall attempts of this nature led to a negativeresult. Before the theory of relativity wasput forward, it was difficult to become rec-onciled to this negative result, for reasonsnow to be discussed. The inherited preju-dices about time and space did not allow anydoubt to arise as to the prime importance ofthe Galilei transformation for changing overfrom one body of reference to another. Now as-suming that the Maxwell-Lorentz equationshold for a reference-body K, we then find thatthey do not hold for a reference-body K ′ mov-ing uniformly with respect to K, if we assumethat the relations of the Galileian transfor-mation exist between the co-ordinates of Kand K ′. It thus appears that of all Galile-ian co-ordinate systems one (K) correspond-ing to a particular state of motion is physicallyunique. This result was interpreted physi-cally by regarding K as at rest with respectto a hypothetical æther of space. On the other

Experience and Special Relativity 69

hand, all co-ordinate systems K ′ moving rela-tively to K were to be regarded as in motionwith respect to the æther. To this motion ofK ′ against the æther (“æther-drift” relative toK ′) were assigned the more complicated lawswhich were supposed to hold relative to K ′.Strictly speaking, such an æther-drift oughtalso to be assumed relative to the earth, andfor a long time the efforts of physicists weredevoted to attempts to detect the existence ofan æther-drift at the earth’s surface.

In one of the most notable of these at-tempts Michelson devised a method which ap-pears as though it must be decisive. Imag-ine two mirrors so arranged on a rigid bodythat the reflecting surfaces face each other. Aray of light requires a perfectly definite timeT to pass from one mirror to the other andback again, if the whole system be at rest withrespect to the æther. It is found by calcula-tion, however, that a slightly different timeT ′ is required for this process, if the body, to-gether with the mirrors, be moving relativelyto the æther. And yet another point: it isshown by calculation that for a given velocityv with reference to the æther, this time T ′ isdifferent when the body is moving perpendic-ularly to the planes of the mirrors from thatresulting when the motion is parallel to theseplanes. Although the estimated difference be-tween these two times is exceedingly small,Michelson and Morley performed an experi-ment involving interference in which this dif-ference should have been clearly detectable.But the experiment gave a negative result—

70 Relativity

a fact very perplexing to physicists. Lorentzand FitzGerald rescued the theory from thisdifficulty by assuming that the motion of thebody relative to the æther produces a contrac-tion of the body in the direction of motion,the amount of contraction being just sufficientto compensate for the difference in time men-tioned above. Comparison with the discussionin Section XII shows that from the standpointalso of the theory of relativity this solution ofthe difficulty was the right one. But on thebasis of the theory of relativity the methodof interpretation is incomparably more satis-factory. According to this theory there is nosuch thing as a “specially favoured” (unique)co-ordinate system to occasion the introduc-tion of the æther-idea, and hence there canbe no æther-drift, nor any experiment withwhich to demonstrate it. Here the contrac-tion of moving bodies follows from the two fun-damental principles of the theory without theintroduction of particular hypotheses; and asthe prime factor involved in this contractionwe find, not the motion in itself, to which wecannot attach any meaning, but the motionwith respect to the body of reference chosenin the particular case in point. Thus for aco-ordinate system moving with the earth themirror system of Michelson and Morley is notshortened, but it is shortened for a co-ordinatesystem which is at rest relatively to the sun.

XVII. Minkowski’sFour-DimensionalSpace

The non-mathematician is seized by a mys-terious shuddering when he hears of “four-dimensional” things, by a feeling not unlikethat awakened by thoughts of the occult. Andyet there is no more common-place statementthan 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 describethe position of a point (at rest) by meansof three numbers (co-ordinates) x, y, z, andthat there is an indefinite number of pointsin the neighbourhood of this one, the posi-tion of which can be described by co-ordinatessuch as x1, y1, z1, which may be as near aswe choose to the respective values of the co-ordinates x, y, z of the first point. In virtue ofthe latter property we speak of a “continuum,”and owing to the fact that there are threeco-ordinates we speak of it as being “three-dimensional.”

71

72 Relativity

Similarly, the world of physical phenom-ena which was briefly called “world” byMinkowski is naturally four-dimensional inthe space-time sense. For it is composed ofindividual events, each of which is describedby four numbers, namely, three space co-ordinates x, y, z and a time co-ordinate, thetime-value t. The “world” is in this sensealso a continuum; for to every event there areas many “neighbouring” events (realised or atleast thinkable) as we care to choose, the co-ordinates x1, y1, z1, t1 of which differ by anindefinitely small amount from those of theevent x, y, z, t originally considered. That wehave not been accustomed to regard the worldin this sense as a four-dimensional continuumis due to the fact that in physics, before the ad-vent of the theory of relativity, time played adifferent and more independent rôle, as com-pared with the space co-ordinates. It is forthis reason that we have been in the habit oftreating time as an independent continuum.As a matter of fact, according to classical me-chanics, time is absolute, i.e. it is independentof the position and the condition of motion ofthe system of co-ordinates. We see this ex-pressed in the last equation of the Galileiantransformation (t′ = t).

The four-dimensional mode of considera-tion of the “world” is natural in the theory ofrelativity, since according to this theory timeis robbed of its independence. This is shownby the fourth equation of the Lorentz trans-

Minkowski’s Four-Dimensional Space 73

formation:t′ =

t− vc2

x√1− v2

c2

Moreover, according to this equation thetime difference ∆t′ of two events with respectto K ′ does not in general vanish, even whenthe time difference ∆t of the same eventswith reference to K vanishes. Pure “space-distance” of two events with respect to K re-sults in “time-distance” of the same eventswith respect to K ′. But the discovery, ofMinkowski, which was of importance for theformal development of the theory of relativ-ity, does not lie here. It is to be found ratherin the fact of his recognition that the four-dimensional space-time continuum of the the-ory of relativity, in its most essential formalproperties, shows a pronounced relationshipto the three-dimensional continuum of Eu-clidean geometrical space.16 In order to givedue prominence to this relationship, however,we must replace the usual time co-ordinatet by an imaginary magnitude

√−1ct propor-

tional to it. Under these conditions, the nat-ural laws satisfying the demands of the (spe-cial) theory of relativity assume mathemati-cal forms, in which the time co-ordinate playsexactly the same rôle as the three space co-ordinates. Formally, these four co-ordinatescorrespond exactly to the three space co-ordinates in Euclidean geometry. It must beclear even to the non-mathematician that, as

16Cf. the somewhat more detailed discussion in Ap-pendix II.

74 Relativity

a consequence of this purely formal additionto our knowledge, the theory perforce gainedclearness in no mean measure.

These inadequate remarks can give thereader only a vague notion of the importantidea contributed by Minkowski. Without itthe general theory of relativity, of which thefundamental ideas are developed in the fol-lowing pages, would perhaps have got no far-ther than its long clothes. Minkowski’s workis doubtless difficult of access to anyone inex-perienced in mathematics, but since it is notnecessary to have a very exact grasp of thiswork in order to understand the fundamen-tal ideas of either the special or the generaltheory of relativity, I shall at present leave ithere, and shall revert to it only towards theend of Part II.

Part II: TheGeneral Theory of

Relativity

75

XVIII. Special andGeneral Principleof Relativity

The basal principle, which was the pivot of allour previous considerations, was the specialprinciple of relativity, i.e. the principle of thephysical relativity of all uniform motion. Letus once more analyse its meaning carefully.

It was at all times clear that, from thepoint of view of the idea it conveys to us, everymotion must only be considered as a relativemotion. Returning to the illustration we havefrequently used of the embankment and therailway carriage, we can express the fact ofthe motion here taking place in the followingtwo forms, both of which are equally justifi-able: The carriage is in motion relative to theembankment. The embankment is in motionrelative to the carriage.

In (a) the embankment, in (b) the carriage,serves as the body of reference in our state-ment of the motion taking place. If it is sim-ply a question of detecting or of describing themotion involved, it is in principle immaterial

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78 Relativity

to what reference-body we refer the motion.As already mentioned, this is self-evident, butit must not be confused with the much morecomprehensive statement called “the princi-ple of relativity,” which we have taken as thebasis of our investigations.

The principle we have made use of notonly maintains that we may equally wellchoose the carriage or the embankment as ourreference-body for the description of any event(for this, too, is self-evident). Our principlerather asserts what follows: If we formulatethe general laws of nature as they are ob-tained from experience, by making use of theembankment as reference-body, the railwaycarriage as reference-body, then these gen-eral laws of nature (e.g. the laws of mechan-ics or the law of the propagation of light invacuo) have exactly the same form in bothcases. This can also be expressed as follows:For the physical description of natural pro-cesses, neither of the reference-bodies K, K ′

is unique (lit. “specially marked out”) as com-pared with the other. Unlike the first, this lat-ter statement need not of necessity hold a pri-ori; it is not contained in the conceptions of“motion” and “reference-body” and derivablefrom them; only experience can decide as to itscorrectness or incorrectness.

Up to the present, however, we have byno means maintained the equivalence of allbodies of reference K in connection with theformulation of natural laws. Our course wasmore on the following lines. In the first place,we started out from the assumption that there

Special and General Relativity 79

exists a reference-body K, whose condition ofmotion is such that the Galileian law holdswith respect to it: A particle left to itself andsufficiently far removed from all other parti-cles moves uniformly in a straight line. Withreference to K (Galileian reference-body) thelaws of nature were to be as simple as pos-sible. But in addition to K, all bodies of ref-erence K ′ should be given preference in thissense, and they should be exactly equivalentto K for the formulation of natural laws, pro-vided that they are in a state of uniform rec-tilinear and non-rotary motion with respectto K; all these bodies of reference are to beregarded as Galileian reference-bodies. Thevalidity of the principle of relativity was as-sumed only for these reference-bodies, but notfor others (e.g. those possessing motion of adifferent kind). In this sense we speak of thespecial principle of relativity, or special theoryof relativity.

In contrast to this we wish to understandby the “general principle of relativity” the fol-lowing statement: All bodies of reference K,K ′, etc., are equivalent for the description ofnatural phenomena (formulation of the gen-eral laws of nature), whatever may be theirstate of motion. But before proceeding farther,it ought to be pointed out that this formula-tion must be replaced later by a more abstractone, for reasons which will become evident ata later stage.

Since the introduction of the special prin-ciple of relativity has been justified, every in-tellect which strives after generalisation must

80 Relativity

feel the temptation to venture the step to-wards the general principle of relativity. But asimple and apparently quite reliable consider-ation seems to suggest that, for the present atany rate, there is little hope of success in suchan attempt. Let us imagine ourselves trans-ferred to our old friend the railway carriage,which is travelling at a uniform rate. As longas it is moving uniformly, the occupant of thecarriage is not sensible of its motion, and it isfor this reason that he can un-reluctantly in-terpret the facts of the case as indicating thatthe carriage is at rest, but the embankmentin motion. Moreover, according to the spe-cial principle of relativity, this interpretationis quite justified also from a physical point ofview.

If the motion of the carriage is nowchanged into a non-uniform motion, as forinstance by a powerful application of thebrakes, then the occupant of the carriage ex-periences a correspondingly powerful jerk for-wards. The retarded motion is manifestedin the mechanical behaviour of bodies rela-tive to the person in the railway carriage.The mechanical behaviour is different fromthat of the case previously considered, andfor this reason it would appear to be impos-sible that the same mechanical laws hold rel-atively to the non-uniformly moving carriage,as hold with reference to the carriage whenat rest or in uniform motion. At all events itis clear that the Galileian law does not holdwith respect to the non-uniformly moving car-riage. Because of this, we feel compelled at

Special and General Relativity 81

the present juncture to grant a kind of abso-lute physical reality to non-uniform motion, inopposition to the general principle of relativ-ity. But in what follows we shall soon see thatthis conclusion cannot be maintained.

