' ill MATHEMATICAL MONOGRAPHS. EDITED BY Mansfield Merriman and Robert S. Woodward. Octavo, Cloth, $1.00 net, each. No. 1. HISTORY OF MODERN MATHEMATICS. By DAVID EUGENE SMITH. No. 2. SYNTHETIC PROJECTIVE GEOMETRY. By GEORGE BRUCE HALSTED. No. 3. DETERMINANTS. By LAENAS GIFFORD WELD. No. 4. HYPERBOLIC FUNCTIONS. By JAMES McMAHON. No. 5. HARMONIC FUNCTIONS. By WILLIAM E. BYERLY. No. 6. QRASSMANN'S SPACE ANALYSIS. By EDWARD W. HYDE. No. 7. PROBABILITY AND THEORY OF ERRORS. By ROBERT S. WOODWARD. No. 8. VECTOR ANALYSIS AND QUATERNIONS. By ALEXANDER MACFARLANE. No. 9. DIFFERENTIAL EQUATIONS. By WILLIAM WOOLSEY JOHNSON. No. 10. THE SOLUTION OF EQUATIONS. By MANSFIELD MERRIMAN. No. 11. FUNCTIONS OF A COMPLEX VARIABLE. By THOMAS S. FISKE. No. 12. THE THEORY OF RELATIVITY. By ROBERT D. CARMICHAEL. No. 13. THE THEORY OF NUMBERS. By ROBERT D. CARMICHAEL. PUBLISHED BY JOHN WILEY & SONS, Inc., NEW YORK. CHAPMAN & HALL, Limited, LONDON. MATHEMATICAL MONOGRAPHS. EDITED BY MANSFIELD MERRIMAN AND ROBERT S. WOODWARD. No. 12. THE THEORY OF RELATIVITY BY ROBERT D. CARMICHAEL, ASSOCIATE PROFESSOR OF MATHEMATICS IN INDIANA UNIVERSITY. FIRST EDITION FIRST THOUSAND. NEW YORK JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITED COPYRIGHT, 1913, BY ROBERT D. CARMICHAEL. THE 8CICNTIFIC PRESS ROBERT DRUMMONO AND COMPANY BROOKLYN. N. Y. EDITORS' PREFACE. THE volume called Higher Mathematics, the third edition of which was published in 1900, contained eleven chapters by eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent to that given in classical and engineering colleges. The publication of that volume was discontinued in 1906, and the chapters have since been issued in separate Monographs, they being generally enlarged by additional articles or appendices which either amplify the former pres- entation or record recent advances. This plan of publication was arranged in order to meet the demand of teachers and the convenience of classes, and it was also thought that it would prove advantageous to readers in special lines of mathe- matical literature. It is the intention of the publishers and editors to add other monographs to the series from time to time, if the demand seems to warrant it. Among the topics which are under con- sideration are those of elliptic functions, the theory of numbers, the group theory, the calculus of variations, and non-Euclidean geometry; possibly also monographs on branches of astronomy, mechanics, and mathematical physics may be included. It is the hope of the editors that this Series of Monographs may tend to promote mathematical study and research over a wider field than that which the former volume has occupied. This number of the series is issued by reason of the widely general contemporary interest in the subject of relativity on the part both of mathematicians and of physicists, and with the hope that it may stimulate further research and progress in this important line of inquiry. May, 1913. AUTHOR'S PREFACE. THIS little book has been written from the point of view of the usefulness of the Theory of Relativity in the develop- ment of physical science. It is principally based on a short course of lectures which I delivered at Indiana University in the Fall of 1912 and on my recent papers in the Physical Review, vol. 35, pp. 153-176, and Second Series, vol. i, pp. 161-197. No attempt is made to give any applications of the theory other than what is incidental to the derivation of the funda- mental results concerning length and time, the transformation of coordinates, mass and energy, and experimental verification. It is believed, however, that the presentation is such as to keep always close to concrete experience, so that the results obtained may be directly useful in suggesting experiments for the laboratory. My indebtedness to other writers is indicated by the refer- ences in my recent papers referred to above. I wish also to speak of the useful suggestions made by my students in their frequent discussion of the Theory of Relativity during the course last Fall. R. D. CARMICHAEL. INDIANA UNIVERSITY, May, 1913. CONTENTS. CHAPTER I. INTRODUCTION. PAGE i. THE FOUNDATIONS OF PHYSICS 7 2. ARE THE LAWS o? NATURE RELATIVE TO THE OBSERVER? 8 3. THE STATE OF THE ETHER 9 4. MOVEMENT OF THE EARTH THROUGH THE ETHER 10 5. THE TEST OF MICHELSON AND MORLEY 10 6. OTHER EXPERIMENTAL INVESTIGATIONS 13 7. THE THEORY OF RELATIVITY is INDEPENDENT OF THE ETHER 14 CHAPTER II. THE POSTULATES OF RELATIVITY. INTRODUCTION 15 SYSTEMS OF REFERENCE 16 THE FIRST CHARACTERISTIC POSTULATE 17 THE SECOND CHARACTERISTIC POSTULATE 17 1 2. THE POSTULATES V AND L 22 13. CONSISTENCY AND INDEPENDENCE OF THE POSTULATES 24 14. OTHER POSTULATES NEEDED 25 CHAPTER III. THE MEASUREMENT OF LENGTH AND TIME. 15. RELATIONS BETWEEN THE TIME UNITS OF Two SYSTEMS 27 16. RELATIONS BETWEEN THE UNITS OF LENGTH OF Two SYSTEMS 32 17. DISCUSSION OF THE NOTION OF LENGTH 34 18. DISCUSSION OF THE MEASUREMENT OF TIME 37 19. SIMULTANEITY OF EVENTS HAPPENING AT DIFFERENT PLACES 40 CHAPTER IV. EQUATIONS OF TRANSFORMATION. 20. TRANSFORMATION OF SPACE AND TIME COORDINATES 44 21. THE ADDITION OF VELOCITIES 46 22. MAXIMUM VELOCITY OF A MATERIAL SYSTEM 47 23. TIME AS A FOURTH DIMENSION 48 5 6 CONTENTS. CHAPTER V. MASS AND ENERGY. PAGE 24. DEPENDENCE OF MASS ON VELOCITY 49 25. ON THE DIMENSIONS OF UNITS 54 26. MASS AND ENERGY 57 27. ON MEASURING THE VELOCITY OF LIGHT 58 28. ON THE PRINCIPLE OF LEAST ACTION 59 29. A MAXIMUM VELOCITY FOR MATERIAL BODIES 60 30. ON THE NATURE OF MASS 60 31. THE MASS OF LIGHT 62 CHAPTER VI. EXPERIMENTAL VERIFICATION OF THE THEORY. 32. Two METHODS OF VERIFICATION 63 33. LOGICAL EQUIVALENTS OF THE POSTULATES 65 34. ESSENTIAL EQUIVALENTS OF THE POSTULATES 65 35. THE BUCHERER EXPERIMENT 68 36. ANOTHER MEANS FOR THE EXPERIMENTAL VERIFICATION OF THE THEORY OF RELATIVITY 70 INDEX 73 THE THEORY OF RELATIVITY. CHAPTER I. INTRODUCTION. 1. THE FOUNDATIONS OF PHYSICS. THOSE who look on physics from the outside not infrequently have the feeling that it has forgotten some of its philosophical foundations. Even among its own workers this condition of the science has not entirely escaped notice. The physicist, who, above all other men, has to deal with space and time, has fallen into certain conventions concerning them of which he is often not aware. It may be true that these conventions are just the ones which he should make. It is certain, however, that they should be made only by one who is fully conscious of their nature as conventions and does not look upon them as fixed realities beyond the power of the investigator to modify. Likewise, a question arises as to what element of conven- tion is involved in our usual conceptions of mass, energy, etc.; that the question is not easily answered becomes apparent on reflection. These and many other considerations suggest the desira- bility of a fresh analysis of the foundations of physical science. Now it is a ground of gratulation for all those interested in this matter that there has arisen within modern physics itself a new movement that associated with the Theory of Rela- tivity which is capable of contributing most effectively to the 7 8 THE THEORY OF RELATIVITY. construction of a more satisfactory foundation for its super- structure of theory. It is at once admitted that the theory of relativity is not yet established on an experimental basis which is satisfactory to all persons; in fact, some of those who dispute its claim to acceptance are among the most eminent men of science of the present time. On the other hand there is an effective body of workers who are pushing forward investigations the inspira- tion for which is afforded by the theory of relativity. This state of affairs will probably give rise to a consider- able controversial literature. If the outcome of this contro- versy is the acceptance in the main of the theory of relativity, then this theory will afford just the means needed to arouse in investigators in the field of physics a lively sense of the phil- osophical foundations of their science. If the conclusions of relativity are refuted this will probably be done by a careful study of the foundations of physical science and a penetrating analysis of the grounds of our confidence in the conclusions which it reaches. This of itself will be sufficient to correct the present tendency to forget the philosophical basis of the science. It follows that in any event the theory of relativity will force a fresh study of the foundations of physical theory. If it accomplishes no more than this it will have done well. 2. ARE THE LAWS OF NATURE RELATIVE TO THE OBSERVER? The fundamental question asked in the theory of relativity is this: In what respect are our enunciated laws of nature rela- tive to us who investigate them and to the earth which serves us as a system of reference? How would they be modified, for instance, by a change in the velocity of the earth? To put the matter more precisely, let us suppose that we have two relatively moving platforms with an observer on each of them. Suppose further that each observer considers a system of reference, say cartesian axes, fixed to his platform, and expresses the laws of nature, as he determines them, by means INTRODUCTION. 9 of mathematical equations involving the cartesian coordinates as variables. To what extent will the laws in the two cases be identical? What transformations of the time and space variables must be carried out in order to go from the equations in one system to those in the other; that is, what relations must exist between the variables on the two platforms in order that the results of observation in the two cases shall be consistent? Any theory which states these relations is a theory of rela- tivity. It is obvious that the questions above must be fundamental in any system of mechanics. In fact, a detailed analysis of the matter would show that such a system is characterized pri- marily by the answers which it gives to these questions. This is the feature which distinguishes between the Newtonian and the various systems of non-Newtonian Mechanics. The theory of relativity, in the sense of this book, belongs to one of the lat- ter. It is developed from a small number of fundamental postulates, or laws, which have been enunciated as the probable teaching of experiment. Some account of these experimental investigations will now be given. 3. THE STATE OF THE ETHER. Those who postulate the existence of an ether as a means of explaining the facts about light, electricity and magnetism have usually been in general agreement as to the conclusion that the parts of this ether have no relative motion among themselves, that is, that the ether may be considered station- ary. Experimental facts, which have to be accounted for, cannot be explained satisfactorily on the hypothesis of a mobile ether. The aberration of light is one of the most conspicuous of those phenomena which seem to require for their explanation the hypothesis of a stationary ether. The experiment of Fizeau, in which a comparison was made between the velocities of light when going with, and against, a stream "of water, was interpreted by Fresnel as indicating 10 THE THEORY OF RELATIVITY. a certain entrainment of the ether; but a later examination of the matter by Lorentz * has led to the conclusion that Fizeau's experiment requires a stationary ether for its explana- tion. A result which leads to a similar conclusion has been obtained in electrodynamics by H. A. Wilson f in measuring the electric force produced by moving an insulator in a magnetic field. 