# On the Theory of Relativity: Analysis of the Postulates

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Physical Review (1913), 35 (3), 153-176, Online
ON THE THEORY OF RELATIVITY: ANALYSIS OF THE POSTULATES.
By R. D. Carmichael.

## Introduction.Introduction.

This analysis of the postulates of relativity was undertaken in order to ascertain on just which of the postulates certain fundamental conclusions of the theory depend. A moment's reflection will convince one of the importance of such an analysis. Some of the conclusions of relativity have been attacked by those who admit just the parts of the postulates from which the conclusions objected to can be derived by purely logical processes. In this paper I have sought to establish some of the most fundamental and most readily accessible conclusions of the theory on the smallest possible foundation from the postulates. This plan of treatment, instead of giving rise to more complicated arguments than those hitherto usually employed, has had the opposite effect of leading to increased simplicity both in the notions which enter and in the arguments by which proofs are reached.
When the work was taken in hand it soon became evident that there was something to be done both on the postulates themselves and on the very first theorems which are to be deduced from them, as the reader will see by reference to the treatment below. It thus appears that some of the most striking conclusions of the theory depend on only a part of the postulates. To bring this fact prominently into view in one's mind is to put the whole subject in a clearer light where one may see better the interactions of its parts and its general relations to the whole body of scientific and philosophical knowledge.
​A certain method has been consistently employed throughout the paper to indicate the postulates on which each theorem depends. Each postulate is designated by a letter. At the end of a theorem and enclosed in parentheses are references (by means of these letters) to the postulates on which the theorem as demonstrated depends. Thus theorem I. depends on postulates M and $R'$.
In carrying out the initial purpose of the paper an important part of the general fundamentals of the theory of relativity come in for a fresh development along lines more or less new. It was observed that the addition to this essential matter of a relatively small amount of material would make the paper as a whole serve as an elementary introduction to the entire theory; and consequently such matter as was necessary to this end has been incorporated. I felt the more disposed to do this in view of the fact that even this additional material is presented in a somewhat novel manner. It is believed that the paper in its present form may be read profitably by one who is making his first acquaintance with the theory and that it will afford an easier introduction than any which has yet been offered.
In every body of doctrine which consists of a finite number of postulates and their logical consequences there are necessarily certain theorems which have the following fundamental relation to the whole body of doctrine: By means of one of these theorems and all the postulates but one that remaining postulate may be demonstrated. That is, one may assume such a theorem in place of one of the postulates and then demonstrate that postulate. When the postulate has thus been proved it may be used in argument as well as the theorem itself; hence it is clear that all the consequences which were obtained from the first set of postulates may now be deduced again, though perhaps in a somewhat different manner. That is, if we consider the whole body of doctrine, composed of postulates and theorems, this totality is the same in the two cases. Two sets of postulates which thus give rise to the same body of doctrine (consisting of postulates and theorems together) are said to be logically equivalent.
The problem of the logical equivalents of a given set of postulates is readily seen to be an important one. Not all sets of postulates logically equivalent to a given one are equally interesting. In fact, some sets may be cumbersome in form and unsatisfactory from an aesthetic point of view. In the present analysis of the postulates of relativity attention has been given to determining some of their important logical equivalents — especial attention being given to those postulates which may replace the so-called ​second postulate of relativity (our postulate R). A remark in this direction has already been made by Tolman. An indication of the results of this character obtained in the present paper is given below in the description of the contents of part II. The principal value of such a matter, from the point of view of physical science, consists in the fact that it affords alternative methods for the experimental proof or disproof of the theory of relativity and that it emphasizes in an effective way the essential difficulties and limitations of such experimental verification in general.
It is the writer's purpose to treat further in a future paper the matter of the logical equivalents of the postulates. In this forthcoming work it is the intention to introduce the general laws of conservation of energy, mass, momentum and electricity, to deduce certain joint consequences of these laws and the principle of relativity, and to determine which of the theorems so obtained may replace the relativity postulates (or one of them) without destroying the equivalence of the resulting totality of doctrine.