82 Relativity

IX. TheGravitational Field

“If we pick up a stone and then let it go, whydoes it fall to the ground?” The usual answerto this question is: “Because it is attracted bythe earth.” Modern physics formulates the an-swer rather differently for the following rea-son. As a result of the more careful study ofelectromagnetic phenomena, we have come toregard action at a distance as a process im-possible without the intervention of some in-termediary medium. If, for instance, a magnetattracts a piece of iron, we cannot be contentto regard this as meaning that the magnetacts directly on the iron through the interme-diate empty space, but we are constrained toimagine—after the manner of Faraday—thatthe magnet always calls into being somethingphysically real in the space around it, thatsomething being what we call a “magneticfield.” In its turn this magnetic field operateson the piece of iron, so that the latter strivesto move towards the magnet. We shall not dis-cuss here the justification for this incidentalconception, which is indeed a somewhat arbi-

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84 Relativity

trary one. We shall only mention that with itsaid electromagnetic phenomena can be theo-retically represented much more satisfactorilythan without it, and this applies particularlyto the transmission of electromagnetic waves.The effects of gravitation also are regarded inan analogous manner.

The action of the earth on the stone takesplace indirectly. The earth produces in its sur-roundings a gravitational field, which acts onthe stone and produces its motion of fall. Aswe know from experience, the intensity of theaction on a body diminishes according to aquite definite law, as we proceed farther andfarther away from the earth. From our pointof view this means: The law governing theproperties of the gravitational field in spacemust be a perfectly definite one, in order cor-rectly to represent the diminution of gravita-tional action with the distance from operativebodies. It is something like this: The body (e.g.the earth) produces a field in its immediateneighbourhood directly; the intensity and di-rection of the field at points farther removedfrom the body are thence determined by thelaw which governs the properties in space ofthe gravitational fields themselves.

In contrast to electric and magnetic fields,the gravitational field exhibits a most remark-able property, which is of fundamental impor-tance for what follows. Bodies which are mov-ing under the sole influence of a gravitationalfield receive an acceleration, which does notin the least depend either on the material oron the physical state of the body. For instance,

The Gravitational Field 85

a piece of lead and a piece of wood fall in ex-actly the same manner in a gravitational field(in vacuo), when they start off from rest orwith the same initial velocity. This law, whichholds most accurately, can be expressed in adifferent form in the light of the following con-sideration.

According to Newton’s law of motion, wehave

(Force) = (inertial mass)× (acceleration),

where the “inertial mass” is a characteristicconstant of the accelerated body. If now gravi-tation is the cause of the acceleration, we thenhave17

(Force) = (gravitational mass)×(gravitational intensity),

where the “gravitational mass” is likewise acharacteristic constant for the body. Fromthese two relations follows:

(acceleration) =(gravitational mass)

(inertial mass)×

(gravitational intensity).

If now, as we find from experience, the ac-celeration is to be independent of the nature

17For editing reasons, in the following two ex-pressions the phrase “intensity of the gravitationalfield” has been replaced by the phrase “gravitationalintensity”—R. Burkey.

86 Relativity

and the condition of the body and always thesame for a given gravitational field, then theratio of the gravitational to the inertial massmust likewise be the same for all bodies. By asuitable choice of units we can thus make thisratio equal to unity. We then have the follow-ing law: The gravitational mass of a body isequal to its inertial mass.

It is true that this important law had hith-erto been recorded in mechanics, but it hadnot been interpreted. A satisfactory interpre-tation can be obtained only if we recognisethe following fact: The same quality of a bodymanifests itself according to circumstances as“inertia” or as “weight” (lit. “heaviness”). Inthe following section we shall show to whatextent this is actually the case, and how thisquestion is connected with the general postu-late of relativity.

XX. The Equality ofInertial andGravitational Massas an Argument forthe GeneralPostulate ofRelativity

We imagine a large portion of empty space,so far removed from stars and other appre-ciable masses that we have before us approx-imately the conditions required by the fun-damental law of Galilei. It is then possibleto choose a Galileian reference-body for thispart of space (world), relative to which pointsat rest remain at rest and points in motioncontinue permanently in uniform rectilinearmotion. As reference-body let us imagine aspacious chest resembling a room with an ob-server inside who is equipped with appara-tus. Gravitation naturally does not exist for

87

88 Relativity

this observer. He must fasten himself withstrings to the floor, otherwise the slightest im-pact against the floor will cause him to riseslowly towards the ceiling of the room.

To the middle of the lid of the chest is fixedexternally a hook with rope attached, and nowa “being” (what kind of a being is immaterialto us) begins pulling at this with a constantforce. The chest together with the observerthen begin to move “upwards” with a uni-formly accelerated motion. In course of timetheir velocity will reach unheard-of values—provided that we are viewing all this from an-other reference-body which is not being pulledwith a rope.

But how does the man in the chest regardthe process? The acceleration of the chest willbe transmitted to him by the reaction of thefloor of the chest. He must therefore take upthis pressure by means of his legs if he doesnot wish to be laid out full length on the floor.He is then standing in the chest in exactly thesame way as anyone stands in a room of ahouse on our earth. If he release a body whichhe previously had in his hand, the accelera-tion of the chest will no longer be transmittedto this body, and for this reason the body willapproach the floor of the chest with an accel-erated relative motion. The observer will fur-ther convince himself that the acceleration ofthe body towards the floor of the chest is al-ways of the same magnitude, whatever kindof body he may happen to use for the experi-ment.

Relying on his knowledge of the gravita-

Inertial and Gravitational Mass 89

tional field (as it was discussed in the preced-ing section), the man in the chest will thuscome to the conclusion that he and the chestare in a gravitational field which is constantwith regard to time. Of course he will be puz-zled for a moment as to why the chest doesnot fall in this gravitational field. Just then,however, he discovers the hook in the middleof the lid of the chest and the rope which is at-tached to it, and he consequently comes to theconclusion that the chest is suspended at restin the gravitational field.

Ought we to smile at the man and say thathe errs in his conclusion? I do not believewe ought if we wish to remain consistent; wemust rather admit that his mode of grasp-ing the situation violates neither reason norknown mechanical laws. Even though it is be-ing accelerated with respect to the “Galileianspace” first considered, we can neverthelessregard the chest as being at rest. We havethus good grounds for extending the princi-ple of relativity to include bodies of referencewhich are accelerated with respect to eachother, and as a result we have gained a pow-erful argument for a generalised postulate ofrelativity.

We must note carefully that the possibilityof this mode of interpretation rests on the fun-damental property of the gravitational fieldof giving all bodies the same acceleration, or,what comes to the same thing, on the lawof the equality of inertial and gravitationalmass. If this natural law did not exist, theman in the accelerated chest would not be

90 Relativity

able to interpret the behaviour of the bodiesaround him on the supposition of a gravita-tional field, and he would not be justified onthe grounds of experience in supposing hisreference-body to be “at rest.”

Suppose that the man in the chest fixes arope to the inner side of the lid, and that he at-taches a body to the free end of the rope. Theresult of this will be to stretch the rope so thatit will hang “vertically” downwards. If we askfor an opinion of the cause of tension in therope, the man in the chest will say: “The sus-pended body experiences a downward force inthe gravitational field, and this is neutralisedby the tension of the rope; what determinesthe magnitude of the tension of the rope isthe gravitational mass of the suspended body.”On the other hand, an observer who is poisedfreely in space will interpret the condition ofthings thus: “The rope must perforce takepart in the accelerated motion of the chest,and it transmits this motion to the body at-tached to it. The tension of the rope is justlarge enough to effect the acceleration of thebody. That which determines the magnitudeof the tension of the rope is the inertial massof the body.” Guided by this example, we seethat our extension of the principle of relativityimplies the necessity of the law of the equal-ity of inertial and gravitational mass. Thuswe have obtained a physical interpretation ofthis law.

From our consideration of the acceleratedchest we see that a general theory of relativ-ity must yield important results on the laws

Inertial and Gravitational Mass 91

of gravitation. In point of fact, the system-atic pursuit of the general idea of relativityhas supplied the laws satisfied by the gravi-tational field. Before proceeding farther, how-ever, I must warn the reader against a mis-conception suggested by these considerations.A gravitational field exists for the man in thechest, despite the fact that there was no suchfield for the co-ordinate system first chosen.Now we might easily suppose that the exis-tence of a gravitational field is always onlyan apparent one. We might also think that,regardless of the kind of gravitational fieldwhich may be present, we could always chooseanother reference-body such that no gravita-tional field exists with reference to it. This isby no means true for all gravitational fields,but only for those of quite special form. Itis, for instance, impossible to choose a bodyof reference such that, as judged from it, thegravitational field of the earth (in its entirety)vanishes.

We can now appreciate why that argumentis not convincing, which we brought forwardagainst the general principle of relativity atthe end of Section XVIII. It is certainly truethat the observer in the railway carriage ex-periences a jerk forwards as a result of theapplication of the brake, and that he recog-nises in this the non-uniformity of motion (re-tardation) of the carriage. But he is compelledby nobody to refer this jerk to a “real” ac-celeration (retardation) of the carriage. Hemight also interpret his experience thus: “Mybody of reference (the carriage) remains per-

92 Relativity

manently at rest. With reference to it, how-ever, there exists (during the period of ap-plication of the brakes) a gravitational fieldwhich is directed forwards and which is vari-able with respect to time. Under the influ-ence of this field, the embankment togetherwith the earth moves non-uniformly in sucha manner that their original velocity in thebackwards direction is continuously reduced.”

XXI. In WhatRespects Are theFoundations ofClassicalMechanics and ofthe Special Theoryof RelativityUnsatisfactory?

We have already stated several times thatclassical mechanics starts out from the follow-ing law: Material particles sufficiently far re-moved from other material particles continueto move uniformly in a straight line or con-tinue in a state of rest. We have also repeat-edly emphasised that this fundamental lawcan only be valid for bodies of reference Kwhich possess certain unique states of motion,and which are in uniform translational mo-tion relative to each other. Relative to other

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reference-bodies K the law is not valid. Bothin classical mechanics and in the special the-ory of relativity we therefore differentiate be-tween reference-bodies K relative to whichthe recognised “laws of nature” can be said tohold, and reference-bodies K relative to whichthese laws do not hold.

But no person whose mode of thought islogical can rest satisfied with this conditionof things. He asks: “How does it come thatcertain reference-bodies (or their states of mo-tion) are given priority over other reference-bodies (or their states of motion)? What is thereason for this preference?” In order to showclearly what I mean by this question, I shallmake use of a comparison.

I am standing in front of a gas range.Standing alongside of each other on the rangeare two pans so much alike that one may bemistaken for the other. Both are half full ofwater. I notice that steam is being emittedcontinuously from the one pan, but not fromthe other. I am surprised at this, even if I havenever seen either a gas range or a pan before.But if I now notice a luminous something ofbluish colour under the first pan but not un-der the other, I cease to be astonished, even ifI have never before seen a gas flame. For I canonly say that this bluish something will causethe emission of the steam, or at least possiblyit may do so. If, however, I notice the bluishsomething in neither case, and if I observethat the one continuously emits steam whilstthe other does not, then I shall remain aston-ished and dissatisfied until I have discovered

Unsatisfactory Foundations 95

some circumstance to which I can attributethe different behaviour of the two pans.

Analogously, I seek in vain for a real some-thing in classical mechanics (or in the spe-cial theory of relativity) to which I can at-tribute the different behaviour of bodies con-sidered with respect to the reference-systemsK and K ′.18 Newton saw this objection and at-tempted to invalidate it, but without success.But E. Mach recognised it most clearly of all,and because of this objection he claimed thatmechanics must be placed on a new basis. Itcan only be got rid of by means of a physicswhich is conformable to the general principleof relativity, since the equations of such a the-ory hold for every body of reference, whatevermay be its state of motion.

18The objection is of importance more especiallywhen the state of motion of the reference-body is ofsuch a nature that it does not require any externalagency for its maintenance, e.g. in the case when thereference-body is rotating uniformly.