4. MOVEMENT OF THE EARTH THROUGH THE ETHER. The theory of a stationary ether leads us to expect certain modifications in the phenomena of light and electricity when there is no relative motion of material bodies, but when both the observer and all his apparatus are carried along through the ether with a velocity v. The effects to be expected are of the order v 2 /c 2 , where c is the velocity of light. Although these effects are very small even when v is the velocity of the earth in its orbit, the possible accuracy of certain optical and electrical experiments is such that these effects could certainly be found if they existed without some compensating effect to mask them. Thus it should be possible for an observer, by making optical and electrical measurements on the earth alone, to detect the motion of the earth relative to the ether. 5. THE TEST OF MICHELSON AND MORLEY. Thus it was predicted that the time which would be required for a beam of light to pass a given distance and return would be different in the two cases when the path of light was parallel to the direction of motion and when it was perpendicular to this direction. Michelson and Morley devised an experiment the object of which was to put this prediction to a crucial test. The experiment was a bold one, seeing that the difference to be measured was so small; but it was carried out in such a * See Lorentz, Versuch einer Theorie der Elektrischen und Optischen Erscheinungen, in Bewegten Korpern, 68. f Proc. Roy. Soc. 73 (1904): 490. t American Journal of Science (3), 34 (1887): 333~345- INTRODUCTION. 11 brilliant way as to permit no serious doubt of the accuracy of the results. The difference of time predicted by theory was found by experiment not to exist; there was not the slightest difference of time in the passage of light along two paths of equal length, one in a direction parallel to the earth's motion and the other in a direction perpendicular to it. Owing to the great importance which this famous experiment has in the theory of relativity some further account of it will be given here. The essential parts of the apparatus used are shown in Fig. i, and the experiment was carried out in the fol- lowing manner: Let a ray of light from a point source S fall on a semi-reflect- ing mirror A, which is set at such an angle that it will reflect p B B' T FIG. i. half the ray to the mirror B and allow the other half to pass on to a third mirror C. The lines AB and AC cross at right angles and the distance AB is made equal to the distance AC. Half of the reflected ray from B will pass through A and on to the telescope T. Also, half of the reflected ray from C will be reflected at A to T. Now the paths A BAT and AC AT are by measurement equal, so that the ray along ABA and the one along AC A should reach T simultaneously, provided that the apparatus is at rest in the ether. Now suppose that the ether is stationary and that the earth is moving through it with little or no disturbance. Then the whole system of apparatus, which is fixed to the earth, will be moving with respect to the ether with the effect indicated in Fig. 2. 12 THE THEORY OF RELATIVITY. While the light is going from mirror A to mirrors B and C and back again to A, the whole apparatus is carried forward in the direction of the incident light to the position A'B'C'. The ray reflected from B, which interferes with a given ray from C along the line A'T, must be considered as traveling along the line AB'A'j the angle BAB' being the angle of aberra- tion. Suppose that the ether remains at rest. Denote by c the velocity of light, and by v the velocity of the apparatus. Let t be the time required for the light to pass from mirror A to mirror C, and let t' be the time required in returning from C to A'. At the time when the reflection takes place at the mirror C, this mirror is approximately half way between C and C of the figure. Let D represent the distance AB or AC . Then ct = D+vt, ct' = D-vt'; whence " 5 v , c v c+v The whole time required for the passage of the light in both directions is 2CD and the distance traveled in this time is (t+f)c = 2D~^ = 2 D^+^J, the terms of fourth order and higher being neglected in the last member. The length of the path ABA' is evidently to the same degree of accuracy as before. The difference of the two lengths is, therefore, approximately DiP/c 2 . If the whole apparatus is now turned through an angle of 90, the difference will be in the opposite direction, and hence INTRODUCTION. . 13 i e displacement of interference fringes along A'T should be This is a very small difference even when v is the velocity of the earth in its orbit; but it is altogether sufficient to be etected and measured if it were present with no other effect to mask it. The result of the experiment was that practically 10 displacement of interference fringes was observed; at most he displacement was less than one-fortieth of that expected. The conclusions which are to be drawn from this experiment ve shall state in the next chapter. 6. OTHER EXPERIMENTAL INVESTIGATIONS. On the electrical side the problem of detecting the move- ment of the earth through the ether has been attacked by Trouton and Noble.* They hung up an electrical condenser by a torsion wire and looked for a torque the presence of which was predicted on the hypothesis of a stationary ether through which the condenser was carried by the motion of the earth. Although the sensitiveness of their electrical arrangement was ample for the observation of the expected effect, no evidence of it was found. Therefore both on the optical and on the electrical side the attempt to detect the motion of the earth through the ether fails; no experiment is known by which it can be put in evidence. In addition to this negative evidence concerning the pre- dicted effect of the earth's motion through the ether there is also the positive evidence which comes from the verification of contrary predictions based on other principles. This will come in incidentally for discussion in our later chapters, and consequently will be dismissed here. The experiments which we have described (and others related to them) are fundamental in the theory of relativity. The postulates of the next chapter are based on them. These postulates are in the nature of generalizations of the facts established by the experiments. * Phil. Trans. Roy. Soc. (A), 20'2 (1004): 165. 14 THE THEORY OF RELATIVITY. 7. THE THEORY OF RELATIVITY is INDEPENDENT OF THE ETHER. In the next chapter we shall begin the systematic develop- ment of the theory of relativity. It will be seen that its fundamental postulates, or laws, are based on the experiments of which we have given a brief account and on others related to them. These experiments have been carried out to test predictions which have been made on the basis of a certain theory of the ether. But the results which have been obtained are of a purely experimental character and can be formulated so as not to depend in any way on a theory of the ether. In other words, the laws stated in the postulates in the next chapter are in no way dependent for their truth on either the existence or the non-existence of the ether or on any of its properties. It is important to keep this in mind on account of the confusion which has sometimes arisen as to the relation between the theory of relativity and the theory of the ether. The postulates, as we shall see, are simply generalizations of exper- imental facts; and, unless an experiment can be devised to show that these generalizations are not legitimate, it is natural and in accordance with the usual procedure in science to accept them as " laws of nature." They are entirely independent of any theory of the ether. CHAPTER II. THE POSTULATES OF RELATIVITY. 8. INTRODUCTION. THERE are two fundamental postulates concerning the nature of space and time which underlie all physical theory. They assert in part that every point of space is like every other point and that every, instant of time is like every otrrer instant. To make the statement of these properties more exact and complete we may say that space is isotropic and homogeneous and three-dimensional, while time is homogeneous and one- dimensional. One important mathematical meaning of this is that the transformations of the space and time coordinates are to be linear. All our theorems will depend directly or indirectly on these two postulates concerning the nature of space and time. Since it is certain that no one will be disposed seriously to call them in question, it is considered unnecessary to give any further statement of them or to make explicit reference to them as part of the basis on which any particular theorem depends, it being understood once for all that they underlie all our work. In the previous chapter we gave some account of the experiments of Michelson and Morley and of Trouton and Noble. There are different points of view from which one may look at these experiments. In the theory of relativity they are taken in the light of an attempt to detect the earth's motion through space by means of the effect of this motion on terrestrial phenomena. So far as the experiments go, they indicate that such motion cannot be detected in this way. Furthermore, no one has yet been able to devise an experiment by means of 15 16 . THE THEORY OF RELATIVITY. which the earth's motion through space can be detected by observations made on the earth alone. The question arises: Is it possible to have any such exper- iment at all? In the theory of relativity this question is answered in the negative. The Michelson-Morley experiment and other experiments have been further generalized into the hypothesis that it is impossible to detect motion through space as such; that is, that the only motion of which we can have any knowledge is the motion of one material body or system of bodies relative to another. A sharp formulation of this conclusion constitutes the first characteristic postulate of relativity. 9. SYSTEMS OF REFERENCE. Before stating the postulate, however, it will be necessary to introduce a definition. In order to be able to deal with such quantities as are involved in the measurement of motion, time, velocity, etc., it is necessary to have some system of reference with respect to which measurements can be made. Let us consider any set of things consisting of objects and any kind of physical quantities whatever * each of which is at rest with reference to each of the others. Let us suppose that among these objects are clocks, to be used for measuring time, and rods or rules, to be used for measuring length. Such a set of objects and quantities, at rest relatively to each other, together with their units for measuring time and length, we shall call a system of reference.! Throughout the book we shall denote such a system by S. In case we have to deal at once with two or more systems of reference we shall denote them by Si, 6*2, 5s, ... or by 5, 5 r , ... Furthermore, it will be assumed that the units of any two systems Si and 6*2 are such that the same numerical result will be obtained in measuring with * As, for instance, charges, magnets, light-sources, telescopes, etc. f If any number of these objects or quantities are absent we shall sometimes refer to what remains as a system of reference. Thus the system might consist of a single light-source alone. THE POSTULATES OF RELATIVITY. , 17 the units of Si a quantity L\ and with the units of 82 a quantity 2 when the relation of LI to Si is precisely the same as that of L 2 to S 2 . 