It is among the possibilities that the wave theory of light itself will in the end be more or less seriously modified. It is even more definitely among the possibilities that the ether will be discarded.]* [Certainly when Lord Kelvin estimates that its mass per cubic centimeter is .000,000,000,000,000,001 gram, while Sir Oliver Lodge insists that the correct figure is 1,000,000,000,000,000 grams, it is quite evident that we know so little about it that it is better to get along without it if we can.]²¹⁶ [But to avoid confusion we must emphasize that Einstein makes no mention whatsoever of the ether; his theory is absolutely independent of any theory of the ether.]¹³⁹ [Save as he forbids us to employ the ether as a standard of absolute motion, Einstein does not in the least care what qualities we assign to it, or whether we retain it at all. His demands are going to be made upon light itself, not upon the alleged medium of light transmission.

When two observers in relative motion to one another measure their velocities with respect to a third material object, they expect to get different results. Their velocities with regard to this object properly differ, for it is no more to be taken as a universal super-observer than either of them. But if they get different results when they come to measure the velocity with which light passes their respective systems, relativity is challenged. Light is with some propriety to be regarded as a universal observer; and if it will measure our velocities against each other we cannot deny it rank as an absolute standard. If we are not prepared to abandon universal relativity, and adopt one of the “fluke” explanations for the Michelson-Morley result, we must boldly postulate that in free space light presents the same velocity C to all observers—whatever the source of the light, whatever the relative motion between source and observer, whatever the relative motion between the several observers. The departure here from the old assumption lies in the circumstance that the old physics with its ether assigned to light a velocity universally constant in this ether; we have stopped talking about the medium and have made the constant C refer to the observer’s measured value of the velocity of light with regard to himself.

We are fortified in this assumption by the Michelson-Morley result and by all other observations bearing directly upon the matter. Nevertheless, as Mr. Francis says in his essay, we feel instinctively that space and time are not so constituted as to make it possible, if I pass you at 100 miles per hour, for the same light-impulse to pass us both at the same speed C.]* [The implicit assumptions underlying this feeling, be they true or false, are now so interwoven with the commonly received notions of space and time that any theory which questions them has all the appearance of a fantastic and unthinkable thing.]¹¹⁵ [We cannot, however, go back on our relativity; so when]* [Einstein shows us that an entirely new set of time and space concepts is necessary to reconcile universe relativity with this fundamental fact of the absolute constancy of the observed velocity of light in vacuo,]¹⁸ [all that is left for us to do is to inquire what revisions are necessary, and submit to them.]*

[The conceptual difficulties of the theory arise principally from attributing to space and time the properties of things. No portion of space can be compared with another, save by convention; it is things which we compare. No interval of time can be compared with another, save by convention. The first has gone when the second becomes “now”.]¹⁴⁹ [It is events that we compare, through the intervention of things. Our measurements are never of space or of time, but only of the things and the events that occupy space and time. And since the measurements which we deal with as though they were of space and of time lie at the foundation of all physical science, while at the same time themselves constituting, as we have seen, the only reality of which we are entitled to speak, it is in order to examine with the utmost care the assumptions underlying them. That there are such assumptions is clear—the very possibility of making measurements is itself an assumption, and every technique for carrying them out rests on an assumption. Let us inquire which of these it is that relativity asks us to revise.]*

[Time is generally conceived as perfectly uniform. How do we judge about it? What tells us that the second just elapsed is equal to the one following? By the very nature of time the superposition of its successive intervals is impossible. How then can we talk about the relative duration of these intervals? It is clear that any relationship between them can only be conventional.]¹⁷⁸ [As a matter of fact, we habitually measure time in terms of moving bodies. The simplest method is to agree that some entity moves with uniform velocity. It will be considered as travelling equal distances in equal intervals of time, the distances to be measured as may be specified by our assumptions governing this department of investigation.]¹⁷⁹ [The motions of the earth through which we ultimately define the length of day and year, the division of the former into 86,400 “equal” intervals as defined by the motions of pendulum or balance wheel through equal distances, are examples of this convention of time measurement. Even when we correct the motions of the earth, on the basis of what our clocks tell us of these motions, we are following this lead; the earth and the clocks fall out, it is plain that one of them does not satisfy our assumption of equal lengths in equal times, and we decide to believe the clock.]*

[The foregoing concerning time may be accepted as inherent in time itself. But concerning lengths it may be thought that we are able to verify absolutely their equality and especially their invariability. Let us have the audacity to verify this statement. We have two lengths, in the shape of two rods, which coincide perfectly when brought together. What may we conclude from this coincidence? Only that the two rods so considered have equal lengths at the same place in space and at the same moment. It may very well be that each rod has a different length at different locations in space and at different times; that their equality is purely a local matter. Such changes could never be detected if they affected all objects in the universe. We cannot even ascertain that both rods remain straight when we transport them to another location, for both can very well take the same curvature and we shall have no means of detecting it.