96 Relativity

XXII. A FewInferences fromthe General Theoryof Relativity

The considerations of Section XX show thatthe general theory of relativity puts us in aposition to derive properties of the gravita-tional field in a purely theoretical manner. Letus suppose, for instance, that we know thespace-time “course” for any natural processwhatsoever, as regards the manner in whichit takes place in the Galileian domain relativeto a Galileian body of reference K. By meansof purely theoretical operations (i.e. simply bycalculation) we are then able to find how thisknown natural process appears, as seen froma reference-body K ′ which is accelerated rela-tively to K. But since a gravitational field ex-ists with respect to this new body of referenceK ′, our consideration also teaches us how thegravitational field influences the process stud-ied.

For example, we learn that a body which

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98 Relativity

is in a state of uniform rectilinear motionwith respect to K (in accordance with the lawof Galilei) is executing an accelerated and ingeneral curvilinear motion with respect to theaccelerated reference-body K ′ (chest). Thisacceleration or curvature corresponds to theinfluence on the moving body of the gravita-tional field prevailing relatively to K ′. It isknown that a gravitational field influences themovement of bodies in this way, so that ourconsideration supplies us with nothing essen-tially new.

However, we obtain a new result of fun-damental importance when we carry out theanalogous consideration for a ray of light.With respect to the Galileian reference-bodyK, such a ray of light is transmitted recti-linearly with the velocity c. It can easilybe shown that the path of the same ray oflight is no longer a straight line when we con-sider it with reference to the accelerated chest(reference-body K ′). From this we conclude,that, in general, rays of light are propagatedcurvilinearly in gravitational fields. In two re-spects this result is of great importance.

In the first place, it can be compared withthe reality. Although a detailed examinationof the question shows that the curvature oflight rays required by the general theory ofrelativity is only exceedingly small for thegravitational fields at our disposal in practice,its estimated magnitude for light rays pass-ing the sun at grazing incidence is neverthe-less 1.7 seconds of arc. This ought to mani-fest itself in the following way. As seen from

Inferences from General Relativity 99

the earth, certain fixed stars appear to be inthe neighbourhood of the sun, and are thuscapable of observation during a total eclipseof the sun. At such times, these stars oughtto appear to be displaced outwards from thesun by an amount indicated above, as com-pared with their apparent position in the skywhen the sun is situated at another part of theheavens. The examination of the correctnessor otherwise of this deduction is a problem ofthe greatest importance, the early solution ofwhich is to be expected of astronomers.19

In the second place our result shows that,according to the general theory of relativity,the law of the constancy of the velocity of lightin vacuo, which constitutes one of the two fun-damental assumptions in the special theory ofrelativity and to which we have already fre-quently referred, cannot claim any unlimitedvalidity. A curvature of rays of light can onlytake place when the velocity of propagationof light varies with position. Now we mightthink that as a consequence of this, the spe-cial theory of relativity and with it the wholetheory of relativity would be laid in the dust.But in reality this is not the case. We can onlyconclude that the special theory of relativitycannot claim an unlimited domain of validity;its result hold only so long as we are able to

19By means of the star photographs of two expedi-tions equipped by a Joint Committee of the Royal andRoyal Astronomical Societies, the existence of the de-flection of light demanded by theory was confirmed dur-ing the solar eclipse of 29th May, 1919. (Cf. AppendixIII.)

100 Relativity

disregard the influences of gravitational fieldson the phenomena (e.g. of light).

Since it has often been contended by op-ponents of the theory of relativity that thespecial theory of relativity is overthrown bythe general theory of relativity, it is per-haps advisable to make the facts of the caseclearer by means of an appropriate compar-ison. Before the development of electrody-namics the laws of electrostatics and the lawsof electricity were regarded indiscriminately.At the present time we know that electricfields can be derived correctly from electro-static considerations only for the case, whichis never strictly realised, in which the electri-cal masses are quite at rest relatively to eachother, and to the co-ordinate system. Shouldwe be justified in saying that for this rea-son electrostatics is overthrown by the field-equations of Maxwell in electrodynamics? Notin the least. Electrostatics is contained inelectrodynamics as a limiting case; the laws ofthe latter lead directly to those of the formerfor the case in which the fields are invariablewith regard to time. No fairer destiny couldbe allotted to any physical theory, than thatit should of itself point out the way to the in-troduction of a more comprehensive theory, inwhich it lives on as a limiting case.

In the example of the transmission of lightjust dealt with, we have seen that the generaltheory of relativity enables us to derive theo-retically the influence of a gravitational fieldon the course of natural processes, the lawsof which are already known when a gravita-

Inferences from General Relativity 101

tional field is absent. But the most attractiveproblem, to the solution of which the generaltheory of relativity supplies the key, concernsthe investigation of the laws satisfied by thegravitational field itself. Let us consider thisfor a moment.

We are acquainted with space-time do-mains which behave (approximately) in a“Galileian” fashion under suitable choice ofreference-body, i.e. domains in which gravita-tional fields are absent. If we now refer such adomain to a reference-body K ′ possessing anykind of motion, then relative to K ′ there existsa gravitational field which is variable with re-spect to space and time.20 The character ofthis field will of course depend on the motionchosen for K ′. According to the general the-ory of relativity, the general law of the grav-itational field must be satisfied for all grav-itational fields obtainable in this way. Eventhough by no means all gravitational fieldscan be produced in this way, yet we may enter-tain the hope that the general law of gravita-tion will be derivable from such gravitationalfields of a special kind. This hope has been re-alised in the most beautiful manner. But be-tween the clear vision of this goal and its ac-tual realisation it was necessary to surmounta serious difficulty, and as this lies deep at theroot of things, I dare not withhold it from thereader. We require to extend our ideas of thespace-time continuum still farther.

20This follows from a generalisation of the discussionin Section XX.

102 Relativity

XXIII. Behaviour ofClocks andMeasuring Rods ona Rotating Body ofReference

Hitherto I have purposely refrained fromspeaking about the physical interpretation ofspace- and time-data in the case of the generaltheory of relativity. As a consequence, I amguilty of a certain slovenliness of treatment,which, as we know from the special theory ofrelativity, is far from being unimportant andpardonable. It is now high time that we rem-edy this defect; but I would mention at theoutset, that this matter lays no small claimson the patience and on the power of abstrac-tion of the reader.

We start off again from quite special cases,which we have frequently used before. Letus consider a space-time domain in whichno gravitational fields exists relative to areference-body K whose state of motion has

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been suitably chosen. K is then a Galileianreference-body as regards the domain consid-ered, and the results of the special theory ofrelativity hold relative to K. Let us supposethe same domain referred to a second body ofreference K ′, which is rotating uniformly withrespect to K. In order to fix our ideas, we shallimagine K ′ to be in the form of a plane circu-lar disc, which rotates uniformly in its ownplane about its centre. An observer who issitting eccentrically on the disc K ′ is sensi-ble of a force which acts outwards in a radialdirection, and which would be interpreted asan effect of inertia (centrifugal force) by anobserver who was at rest with respect to theoriginal reference-body K. But the observeron the disc may regard his disc as a reference-body which is “at rest”; on the basis of thegeneral principle of relativity he is justifiedin doing this. The force acting on himself,and in fact on all other bodies which are atrest relative to the disc, he regards as the ef-fect of a gravitational field. Nevertheless, thespace-distribution of this gravitational field isof a kind that would not be possible on New-ton’s theory of gravitation.21 But since the ob-server believes in the general theory of rela-tivity, this does not disturb him; he is quite inthe right when he believes that a general lawof gravitation can be formulated—a law whichnot only explains the motion of the stars cor-

21The field disappears at the centre of the disc andincreases proportionally to the distance from the centreas we proceed outwards.

Rotating Clocks and Measuring Rods 105

rectly, but also the field of force experiencedby himself.

The observer performs experiments on hiscircular disc with clocks and measuring-rods.In doing so, it is his intention to arrive atexact definitions for the signification of time-and space-data with reference to the circulardisc K ′, these definitions being based on hisobservations. What will be his experience inthis enterprise?

To start with, he places one of two iden-tically constructed clocks at the centre of thecircular disc, and the other on the edge of thedisc, so that they are at rest relative to it.We now ask ourselves whether both clocks goat the same rate from the standpoint of thenon-rotating Galileian reference-body K. Asjudged from this body, the clock at the centreof the disc has no velocity, whereas the clockat the edge of the disc is in motion relativeto K in consequence of the rotation. Accord-ing to a result obtained in Section XII, it fol-lows that the latter clock goes at a rate perma-nently slower than that of the clock at the cen-tre of the circular disc, i.e. as observed fromK. It is obvious that the same effect would benoted by an observer whom we will imaginesitting alongside his clock at the centre of thecircular disc. Thus on our circular disc, or, tomake the case more general, in every gravita-tional field, a clock will go more quickly or lessquickly, according to the position in which theclock is situated (at rest). For this reason itis not possible to obtain a reasonable defini-tion of time with the aid of clocks which are

106 Relativity

arranged at rest with respect to the body ofreference. A similar difficulty presents itselfwhen we attempt to apply our earlier defini-tion of simultaneously in such a case, but I donot wish to go any farther into this question.

Moreover, at this stage the definition ofthe space co-ordinates also presents unsur-mountable difficulties. If the observer applieshis standard measuring-rod (a rod which isshort as compared with the radius of the disc)tangentially to the edge of the disc, then, asjudged from the Galileian system, the lengthof this rod will be less than 1, since, accordingto Section XII, moving bodies suffer a short-ening in the direction of the motion. On theother hand, the measuring-rod will not expe-rience a shortening in length, as judged fromK, if it is applied to the disc in the directionof the radius. If, then, the observer first mea-sures the circumference of the disc with hismeasuring-rod and then the diameter of thedisc, on dividing the one by the other, he willnot obtain as quotient the familiar numberπ = 3.14 . . ., but a larger number,22 whereasof course, for a disc which is at rest with re-spect to K, this operation would yield π ex-actly. This proves that the propositions of Eu-clidean geometry cannot hold exactly on therotating disc, nor in general in a gravitationalfield, at least if we attribute the length 1 to the

22Throughout this consideration we have to use theGalileian (non-rotating) system K as reference-body,since we may only assume the validity of the resultsof the special theory of relativity relative to K (relativeto K ′ a gravitational field prevails).

Rotating Clocks and Measuring Rods 107

rod in all positions and in every orientation.Hence the idea of a straight line also loses itsmeaning. We are therefore not in a positionto define exactly the co-ordinates x, y, z rela-tive to the disc by means of the method usedin discussing the special theory, and as long asthe co-ordinates and times of events have notbeen defined we cannot assign an exact mean-ing to the natural laws in which these occur.

Thus all our previous conclusions based ongeneral relativity would appear to be called inquestion. In reality we must make a subtledetour in order to be able to apply the postu-late of general relativity exactly. I shall pre-pare the reader for this in the following para-graphs.

108 Relativity

XXIV. Euclideanand Non-EuclideanContinuum

The surface of a marble table is spread out infront of me. I can get from any one point onthis table to any other point by passing con-tinuously from one point to a “neighbouring”one, and repeating this process a (large) num-ber of times, or, in other words, by going frompoint to point without executing jumps. I amsure the reader will appreciate with sufficientclearness, what I mean here by “neighbour-ing” and by “jumps” (if he is not too pedantic).We express this property of the surface by de-scribing the latter as a continuum.

Let us now imagine that a large numberof little rods of equal length have been made,their lengths being small compared with thedimensions of the marble slab. When I saythey are of equal length, I mean that one canbe laid on any other without the ends over-lapping. We next lay four of these little rodson the marble slab so that they constitutea quadrilateral figure (a square), the diago-

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nals of which are equally long. To ensure theequality of the diagonals, we make use of alittle testing-rod. To this square we add simi-lar ones, each of which has one rod in commonwith the first. We proceed in like manner witheach of these squares until finally the wholemarble slab is laid out with squares. The ar-rangement is such, that each side of a squarebelongs to two squares and each corner to foursquares.