10. THE FIRST CHARACTERISTIC POSTULATE. With this definition before us we are now able to state the first characteristic postulate of relativity: POSTULATE M. The unaccelerated motion of a system of reference S cannot be detected by observations made on S alone } the units of measurement being those belonging to S. The postulate, as stated, is a direct generalization from experiment. None of the actually existing experimental evidence is opposed to it. The conviction that future evidence will continue to corroborate it is so strong that objection has seldom or never been offered to this postulate by either the friends or the foes of relativity. No means at present known will enable the observer to detett motion through space or through any sort of medium which may be supposed to pervade space. Furthermore, in every case where the usual theories have predicted the possibility of detecting such motion and where sufficiently exact observations have been made, it has turned out that no such motion was detected. Moreover, one at least of these contradictions of theory the Michelson-Morley experiment has been outstanding for a period of twenty- five years and no satisfactory explanation has been offered unless one is willing to accept the law stated in postulate M above. It would appear, therefore, that the experimental evidence foi? the postulate is to be considered of strong character. 11. THE SECOND CHARACTERISTIC POSTULATE. The so-called second postulate of relativity, in the form in which it has frequently been stated,* involves two entirely distinct parts. To the present writer it appears that no incon- siderable part of the difficulty which has been felt concerning * See postulate R below and the remarks which lead up to it. 18 THE THEORY OF RELATIVITY. this second postulate has been due to a failure to perceive the interdependence of these two parts and of postulate M above. Precisely that part of the second postulate to which most objec- tion has been raised is a logical consequence of M and of the other part, the part last mentioned being a statement of a law which for a long time has been accepted by physicists. Consequently, we shall state separately the two parts of the second postulate and bring out with care the interdependence of these and of postulate M above. The part which we shall give first states a principle which has long been familiar in the theory of light, namely, that the velocity of light is unaffected by the velocity of the source. Stated in exact language this postulate is as follows: POSTULATE R'. The 'velocity of light in free space, measured on an nonaccelerated system of reference S by means of units belong- ing to S, is independent of the unaccelerated velocity of the source of light. The law stated in this postulate is a conclusion which follows readily from the usual undulatory theory of light and will therefore be accepted by any one who holds to that theory. But it should be emphasized that R' does not depend for its truth on any theory of light. It is a matter for direct experi- mental verification or disproof, and this should be made in such a way as to be independent, as far as possible, of all general theories of light, at least insofar as they are not supported by direct experimental evidence. So far as the writer is aware, there is no experimental evidence which is undoubtedly opposed to postulate M, while on the other hand there is direct experi- mental evidence which is believed by some to be definitely in its favor. Tolman,* in particular has considered this matter in relation to the Doppler effect and to the velocity of light from the two limbs of the sun; and has concluded that experiment bears out the postulate. Stewart, f on the other hand, has examined the same experiments and has found an explanation * Physical Review, 31 (1910): 26-40. ^Physical Review 1 , 32 (1911): 418-428. THE POSTULATES OF RELATIVITY. 19 for them in Thomson's electromagnetic emission theory of light. According to Stewart these experiments are in agreement with our postulate M but are opposed to our postulate R' '. All other attempted proof or disproof of the postulate appears to be in the same state; it is capable of twp^ interpretations which are directly opposed to each other with respect to their conclusions as to the validity of R'. Thus at present there is no undoubted experimental evidence for or against postu- late R'. If the assumption is to be proved at all, either new experiments must be devised or it must be proved by indirect means by showing that it is a consequence of experiment and accepted laws. Now any one who accepts postulates M and R f will perforce accept also all the logical consequences which necessarily flow from them. Of these logical consequences we shall now prove one which is of great importance in the theory of relativity: THEOREM I. The velocity of light in free space, measured on an unaccelerated system of reference S by means of units belonging to S, is independent of the direction of motion of S(MR').* Since by R' the velocity of light is independent of that of the light-source we may suppose that the light-source belongs to the system of reference S. Now let the velocity of light , as it is emitted from this source in various directions, be observed and tabulated. On account of the homogeneity and isotropy of space mere direction through space will have no effect on these observed velocities; and therefore if they differ at all, the difference will be due to the velocity of S. Now if there were a difference due to the direction of motion of S this difference would put in evidence the motion of 5. But by M it is impos- sible to detect such motion in this way. Hence the observed velocity must be the same in all directions. In other words, it is independent of the direction of motion of S; and thus the theorem is proved. It is clear, however, that we cannot take the next step and * Letters attached to a theorem in this way indicate those of the postulates on which the theorem depends. 20 THE THEORY OF RELATIVITY. prove that this observed velocity of light is independent of the numerical value of the velocity of 5. To see this clearly, let us suppose that the numerical value of the velocity of 6 1 does effect the observed velocity of light. On account of R f it will have the same effect on the observed velocity of light whatever may be the unaccelerated motion of the light-source. Hence, from all possible observations, the experimenter will have only a single datum from which to determine the effect of one phe- nomenon on another; namely, a datum in which the two phe- nomena are connected in a certain definite way. It is obvious then that he cannot determine the effect of one of the phenomena on the other; for he can never observe the one without the other being present also and the connection which exists between them is always the same however he may vary the experiment. And if the observer cannot determine an existing effect it is clear that he cannot prove the absence of any effect whatever. But, although the absence of this effect cannot be proved, it is probably impossible to conceive any satisfactory way in which it could be present. Physical intuition is emphatic in asserting that if the direction of the velocity of 5 has no effect on the observed velocity of light then the numerical value of the veloc- ity of S has no effect on such observed velocity. But this does not constitute a proof. There is in this, however, nothing to invalidate the naturalness of the assumption of such independence of the two velocities; in fact, it would be unscientific to make a different assumption (which would necessarily introduce greater complications) unless we were forced to it by unques- tioned experimental fact. Accordingly, we shall make the assumption and shall state it as postulate R": POSTULATE R". The velocity of light in free space, measured on an unaccelerated system of reference S by means of units belong- ing to 5, is independent of the numerical value of the velocity of S. POSTULATE R. The postulate obtained by combining R' and R" will, for convenience, often be referred to as postulate R. Now since unaccelerated velocity is completely determined when the numerical value of the velocity and the direction of THE POSTULATES OF RELATIVITY. 21 the motion are given the truth of the following theorem is an immediate consequence of theorem I and postulate R" : THEOREM II. The velocity of light in free space, measured on an unaccelerated system of reference S by means of units belong- ing to S, is independent of the velocity of S (MR). The second postulate of relativity has usually been stated in a form different from that given above in R' and R" or R. In fact, the truth of theorem I has often been taken as part of the assumption in this postulate, notwithstanding that I can be derived from M and R' . Now, it is precisely the assump- tion of I that has given most difficulty to some persons. It is believed that a part of this difficulty will disappear in view of the fact that I is here demonstrated by means of M and R! '. For the sake of convenience in future discussion one of the customary formulations of the second postulate is appended here. It must be remembered, however, that it is not a separate constituent part of our present body of doctrine but is already contained in M and R, in part directly and in part as a nec- essary consequence of these postulates. POSTULATE R. The velocity of light in free space, measured on an unaccelerated system of reference S by means of units belonging to S, is independent of the velocity of S and of the unaccelerated velocity of the light-source. From the very nature of the postulate R" it is difficult to obtain direct experimental evidence for or against it. It seems, however, as we have previously pointed out, that one who accepts theorem I can hardly refuse to assume R". But theorem I is a logical consequence of postulates M and R 1 ', as we have shown. Moreover, from what follows it will be seen that we have occasion to make no further assumptions which can in any way run counter to currently accepted notions. Consequently, it would seem that the experimental evidence for or against the whole theory of relativity must center around postulates M and R' '. We have already given some account of the experimental evidence for these postulates. In connec- tion with theorems to be derived later further reference will 22 THE THEORY OF RELATIVITY. be given to the existing experimental evidence and some other possible lines of research in this direction will be pointed out. It is generally conceded that the strange conclusions which are obtained in the theory of relativity are due to postulate R (or to postulate R in the customary formulation). In view of theorem I above and the discussion of its consequences, it is now clear that the strangeness in the conclusions of relativity is due to that part of R which is contained in R'. It is important therefore to have a careful analysis of this pos- tulate and especially to know alternative forms, which, in view of the other postulates, are logically equivalent to it. We shall return to this matter in Chapter VI. 12. THE POSTULATES V AND L. It has been customary for writers on relativity to state explicitly only the postulates M and R. But every one," as a matter of fact, has made further assumptions concerning the relations of the two systems. These assumptions in some form are essential to the initial arguments and to the conclusions which are drawn by means of them. To the present writer it seems preferable to have these assumptions explicitly stated. Among several forms, any one of which might be chosen, there is one which seems to be decidedly simpler than any of the others; and it is this one which w r e shall employ here. We state the postulates V and L as follows: POSTULATE V. If the velocity of a system of reference 82 relative to a system of reference Si is measured by means of the units belonging to Si and if the velocity of Si relative to S-2 is measured by means of the units belonging to 82 the two results will agree in numerical value. This velocity we shall call the relative velocity of the two systems. The direction line of this velocity will be called the line of relative motion of the two systems. POSTULATE L. If two systems of reference Si and S-z move with unaccelerated relative velocity and if a line segment I is per- THE POSTULATES OF RELATIVITY. 