Part I. of the present paper is devoted to a general statement and preliminary analysis of the postulates of relativity. In § 1 I give the fundamental homogeneity postulates of space and time which underlie all physical theory. In § 2 the first characteristic postulate of relativity is given, while § 3 contains a statement of the second postulate. It is shown that a part of this postulate (in the form in which the postulate is usually stated) is a consequence of the other part and of the first postulate, together with the fundamental homogeneity postulates.
Writers on relativity have usually stated only these two postulates. But as a matter of fact every one has made further assumptions; in some cases it appears to have been done unconsciously. To the present writer it seems desirable that these assumptions should be brought into the light as postulates. Accordingly, in § 4, 1 give those additional postulates which I shall use. One familiar with the theory will see that these assumptions are different from those usually employed (without explicit statement as postulates), as by Einstein for instance. The choice has been made in the interest of simplicity in postulates and in proofs. The writer believes that this innovation in the statement of the postulates is important. It leads to new and simpler proofs than those ordinarily employed; and this, it is hoped, will in large measure remove the feeling of vagueness which many persons experience in approaching the theory of relativity for the first time.
​Further remarks (of a general nature) on the postulates are added in §§ 5 and 6.
In part II. I give a discussion of the relative measurements of space and time in two systems of reference which move with respect to each other and obtain Einstein's formulae of transformation. In §§ 7 and 8 it is shown that the most remarkable part of the conclusions of relativity concerning the time and space units is due to a part of the second postulate along with the other postulates; compare theorems III. and IV. In § 9 I treat the question of simultaneity of events happening at different places. In §§ 10 and 11 I obtain Einstein's formulae of transformation from one system of reference to another and also the addition theorem of velocities. These results are applied in § 12 to the problem of finding logical equivalents for the postulate R.

## I. The Postulates of Relativity.I. The Postulates of Relativity.

§ I. Postulates of Homogeneity. — 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 other instant. For our present purpose these postulates can best be given the following more exact and complete statement.
Postulate $H_{1}$. Space is homogeneous and three-dimensional.
Postulate $H_{2}$. Time is homogeneous and one-dimensional.
One important meaning of these postulates, mathematically, 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, those concerning space depending on $H_{1}$ and those concerning time depending on $H_{2}$, Moreover, it is certain that no one will be disposed seriously to call these postulates in question. On account of these facts we shall consider it unnecessary to give any explicit reference to these postulates as part of the basis on which any particular theorem depends, it being understood once for all that they underlie all our work.
§ 2. The First Characteristic Postulate. — 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 this ether is stationary. Experimental facts, which have to be accounted for, cannot be explained satisfactorily on the hypothesis of a mobile ether. The theory of a stationary ether leads naturally to the conclusion that it would be possible for an observer to detect and measure ​his absolute motion with respect to the ether. In this way 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 the direction of motion.
But a classical experiment of Michelson and Morley, in which the ray-path was wholly in air, put this prediction to a crucial test; and not the slightest difference of time was found in the passage of light along the two paths. The extreme precision of their methods leaves no doubt as to the accuracy of the results.
In a similar manner the theory of a stationary ether gave rise to the prediction that a charged condenser suspended by a wire would exhibit a torsional effect due to the earth's motion. This prediction was tested in the crucial experiment of Trouton and Noble with the result of showing that no such torsional effect is present.
These results are in perfect agreement with the hypothesis that it is impossible to detect absolute translatory motion through space; and as a matter of fact they have been generalized into this hypothesis. A sharp formulation of this conclusion constitutes the first characteristic postulate of relativity.
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 paper 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 $S_{1},S_{2},\dots$ Furthermore, it will be assumed that the units of any two systems $S_{1}$ and $S_{2}$ are such that the same numerical result will be obtained in measuring with the units of $S_{1}$ a quantity $L_{1}$ and with the units of $S_{2}$ a quantity $L_{1}$ when the relation of $L_{1}$ to $S_{1}$ is precisely the same as that of $L_{2}$ to $S_{2}$.