It is a veritable wonder that we can carryour this business without getting into thegreatest difficulties. We only need to thinkof the following. If at any moment threesquares meet at a corner, then two sides of thefourth square are already laid, and as a con-sequence, the arrangement of the remainingtwo sides of the square is already completelydetermined. But I am now no longer able toadjust the quadrilateral so that its diagonalsmay be equal. If they are equal of their ownaccord, then this is an especial favour of themarble slab and of the little rods about whichI can only be thankfully surprised. We mustneeds experience many such surprises if theconstruction is to be successful.

If everything has really gone smoothly,then I say that the points of the marble slabconstitute a Euclidean continuum with re-spect to the little rod, which has been used asa “distance” (line-interval). By choosing onecorner of a square as “origin,” I can charac-terise every other corner of a square with ref-erence to this origin by means of two num-bers. I only need state how many rods I

Euclidean & Non-Euclidean Continuum 111

must pass over when, starting from the origin,I proceed towards the “right” and then “up-wards,” in order to arrive at the corner of thesquare under consideration. These two num-bers are then the “Cartesian co-ordinates” ofthis corner with reference to the “Cartesianco-ordinate system” which is determined bythe arrangement of little rods.

By making use of the following modifica-tion of this abstract experiment, we recognisethat there must also be cases in which theexperiment would be unsuccessful. We shallsuppose that the rods “expand” by an amountproportional to the increase of temperature.We heat the central part of the marble slab,but not the periphery, in which case two ofour little rods can still be brought into coin-cidence at every position on the table. But ourconstruction of squares must necessarily comeinto disorder during the heating, because thelittle rods on the central region of the table ex-pand, whereas those on the outer part do not.

With reference to our little rods—definedas unit lengths—the marble slab is no longera Euclidean continuum, and we are also nolonger in the position of defining Cartesianco-ordinates directly with their aid, since theabove construction can no longer be carriedout. But since there are other things whichare not influenced in a similar manner to thelittle rods (or perhaps not at all) by the tem-perature of the table, it is possible quite nat-urally to maintain the point of view that themarble slab is a “Euclidean continuum.” Thiscan be done in a satisfactory manner by mak-

112 Relativity

ing a more subtle stipulation about the mea-surement or the comparison of lengths.

But if rods of every kind (i.e. of every ma-terial) were to behave in the same way as re-gards the influence of temperature when theyare on the variably heated marble slab, andif we had no other means of detecting the ef-fect of temperature than the geometrical be-haviour of our rods in experiments analogousto the one described above, then our best planwould be to assign the distance one to twopoints on the slab, provided that the ends ofone of our rods could be made to coincide withthese two points; for how else should we definethe distance without our proceeding being inthe highest measure grossly arbitrary? Themethod of Cartesian co-ordinates must thenbe discarded, and replaced by another whichdoes not assume the validity of Euclidean ge-ometry for rigid bodies.23 The reader will no-tice that the situation depicted here corre-sponds to the one brought about by the gen-eral postulate of relativity (Section XXIII).

NOTE:—Gauss undertook the task oftreating this two-dimensional geometry fromfirst principles, without making use of the factthat the surface belongs to a Euclidean contin-uum of three dimensions. If we imagine con-structions to be made with rigid rods in thesurface (similar to that above with the mar-

23Mathematicians have been confronted with ourproblem 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 ge-ometry, just as much as for a plane surface.

Euclidean & Non-Euclidean Continuum 113

ble slab), we should find that different lawshold for these from those resulting on the ba-sis of Euclidean plane geometry. The surfaceis not a Euclidean continuum with respect tothe rods, and we cannot define Cartesian co-ordinates in the surface. Gauss indicated theprinciples according to which we can treat thegeometrical relationships in the surface, andthus pointed out the way to the method ofRiemann of treating multi-dimensional, non-Euclidean continua. Thus it is that mathe-maticians long ago solved the formal problemsto which we are led by the general postulate ofrelativity.

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XXV. GaussianCo-ordinates

FIG. 4.

According to Gauss, this combined ana-lytical and geometrical mode of handling theproblem can be arrived at in the followingway. 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 indi-cate each of them by means of a number. The

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curves u = 1, u = 2 and u = 3 are drawn in thediagram. Between the curves u = 1 and u = 2we must imagine an infinitely large number tobe drawn, all of which correspond to real num-bers lying between 1 and 2. We have then asystem of u-curves, and this “infinitely dense”system covers the whole surface of the table.These u-curves must not intersect each other,and through each point of the surface one andonly one curve must pass. Thus a perfectlydefinite value of u belongs to every point onthe surface of the marble slab. In like man-ner we imagine a system of v-curves drawnon the surface. These satisfy the same condi-tions as the u-curves, they are provided withnumbers in a corresponding manner, and theymay likewise be of arbitrary shape. It followsthat a value of u and a value of v belong toevery point on the surface of the table. Wecall these two numbers the co-ordinates of thesurface of the table (Gaussian co-ordinates).For example, the point P in the diagram hasthe Gaussian co-ordinates u = 3, v = 1. Twoneighbouring points P and P ′ on the surfacethen 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 dis-tance (line-interval) between P and P ′, asmeasured with a little rod, by means of thevery small number ds. Then according toGauss we have

Gaussian Co-ordinates 117

ds2 = g11 + 2g12du dv + g22dv2,

where g11, g12, g22, are magnitudes which de-pend in a perfectly definite way on u and v.The magnitudes g11, g12, and g22 determine thebehaviour of the rods relative to the u-curvesand v-curves, and thus also relative to thesurface of the table. For the case in whichthe points of the surface considered form aEuclidean continuum with reference to themeasuring-rods, but only in this case, it is pos-sible to draw the u-curves and v-curves and toattach numbers to them, in such a manner,that we simply have:

ds2 = du2 + dv2.

Under these conditions, the u-curves andv-curves are straight lines in the sense of Eu-clidean geometry, and they are perpendicularto each other. Here the Gaussian co-ordinatesare simply Cartesian ones. It is clear thatGauss co-ordinates are nothing more than anassociation of two sets of numbers with thepoints of the surface considered, of such anature that numerical values differing veryslightly from each other are associated withneighbouring points “in space.”

So far, these considerations hold for a con-tinuum of two dimensions. But the Gaus-sian method can be applied also to a contin-uum of three, four or more dimensions. If,for instance, a continuum of four dimensionsbe supposed available, we may represent it

118 Relativity

in the following way. With every point ofthe continuum we associate arbitrarily fournumbers, x1, x2, x3, x4, which are known as “co-ordinates.” Adjacent points correspond to ad-jacent values of the co-ordinates. If a distanceds is associated with the adjacent points Pand P ′, this distance being measurable andwell-defined from a physical point of view,then the following formula holds:

ds2 = g11dx21 + 2g12dx1dx2 + . . . + g44dx2

4,

where the magnitudes g11 etc., have valueswhich vary with the position in the contin-uum. Only when the continuum is a Eu-clidean one is it possible to associate the co-ordinates x1 . . . x4 with the points of the con-tinuum so that we have simply

ds2 = dx21 + dx2

2 + dx23 + dx2

4.

In this case relations hold in the four-dimensional continuum which are analogousto those holding in our three-dimensionalmeasurements.

However, the Gauss treatment for ds2

which we have given above is not always pos-sible. It is only possible when sufficientlysmall regions of the continuum under consid-eration may be regarded as Euclidean con-tinua. For example, this obviously holds inthe case of the marble slab of the table andlocal variation of temperature. The tempera-ture is practically constant for a small part ofthe slab, and thus the geometrical behaviourof the rods is almost as it ought to be according

Gaussian Co-ordinates 119

to the rules of Euclidean geometry. Hence theimperfections of the construction of squares inthe previous section do not show themselvesclearly until this construction is extended overa considerable portion of the surface of the ta-ble.

We can sum this up as follows: Gauss in-vented a method for the mathematical treat-ment of continua in general, in which “size-relations” (“distances” between neighbouringpoints) are defined. To every point of acontinuum are assigned as many numbers(Gaussian co-ordinates) as the continuum hasdimensions. This is done in such a way,that only one meaning can be attached tothe assignment, and that numbers (Gaussianco-ordinates) which differ by an indefinitelysmall amount are assigned to adjacent points.The Gaussian co-ordinate system is a logi-cal generalisation of the Cartesian co-ordinatesystem. It is also applicable to non-Euclideancontinua, but only when, with respect to thedefined “size” or “distance,” small parts of thecontinuum under consideration behave morenearly like a Euclidean system, the smallerthe part of the continuum under our notice.

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XXVI. TheSpace-TimeContinuum of theSpecial Theory ofRelativityConsidered as aEuclideanContinuum

We are now in a position to formulate more ex-actly the idea of Minkowski, which was onlyvaguely indicated in Section XVII. In accor-dance with the special theory of relativity, cer-tain co-ordinate systems are given preferencefor the description of the four-dimensional,space-time continuum. We called these “Gali-leian co-ordinate systems.” For these systems,the four co-ordinates x, y, z, t, which deter-mine an event or—in other words—a point ofthe four-dimensional continuum, are defined

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physically in a simple manner, as set forth indetail in the first part of this book. For thetransition from one Galileian system to an-other, which is moving uniformly with refer-ence to the first, the equations of the Lorentztransformation are valid. These last form thebasis for the derivation of deductions from thespecial theory of relativity, and in themselvesthey are nothing more than the expression ofthe universal validity of the law of transmis-sion of light for all Galileian systems of refer-ence.

Minkowski found that the Lorentz trans-formations satisfy the following simple con-ditions. Let us consider two neighbouringevents, the relative position of which in thefour-dimensional continuum is given with re-spect to a Galileian reference-body K by thespace co-ordinate differences dx, dy, dz and thetime-difference dt. With reference to a secondGalileian system we shall suppose that thecorresponding differences for these two eventsare dx′, dy′, dz′, dt′. Then these magnitudes al-ways fulfil the condition24

dx2 +dy2 +dz2−c2dt2 = dx′2 +dy′2 +dz′2−c2dt′2.

The validity of the Lorentz transformationfollows from this condition. We can expressthis as follows: The magnitude

ds2 = dx2 + dy2 + dz2 − c2dt2,

24Cf. Appendices I and II. The relations which are de-rived there for the co-ordinates themselves are validalso for co-ordinate differences, and thus also for co-ordinate differentials (indefinitely small differences).

Euclidean Special Relativity 123

which belongs to two adjacent points of thefour-dimensional space-time continuum, hasthe same value for all selected (Galileian)reference-bodies. If we replace x, y, z,

√−1ct,

by x1, x2, x3, x4, we also obtain the result that

ds2 = dx21 + dx2

2 + dx23 + dx2

4

is independent of the choice of the body ofreference. We call the magnitude ds the“distance” apart of the two events or four-dimensional points.

Thus, if we choose as time-variable theimaginary variable

√−1ct instead of the real

quantity t, we can regard the space-timecontinuum—in accordance with the specialtheory of relativity—as a “Euclidean” four-dimensional continuum, a result which fol-lows from the considerations of the precedingsection.

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XXVII. TheSpace-TimeContinuum of theGeneral Theory ofRelativity is not aEuclideanContinuum

In the first part of this book we were able tomake use of space-time co-ordinates which al-lowed of a simple and direct physical interpre-tation, and which, according to Section XXVI,can be regarded as four-dimensional Carte-sian co-ordinates. This was possible on the ba-sis of the law of the constancy of the velocity oflight. But according to Section XXI, the gen-eral theory of relativity cannot retain this law.On the contrary, we arrived at the result thataccording to this latter theory the velocity oflight must always depend on the coordinateswhen a gravitational field is present. In con-

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nection with a specific illustration in SectionXXIII, we found that the presence of a gravi-tational field invalidates the definition of theco-ordinates and the time, which led us to ourobjective in the special theory of relativity.