23 pendicular to the line of relative motion of Si and 5*2 and is fixed to one of these systems, then the length of I measured by means of the units belonging to S\ will be the same as its length measured by means of the units belonging to S%. The essential content of these two postulates may be stated in simpler terms (but less accurately) if one allows the explicit introduction of the observer. Thus V is roughly equivalent to the following statement: Two observers whose relative motion is uniform will agree in their measurement of that uniform relative motion. As an approximate equivalent of L we have: Two observers whose relative motion is uniform will agree in their measurement of length in a line perpendicular to their line of relative motion. It will be observed that these two postulates are nothing more than explicit statements of notions which underlie the classic theories of mechanics. The first is assumed in suppos- ing that there exists such a thing as the relative motion of two bodies which are not at rest relatively to each other. The second is nothing more than the statement of a portion of the idea which lies at the bottom of our conception of such a thing as the length of a rod or other object. Since these two postulates are universally accepted, the question might naturally arise, Why state them at all? Is it not enough simply to take them for granted? The answer is that there are other notions which have heretofore met with the same universal acceptance and which do not agree with the postulates of relativity. Therefore it seems to be desirable in fact, to be essential to proper logical procedure to state explicitly just those assumptions concerning the rela- tion of the two systems of reference which we shall have occasion to employ in argument. Only in this way is one able to see exactly on what basis our strange conclusions rest. We shall make a digression here to say one further word about postulate L. In the next chapter we shall draw the conclusion that length in the line of motion is not independent of the velocity with which the system is moving. In view of 24 THE THEORY OF RELATIVITY. this the question arises as to why we must assume that length in a line perpendicular to the line of motion is independent of the motion. The answer is that we are under no such ne- cessity, that we are at liberty to assume that length in a line perpendicular to the line of motion is dependent on the velocity of such motion. In fact, the general formulation of such an hypothesis has already been made by E. Riecke.* This hypoth- esis, however, is undoubtedly more complicated and less elegant than the one which we have made; and the latter, as we shall see, is in conflict with no known experimental facts. Therefore, following that instinct which has always wisely guided the physicist, we make the simplest hypothesis which is in agreement with and explanatory of the totality of exper- imental facts at present known. If at any time experiments are set forth which do not agree with the theory developed on the basis of the above postulates, then will be the time to con- sider the question of introducing a more complicated postulate in place of our postulate L above. 13. CONSISTENCY AND INDEPENDENCE OF THE POSTULATES. Throughout our treatment it will be assumed that the postulates as stated above are consistent; that is to say, no attempt will be made to prove their consistency. The fact that no contradictory conclusions have been drawn from them will be accepted as (partial) evidence that they are mutually consistent. Moreover, from their very nature and from the differing range of applicability of the several postulates it is difficult to conceive how any of them can possibly contradict conclusions which may be drawn from the others. There is another question also which it is our purpose to pass over without discussion, namely, the question of the logical independence of the postulates. Is any postulate or a part of any postulate a logical consequence of the remaining postulates? This question is important from the point of view * Gottinger Nachrichten, Math. Phys., 1911. pp. 271-277. THE POSTULATES OF RELATIVITY. 25 of formal logic, but in the present case its value to physical science is probably small. 14. OTHER POSTULATES NEEDED. From the postulates stated above it is possible to draw only those conclusions of the theory of relativity which are of a general nature and have to do merely with the measurement of time and space. They alone are employed in Chapters III and IV. If it is desired to study the nature of mass or the relation of mass and energy in the theory of relativity, it is necessary to have some assumption concerning mass in the first case and concerning both mass and energy in the second case. Thus we might assume the conservation laws of energy, electricity and momentum and deduce the joint consequences of these assumptions and those given above. It is our purpose to take up these matters in Chapters V and VI. It is convenient to state the postulates here; and this we do, after giving some necessary definitions. If m, M and v are respectively the mass, momentum and velocity of a body we shall assume (as in the classical mechanics) that they are connected by a relation of the form M = mv. We shall take mass and velocity to be the fundamental quanti- ties and shall define momentum in terms of them by the above relation. Likewise we shall define the kinetic energy E of a moving body by means of the usual relation Later we shall see that " mass " is variable and is not in general independent of the direction in which it is measured; conse- quently, we must take for m in the above formulas the mass of the body in the direction of its motion. We shall take for granted the following laws of conservation of momentum and energy and electricity: 26 THE THEORY OF RELATIVITY. POSTULATE C\. The sum total of momentum in any isolated system remains unaltered, whatever changes may take place in the system, provided that it is not affected by any forces from without. POSTULATE 2- The sum total of energy in any isolated system remains unaltered, whatever changes may take place in the system, provided that it is not affected by any forces from without. POSTULATE 3. The sum total of electricity in any isolated system remains unaltered, whatever changes may take place in the system, provided that the system as a whole neither receives elec- tricity from nor gives out electricity to bodies not belonging to the system. The " action " of a moving body in passing from one posi- tion to another may be denned as the space integral of the momentum taken over the path of motion. If we denote this action by A we have therefore A = jMds = jmvds. Now ds = vdt, so that we have also A=jmv 2 dt. If several bodies are involved we have A = 2. (mvds = ^> \mv 2 dt, where the summation is for the various bodies in the system. We may state the fundamental principle of least action in the following form: PRINCIPLE OF LEAST ACTION. The free motion of a con- servative system between any two given configurations has the property that the action A is a minimum, the admissible values A of the action with which A is compared being obtained from varied motions in which the total energy has the same constant value as in the actual free motion. CHAPTER III. THE MEASUREMENT OF LENGTH AND TIME. 15. RELATIONS BETWEEN THE TIME UNITS or Two SYSTEMS. LET us consider three systems of reference S, Si and 6*2 related to each other in the following manner: The lines of relative motion of 5 and Si, of S and -5*2, of Si and 82 are all parallel; Si and 62 have a relative velocity * z>; S and Si have a relative velocity \D in one sense and S and 2 have a relative velocity \v in the opposite sense. The system S con- sists of a single light-source, and this source is symmetrically placed with respect to two points of which one is fixed to Si and the other is fixed to -5*2 . This is possible as a permanent relation on account of the relative motions of the three systems. For convenience, let us assume S to be at rest. We shall now suppose that observers on the systems 6*1 and 5*2 measure the velocity of light as it emanates from the source S. Let a point A .on Si and a point B on 62, which are symmetrically placed with respect to the light-source S, move along the lines h and h] these lines are parallel. FlG 3> From postulate L it follows that the observers on Si and 6*2 will obtain the same measure- ment of the distance between li and /2. Denote this distance by d. From postulate M it follows that neither observer is able to detect his motion. Therefore he will make his observa- tions on the assumption that his system is at rest; that is to say, his measurements will be made by means of the units * Note that postulate V is required to make this hypothesis legitimate. 