​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 detect absolute motion or motion through any sort of medium which may be assumed to pervade space. Furthermore, in every case where the heretofore accepted theory has 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 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 in the present state of knowledge, the experimental evidence for the postulate should be considered of strong character.
One additional remark should be made here. The direct experimental evidence which led to the formulation of postulate M was undertaken on account of predictions made on the basis of a theory of the ether as the vehicle of light and electricity. But the result which has been obtained is of a purely experimental character and does not in any way depend on a theory of the ether. In other words, the law stated in postulate M is in no way dependent either on the existence or the non-existence of the ether. It is important to keep this in mind on account of the confusion of thought which has arisen in some quarters as to the relation between the theory of relativity and the theory of the ether. The postulate is simply a generalization of experimental fact; and, unless an experiment can be devised to show that this generalization is not legitimate, it is natural and in accordance with the usual procedure in physics to accept it as a "law of nature." Theory, then, must be made to agree with it and not it with theory.
§ 3. 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 inconsiderable part of the difficulty which has been felt concerning 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 objection 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 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. In other words the velocity with which light passes an observer is not increased if the light-source moves toward the observer or decreased if it recedes from him. Stated in exact language this postulate is as follows:
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 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 experimental verification or disproof, and this should be made in such 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 experimental evidence which by some is believed 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, on the other hand, has examined the same experiments and has found an explanation 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 two 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 postulate $R'$. 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 $R'$ 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 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 of space mere direction through space will have no effect on these observed velocities; and therefore it 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 S. But by M it is impossible 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 prove that this observed velocity of light is independent of the absolute value of the velocity of S, as Tolman appears to conclude. To see this clearly, let us suppose that the absolute value of the velocity of S does affect the observed velocity of light. On account of $R'$ 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 phenomenon on another; namely, a datum in which the two phenomena 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 his 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 may in which it could be present. Physical intuition is emphatic (and it may be that this is what Tolman intended to say in the passage cited) in asserting that if the direction of the velocity of S has no effect on the observed velocity of light then the absolute value of the velocity 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 unquestioned 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 belonging to S, is independent of the absolute 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 absolute value of the velocity and the direction of 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 belonging 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 assumption 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 necessary consequence of these postulates.
Postulate ${\overline {R}}$. 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 ​already 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 $R'$, 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 connection with theorems to be derived later (in this paper and in another which the writer has in preparation) further reference will 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 flow from the theory of relativity are due to postulate R (or to postulate ${\overline {R}}$, in the customary formulation). In view of the theorem I. above and our 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 postulate and especially to know alternative forms, which, in view of the other postulates, are logically equivalent to it. One such form has already been given by Tolman (l. c., p. 36), who has also urged the importance of the general problem. In the second paper of this series the alternative form due to Tolman will be subjected to a fresh analysis. As already pointed out in the Introduction, other alternative forms will also be given.
§ 4. 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 the several forms, any one of which might be chosen, there is one which seems to us to be decidedly simpler than any of the others; and it is this one which we shall employ here. We state the postulates V and L as follows:
Postulate V. If the velocity of a system of reference $S_{2}$ relative to a system of reference $S_{1}$ is measured by means of the units belonging to $S_{1}$ and if the velocity of $S_{1}$ relative to $S_{2}$ is measured by means of the units belonging to $S_{2}$ the two results will agree in absolute 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 $S_{1}$ and $S_{2}$ move with unaccelerated relative velocity and if a line segment l is perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$ and is fixed to one of these systems, then the length of l measured by means of the units belonging to $S_{1}$ will be the same as its length measured by means of the units belonging to $S_{2}$.