In view of the results of these considera-tions we are led to the conviction that, accord-ing to the general principle of relativity, thespace-time continuum cannot be regarded asa Euclidean one, but that here we have thegeneral case, corresponding to the marble slabwith local variations of temperature, and withwhich we made acquaintance as an exampleof a two-dimensional continuum. Just as itwas there impossible to construct a Cartesianco-ordinate system from equal rods, so here itis impossible to build up a system (reference-body) from rigid bodies and clocks, which shallbe of such a nature that measuring-rods andclocks, arranged rigidly with respect to oneanother, shall indicate position and time di-rectly. Such was the essence of the diffi-culty with which we were confronted in Sec-tion XXIII.

But the considerations of Sections XXVand XXVI show us the way to surmountthis difficulty. We refer the four-dimensionalspace-time continuum in an arbitrary man-ner to Gauss co-ordinates. We assign to everypoint of the continuum (event) four numbers,x1, x2, x3, x4 (co-ordinates), which have not theleast direct physical significance, but onlyserve the purpose of numbering the pointsof the continuum in a definite but arbitrarymanner. This arrangement does not even

Non-Euclidean General Relativity 127

need to be of such a kind that we must re-gard x1, x2, x3 as “space” co-ordinates and x4

as a “time” co-ordinate.The reader may think that such a descrip-

tion of the world would be quite inadequate.What does it mean to assign to an eventthe particular co-ordinates x1, x2, x3, x4, if inthemselves these co-ordinates have no signif-icance? More careful consideration shows,however, that this anxiety is unfounded. Letus consider, for instance, a material point withany kind of motion. If this point had only amomentary existence without duration, thenit would be described in space-time by a sin-gle system of values x1, x2, x3, x4. Thus its per-manent existence must be characterised byan infinitely large number of such systemsof values, the co-ordinate values of which areso close together as to give continuity; corre-sponding to the material point, we thus have a(uni-dimensional) line in the four-dimensionalcontinuum. In the same way, any such lines inour continuum correspond to many points inmotion. The only statements having regard tothese points which can claim a physical exis-tence are in reality the statements about theirencounters. In our mathematical treatment,such an encounter is expressed in the fact thatthe two lines which represent the motions ofthe points in question have a particular sys-tem of co-ordinate values, x1, x2, x3, x4, in com-mon. After mature consideration the readerwill doubtless admit that in reality such en-counters constitute the only actual evidenceof a time-space nature with which we meet in

128 Relativity

physical statements.When we were describing the motion of

a material point relative to a body of refer-ence, we stated nothing more than the en-counters of this point with particular points ofthe reference-body. We can also determine thecorresponding values of the time by the obser-vation of encounters of the body with clocks,in conjunction with the observation of the en-counter of the hands of clocks with particu-lar points on the dials. It is just the same inthe case of space-measurements by means ofmeasuring-rods, as a little consideration willshow.

The following statements hold generally:Every physical description resolves itself intoa number of statements, each of which refersto the space-time coincidence of two events Aand B. In terms of Gaussian co-ordinates, ev-ery such statement is expressed by the agree-ment of their four co-ordinates x1, x2, x3, x4.Thus in reality, the description of the time-space continuum by means of Gauss co-ordinates completely replaces the descriptionwith the aid of a body of reference, withoutsuffering from the defects of the latter modeof description; it is not tied down to the Eu-clidean character of the continuum which hasto be represented.

XXVIII. ExactFormulation of theGeneral Principleof Relativity

We are now in a position to replace the pro-visional formulation of the general principleof relativity given in Section XVIII by an ex-act formulation. The form there used, “Allbodies of reference K, K ′, etc., are equiva-lent for the description of natural phenomena(formulation of the general laws of nature),whatever may be their state of motion,” can-not be maintained, because the use of rigidreference-bodies, in the sense of the methodfollowed in the special theory of relativity, isin general not possible in space-time descrip-tion. The Gauss co-ordinate system has totake the place of the body of reference. Thefollowing statement corresponds to the funda-mental idea of the general principle of relativ-ity: “All Gaussian co-ordinate systems are es-sentially equivalent for the formulation of thegeneral laws of nature.”

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130 Relativity

We can state this general principle of rel-ativity in still another form, which renders ityet more clearly intelligible than it is whenin the form of the natural extension of thespecial principle of relativity. According tothe special theory of relativity, the equationswhich express the general laws of nature passover into equations of the same form when,by making use of the Lorentz transformation,we replace the space-time variables x, y, z, t, ofa (Galileian) reference-body K by the space-time variables x′, y′, z′, t′, of a new reference-body K ′. According to the general theory ofrelativity, on the other hand, by application ofarbitrary substitutions of the Gauss variablesx1, x2, x3, x4, the equations must pass over intoequations of the same form; for every transfor-mation (not only the Lorentz transformation)corresponds to the transition of one Gauss co-ordinate system into another.

If we desire to adhere to our “old-time”three-dimensional view of things, then we cancharacterise the development which is beingundergone by the fundamental idea of thegeneral theory of relativity as follows: Thespecial theory of relativity has reference toGalileian domains, i.e. to those in which nogravitational field exists. In this connectiona Galileian reference-body serves as body ofreference, i.e. a rigid body the state of motionof which is so chosen that the Galileian lawof the uniform rectilinear motion of “isolated”material points holds relatively to it.

Certain considerations suggest that weshould refer the same Galileian domains to

Exact Formulation of General Relativity 131

non-Galileian reference-bodies also. A gravi-tational field of a special kind is then presentwith respect to these bodies (cf. Sections XXand XXIII).

In gravitational fields there are no suchthings as rigid bodies with Euclidean proper-ties; thus the fictitious rigid body of referenceis of no avail in the general theory of relativ-ity. The motion of clocks is also influenced bygravitational fields, and in such a way that aphysical definition of time which is made di-rectly with the aid of clocks has by no meansthe same degree of plausibility as in the spe-cial theory of relativity.

For this reason non-rigid reference-bodiesare used which are as a whole not only mov-ing in any way whatsoever, but which alsosuffer alterations in form ad lib. during theirmotion. Clocks, for which the law of mo-tion is any kind, however irregular, serve forthe definition of time. We have to imagineeach of these clocks fixed at a point on thenon-rigid reference-body. These clocks sat-isfy only the one condition, that the “read-ings” which are observed simultaneously onadjacent clocks (in space) differ from eachother by an indefinitely small amount. Thisnon-rigid reference-body, which might appro-priately be termed a “reference-mollusk,” isin the main equivalent to a Gaussian four-dimensional co-ordinate system chosen arbi-trarily. That which gives the “mollusk” acertain comprehensibleness as compared withthe Gauss co-ordinate system is the (reallyunqualified) formal retention of the separate

132 Relativity

existence of the space co-ordinate. Every pointon the mollusk is treated as a space-point,and every material point which is at rest rel-atively to it as at rest, so long as the molluskis considered as reference-body. The generalprinciple of relativity requires that all thesemollusks can be used as reference-bodies withequal right and equal success in the formula-tion of the general laws of nature; the lawsthemselves must be quite independent of thechoice of mollusk.

The great power possessed by the generalprinciple of relativity lies in the comprehen-sive limitation which is imposed on the lawsof nature in consequence of what we have seenabove.

XXIX. The Solutionof the Problem ofGravitation on theBasis of theGeneral Principleof Relativity

If the reader has followed all our previous con-siderations, he will have no further difficultyin understanding the methods leading to thesolution of the problem of gravitation.

We start off from a consideration of a Gal-ileian domain, i.e. a domain in which thereis no gravitational field relative to the Gal-ileian reference-body K. The behaviour ofmeasuring-rods and clocks with reference toK is known from the special theory of relativ-ity, likewise the behaviour of “isolated” mate-rial points; the latter move uniformly and instraight lines.

Now let us refer this domain to a randomGauss co-ordinate system or to a “mollusk” as

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134 Relativity

reference-body K ′. Then with respect to K ′

there is a gravitational field G (of a particularkind). We learn the behaviour of measuring-rods and clocks and also of freely-moving ma-terial points with reference to K ′ simply bymathematical transformation. We interpretthis behaviour as the behaviour of measuring-rods, clocks and material points under the in-fluence of the gravitational field G. Hereuponwe introduce a hypothesis: that the influenceof the gravitational field on measuring-rods,clocks and freely-moving material points con-tinues to take place according to the samelaws, even in the case when the prevailinggravitational field is not derivable from theGalileian special case, simply by means of atransformation of co-ordinates.

The next step is to investigate the space-time behaviour of the gravitational field G,which was derived from the Galileian spe-cial case simply by transformation of the co-ordinates. This behaviour is formulated in alaw, which is always valid, no matter how thereference-body (mollusk) used in the descrip-tion may be chosen.

This law is not yet the general law ofthe gravitational field, since the gravitationalfield under consideration is of a special kind.In order to find out the general law-of-field ofgravitation we still require to obtain a gener-alisation of the law as found above. This canbe obtained without caprice, however, by tak-ing into consideration the following demands:The required generalisation must likewisesatisfy the general postulate of relativity. If

Solution of the Problem of Gravitation 135

there is any matter in the domain under con-sideration, only its inertial mass, and thus ac-cording to Section XV only its energy is of im-portance for its effect in exciting a field. Grav-itational field and matter together must sat-isfy the law of the conservation of energy (andof impulse).

Finally, the general principle of relativ-ity permits us to determine the influence ofthe gravitational field on the course of allthose processes which take place accordingto known laws when a gravitational field isabsent, i.e. which have already been fittedinto the frame of the special theory of relativ-ity. In this connection we proceed in princi-ple according to the method which has alreadybeen explained for measuring-rods, clocks andfreely-moving material points.

The theory of gravitation derived in thisway from the general postulate of relativityexcels not only in its beauty; nor in remov-ing the defect attaching to classical mechanicswhich was brought to light in Section XXI; norin interpreting the empirical law of the equal-ity of inertial and gravitational mass; but ithas also already explained a result of obser-vation in astronomy, against which classicalmechanics is powerless.

If we confine the application of the theoryto the case where the gravitational fields canbe regarded as being weak, and in which allmasses move with respect to the co-ordinatesystem with velocities which are small com-pared with the velocity of light, we then ob-tain as a first approximation the Newtonian

136 Relativity

theory. Thus the latter theory is obtained herewithout any particular assumption, whereasNewton had to introduce the hypothesis thatthe force of attraction between mutually at-tracting material points is inversely propor-tional to the square of the distance betweenthem. If we increase the accuracy of the cal-culation, deviations from the theory of New-ton make their appearance, practically all ofwhich must nevertheless escape the test of ob-servation owing to their smallness.

We must draw attention here to one ofthese deviations. According to Newton’s the-ory, a planet moves round the sun in an el-lipse, which would permanently maintain itsposition with respect to the fixed stars, if wecould disregard the motion of the fixed stars,themselves and the action of the other planetsunder consideration. Thus, if we correct theobserved motion of the planets for these twoinfluences, and if Newton’s theory be strictlycorrect, we ought to obtain for the orbit of theplanet an ellipse, which is fixed with referenceto the fixed stars. This deduction, which canbe tested with great accuracy, has been con-firmed for all the planets save one, with theprecision that is capable of being obtained bythe delicacy of observation attainable at thepresent time. The sole exception is Mercury,the planet which lies nearest the sun. Sincethe time Leverrier, it has been known thatthe ellipse corresponding to the orbit of Mer-cury, after it has been corrected for the influ-ences mentioned above, is not stationary withrespect to the fixed stars, but that it rotates

Solution of the Problem of Gravitation 137

exceedingly slowly in the plane of the orbitand in the sense of the orbital motion. Thevalue obtained for this rotary movement ofthe orbital ellipse was 43 seconds of arc percentury, an amount ensured to be correct towithin a few seconds of arc. This effect canbe explained by means of classical mechanicsonly on the assumption of hypotheses whichhave little probability, and which were de-vised solely for this purpose.