27 28 THE THEORY OF RELATIVITY. belonging to his system and no corrections will be made on account of the motion of the system. Let the observer on Si reflect a ray of light 5^4 from a point A to a point C on h and back to A\ and let the observed time of passage of the light from A to C and back to A be /. Since the observer assumes his system to be at rest he will suppose that the ray of light passes (in both directions) along the line AC which is perpendic- ular to l\ and /2. His measurement of the distance traversed by the ray of light in time / will therefore be 2d. Hence he will obtain as a result 2d t ==C > where c is his observed velocity of light. Similarly, an observer on 52, supposing his system to be at rest finds the time /i which it requires for a ray of light to pass from B to D and return, the ray employed being gotten by reflecting a ray SB at B. Thus the second observer obtains 2d 7T C1 > where c\ is his observed velocity of light. Now, from the assumed relations among the systems 5, Si and 5*2 and from the homogeneity of space it follows that the two observations which we have supposed to be made must lead to the same estimate for the velocity of light. This is readily seen from the fact that the observations were made in such a way that the effect due to either the numerical value or the direction of the motion of the systems Si and 6*2 is the same in the two cases. In other words, if we denote by LI and L% the quantities measured on Si and 6*2 respectively, then the relation of LI to Si is precisely the same as that of L% to 6*2; and hence the numerical results are equal, as one sees from the definition of systems of reference. Therefore we have ci=c. Let us now suppose that the observer at A is watching the experiment at B. To him it appears that B is moving with a velocity v, since by hypothesis the two systems have the relative velocity v and A and B measure this velocity alike. We shall THE MEASUREMENT OF LENGTH AND TIME. 29 assume that the apparent motion is in the direction indicated by the arrow in the figure. To the observer at B it appears that the ray of light traverses BD from B to D and returns along the same line to B. To the observer at A it appears that the ray traverses the line BEF, F being the point which B has reached by the time that the ray has returned to the observer at this point. If EG is perpendicular to 1 2 and d\ is the length of EF as measured by means of units belonging to Si, then, evidently, GF (when measured in the same units) is $dij where $=v/c and c is the (apparent) velocity of light as estimated in this case by the observer at A. From the right triangle EFG it follows at once that we have a d v^-T Now, if f is the time which is required, according to the observer at A, for the light to traverse the path BEF, then we have = c. t tVi-^ So far in our argument in this section we have employed only those of our postulates which are generally accepted by both the friends and the foes of relativity. Now we come to the place where the men of the two camps must part company. Let us introduce for the moment the following additional hypothesis : ASSUMPTION A. The two estimates c and c of the velocity of light obtained as above by the observer at A are equal. Now we have shown that c is equal to c\. Hence we may equate the values of c\ and c given above; thus we have 2d or /i=^v i (j 2 . But t\ and t are measures of the same interval of time, t\ being in units belonging to 6*2 and t being in units belonging to Si. Hence, to the observer on Si, the ratio of his time unit to that of 30 THE THEORY OF RELATIVITY. if [$fMpltt$&*tt$ the system 62 appears to^ be Vi 2 : i. On the other hand, it may be shown in exactly the same way that to the observer on 6*2 the ratio of his time unit to that of the system Si appears to be Vi 2 : i. That is, the time units of the two systems are different and each observer comes to the same conclusion as to the relation which the unit of the other system bears to his own. This important and striking result may be stated in the following theorem: THEOREM III. If two systems of reference Si and 62 move with a relative velocity v, and $ is defined as the ratio of v to the velocity of light estimated in the manner indicated above, then to an observer on Si the time unit of Si appears to be in the ratio Vi 2 : i to that of 6*2 while to an observer on 6*2 the time unit of 5*2 appears to be in the ratio Vi p 2 : i to that of Si (MVLA). Let us now bring into play our postulate R'. In theorem I we have already seen that a logical consequence of M and R' is that the velocity of light, as observed on a system of reference, is independent of the direction of motion of that system. Now, if c and c as estimated above differ at all, that difference can be due only to the direction of motion of Si. as one sees readily from postulate R f and the method of determining these quan- tities. Hence the statement which we made above as assump- tion A is a logical consequence of postulates M and R f . There- fore we are led to the following corollary of the above theorem: COROLLARY. Theorem III may be stated as depending on (MVLR') instead of on (MVLA). Let us now go a step further and employ postulate R" '. From theorem I and postulates R' and R" it follows that the observed velocity of light is a pure constant for all admissible methods of observation. If we make use of this fact the preceding result may be stated in the following simpler form: THEOREM IV. If two systems of reference Si and 62 move with a relative velocity v, and $ is the ratio of v to the velocity of light, then to an observer on Si the time unit of Si appears to be in the ratio V i 2 : i /0 that of 62 while to an observer on S2 the time THE MEASUREMENT OF LENGTH AND TIME. 31 unit of 82 appears to be in the ratio Vi g 2 : i to that of Si (MVLR). Let us subject these remarkable results to a further analysis. Theorem III, its corollary and theorem IV all agree in the extraordinary conclusion that the time units of the two systems of reference 6*1 and 62 are of different lengths. Just how much they differ is a secondary matter; that they differ at all is the surprising and important thing. As postulates M, V, L are generally accepted and have not elsewhere led to such strange conclusions it is natural to suppose that the strangeness here is not due to them. Referring to the argument carried out above, we see that no unusual conclusions were reached until we had introduced and made use of assumption A. Moreover we have seen that this assumption itself is a logical consequence of M and R'. Further, R" is not involved either in theorem III or in its corollary. But these already contain the strange features of our results. Hence the conclusion is irresistible that the extraordinary element in these results is due to postulate R f or to speak more accurately, to just that part of it which it is necessary to use in connection with M in order to prove A as a theorem. This result is important, as the following considerations show. Postulates V and L state laws which have been univer- sally accepted in the classical mechanics. Postulate M is a direct generalization from experiment, and the generalization is legitimate according to the usual procedure of physicists in like situations. Postulate R' is a statement of a principle which has long been familiar in the theory of light and has met with wide acceptance. Thus we see that no one of these postulates, in itself, runs counter to currently accepted physical notions. And yet just these postulates alone are sufficient to enable us to conclude that corresponding time units in two systems of reference are of different magnitude. In the next section we shall show on the basis of the same postulates that the corresponding units of length in the two systems are also different. Thus the most remarkable elements in the 32 THE THEORY OF RELATIVITY. conclusions of the theory of relativity are deducible from postulates M, V, L, R' alone; and yet these are either generaliza- tions from experiment or statements of laws which have usually been accepted. Hence we conclude: The theory of relativity, in its most characteristic elements, is a logical consequence of certain generalizations from experiment together with certain laws which have for a long time been accepted. One other remark, of a totally different nature, should be made with reference to the characteristic result of theorem IV. It has to do with the relation between the time units of the two systems. This relation is intimately associated with the fact that each observer makes his measurements on the hypothesis that his own system is at rest, while the other sys- tem is moving past him with the velocity v. If both observers should agree to call S fixed and if further in this modified " universe " our postulates V, L, R were still valid it would turn out that the two observers would find their time units in agreement. But, in viewdf M, the choice of S as fixed would undoubtedly seem perfectly arbitrary to both observers; and the content of the modified postulate R would be essentially different from that of the postulate as we have employed it. Hence, if we accept R as it stands or, indeed, even a certain part of it, as we have shown above we must conclude that the time units in the two systems are not in agreement, in fact, that their ratio is that stated in the theorems above. 16. RELATIONS BETWEEN THE UNITS OF LENGTH OF Two SYSTEMS. Let us consider three systems of reference S, Si and 52 related in the same manner as in the preceding section except that now the two lines l\ and /2 coincide. We suppose that S\ is moving in the direction indicated by the arrow at A and that 5*2 is moving in the direction indicated by the arrow at B. We suppose that observers at A and B again measure the velocity of light as it emanates from 5, this time in the direc- tion of the line of motion. Iiach will carry out his observations THE MEASUREMENT OF LENGTH AND TIME. 33 on the supposition that his system is at rest, for from M it fol- lows that he cannot detect the motion of his system. The observer at A measures the time ti of passage of a ray of Zl c D lo light from A to C and return s, **"<& ^ & *"*" to A, the length of AC being FIG. 4. d when the measurement is made with a unit belonging to Si. Likewise, the observer at B measures the time fe of passage of a ray of light from B to D and return to B, the length BD being d when measured with a unit belonging to S%. Just as in the preceding case it may be shown that the two observers must obtain the same estimate for the velocity of light. But the estimate of the observer at A is zd/t\ while that of the observer at B is 2d/t2. Hence *ife; that is, the number of units of time required for the passage of the ray at A and of the ray at B is the same, the former being measured on Si and the latter on 62. Moreover, the measure of length is the same in the two cases. But the units of time, as we saw in the preceding section, do not have the same magni- tude. Hence the units of length of the two systems along their line of motion do not have the same magnitude; and the ratio of units of length is the same as the ratio of units of time. Combining this result with theorem III, its corollary and theorem IV we have the following three results: THEOREM V. If two systems of reference Si and S2 move with a relative velocity v and $ is defined as the ratio of v to the velocity of light estimated in the manner indicated in the first part of 15, then to an observer on Si the unit of length of Si along the line of relative motion appears to be in the ratio Vi (3 2 : i to that of 82 while to an observer on 82 the unit of length of 82 along the line of relative motion appears to be in the ratio Vi g 2 : i to that of Si(MVLA). COROLLARY. Theorem V may be stated as depending on (MVLR f ) instead of on (MVLA). 