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 supposing 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 relation 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 part II. 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 this the question arises as to why we must assume that length in a line perpendicular to the motion is independent of the motion. The answer is that we are under no such necessity, that we are at liberty to assume that length in a line perpendicular to the 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 hypothesis, 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 experimental 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 consider the question of introducing a more complicated postulate in place of our postulate L above.
§ 5. Consistency and Independence of the Postulates. — Throughout the paper it will be assumed that the postulates as stated 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 the postulates 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 one 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 of formal logic, but in the present case its value to physical science is probably small.
§ 6. 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. If, for instance, it is desired to study the nature of mass or the relation of mass and energy in this theory, 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 mass, energy, electricity, and momentum and deduce the joint consequences of these assumptions and those given above. It is our purpose to return to this matter in a future paper. For the present we are concerned only with the postulates above stated and their consequences.

## II. Relative Measurements of Time and Space in Two Systems of Reference. Transformations.II. Relative Measurements of Time and Space in Two Systems of Reference. Transformations.

§ 7. Relations Between the Time Units. — Let us consider three systems of reference $S,S_{1}$ and $S_{2}$ related to each other in the following manner: The lines of relative motion of S and $S_{1}$, of S and $S_{2}$, of $S_{1}$ and $S_{2}$ are all ​parallel; $S_{1}$ and $S_{2}$ have a relative velocity v; S and $S_{1}$ have a relative velocity ½v in one sense and S and $S_{2}$ have a relative velocity ½v in the opposite sense. The system S consists of a single light-source, and this source is symmetrically placed with respect to two points of which one is fixed to $S_{1}$ and the other is fixed to $S_{2}$. 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 $S_{1}$ and $S_{2}$ measure the velocity of light as it emanates from the source S. Let a point A in $S_{1}$ and a point B in $S_{2}$ which are symmetrically placed with respect to the light-source S, move along the lines $l_{1}$ and $l_{2}$; these lines are parallel. From postulate L it follows that observers on $S_{1}$ and $S_{2}$ will obtain the same measurement of the distance between $l_{1}$ and $l_{2}$. Denote this distance by d. On account of postulate M neither observer is able to detect his motion. Therefore he will make his observations on the assumption that his system is at rest; that is to say, his measurements will be made by means of the units belonging to his system and no corrections will be made on account of the motion of the system. Let the observer on $S_{1}$ reflect a beam of light SA from the point A to a point C on $l_{w}$ and back to A; and let the observed time of passage of the light from A to C and back to A be t. 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 perpendicular to $l_{1}$ and $l_{2}$. His measurement of the distance traversed by the ray of light in time t will therefore be 2d. Hence he will obtain as a result
${\frac {2d}{t}}=c$,
where c is his observed velocity of light.
Similarly, an observer on $S_{2}$, supposing his system to be at rest, finds the time $t_{1}$ 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 the result
${\frac {2d}{t_{1}}}=c_{1}$,
where $c_{1}$ is his observed velocity of light.
​Now, from the assumed relations among the systems S, $S_{1}$ and $S_{2}$ 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 absolute value or the direction of the motion of the systems $S_{1}$ and $S_{2}$ is the same in the two cases. In other words, if we denote by $L_{1}$ and $L_{2}$ the quantities measured on $S_{1}$ and $S_{2}$ respectively, then the relation of $L_{1}$ to $S_{1}$ is precisely the same as that of $L_{2}$ to $S_{2}$; and hence the numerical results are identical, as one sees from the definition of systems of reference. Therefore we have $c_{1}=c$.
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; we shall assume that the apparent motion is in the direction indicated by the arrow. 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 $l_{2}$ and $d_{1}$ is the length of EF as measured by means of units belonging to $S_{1}$, then, evidently, GF (when measured in the same units) is $\beta d_{1}$ where $\beta =v/{\overline {c}}$ and ${\overline {c}}$ 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
$d_{1}={\frac {d}{\sqrt {1-\beta ^{2}}}}$
Now, if ${\overline {t}}$ is the time which is required, according to the observer at A, for the light to traverse the path BEF, then we have
${\frac {2d_{1}}{\overline {t}}}={\frac {2d}{{\overline {t}}{\sqrt {1-\beta ^{2}}}}}={\overline {c}}$
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 ${\overline {c}}$ of the velocity of light obtained as above by the observer at A are equal.