On the basis of the general theory of rel-ativity, it is found that the ellipse of everyplanet round the sun must necessarily rotatein the manner indicated above; that for all theplanets, with the exception of Mercury, thisrotation is too small to be detected with thedelicacy of observation possible at the presenttime; but that in the case of Mercury it mustamount to 43 seconds of arc per century, a re-sult which is strictly in agreement with obser-vation.

Apart from this one, it has hitherto beenpossible to make only two deductions from thetheory which admit of being tested by obser-vation, to wit, the curvature of light rays bythe gravitational field of the sun,25 and a dis-placement of the spectral lines of light reach-ing us from large stars, as compared withthe corresponding lines for light produced inan analogous manner terrestrially (i.e. by thesame kind of molecule). I do not doubt thatthese deductions from the theory will be con-

25Observed by Eddington and others in 1919. (Cf. Ap-pendix III.)

138 Relativity

firmed also.

Part III:Considerations onthe Universe as a

Whole

139

XXX. CosmologicalDifficulties ofNewton’s Theory

Apart from the difficulty discussed in Sec-tion XXI, there is a second fundamental diffi-culty attending classical celestial mechanics,which, to the best of my knowledge, was firstdiscussed in detail by the astronomer Seeliger.If we ponder over the question as to how theuniverse, considered as a whole, is to be re-garded, the first answer that suggests itself tous is surely this: As regards space (and time)the universe is infinite. There are stars ev-erywhere, so that the density of matter, al-though very variable in detail, is neverthelesson the average everywhere the same. In otherwords: However far we might travel throughspace, we should find everywhere an attenu-ated swarm of fixed stars of approximately thesame kind and density.

This view is not in harmony with the the-ory of Newton. The latter theory rather re-quires that the universe should have a kindof centre in which the density of the stars is

141

142 Relativity

a maximum, and that as we proceed outwardsfrom this centre the group-density of the starsshould diminish, until finally, at great dis-tances, it is succeeded by an infinite region ofemptiness. The stellar universe ought to be afinite island in the infinite ocean of space.26

This conception is in itself not very satis-factory. It is still less satisfactory because itleads to the result that the light emitted bythe stars and also individual stars of the stel-lar system are perpetually passing out into in-finite space, never to return, and without everagain coming into interaction with other ob-jects of nature. Such a finite material uni-verse would be destined to become graduallybut systematically impoverished.

In order to escape this dilemma, Seeligersuggested a modification of Newton’s law, inwhich he assumes that for great distances theforce of attraction between two masses dimin-ishes more rapidly than would result from theinverse square law. In this way it is possi-ble for the mean density of matter to be con-

26Proof.—According to the theory of Newton, thenumber of “lines of force” which come from infinityand terminate in a mass m is proportional to the massm. If, on the average, the mass-density P0 is constantthroughout the universe, then a sphere of volume Vwill enclose the average mass P0V . Thus the numberof lines of force passing through the surface F of thesphere into its interior is proportional to P0V . For unitarea of the surface of the sphere the number of linesof force which enters the sphere is thus proportional toP0 · V/F or P0R. Hence the intensity of the field at thesurface would ultimately become infinite with increas-ing radius R of the sphere, which is impossible.

Cosmological Difficulties of Newton 143

stant everywhere, even to infinity, without in-finitely large gravitational fields being pro-duced. We thus free ourselves from the dis-tasteful conception that the material universeought to possess something of the nature ofcentre. Of course we purchase our emanci-pation from the fundamental difficulties men-tioned, at the cost of a modification and com-plication of Newton’s law which has neitherempirical nor theoretical foundation. We canimagine innumerable laws which would servethe same purpose, without our being able tostate a reason why one of them is to be pre-ferred to the others; for any one of these lawswould be founded just as little on more gen-eral theoretical principles as is the law ofNewton.

144 Relativity

XXXI. ThePossibility of a“Finite” and Yet“Unbounded”Universe

But speculations on the structure of the uni-verse also move in quite another direction.The development of non-Euclidean geome-try led to the recognition of the fact, thatwe can cast doubt on the infiniteness of ourspace without coming into conflict with thelaws of thought or with experience (Riemann,Helmholtz). These questions have alreadybeen treated in detail and with unsurpassablelucidity by Helmholtz and Poincaré, whereasI can only touch on them briefly here.

In the first place, we imagine an existencein two-dimensional space. Flat beings withflat implements, and in particular flat rigidmeasuring-rods, are free to move in a plane.For them nothing exists outside of this plane:that which they observe to happen to them-

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146 Relativity

selves and to their flat “things” is the all-inclusive reality of their plane. In particu-lar, the constructions of plane Euclidean ge-ometry can be carried out by means of therods, e.g. the lattice construction, consideredin Section XXIV. In contrast to ours, the uni-verse of these beings is two-dimensional; but,like ours, it extends to infinity. In their uni-verse there is room for an infinite number ofidentical squares made up of rods, i.e. its vol-ume (surface) is infinite. If these beings saytheir universe is “plane,” there is sense in thestatement, because they mean that they canperform the constructions of plane Euclideangeometry with their rods. In this connectionthe individual rods always represent the samedistance, independently of their position.

Let us consider now a second two-dimensional existence, but this time on aspherical surface instead of on a plane. Theflat beings with their measuring-rods andother objects fit exactly on this surface andthey are unable to leave it. Their whole uni-verse of observation extends exclusively overthe surface of the sphere. Are these beingsable to regard the geometry of their universeas being plane geometry and their rods withalas the realisation of “distance”? They can-not do this. For if they attempt to realise astraight line, they will obtain a curve, whichwe “three-dimensional beings” designate as agreat circle, i.e. a self-contained line of defi-nite finite length, which can be measured upby means of a measuring-rod. Similarly, thisuniverse has a finite area, that can be com-

“Finite” and Yet “Unbounded” Universe 147

pared with the area of a square constructedwith rods. The great charm resulting fromthis consideration lies in the recognition of thefact that the universe of these beings is finiteand yet has no limits.

But the spherical-surface beings do notneed to go on a world-tour in order to perceivethat they are not living in a Euclidean uni-verse. They can convince themselves of thison every part of their “world,” provided theydo not use too small a piece of it. Starting froma point, they draw “straight lines” (arcs of cir-cles as judged in three-dimensional space) ofequal length in all directions. They will callthe line joining the free ends of these lines a“circle.” For a plane surface, the ratio of thecircumference of a circle to its diameter, bothlengths being measured with the same rod, is,according to Euclidean geometry of the plane,equal to a constant value π, which is indepen-dent of the diameter of the circle. On theirspherical surface our flat beings would find forthis ratio the value

πsin( r

R)

( rR)

,

i.e. a smaller value than π, the difference be-ing the more considerable, the greater is theradius of the circle in comparison with the ra-dius R of the “world-sphere.” By means ofthis relation the spherical beings can deter-mine the radius of their universe (“world”),even when only a relatively small part of theirworld-sphere is available for their measure-ments. But if this part is very small indeed,

148 Relativity

they will no longer be able to demonstratethat they are on a spherical “world” and not ona Euclidean plane, for a small part of a spher-ical surface differs only slightly from a pieceof a plane of the same size.

Thus if the spherical-surface beings areliving on a planet of which the solar systemoccupies only a negligibly small part of thespherical universe, they have no means of de-termining whether they are living in a finiteor in an infinite universe, because the “pieceof universe” to which they have access is inboth cases practically plane, or Euclidean. Itfollows directly from this discussion, that forour sphere-beings the circumference of a circlefirst increases with the radius until the “cir-cumference of the universe” is reached, andthat it thenceforward gradually decreases tozero for still further increasing values of theradius. During this process the area of the cir-cle continues to increase more and more, untilfinally it becomes equal to the total area of thewhole “world-sphere.”

Perhaps the reader will wonder why wehave placed our “beings” on a sphere ratherthan on another closed surface. But thischoice has its justification in the fact that, ofall closed surfaces, the sphere is unique inpossessing the property that all points on itare equivalent. I admit that the ratio of thecircumference C of a circle to its radius r de-pends on r, but for a given value of r it is thesame for all points of the “world-sphere”; inother words, the “world-sphere” is a “surfaceof constant curvature.”

“Finite” and Yet “Unbounded” Universe 149

To this two-dimensional sphere-universethere is a three-dimensional analogy, namely,the three-dimensional spherical space whichwas discovered by Riemann. Its points arelikewise all equivalent. It possesses a finitevolume, which is determined by its “radius”(2π2R3). Is it possible to imagine a spheri-cal space? To imagine a space means nothingelse than that we imagine an epitome of our“space” experience, i.e. of experience that wecan have in the movement of “rigid” bodies. Inthis sense we can imagine a spherical space.

Suppose we draw lines or stretch strings inall directions from a point, and mark off fromeach of these the distance r with a measuring-rod. All the free end-points of these lengths lieon a spherical surface. We can specially mea-sure up the area (F = 4πr2) of this surface bymeans of a square made up of measuring-rods.If the universe is Euclidean, then F = 4πr2;if it is spherical, then F is always less than4πr2. With increasing values of r, F increasesfrom zero up to a maximum value which isdetermined by the “world-radius,” but for stillfurther increasing values of r, the area grad-ually diminishes to zero. At first, the straightlines which radiate from the starting point di-verge farther and farther from one another,but later they approach each other, and fi-nally they run together again at a “counter-point” to the starting point. Under such con-ditions they have traversed the whole spher-ical space. It is easily seen that the three-dimensional spherical space is quite analo-gous to the two-dimensional spherical surface.

150 Relativity

It is finite (i.e. of finite volume), and has nobounds.

It may be mentioned that there is yet an-other kind of curved space: “elliptical space.”It can be regarded as a curved space in whichthe two “counter-points” are identical (indis-tinguishable from each other). An ellipticaluniverse can thus be considered to some ex-tent as a curved universe possessing centralsymmetry.

It follows from what has been said, thatclosed spaces without limits are conceivable.From amongst these, the spherical space (andthe elliptical) excels in its simplicity, since allpoints on it are equivalent. As a result of thisdiscussion, a most interesting question arisesfor astronomers and physicists, and that iswhether the universe in which we live is in-finite, or whether it is finite in the manner ofthe spherical universe. Our experience is farfrom being sufficient to enable us to answerthis question. But the general theory of rela-tivity permits of our answering it with a mod-erate degree of certainty, and in this connec-tion the difficulty mentioned in Section XXXfinds its solution.

XXXII. TheStructure of SpaceAccording to theGeneral Theory ofRelativity

According to the general theory of relativity,the geometrical properties of space are not in-dependent, but they are determined by mat-ter. Thus we can draw conclusions about thegeometrical structure of the universe only ifwe base our considerations on the state of thematter as being something that is known. Weknow from experience that, for a suitably cho-sen co-ordinate system, the velocities of thestars are small as compared with the veloc-ity of transmission of light. We can thus as arough approximation arrive at a conclusion asto the nature of the universe as a whole, if wetreat the matter as being at rest.

We already know from our previous discus-sion that the behaviour of measuring-rods andclocks is influenced by gravitational fields, i.e.

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152 Relativity

by the distribution of matter. This in itselfis sufficient to exclude the possibility of theexact validity of Euclidean geometry in ouruniverse. But it is conceivable that our uni-verse differs only slightly from a Euclideanone, and this notion seems all the more prob-able, since calculations show that the metricsof surrounding space is influenced only to anexceedingly small extent by masses even ofthe magnitude of our sun. We might imag-ine that, as regards geometry, our universebehaves analogously to a surface which is ir-regularly curved in its individual parts, butwhich nowhere departs appreciably from aplane: something like the rippled surface ofa lake. Such a universe might fittingly becalled a quasi-Euclidean universe. As regardsits space it would be infinite. But calcula-tion shows that in a quasi-Euclidean universethe average density of matter would neces-sarily be nil. Thus such a universe couldnot be inhabited by matter everywhere; itwould present to us that unsatisfactory pic-ture which we portrayed in Section XXX.