Euclidean geometry assumes that geometrical objects have sizes and shapes independent of position and of orientation in space, and equally invariable in time. But the properties thus presupposed are only conventional and in no way subject to direct verification. We cannot even ascertain space to be independent of time, because when comparing geometrical objects we have to conceive them as brought to the same place in space and in time.]¹⁷⁸ [Even the statement that when they are made to coincide their lengths are equal is, after all, itself an assumption inherent in our ideas of what constitutes length. And certainly the notion that we can shift them from place to place and from moment to moment, for purposes of comparison, is an assumption; even Euclid, loose as he was from modern standards in this business of “axioms,” knew this and included a superposition axiom among his assumptions.

As a matter of fact, this procedure for determining equality of lengths is not always available. It assumes, it will be noted, that we have free access to the object which is to be measured—which is to say, it assumes that this object is at rest with respect to us. If it is not so at rest, we must employ at least a modification of this method; a modification that will in some manner involve the sending of signals. Even when we employ the Euclidean method of superposition directly, we must be assured that the respective ends of the lengths under comparison coincide at the same time. The observer cannot be present at both ends simultaneously; at best he can only be present at one end and receive a signal from the other end.

Accordingly, in making the necessary assumptions to cover the matter of measuring lengths, we must make one with regard to the character of the signals which are to be employed for this purpose. If we could assume a system of signalling that would consume no time in transmission all would be simple enough. But we have no experience with such a system. Even if we believe that it ought to be possible thus to transmit signals at infinite velocity, we may not, in the absence of our present ability to do this, assume that it is possible. So we may only assume, with Einstein, that for our signals we shall employ the speediest messenger with which we are at present acquainted. This of course is light, the term including any of the electromagnetic impulses that travel at the speed C.

Of course in the vast majority of cases the distance that any light signal in which we are interested must go to reach us is so small that the time taken by its transmission can by no means be measured. We are then, to all intents and purposes, at both places—the point of origin of the signal and the point of receipt—simultaneously. But this is not the question at all. Waiving the fact that in astronomical investigations this approximation no longer holds, the fact remains that it is, in every case, merely an approximation. Approximations are all right in observations, where we know that they are approximations and act accordingly. But in the conceptual universe that parallels the external reality, computation is as good an agent of observation as visual or auditory or tactile sensation; if we can compute the error involved in a wrong procedure the error is there, regardless of whether we can see it or not. We must have methods which are conceptually free from error; and if we attempt to ignore the velocity of our light signals we do not meet this condition.

The measurement of lengths demands that we have a criterion of simultaneity between two remote points—remote in inches or remote in light-years, it does not matter which. There is no difficulty in defining simultaneity of two events that fall in the same point—or rather, in agreeing that we know what we mean by such simultaneity. But with regard to two events that occur in remote places there may be a question. A scientific definition differs from a mere description in that it must afford us a means of testing whether a given item comes under the definition or not. There is some difficulty in setting up a definition of simultaneity between distant events that satisfies this requirement. If we try simply to fall back upon our inherent ideas of what we mean by “the same instant” we see that this is not adequate. We must lay down a procedure for determining whether two events at remote points occur at “the same instant,” and check up alleged simultaneity by means of this procedure.