Now we have shown that c is equal to $c_{1}$, Hence we may equate the values of $c_{1}$ and ${\overline {c}}$ given above; thus we have
${\frac {2d_{1}}{t_{1}}}={\frac {2d}{{\overline {t}}{\sqrt {1-\beta ^{2}}}}}$
$t_{1}={\overline {t}}{\sqrt {1-\beta ^{2}}}$
But $t_{1}$ and ${\overline {t}}$ are measures of the same interval of time, $t_{1}$ being in units belonging to $S_{2}$ and ${\overline {t}}$ being in units belonging to $S_{1}$. Hence to the observer on $S_{1}$ the ratio of his time unit to that of the system $S_{2}$ appears to be ${\sqrt {1-\beta ^{2}}}:1$. On the other hand, it may be shown in exactly the same way that to the observer on $S_{2}$ the ratio of his time unit to that of the system $S_{1}$ appears to be ${\sqrt {1-\beta ^{2}}}:1$. 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 $S_{1}$ and $S_{2}$ move with a relative velocity v and β is defined as the ratio of v to the velocity of light estimated in the manner indicated above, then to an observer on $S_{1}$ the time unit of $S_{3}$ appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{2}$ while to an observer on $S_{2}$ the time unit of $S_{2}$ appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{1}$ (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 ${\overline {c}}$ as estimated above differ at all, that difference can be due only to the direction of motion of $S_{1}$, as one sees readily from postulate $R'$ and the method of determining these quantities. Hence the statement which we made above as assumption A is a, logical consequence of postulates M and $R'$. Therefore 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 $S_{1}$ and $S_{2}$ move with a relative velocity v and β is the ratio of v to the velocity of light, then to an observer on $S_{1}$ the time unit of $S_{1}$ appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{2}$ while to an observer on $S_{2}$ the time unit of $S_{2}$ appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{1}$ (MVLR).
Let us subject these remarkable results to a further analysis. Theorem III., its corollary and theorem IV. all agree in the extraordinary conclusions ​that the time units of the two systems of reference $S_{1}$ and $S_{2}$ 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 involve the strange features of our results. Hence the conclusion is irresistible that the extraordinary element in these results is due to postulate $R'$ — 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 universally 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 the 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 conclusions of the theory of relativity are deducible from postulates M, V, L, $R'$ alone; and yet these are either generalizations from experiment or statement 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 experiments 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 system is moving past him with a 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 view of 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.
§ 8. Relation Between the Units of Length. — Let us consider three systems of reference S, $S_{1}$ and $S_{2}$ related in the same manner as in the preceding section except that now the two lines $l_{1}$ and $l_{2}$ coincide. We suppose that $S_{1}$ is moving in the direction indicated by the arrow at A and that $S_{2}$ 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 S, this time in the direction of the line of motion. Each will carry out his observations on the supposition that his system is at rest, for from M it follows that he cannot detect the motion of his system. The observer at A measures the time $t_{1}$ of passage of a ray of light from A to C and return to A, the length of AC being d when the measurement is made with a unit belonging to $S_{1}$. Likewise, the observer at B measures the time $t_{2}$ of passage of a ray of light from B to D and return to B, the length of BD being d when measured with a unit belonging to $S_{2}$.
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 $2d/t_{1}$ while that of the observer at B is $2d/t_{2}$. Hence
$t_{1}=t_{2}$
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 $S_{1}$ and the latter on $S_{3}$. 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 magnitude. 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 $S_{1}$ and $S_{2}$ move with a relative ​velocity v and β is defined as the ratio of v to the velocity of light estimated in the manner indicated in the first part of § 7, then to an observer on $S_{1}$ the unit of length of $S_{1}$ along the line of relative motion appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{2}$ while to an observer on $S_{3}$ the unit of length of $S_{2}$ along the line of relative motion appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{1}$ (MVLA).