If we are to have in the universe an aver-age density of matter which differs from zero,however small may be that difference, thenthe universe cannot be quasi-Euclidean. Onthe contrary, the results of calculation indi-cate that if matter be distributed uniformly,the universe would necessarily be spherical(or elliptical). Since in reality the detaileddistribution of matter is not uniform, the realuniverse will deviate in individual parts fromthe spherical, i.e. the universe will be quasi-

The Structure of Space 153

spherical. But it will be necessarily finite.In fact, the theory supplies us with a simpleconnection27 between the space-expanse of theuniverse and the average density of matter init.

27For the “radius” R of the universe we obtain theequation

R2 =2κρ

The use of the C.G.S. system in this equation gives isthe average density of the matter.

154 Relativity

Appendices

155

I. SimpleDerivation of theLorentzTransformation

For the relative orientation of the co-ordinatesystems indicated in Fig. 2, the x-axes of bothsystems permanently coincide. In the presentcase we can divide the problem into partsby considering first only events which are lo-calised on the x-axis. Any such event is repre-sented with respect to the co-ordinate systemK by the abscissa x and the time t, and withrespect to the system K ′ by the abscissa x′ andthe time t′, when x and t are given.

A light-signal, which is proceeding alongthe positive axis of x, is transmitted accord-ing to the equation

x = ct

or

x− ct = 0. (3)

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158 Relativity

Since the same light-signal has to be trans-mitted relative to K ′ with the velocity c, thepropagation relative to the system K ′ will berepresented by the analogous formula

x′ − ct′ = 0. (4)

Those space-time points (events) which sat-isfy (3) must also satisfy (4). Obviously thiswill be the case when the relation

(x′ − ct′) = λ · (x− ct) (5)

is fulfilled in general, where λ indicates aconstant; for, according to (5), the disappear-ance of (x − ct) involves the disappearance of(x′ − ct′).

If we apply quite similar considerations tolight rays which are being transmitted alongthe negative x-axis, we obtain the condition

(x′ + ct′) = µ · (x + ct). (6)

By adding (or subtracting) equations (5)and (6), and introducing for convenience theconstants a and b in place of the constants λand µ where

a =λ + µ

2and

b =λ− µ

2

we obtain the equations

x′ = ax− bct (7)ct′ = act− bx. (8)

Derivation of Lorentz Transformation 159

We should thus have the solution of ourproblem, if the constants a and b were known.These result from the following discussion.

For the origin of K ′ we have permanentlyx′ = 0, and hence according to the first of theequations (7) and (8)

x =bc

at.

If we call v the velocity with which the ori-gin of K ′ is moving relative to K, we then have

v =bc

a. (9)

The same value v can be obtained fromequations (7) and (8), if we calculate the veloc-ity of another point of K ′ relative to K, or thevelocity (directed towards the negative x-axis)of a point of K with respect to K ′. In short, wecan designate v as the relative velocity of thetwo systems.

Furthermore, the principle of relativityteaches us that, as judged from K, the lengthof a unit measuring-rod which is at rest withreference to K ′ must be exactly the sameas the length, as judged from K ′, of a unitmeasuring-rod which is at rest relative to K.In order to see how the points of the x′-axisappear as viewed from K, we only require totake a “snapshot” of K ′ from K; this meansthat we have to insert a particular value of t(time of K), e.g. t = 0. For this value of t wethen obtain from the equation (7)

x′ = ax.

160 Relativity

Two points of the x′-axis which are sepa-rated by the distance x′ = 1 when measuredin the K ′ system are thus separated in our in-stantaneous photograph by the distance

∆x =1

a. (10)

But if the snapshot be taken from K ′ (t′ =0), and if we eliminate t from the equations(7) and (8), taking into account the expression(9), we obtain

x′ = a · (1− v2

c2) · x.

From this we conclude that two points onthe x-axis and separated by the distance 1(relative to K) will be represented on oursnapshot by the distance

∆x′ = a · (1− v2

c2). (11)

But from what has been said, the twosnapshots must be identical; hence ∆x in (10)must be equal to ∆x′ in (11), so that we obtain

a2 =1

1− v2

c2

. (12)

The equations (9) and (12) determine theconstants a and b. By inserting the values ofthese constants in (7) and (8), we obtain thefirst and the fourth of the equations given inSection XI.

x′ =x− vt√1− v2

c2

(13)

t′ =t− v

c2x√

1− v2

c2

. (14)

Derivation of Lorentz Transformation 161

Thus we have obtained the Lorentz trans-formation for events on the x-axis. It satisfiesthe condition

x′2 − c2t′2 = x2 − c2t2 (15)

The extension of this result, to includeevents which take place outside the x-axis, isobtained by retaining equations (13) and (14)and supplementing them by the relations

y′ = yz′ = z.

}(16)

In this way we satisfy the postulate of the con-stancy of the velocity of light in vacuo for raysof light of arbitrary direction, both for the sys-tem K and for the system K ′. This may beshown in the following manner.

We suppose a light-signal sent out from theorigin of K at the time t = 0. It will be propa-gated according to the equation

r =√

x2 + y2 + z2 = ct,

or, if we square this equation, according to theequation

x2 + y2 + z2 − c2t2 = 0. (17)

It is required by the law of propagation oflight, in conjunction with the postulate of rel-ativity, that the transmission of the signal inquestion should take place—as judged fromK ′—in accordance with the corresponding for-mula

r′ = ct′

162 Relativity

or,

x′2 + y′2 + z′2 − c2t′2 = 0. (18)

In order that equation (18) may be a conse-quence of equation (17), we must have

x′2+y′2+z′2−c2t′2 = σ ·(x2+y2+z2−c2t2). (19)

Since equation (15) must hold for points onthe x-axis, we thus have σ = 1; for (19) is aconsequence of (15) and (16), and hence alsoof (13) and (16). We have thus derived theLorentz transformation.

The Lorentz transformation representedby (13) and (16) still requires to be gener-alised. Obviously it is immaterial whetherthe axes of K ′ be chosen so that they are spa-tially parallel to those of K. It is also not es-sential that the velocity of translation of K ′

with respect to K should be in the directionof the x-axis. A simple consideration showsthat we are able to construct the Lorentztransformation in this general sense from twokinds of transformations, viz. from Lorentztransformations in the special sense and frompurely spatial transformations, which corre-sponds to the replacement of the rectangularco-ordinate system by a new system with itsaxes pointing in other directions.

Mathematically, we can characterise thegeneralised Lorentz transformation thus: Itexpresses x′, y′, z′, t′, in terms of linear homo-geneous functions of x, y, z, t, of such a kind

Derivation of Lorentz Transformation 163

that the relation

x′2 + y′2 + z′2 − c2t′2 = x2 + y2 + z2 − c2t2 (20)

is satisfied identically. That is to say: Ifwe substitute their expressions in x, y, z, t,in place of x′, y′, z′, t′, on the left-hand side,then the left-hand side of (20) agrees with theright-hand side.

164 Relativity

II. Minkowski’sFour-DimensionalSpace (“World”)

We can characterise the Lorentz transfor-mation still more simply if we introducethe imaginary

√−1ct in place of t, as time-

variable. If, in accordance with this, we insert

x1 = x

x2 = y

x3 = z

x4 =√−1ct

and similarly for the accented system K ′, thenthe condition which is identically satisfied bythe transformation can be expressed thus:

x′21 + x′2

2 + x′23 + x′2

4 = x21 + x2

2 + x23 + x2

4. (21)

That is, by the afore-mentioned choice of“co-ordinates” (20) is transformed into thisequation.

We see from (21) that the imaginary timeco-ordinate x4 enters into the condition of

165

166 Relativity

transformation in exactly the same way as thespace co-ordinates x1, x2, x3. It is due to thisfact that, according to the theory of relativity,the “time” x4 enters into natural laws in thesame form as the space co-ordinates x1, x2, x3.

A four-dimensional continuum describedby the “co-ordinates” x1, x2, x3, x4, was called“world” by Minkowski, who also termed apoint-event a “world-point.” From a “happen-ing” in three-dimensional space, physics be-comes, as it were, an “existence” in the four-dimensional “world.”

This four-dimensional “world” bears aclose similarity to the three-dimensional“space” of (Euclidean) analytical geometry. Ifwe introduce into the latter a new Cartesianco-ordinate system (x′

1, x′2, x

′3) with the same

origin, then x′1, x

′2, x

′3, are linear homogeneous

functions of x1, x2, x3, which identically satisfythe equation

x′21 + x′2

2 + x′23 = x2

1 + x22 + x2

3.

The analogy with (21) is a complete one.We can regard Minkowski’s “world” in a for-mal manner as a four-dimensional Euclideanspace (with imaginary time co-ordinate); theLorentz transformation corresponds to a “ro-tation” of the co-ordinate system in the four-dimensional “world.”

III. TheExperimentalConfirmation ofthe General Theoryof Relativity

From a systematic theoretical point of view,we may imagine the process of evolution ofan empirical science to be a continuous pro-cess of induction. Theories are evolved, andare expressed in short compass as statementsof a large number of individual observationsin the form of empirical laws, from which thegeneral laws can be ascertained by compari-son. Regarded in this way, the development ofa science bears some resemblance to the com-pilation of a classified catalogue. It is, as itwere, a purely empirical enterprise.

But this point of view by no means em-braces the whole of the actual process; for itslurs over the important part played by in-tuition and deductive thought in the develop-ment of an exact science. As soon as a science

167

168 Relativity

has emerged from its initial stages, theoret-ical advances are no longer achieved merelyby a process of arrangement. Guided by em-pirical data, the investigator rather developsa system of thought which, in general, is builtup logically from a small number of funda-mental assumptions, the so-called axioms. Wecall such a system of thought a theory. Thetheory finds the justification for its existencein the fact that it correlates a large numberof single observations, and it is just here thatthe “truth” of the theory lies.

Corresponding to the same complex of em-pirical data, there may be several theories,which differ from one another to a consider-able extent. But as regards the deductionsfrom the theories which are capable of beingtested, the agreement between the theoriesmay be so complete, that it becomes difficultto find such deductions in which the two the-ories differ from each other. As an example,a case of general interest is available in theprovince of biology, in the Darwinian theoryof the development of species by selection inthe struggle for existence, and in the theory ofdevelopment which is based on the hypothe-sis of the hereditary transmission of acquiredcharacters.

We have another instance of far-reachingagreement between the deductions from twotheories in Newtonian mechanics on the onehand, and the general theory of relativity onthe other. This agreement goes so far, that upto the present we have been able to find onlya few deductions from the general theory of

The Experimental Confirmation 169

relativity which are capable of investigation,and to which the physics of pre-relativity daysdoes not also lead, and this despite the pro-found difference in the fundamental assump-tions of the two theories. In what follows, weshall again consider these important deduc-tions, and we shall also discuss the empiri-cal evidence appertaining to them which hashitherto been obtained.

(a) MOTION OF THEPERIHELION OFMERCURYAccording to Newtonian mechanics and New-ton’s law of gravitation, a planet which is re-volving round the sun would describe an el-lipse round the latter, or, more correctly, roundthe common centre of gravity of the sun andthe planet. In such a system, the sun, or thecommon centre of gravity, lies in one of thefoci of the orbital ellipse in such a mannerthat, in the course of a planet-year, the dis-tance sun-planet grows from a minimum to amaximum, and then decreases again to a min-imum. If instead of Newton’s law we inserta somewhat different law of attraction intothe calculation, we find that, according to thisnew law, the motion would still take place insuch a manner that the distance sun-planetexhibits periodic variations; but in this casethe angle described by the line joining sun andplanet during such a period (from perihelion—

170 Relativity

closest proximity to the sun—to perihelion)would differ from 360◦. The line of the or-bit would not then be a closed one, but in thecourse of time it would fill up an annular partof the orbital plane, viz. between the circle ofleast and the circle of greatest distance of theplanet from the sun.