34 THE THEORY OF RELATIVITY. J( THEOREM VI. If two systems of reference Si and 62 move with a relative velocity v.and (i is the ratio of v to the velocity of light, then to an observer on Si the unit of length of Si along the line of relative motion appears to be in the ratio Vi (3 2 : i?to that of 62 while to an observer on 62 the unit of length of 62 along the line of relative motion appears to be in the ratio Vi $ 2 : if to that of Si (MVLR). We might make an analysis of these results similar to that which we gave for the corresponding results in the preceding section. But it would be largely a repetition. It is sufficient to point out that the remarkable conclusions as to units of length in the two systems rest on just those postulates which led to the strange results as to the units of time. 17. DISCUSSION OF THE NOTION OF LENGTH. In the preceding section we saw that two observers A and B on relatively moving systems of reference Si and 2 respectively are in a very peculiar disagreement as to units of length along a line I parallel to their line of relative motion. To A it appears that B's units are longer than his own. On the other hand, it seems to B that his units are shorter than A's. In the two cases the apparent ratio is the same; more precisely, the unit which appears to either observer to be the shorter seems to him to have the ratio Vi g 2 : i to that which appears to him to be the longer. Although they are thus in disagreement there is yet a certain symmetry in the way in which their opinions diverge. Let us suppose that these two observers now undertake to bring themselves into a closer agreement in measurements of length along the line /. Suppose that B agrees arbitrarily to shorten his unit so that it will appear to A that the units of A and B are of the same length. Then, so far as A is concerned, all difficulty has disappeared. How is B affected by this change? We see that the difficulty which he experienced is not disposed of; on the other hand it is greater than before. Already, it seemed to him that his unit was shorter than A's. Now, since THE MEASUREMENT OF LENGTH AND TIME. 35 he has shortened his unit, the divergence appears to him to be increased. Moreover, the symmetry which we found in the former case is now absent. Furthermore, if any other changes in the units of A and B are made we shall always find difficulties as great as or greater than those which we encountered in the initial case. There is no other conclusion than this : We are face to face with an essen- tial difficulty one that is not to be removed by any mere artifice. What account of it shall we render to ourselves? This much is already obvious: The length of an object depends in an essential way upon the measurer and the system to which he belongs. We have certain intuitive notions concerning the nature of matter which it is necessary for us to examine if we are to dis- cuss adequately the notion of length. We have usually supposed that to revolve a steel bar, for instance, through an angle of ninety degrees has no effect upon its length. Let us suppose for the moment that this is not so; but that the bar is shorter when pointing in some directions than in others, so that its length is the product of two factors one of which is its length in a cer- tain initial position and the other of which is a function of the direction in which the body points relative to that in the initial position. Suppose that at the same time all other objects experience precisely the same change for varying directions. It is obvious that in this case we should have no means of ascertaining this dependence of length upon the direction in which the body points. To an observer placed in a situation like this it would be natural to assume that the length of the steel bar is the same in all directions. In other words, in arriving at his definition of length he would make certain conventions to suit his con- venience. Now suppose that the system of such an observer is set in motion with a uniform velocity v relative to the previous state of the system; and that at the same time all bodies on his system undergo simultaneously a continuous dilatation or contrac- 36 THE THEORY OF RELATIVITY. tion. This observer would have no means of ascertaining that fact; and accordingly he would suppose that his steel bar had the same length as before. In other words, he would unconsciously introduce a new convention concerning his measurement of length. There is no a priori reason why our actual universe should not be such as the hypothetical one just described. To sup- pose it so unless our experience demands such a supposition would be unnatural; because it would introduce an unnecessary inconvenience. But suppose that in our growing knowledge of the universe there should come a time when we could more con- veniently represent to ourselves the actual facts of experience by supposing that all material things are subject to some such deformations as those which we have indicated above; there is certainly no a priori reason why we should not conclude that such is the essential nature of the structure of the universe. Naturally we would not come to this conclusion without due consideration. We would first enquire carefully if there is not some more convenient way by which we can reconcile all experimental facts; and only in the event of a failure to find such a way would we be willing to modify so profoundly our views of the material world. Now, if we agree to suppose that our actual universe is subject to a certain (appropriately defined) deformation of the general type discussed above it would follow that observers A and B on the respective systems Si and $2 would be in a dis- agreement as to units of length similar to that which exists, according to the theory of relativity. Therefore, that which at the outset seemed to be of such essential difficulty is explained easily enough, if we are willing to modify so profoundly our conception of the nature of material bodies. Whether in the present state of science experimental facts demand such a radical procedure is a question which will be answered differently by different minds. To one who accepts the postulates of relativity there is indeed no other recourse; one who refuses to accept them must find some other THE MEASUREMENT OF LENGTH AND TIME. 37 satisfactory way to account for experimental facts. The Lorentz theory of electrons gives striking evidence in favor of supposing that matter is subject to some such deforma- tions as those mentioned above; and this evidence is the more important and interesting in that the deformations (as con- ceived in this theory) were assumed to exist simply in order to be able to account directly for experimental facts. 18. DISCUSSION OF THE MEASUREMENT OF TIME.* That two observers in relative motion are in hopeless dis- agreement as to the measurement of length in their line of relative motion is a conclusion which is probably (at first) sufficiently disconcerting to most of us; but it is an even greater shock to intuition to conclude, as we are forced to do accord- ing to the theory of relativity, that there is a like ineradicable disagreement in the measurement of time. A discussion similar to that in the preceding section brings out the fact that our observers A and B cannot possibly arrive at con- sistent means of measuring intervals of time. The treatment is so far similar to the preceding discussion for length that we need not repeat it; we shall content ourselves with a brief discussion of conclusions to be drawn from the matter. Why is this inability of A and B to agree in measuring time received in our minds with such a distinct feeling of surprise and shock? It is doubtless because we have such a lively sense of the passage of time. It seems to be a thing which we know directly, and the conclusion in question is contrary to our unsophisticated intuition concerning the nature of time. But what is it that we know directly? We have an imme- diate perception of what it is for two conscious phenomena to coexist in our mind, and consequently we perceive imme- * In connection with this section and the following one the reader should compare the excellent and interesting treatment of the problem of measuring time to be found in Chapter II of Poincare's Value of Science (translated into English by Halsted). 38 THE THEORY OF RELATIVITY. diately the simultaneity of events in our mind. Further, we have a perfectly clear sense of the order of succession of events in our own consciousness. Is not that all that we know directly? The difficulties which A and B experience in correlating their measurements of time grow out of two things, of neither of which we have direct perception. In the first place there are two consciousnesses involved; and what reason have we to suppose that succession of events is the same for these two? This question we shall not treat, assuming that the principal matter can be put into such imper- sonal form as to obviate this difficulty altogether. (As a mat- ter of fact, so far as anything characteristic of the theory of relativity is concerned this can be done.) The other difficulty has to do with the measurement of time as opposed to the mere psychological experience of its passage. In this matter we are entirely without any direct intuition to guide us. We have no immediate sense of the equality of two intervals of time. Therefore, whatever definition we employ for such equality will necessarily have in it an important ele- ment of convention. To keep this well in mind will facilitate our discussion. Our problem is this: How shall we assign a numerical measure of length to a given time interval; say to an interval in which a given physical phenomenon takes place? We shall arrive at the answer by asking another question: Why should we seek to measure time intervals at all, seeing that we have no immediate consciousness of the equality of such intervals? There can be only one answer: we seek to measure time as a matter of convenience to us in representing to our- selves our experiences and the phenomena of which we are witnesses. In such a way we can render to ourselves a better account of the world in which we live and of our relation to it. Now, since our only reason for attempting to measure time is in a matter of convenience, the way in which we measure it will be determined by the dictates of that convenience. The system of time measurement which we shall adopt is just that THE MEASUREMENT OF LENGTH AND TIME. 39 system by means of which the laws of nature may be stated in the simplest form for our comprehension. Let us return to the case of the two observers A and B of the preceding section. Suppose that each of them has chosen a system of measuring time that suits his convenience in the interpretation of the laws of nature ori his system. There is no a priori reason why the two observers should measure time intervals in the same way. In fact, since there is an arbitrary element in the case of each method of measurement and since the two systems are in a state of relative motion, it is not at all unnatural that the units of A and B should differ. Now it is to be noticed that each of the observers A and B is in just the situation in which we find ourselves. We have chosen a method of measuring time which seems to us conven- ient. Insofar as that method depends on convenience it is rela- tive to us who are observers, and therefore it has in it something which is arbitrary. There is no doubt that it would be desir- able for us to know what it is which is arbitrary, which is relative to us who observe; but it is equally obvious that it must be difficult for us to determine what this arbitrary element is. The theory of relativity makes a contribution to the solu- tion of this problem. We suppose that two observers on dif- ferent systems find the laws of nature the same as we find them ; or, more exactly, we suppose that they find certain specific laws the same as we find them. Then we inquire as to their agreement in measuring time and see that they differ in a certain definite way. This difference is due to things which are relative to the two observers; and thus we begin to get some insight into the ultimate basis of our own method of measurement. It is obviously an important service which the theory of relativity renders to us when it enables us to to make an advance towards a better understanding of such a fundamental matter as this. This matter will become clearer if we speak of the simulta- neity of events which happen at different places ; and therefore we turn to a discussion of this topic. 