Corollary. Theorem V. may be stated as depending on (MVLR') instead of on {MVLA).
Theorem VI. If two systems of reference $S_{1}$ and $S_{2}$ move with a relative velocity v and if β is the ratio of v to the velocity of light, then to an observer on $S_{1}$ the unit of length of $S_{1}$ along the line of relative motion appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{2}$ while to an observer on $S_{2}$ the unit of length of $S_{2}$ along the line of relative motion appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{1}$ (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 assumptions which led to the strange results as to the units of time.
§ 9. Simultaneity of Events Happening at Different Places. — Let us now assume two systems of reference S and $S'$ 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 indicate the same time. 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 $S'$ 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 has assistants at A and B. A single light signal is flashed from 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 beforehand. The observer on $S'$ 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.
$S'$
${\frac {1}{2}}d{\sqrt {1-\beta ^{2}}}$
instead of ½d. Moreover, since the velocity of light is independent 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 light 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
${\frac {{\frac {1}{2}}d{\sqrt {1-\beta ^{2}}}}{c-v}}-{\frac {{\frac {1}{2}}d{\sqrt {1-\beta ^{2}}}}{c+v}}={\frac {vd{\sqrt {1-\beta ^{2}}}}{c^{2}-v^{2}}}$
But since $\beta =v/c$ the last expression is readily found to be equal to
${\frac {v}{c^{2}}}\cdot {\frac {d}{\sqrt {1-\beta ^{2}}}}$.
Therefore, to an observer on S the clocks on $S'$ 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' have a uniform relative velocity v. Let an observer on S' place two clocks at a distance d apart in the line of relative motion of S and S' and adjust them so that they appear to him to mark simultaneously the same time. 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
${\frac {v}{c^{2}}}\cdot {\frac {d}{\sqrt {1-\beta ^{2}}}}$,
where c is the velocity of light and $\beta =v/c$ (MVLR).
It should be emphasized that the clocks on $S'$ are in agreement 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 simultaneity is to be determined. In other words, there is no such thing as the absolute simultaneity of events happening at different places.
§ 10. 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 S and $S'$ have the relative velocity v in the line l. Let systems of rectangular coordinates be attached to the systems of reference S and $S'$ in such way that the x-axis of each system is in the line l, and let the y-axis and the z-axis of one system be parallel to the y-axis and the z-axis respectively of the other system. Let the origins of the two systems coincide at the time t = 0. Furthermore, for the sake of distinction, denote the coordinates on S by x, y, z, t and those on $S'$ by x', y', z', t'. We require to find the value of the latter coordinates in terms of the former.
From postulate L it follows at once that y' = y and z' = z. Let an observer on S consider a point which at the time t = 0 appears to him to be at distance x from the $y'z'$-plane; at time t = t 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
$x'{\sqrt {1-\beta ^{2}}}=x-vt$
Now consider a point at the distance x from the yz-plane at time t = t in units of system S. From theorem VII. it follows 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
${\frac {v}{c^{2}}}x$,
where c is the velocity of light. That is, in units of S this clock would register the time
$t-{\frac {v}{c^{2}}}x$.
Hence, by means of theorem IV., we have at once the result
$t'{\sqrt {1-\beta ^{2}}}=t-{\frac {v}{c^{2}}}x$.
Solving the two equations involving $x'$ and $t'$ and collecting results, we have
| (A) | ${\begin{array}{llc}t'=&{\frac {1}{\sqrt {1-\beta ^{2}}}}\left(t-{\frac {v}{c^{2}}}x\right),\\\\x'=&{\frac {1}{\sqrt {1-\beta ^{2}}}}(x-vt),&(MVLR)\\\\y'=&y\\\\z'=&z,\end{array}}$ |
where $\beta =v/c$ and c is the velocity of light.