According also to the general theory of rel-ativity, which differs of course from the the-ory of Newton, a small variation from theNewton-Kepler motion of a planet in its orbitshould take place, and in such a way, that theangle described by the radius sun-planet be-tween one perihelion and the next should ex-ceed that corresponding to one complete revo-lution by an amount given by

+24π3a2

T 2c2(1− e2).

(N.B.—One complete revolution corre-sponds to the angle 2π in the absolute angu-lar measure customary in physics, and theabove expression gives the amount by whichthe radius sun-planet exceeds this angle dur-ing the interval between one perihelion andthe next.) In this expression a represents themajor semi-axis of the ellipse, e its eccentric-ity, c the velocity of light, and T the period ofrevolution of the planet. Our result may alsobe stated as follows: According to the generaltheory of relativity, the major axis of the el-lipse rotates round the sun in the same senseas the orbital motion of the planet. Theoryrequires that this rotation should amount to43 seconds of arc per century for the planet

The Experimental Confirmation 171

Mercury, but for the other planets of our solarsystem its magnitude should be so small thatit would necessarily escape detection.28

In point of fact, astronomers have foundthat the theory of Newton does not sufficeto calculate the observed motion of Mercurywith an exactness corresponding to that ofthe delicacy of observation attainable at thepresent time. After taking account of allthe disturbing influences exerted on Mer-cury by the remaining planets, it was found(Leverrier—1859—and Newcomb—1895) thatan unexplained perihelial movement of the or-bit of Mercury remained over, the amount ofwhich does not differ sensibly from the above-mentioned +43 seconds of arc per century. Theuncertainty of the empirical result amounts toa few seconds only.

(b) DEFLECTION OFLIGHT BY AGRAVITATIONAL FIELDIn Section XXII it has been already mentionedthat, according to the general theory of rela-tivity, a ray of light will experience a curva-ture of its path when passing through a grav-itational field, this curvature being similar tothat experienced by the path of a body whichis projected through a gravitational field. As

28Especially since the next planet Venus has an or-bit that is almost an exact circle, which makes it moredifficult to locate the perihelion with precision.

172 Relativity

a result of this theory, we should expect thata ray of light which is passing close to a heav-enly body would be deviated towards the lat-ter. For a ray of light which passes the sun ata distance of ∆ sun-radii from its centre, theangle of deflection (α) should amount to

α =1.7 seconds of arc

∆.

It may be added that, according to the the-ory, half of this deflection is produced by theNewtonian field of attraction of the sun, andthe other half by the geometrical modification(“curvature”) of space caused by the sun.

The Experimental Confirmation 173

FIG. 5.

This result admits of an experimental testby means of the photographic registration ofstars during a total eclipse of the sun. Theonly reason why we must wait for a totaleclipse is because at every other time the at-mosphere is so strongly illuminated by thelight from the sun that the stars situated nearthe sun’s disc are invisible. The predicted ef-fect can be seen clearly from the accompany-ing diagram. If the sun (S) were not present,a star which is practically infinitely distantwould be seen in the direction D1, as observedfrom the earth. But as a consequence of thedeflection of light from the star by the sun, thestar will be seen in the direction D2, i.e. at asomewhat greater distance from the centre ofthe sun than corresponds to its real position.

In practice, the question is tested in the fol-lowing way. The stars in the neighbourhoodof the sun are photographed during a solareclipse.

In addition, a second photograph of thesame stars is taken when the sun is situ-ated at another position in the sky, i.e. a fewmonths earlier or later. As compared withthe standard photograph, the positions of thestars on the eclipse-photograph ought to ap-pear displaced radially outwards (away fromthe centre of the sun) by an amount corre-sponding to the angle α.

We are indebted to the Royal Society andto Royal Astronomical Society for the inves-tigation of this important deduction. Un-

174 Relativity

daunted by the war and by difficulties of botha material and a psychological nature arousedby the war, these societies equipped twoexpeditions—to Sobral (Brazil) and to the is-land of Principe (West Africa)—and sent sev-eral of Britain’s most celebrated astronomers(Eddington, Cottingham, Crommelin, David-son), in order to obtain photographs of the so-lar eclipse of 29th May, 1919. The relative de-screpancies to be expected between the stel-lar photographs obtained during the eclipseand the comparison photographs amounted toa few hundredths of a millimetre only. Thusgreat accuracy was necessary in making theadjustments required for the taking of thephotographs, and in their subsequent mea-surement.

The results of the measurements con-firmed the theory in a thoroughly satisfactorymanner. The rectangular components of theobserved and of the calculated deviations ofthe stars (in seconds of arc) are set forth inthe following table of results:29

1st Co-ordinate 2nd Co-ordinateStar Obs. Calc. Obs. Calc.#11 -0.19 -0.22 +0.16 +0.02#5 +0.29 +0.31 -0.46 -0.43#4 +0.11 0.10 +0.83 +0.74#3 +0.20 +0.12 +1.00 +0.87#6 +0.10 +0.04 +0.57 +0.40#10 -0.08 +0.09 +0.35 +0.32#2 +0.95 +0.85 -0.27 -0.09

29For editorial reasons, the format of the table hasbeen slightly modified—Ron Burkey.

The Experimental Confirmation 175

(c) DISPLACEMENT OFSPECTRAL LINESTOWARDS THE REDIn Section XXIII it has been shown that in asystem K ′ which is in rotation with regardto a Galileian system K, clocks of identicalconstruction and which are considered at restwith respect to the rotating reference-body, goat rates which are dependent on the positionsof the clocks. We shall now examine this de-pendence quantitatively. A clock, which issituated at a distance r from the centre ofthe disc, has a velocity relative to K which isgiven by

v = ωr,

where ω represents the velocity of rotation ofthe disc K ′ with respect to K. If v0 representsthe number of ticks of the clock per unit time(“rate” of the clock) relative to K when theclock is at rest, then the “rate” of the clock (v)when it is moving relative to K with a velocityv, but at rest with respect to the disc, will, inaccordance with Section XII, be given by

v = v0

√1− v2

c2

or with sufficient accuracy by

v = v0(1−1

2

v2

c2).

This expression may also be stated in the fol-lowing form:

176 Relativity

v = v0(1−1

c2

ω2r2

2).

If we represent the difference of potentialof the centrifugal force between the positionof the clock and the centre of the disc by φ, i.e.the work, considered negatively, which mustbe performed on the unit of mass against thecentrifugal force in order to transport it fromthe position of the clock on the rotating disc tothe centre of the disc, then we have

φ = −ω2r2

2.

From this it follows that

v = v0(1 +φ

c2).

In the first place, we see from this expressionthat two clocks of identical construction willgo at different rates when situated at differ-ent distances from the centre of the disc. Thisresult is also valid from the standpoint of anobserver who is rotating with the disc.

Now, as judged from the disc, the latter isin a gravitational field of potential φ, hencethe result we have obtained will hold quitegenerally for gravitational fields. Further-more, we can regard an atom which is emit-ting spectral lines as a clock, so that the fol-lowing statement will hold:

An atom absorbs or emits light ofa frequency which is dependent onthe potential of the gravitationalfield in which it is situated.

The Experimental Confirmation 177

The frequency of an atom situated on the sur-face of a heavenly body will be somewhat lessthan the frequency of an atom of the same el-ement which is situated in free space (or onthe surface of a smaller celestial body). Nowφ = −K ·M/r where K is Newton’s constant ofgravitation, and M is the mass of the heavenlybody. Thus a displacement towards the redought to take place for spectral lines producedat the surface of stars as compared with thespectral lines of the same element producedat the surface of the earth, the amount of thisdisplacement being

v0 − v

v0

=K

c2

M

r.

For the sun, the displacement towards thered predicted by theory amounts to about twomillionths of the wave-length. A trustworthycalculation is not possible in the case of thestars, because in general neither the mass Mnor the radius r is known.

It is an open question whether or not thiseffect exists, and at the present time as-tronomers are working with great zeal to-wards the solution. Owing to the smallnessof the effect in the case of the sun, it is dif-ficult to form an opinion as to its existence.Whereas Grebe and Bachem (Bonn), as a re-sult of their own measurements and those ofEvershed and Schwarzschild on the cyanogenbands, have placed the existence of the effectalmost beyond doubt, other investigators, par-ticularly St. John, have been led to the oppo-site opinion in consequence of their measure-

178 Relativity

ments.Mean displacements of lines towards the

less refrangible end of the spectrum are cer-tainly revealed by statistical investigationsof the fixed stars; but up to the present theexamination of the available data does notallow of any definite decision being arrivedat, as to whether or not these displacementsare to be referred in reality to the effect ofgravitation. The results of observation havebeen collected together, and discussed in de-tail from the standpoint of the question whichhas been engaging our attention here, in a pa-per by E. Freundlich entitled “Zur Prüfungderallgemeinen Relativitäts-Theorie” (Die Natur-wissenschaften, 1919, No. 35, p. 520: JuliusSpringer, Berlin).

At all events, a definite decision will bereached during the next few years. If the dis-placement of spectral lines towards the redby the gravitational potential does not exist,then the general theory of relativity will beuntenable. On the other hand, if the cause ofthe displacement of spectral lines be definitelytraced to the gravitational potential, then thestudy of this displacement will furnish us withimportant information as to the mass of theheavenly bodies.

Bibliography

INTRODUCTORYThe Foundations of Einstein’s Theory of

Gravitation: Erwin Freundlich (translationby H. L. Brose). Camb. Univ. Press, 1920.

Space and Time in Contemporary Physics:Moritz Schlick (translation by H. L. Brose).Clarendon Press, Oxford, 1920.

THE SPECIAL THEORYThe Principle of Relativity: E. Cunning-

ham. Camb. Univ. Press.Relativity and the Electron Theory: E.

Cunningham, Monographs on Physics. Long-mans, Green & Co.

The Theory of Relativity: L. Silberstein.Macmillan & Co.

“The Space-Time Manifold of Relativity”:E. B. Wilson and G. N. Lewis, Proc. Amer. Soc.Arts & Science, vol. xlviii., No. 11, 1912.

179

180 Relativity

THE GENERAL THEORYReport on the Relativity Theory of Gravita-

tion: A. S. Eddington. Fleetway Press Ltd.,Fleet Street, London.

“On Einstein’s Theory of Gravitation andits Astronomical Consequences”: W. de Sitter,M. N. Roy. Astron. Soc., lxxvi. p. 699, 1916;lxxvii. p. 155, 1916; lxxviii. p. 3, 1917.

“On Einstein’s Theory of Gravitation”: H.A. Lorentz, Proc. Amsterdans Acad., vol. xix.p. 1341, 1917.

Space, Time and Gravitation: W. de Sitter:The Observatory, No. 505, p. 412. Taylor &Francis, Fleet Street, London.

“The Total Eclipse of 29th May 1919, andthe Influence of Gravitation on Light”: A. S.Eddington, ibid., March, 1919.

“Discussion on the Theory of Relativity”:M. N. Roy. Astron. Soc., vol. lxxx., No. 2.,p. 96, December 1919.

“The Displacement of Spectrum Lines andthe Equivalence Hypothesis”: W. G. Duffield,M. N. Roy. Astron. Soc., vol. lxxx.; No. 3, p.262, 1920.

Space, Time and Gravitation: A. S. Ed-dington. Camb. Univ. Press, 1920.

ALSO, CHAPTERS INThe Mathematical Theory of Electricity

and Magnetism: J. H. Jeans (4th edition).Camb. Univ. Press, 1920.

Bibliography 181

The Electron Theory of Matter: O. W.Richardson. Camb. Univ. Press.