40 THE THEORY OF RELATIVITY. 19. SIMULTANEITY OF EVENTS HAPPENING AT DIFFERENT PLACES. Let us now assume two systems of reference 5" and S f moving with a uniform relative velocity v. Let an observer on S' undertake to adjust two clocks at different places so that they shall simultaneously mark the same hour. We will suppose that he does this in the following very natural manner: Two stations A and B are chosen in the line of relative motion of S and 5" and at a distance d apart. The point C midway between these two stations is found by measurement. The observer is himself stationed at C and A Hd y*d B k as ass i stants at A and B. A c *-* single light signal is flashed from FIG. 5. C to A and to B, and as soon as the light ray reaches each station the clock there is set at an hour agreed upon before- hand. The observer on S f now concludes that his two clocks, the one at A and the other at B, are simultaneously marking the same hour; for, in his opinion (since he supposes his system to be at rest) the light has taken exactly the same time to travel from C to A as to travel from C to B. Now let us suppose that an observer on the system 5 has watched the work of regulating these clocks on S f . The dis- tances CA and CB appear to him to be instead of %d. Moreover, since the velocity of light is independ- ent of the velocity of the source, it appears to him that the light ray proceeding from C to A has approached A at the velocity c+v, where c is the velocity of light, while the ray going from C to B has approached B at the velocity c v. Thus to him it appears that the light has taken longer to go from C to B than from C to A by the amount C + V C 2 -V 2 THE MEASUREMENT OF LENGTH AND TIME. 41 But since $ = v/c the last expression is readily found to be equal to d Therefore, to an observer on 5 the clocks of S f appear to mark different times; and the difference is that given by the last expression above. Thus we have the following conclusion: THEOREM VII. Let two systems of reference S and S f have a uniform relative velocity v. Let an observer on S' place iwo clocks at a distance d apart in the line of relative motion of S and S f and adjust them so that they appear to him to mark simultaneously the same hour. Then to an observer on S the clock on S' which is forward in point of motion appears to be behind in point of time by the amount v d where c is the velocity of light and $=v/c (MVLR). It should be emphasized that the clocks on S' are in agree- ment in the only sense in which they can be in agreement for an observer on that system who supposes (as he naturally will) that his own system is at rest notwithstanding the fact that to an observer on the other system there appears to be an irreconcilable disagreement depending for its amount directly on the distance apart of the two clocks. According to the result of the last theorem the notion of simultaneity of events happening at different places is indefinite in meaning until some convention is adopted as to how simul- taneity is to be determined. In other words, there is no such thing as the absolute simultaneity of events happening Mt different places. How shall we adjust this remarkable conclusion to our ordinary intuitions concerning the nature of time? We shall probably most readily get an answer to this question by inquir- 42 THE THEORY OF RELATIVITY. ing further: What shall we mean by saying that two events which happen at different places are simultaneous? First of all it should be noticed that we have no direct sense of what such simultaneity should mean. I have a direct per- ception of the simultaneity of two events in my own con- sciousness. I consider them simultaneous because they are so interlocked that I cannot separate them without mutilating them. If two things happen which are far removed from each other I do not have a direct perception of both of them in such way that I perceive them as simultaneous. When should I consider such events to be simultaneous? To answer this question we are forced to the same consider- ations as those which we met in the preceding section. There can be no absolute criterion by which we shall be able to fix upon any definition as the only appropriate one. We must be guided by the demands of convenience, and by this alone. In view of these considerations there is nothing unthink- able about the conclusion concerning simultaneity which we have obtained above. An observer A on one system of refer- ence regulates clocks so that they appear to him to be simulta- neous. It is apparent that to him the notion of simultaneity appears to be entirely independent of position in space. His clocks, even though they are separated by space, appear to him to be running together, that is, to be together in a sense which is entirely independent of all considerations of space. But when B from another system of reference observes the clocks of A 's system they do not appear to him to be mark- ing simultaneously the same hour; and their lack of agreement is proportional to their distance apart, the factor of propor- tionality being a function of the relative velocity of the two systems. Thus instants of time at different places which appear to A to be simultaneous in a sense which is entirely independent of all considerations of space appear to B in a very different light; namely, as if they were different instants of time, the one preceding the other by an amount directly proportional THE MEASUREMENT OF LENGTH AND TIME. 43 to the distance between the points in space at which events occur to mark these instants. Even the order of succession of events is in certain cases different for the two observers, as one can readily verify. It thus appears that the notion of simultaneity at different places is relative to the system on which it is determined. The only meaning which it can have is that which is given to it by convention. CHAPTER IV. EQUATIONS OF TRANSFORMATION. 20. TRANSFORMATION OF SPACE AND TIME COORDINATES. IT is now an easy matter to derive the Einstein formulae for the transformation of space and time coordinates. Let two systems of reference 5 and S' have the relative velocity v in the line /. Let systems of rectangular coordinates be attached to the systems of reference 5 and S' in such a way 'that the #-axis of each system is in the line /, and let the ^-axis and the z-axis of one system be parallel to the ^-axis and the 2-axis respectively of the other system. Let the origins of the two systems coincide at the time t = o. Furthermore, for the sake of distinction, denote the coordinates on 5 by x, y, z, t and those on S' by x', /, z' , t f . We require to find the value of the latter coordinates in terms of the former. From postulate L it follows at once that y f y and z' = z. Let an observer on S consider a point which at time / = o appears to him to be at distance * x from the ;yV-plane; at time / = / it will appear to him to be at the distance x vt from the y'z'- plane. Now, by an observer on S' this distance is denoted by x'. Then from theorem VI we have Now consider a point at the distance x from the yz-piane at time / = / in units of system S. From theorem VII it follows * The algebraic sign of the distance is supposed to be taken into account in the value of x. 44 EQUATIONS OF TRANSFORMATION. 45 that to an observer on S the clock on S' at the same distance x from the ;yz-plane will appear behind by the amount where c is the velocity of light. That is to say, in units of 5 1 this clock would register the time Hence, by means of theorem IV, we have at once the result c Solving the two equations involving x' and t' and collecting results, we have (A) where $ = v/c and c is the velocity of light. In the same way we may obtain the equations which express /, x, y, z in terms of /', x', y', z f . But these can be found more easily by solving equations (.4) for /, x, y, z. Thus we have VI- y=y', (MVLR) These two sets of equations (A) and (A\) are identical in form except for the sign of v. This symmetry in the transforma- tions constitutes one of their chief points of interest. 46 THE THEORY OF RELATIVITY. 21. THE ADDITION OF VELOCITIES. We shall now derive the formulae for the addition of veloc- ities. Let the velocity of a point in motion be represented in units belonging to S' and to S by means of the equations x = u x t, y = u v t j z = u z t, respectively. In the first of these substitute for t 1 ', x', y', z' their values given by (X), solve for x/t, y/t, z/t and replace these quantities by their equals u x , u y , u z respectively. Thus we have (B) u v = - u v> (MVLR) From these results it follows that the law of the parallelo- gram of velocities is only approximate. This conclusion of the theory of relativity has given rise, in the minds of some persons, to the most serious objections to the entire theory. Suppose that both the velocities considered above are in the line of relative motion of S and S'. Then we have v+u' u = ,. vu I+ ^ This equation gives rise to the following theorem: THEOREM VIII. If two velocities, each of which is less than Cf are combined the resultant velocity is also less than c(MVLR). EQUATIONS OF TRANSFORMATION. 47 To prove this we substitute in the preceding equation for v and u' the values where each of the numbers k and / is positive and less than c. Then the equation becomes 2C~k-l u c kr 2C-k-l + The second member is evidently less than c. Hence the theorem. If, however, either one (or both) of the velocities v and u' is equal to c and hence k or I (or both) is equal to zero we see at once from the last equation that u = c. Hence, we have the following result: THEOREM IX. If a velocity c is compounded with a velocity equal to or less than c, the resultant velocity is c(MVLR). 