In the same way we may obtain the equations which express t, x, y, z ​in terms of t', x', y', z'. But these can be found more easily by solving equations (A) for t', x', y', z'. Thus we have
| $(A_{1})$ | ${\begin{array}{llc}t=&{\frac {1}{\sqrt {1-\beta ^{2}}}}\left(t'+{\frac {v}{c^{2}}}x'\right),\\x=&{\frac {1}{\sqrt {1-\beta ^{2}}}}(x'+vt'),&(MVLR)\\y=&y'\\z=&z',\end{array}}$ |
These two sets of equations (A) and ($A_{1}$) are identical in form except for the sign of v. This symmetry in the transformations constitutes one of their chief points of interest.
Our method of proof of these formulae is very different from that of Einstein, as a comparison will readily show. The difference is due primarily to our use of postulates V and L instead of the assumptions of Einstein.
In a paper entitled "The Common Sense of Relativity" Campbell has made some interesting remarks concerning these transformations.
§ II. The Addition of Velocities. — For the sake of completeness in the presentation of the fundamental results of relativity and for use in the next section we derive here the formulae for addition of velocities due to Einstein.
Let the velocity of a point in motion be represented in units belonging to $S'$ and to $S$ by means of the equations
${\begin{array}{lll}x'=u_{x'}t,&y'=u_{y'}t,&z'=u_{z'}t,\\x=u_{x}t,&y=u_{y}t,&z=u_{z}t,\end{array}}$
respectively. In the first of these substitute for $t',x',y',z'$ their values given by (A), 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) | ${\begin{array}{lc}u_{x}={\frac {u_{x'}+v}{1+{\frac {vu_{x'}}{c^{2}}}}}\\\\u_{y}={\frac {\sqrt {1-\beta ^{2}}}{1+{\frac {vu_{x'}}{c^{2}}}}}u_{y'},&(MVLR)\\\\u_{z}={\frac {\sqrt {1-\beta ^{2}}}{1+{\frac {vu_{x'}}{c^{2}}}}}u_{z'}\end{array}}$ |
Suppose that both the velocities considered above are in the line of relative motion of S and $S'$. Then we have
$u={\frac {v+u}{1+{\frac {vu'}{c^{2}}}}}$
This equation gives rise to the following theorem:
Theorem VIII. If two velocities, each of which is less than c, are combined the resultant velocity is also less than c (MVLR).
To prove this we substitute in the preceding equation for v and $u'$ the values
$v=c-k,\ u'=c-l,$
where each of the numbers k and l is positive and less than c. Then the equation becomes
$u=c{\frac {2c-k-l}{2c-k-l+{\frac {kl}{c}}}}.$
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 l (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).
Remark. — A conclusion of importance is implicitly involved in the results obtained in §§ 7-11. 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 β is in absolute value greater than 1 — whence $1-\beta ^{2}$ 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 absolute value of β is equal to or less than 1. If it is one, then any length in one system, however short, would be measured in the other as infinite; and a like result holds for time. Hence β is less than 1. But $\beta =v/c$, the ratio of the relative velocity of the two systems to that of light. Hence, the velocity of light is a maximum ​which the relative velocity of two systems may approach but can never reach. This may be formulated in the following theorem:
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 from theorem IX, as one can readily show. This fact will be useful in the next section.
§12. Logical Equivalents of the Postulates. — We shall now show that theorem IX. is in a certain sense a logical equivalent of R. From IX. it follows, as we have seen in theorem X., that the velocity of a material body is less than the velocity of light. But the source of light is always a material body; and therefore no light source can have a velocity as great as that of light. Now, the following is a natural hypothesis:
Postulate B. The velocity of the light source cannot add to the velocity of light a greater velocity than that of the source itself. Likewise, the velocity of a system of reference cannot add to the velocity of light a greater velocity than that of the source itself.
Now, from theorem IX. it follows that if any velocity less than that of light is compounded with that of light the resultant is the velocity of light. Hence, if we assume theorem IX. and postulate B we can conclude as a consequence postulates $R'$ and $R''$. Hence we have the following result:
Theorem XI. Postulates (MVLB) and theorem IX. are a logical equivalent of postulates (MVLR).
That is, in our system of postulates R may be replaced by theorem IX. and postulate B, and the resulting total body of postulates and theorems will be unaltered.
Now theorem IX. was proved by means of formulae (B) alone; and formulae (B) are a direct consequence of formulae (A) above. Hence postulate R may be proved solely from postulate B and formulae (A) if these are assumed to be true. Further, the third and fourth equations in (A) are equivalent to postulate L. We shall show that a special case of the first two formulae (A) is postulate V. Putting x' = 0 we have x/t = v. That is, to an observer on S the system $S'$ appears to move with the velocity v. Now the first two equations in ($A_{1}$) may be obtained algebraically by solving the first two in (A) . Then in the second equation of ($A_{1}$) put x = 0; thus we have x't' = -v. That is, to an observer on $S'$ the system S appears to move with the velocity -v. These two results together constitute our postulate V. Combining the several conclusions thus reached we have the following theorem:
​Theorem XII. Postulates (MB) and formulae (A) are a logical equivalent of postulates (MVLR).
An analysis of the proof of formulae (A) will show that they follow directly from theorems IV., VI., VII. and postulate V. Hence we have the following theorem as a corollary of the preceding:
Theorem XIII. Postulates (MVB) and theorems IV., VI., VII. are a logical equivalent of postulates (MVLR).
    Indiana University,
       June, 1912.
* ↑ In the theory of relativity the word "postulate" has been used in the sense in which one is accustomed to employ the term "law of nature."
* ↑ This method has been employed by Veblen and Young in their Projective Geometry, 1910.
* ↑ Physical Review, 31 (1910): 26-40.
* ↑ Jahrbuch der Radioaktivität, 4 (1907): 411-462. See the assumptions stated in a footnote on p. 420.
* ↑ See. however, a presentation of the opposing view by H. A. Wilson, Phil. Mag., 19 (1910); 809-817.
* ↑ Am. Joum. Science (3). 34 (1887): 333-345.
* ↑ Phil. Trans. Roy. Soc. (A). 202 (1904): 165.
* ↑ As, for instance, charges, magnets, light-sources, telescopes, etc.
* ↑ 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.
* ↑ Reference here is to the Michelson and Morley experiment.
* ↑ See the postulate ${\overline {R}}$ below and the remarks which lead up to it.
* ↑ Physical Review. 31 (1910): 26-40.
* ↑ Physical Review, 33 (1911): 418-438.
* ↑ Letters attached to a theorem in this way indicate those of the postulates (exclusive of $H_{1}$ and $H_{2}$) on which the theorem depends. See Introduction and §1.
* ↑ Physical Review. 31 (1910), p. 27.
* ↑ This has already been pointed out by Tolman, l. c., pp. 27-28.
* ↑ Göttinger Nachrichten, Math. Phys., 1911. pp. 271-277.
* ↑ This hypothesis is in agreement with Einstein's theory of relativity.
* ↑ Note that postulate V is required to make this hypothesis legitimate.
* ↑ Compare Comstock, Science, N. S.. 31 (1900): 767-772.
* ↑ Jahrbuch der Radioaktivität, 4 (1907): 418-420.
* ↑ The algebraic sign of the distance is supposed to be taken into account in the value of x.
* ↑ Einstein, l. c. p. 420, footnote.
* ↑ Phil. Mag.. 21 (1911): 502-517; see esp. pp. 505-507.
* ↑ See remarks in the Introduction.
* ↑ Einstein, l. c, pp. 422-424.
* ↑ Addition is defined by saying that the sum of two velocities is the result of compounding them.
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