22. MAXIMUM VELOCITY OF A MATERIAL SYSTEM. A conclusion of importance is implicity involved in the pre- ceding results. It can probably be seen in the simplest way by reference to the first two equations (A), these being nothing more nor less than an analytic formulation of theorems IV and VI. If is in numerical value greater than i whence i 2 is negative the transformation of time coordinates from one system to the other gives an imaginary result for the time in one system if the time in the other system is real. Likewise, measurement of length in the direction of motion is imaginary in one system if it is real in the other. Both of these conclusions are absurd and hence the numerical value of is equal to or less than i. If it is i, then any length in one system, however short, would be measured in the other as infin- ite; and a like result holds for time. Hence g is numerically less than i. But $=v/c, the ratio of the relative velocity of the two systems to the velocity of light. Hence: 48 THE THEORY OF RELATIVITY. THEOREM X. The velocity of light is a maximum which the velocity of a material system may approach but can never reach (MVLR). It should be pointed out that this theorem may also be proved directly by means of theorem IX. 23. TIME AS A FOURTH DIMENSION. I have no intention of asserting that time is a fourth dimen- sion of space in the sense in which we ordinarily employ the word " dimension"; such a statement would have no meaning. I wish to point out rather that it is in some measure connected with space, and that in many formulae it must enter as it would if it were essentially and only a fourth dimension. We shall see this readily if we examine the formulae (^4 )/-' of transformation from one system of reference to another. Here the time variable / enters in a way precisely analogous to that in which the space variables x, y, z enter. Suppose now that the law of some phenomenon as observed on S f is given by the equation F(x', y', z'. t' )=o and we desire to know the expression of this law on S. We substitute for x f , y f , z' ', /' their values in terms of x, y, z, t given in (A); and thus we obtain an equation stating the law in question. From these considerations it appears that in many of our problems, namely in those which have to do at once with two or more systems of reference, the time and space variables taken together play the role of four variables each having to do with one dimension of a four-dimensional continuum. This conclusion raises philosophical questions of profound importance concerning the nature of space and time; but into these we cannot enter here. CHAPTER V. MASS AND ENERGY. 24. DEPENDENCE OF MASS ON VELOCITY. SUPPOSE that we have two systems of reference Si and $2 moving with a relative velocity v. We inquire as to whether, and in what way, the mass of a body as measured on the two systems depends on v. Will a given body have the same measure of mass when that mass is estimated in units of Si and in units of 52? And will the mass of a body depend on the direc- tion of its motion by means of which that mass is measured? Our purpose in this section is to answer these two questions. The two most important directions in which to measure the mass of a body are, first, that perpendicular to the line of relative motion of Si and 2, and, secondly, that parallel to this line of motion. For convenience in distinguishing these we shall speak of the " transverse mass " of a body as that with which we have to deal when we are concerned with the motion of the body in a direction perpendicular to the line of relative motion of Si and 6*2; when the motion is parallel to this line we shall speak of the " longitudinal mass " of the body. Lewis and Tolman (Phil. Mag. 18: 510-523) determine what they call the " mass of a body in motion," employing for this purpose a very simple and elegant method. This " mass " is what we have just defined as the transverse mass of the body. We employ the excellent method of these authors in deriving the formula for transverse mass. Suppose that an experimenter A on the system 6*1 constructs a ball Bi of some rigid elastic material, with unit volume, and 49 50 THE THEORY OF RELATIVITY. puts it in motion with unit velocity in a direction perpendicular to the line of relative motion of 6*1 and 52, the units of measure- ment employed being those belonging to Si. Likewise sup- pose that an experimenter C on 52 constructs a ball #2 of the same material, also of unit volume, and puts it in motion with unit velocity in a direction perpendicular to the line of relative motion of Si and 52; we suppose that the measurements made by C are with units belonging to 52. Assume that the exper- iment has been so planned that the balls will collide and rebound over their original paths, the path of each ball being thought of as relative to the system to which it belongs. Now the relation of the ball 2 to the system 5i is the same as that of the ball BI to the system 52, on account of the perfect symmetry which exists between the two systems of reference in accordance with previous results. Therefore the change of velocity of 2 relative to its starting point on 52 as measured by A is equal to the change of velocity of BI relative to its starting point on 5i as measured by C. Now velocity is equal to the ratio of distance to time : and in the direction perpendicular to the line of relative motion of the two systems the units of length are equal; but the units of time are unequal. Hence to either of the observers the change of velocity of the two balls, each with respect to its starting point on its own system, will appear to be unequal. To A the time unit on 52 appears to be longer than his own in the ratio i : Vi g 2 (see theorem IV). Hence to A it must appear that the change in velocity of 2 relative to its starting point is smaller than that of BI relative to its starting point in the ratio Vi g 2 : i. But the change in velocity of each ball multiplied by its mass gives its change in momentum. From postulate Ci it follows that these two changes of momentum are equal. Hence to A it appears that the mass of the ball BI is smaller than that of the ball B 2 in the ratio Vi g 2 : i. Similarly, it may be shown that to C it appears that the mass of the ball B% is smaller than that of BI in the ratio MASS AND ENERGY. 51 From our general results concerning the measurement of length it follows that if the baH which has been constructed by A were transferred to C's system it would be impossible for C to distinguish A 's ball from his own by any considerations of shape and size. Likewise, as A looks at them from his own system he is similarly unable to distinguish them. It is there- fore natural to take the mass of C's ball as that which A's would have if it had the velocity v with respect to Si of the system 2. Thus we obtain a relation existing between the mass of a body in motion and at rest. Now, " mass " as we have measured it above is the trans- verse mass of our definition. From the argument just carried out we are forced to conclude that the transverse mass of a body in motion depends (in a certain definite way) on the velocity of that motion. The result may be formulated as follows: THEOREM XI. Let mo denote the mass of a body when at rest relative to a system of reference S. When it is moving with a velocity v relative to S denote by t(m v ) its transverse mass, that is, its mass in a direction perpendicular to its line of motion. Then we have where $=v/c and c is the velocity of light (MVLRCi). In the statement of this theorem we have tacitly assumed that the mass of a body at rest relative to S, when measured by means of units belonging to 5, is independent of the direc- tion in which it is measured. If this assumption were not true we should have a means of detecting the motion of S, a conclusion which is in contradiction to postulate M. In order to find the longitudinal mass of a moving body we first find the relation which exists between longitudinal mass and transverse mass. We employ for this purpose the elegant method of Bumstead (Am. Journ. Science (4) 26: 498-500). Let us as usual consider two systems of reference Si and 6*2 moving with a relative velocity v, observers A and B being 52 THE THEORY OF RELATIVITY. stationed on Si and 62 respectively. Suppose that B per- forms the following experiment: He takes a rod of two units length, whose mass is so small as to be negligible, and attaches to its ends two balls of equal mass. Then he suspends this rod by a wire so as to form a torsion pendulum. We assume that the line of relative motion of the two systems is perpendic- ular to the line of this wire. Let us consider the period of this torsion pendulum in the two cases when the rod is clamped to the wire so as to be in equilibrium in each of the following two positions: (i) With its length perpendicular to the line of relative motion of Si and 52; (2) with its length parallel to this line of motion. As B observes it the period must be the same in the two cases; for, otherwise, he would have a means of detecting his motion by observations made on his system alone, contrary to postulate M. Then from the relation of time units on Si and 6*2 it follows that the two periods will also appear the same to A. As observed by B the apparent mass of the balls is the same in both cases. We inquire as to how they appear to A. Let mi and mi be the apparent masses, as observed by A, in the first and second cases respectively. It is obvious that m\ is the longitudinal mass and W2 the transverse mass of the balls in question. When the pendulum is in motion it appears to B that each ball traces a circular arc. From the relations between the units of length in the two systems it follows that to A it appears that the balls trace arcs of an ellipse whose semiaxes are i and Vi @ 2 and lie perpendicular and parallel, respectively, to the line of relative motion of the two systems. Let us now determine the period of each of these two pendulums as they are observed by A . By equating the expres- sions for these periods we shall find the relation which exists between mi and mz. Let x and y be the cartesian coordinates of a point as deter- mined by A. the axes of reference being the major and minor axes of the ellipse in which the balls move. Let x' and y be MASS AND ENERGY. 53 the coordinates of the same point as determined by B. Then the circular path of motion, as determined by B, has the equations #' = cos 6, y = sin 6, the angle 6 being measured from the major axis of the ellipse. The equations of the ellipse, as determined by A , are x = cos 0, y = Vi 2 sin 0. In the first case when the rod is perpendicular to the line of relative motion of Si and 2 the amount of twisting in the wire when the ball is in a given position is the numerical value of the corresponding angle 0; and therefore the potential energy * is proportional to 2 , say that it is ^&0 2 . Now from the values of y and x above we have y = xi-$ 2 tan 0. For small oscillations we have x = i and tan = 0; and therefore Hence the potential energy is and the equation of motion of the particle becomes d?y_ k Hence the period T\ of oscillation is TI-: * That the potential energy is proportional to O 2 when measured by B is obvious. Since A observes a different apparent angle 6' (say) corresponding to 's observed angle 6 it might at first sight appear that the potential energy as observed by A is proportional to 6' 2 ; that this is not the case is seen from the fact that for a given twist in the wire 6' depends on the direction of equilibrium of the bar, that is, it depends on the way in which the bar is attached to the wire; hence, if the potential energy as observed by A were proportional to O' 2 , it would depend on the way in which the bar is attached. Since this is obviously not the case we conclude that the potential energy is proportional to 6 2 . 54 THE THEORY OF RELATIVITY. In the second case when the rod is parallel to the line of relative motion of Si and 62 the amount of twisting in the wire for a given position of the balls is the numerical value of 0. The potential energy is | &( j . We have x= . cot 0. Vl-g2 For small oscillations we have cot = tan{ -6)= -0. 2 2 Hence the potential energy is \kx 2 , and the period T^ of oscillation is therefore Equating the two periods of oscillation found above we have Remembering that m\ and m