Einstein says, and we must agree with him, that he can find but one reasonable definition to cover this ground. An observer can tell whether he is located half way between two points of his observation; he can have mirrors set up at these points, send out light-signals, and note the time at which he gets back the reflection. He knows that the velocity of both signals, going and coming, is the same; if he observes that they return to him together so that their time of transit for the round trip is the same, he must accept the distances as equal. He is then at the mid-point of the line joining the two points under observation; and he may define simultaneity as follows, without introducing anything new or indeterminate: Two events are simultaneous if an observer midway between them sees them at the same instant, by means, of course, of light originating at the points of occurrence.]*

[It is this definition of simultaneity, coupled with the assumption that all observers, on whatever uniformly moving systems, would obtain the same experimental value for the velocity of light, that leads to the apparent paradoxes of the Special Theory of Relativity. If it be asked why we adopt it, we must in turn ask the inquirer to propose a better system for defining simultaneous events on different moving bodies.]¹⁹⁸

[There is nothing in this definition to indicate, directly, whether simultaneity persists for all observers, or whether it is relative, so that events simultaneous to one observer are not so to another. The question must then be investigated; and the answer, of course, will hinge upon the possibility of making proper allowances for the time of transit of the light signals that may be involved. It seems as though this ought to be possible; but a simple experiment will indicate that it is not, unless the observers involved are at rest with respect to one another.

Let us imagine an indefinitely long, straight railroad track, with an observer located somewhere along it at the point M. According to the convention suggested above, he has determined points A and B in opposite directions from him along the

track, and equally distant from him. We shall imagine, further, than a beneficent Providence supplies two lightning flashes, one striking at A and one at B, in such a way that observer M finds them to be simultaneous.

While all this is going on, a train is passing—a very long train, amply long enough to overlap the section AMB of the track. Among the passengers there is one, whom we may call M′, who is directly opposite M at the instant when, according to M, the lightning strikes. Observe he is not opposite M when M sees the flashes, but a brief time earlier—at the instant when, according to M’s computation, the simultaneous flashes occurred. At this instant there are definitely determined the points A′ and B′, on the train; and since we may quite well think of the two systems—train-system and track-system—as in coincidence at this instant, M′ is midway between A′ and B′, and likewise is midway between A and B.

Now if we think of the train as moving over the track in the direction of the arrow, we see very easily that M′ is running away from the light from A and toward that from B, and that, despite—or if you prefer because of—the uniform velocity of these light signals, the one from B reaches him, over a slightly shorter course, sooner than the one from A, over the slightly longer course. When the light signals reach M, M′ is no longer abreast of him but has moved along a wee bit, so that at this instant when M has the two signals, one of these has passed M′ and the other has yet to reach him. The upshot is that the events which were simultaneous to M are not so to M′.

It will probably be felt that this result is due to our having, somewhat unjustifiably and inconsistently, localized on the train the relative motion between train and track. But if we think of the track as sliding back under the train in the direction opposite to the arrow, and carrying with it the points A and B; and if we remember that this in no way affects M’s observed velocity of light or the distances AM and BM as he observes them: we can still accept his claim that the flashes were simultaneous. Then we have again the same situation: when the flashes from A and from B reach M at the same moment, in his new position a trifle to the left of his initial position of the diagram, the flash from A has not yet reached M′ in his original position while that from B has passed him. Regardless of what assumption we make concerning the motion between train-system and track-system, or more elegantly regardless of what coordinate system we use to define that motion, the event at B precedes that at A in the observation of M′. If we introduce a second train moving on the other track in the opposite direction, the observer on it will of course find that the flash at A precedes that at B—a disagreement not merely as to simultaneity but actually as to the order of two events! If we conceive the lightning as striking at the points A′ and B′ on the train, these points travel with M′ instead of with M; they are fixed to his coordinate system instead of to the other. If you carry out the argument now, you will find that when the flashes are simultaneous to M′, the one at A precedes that at B in M’s observation.

A large number of experiments more or less similar in outline to this one can be set up to demonstrate the consequences, with regard to measured values of time and space, of relative motion between two observers. I do not believe that a multiplicity of such demonstrations contributes to the intelligibility of the subject, and it is for this reason that I have cut loose from immediate dependence upon the essayists in this part of the discussion, concentrating upon the single experiment to which Einstein himself gives the place of importance.

We may permit Mr. Francis to remind us here that neither M nor M′ may correct his observation to make it accord with the other fellow’s. The one who does this is admitting that the other is at absolute rest and that he is himself in absolute motion; and this cannot be. They are simply in disagreement as to the simultaneity of two events, just as two observers might be in disagreement about the distance or the direction of a single event. This can mean nothing else than that, under the assumptions we have made, simultaneity is not an absolute characteristic as we had supposed it to be, but, like distance and direction, is in fact merely a relation between observer and objective, and therefore depends upon the particular observer who happens to be operating and upon the reference frame he is using.

But this is serious. My time measurements depend ultimately upon my space measurements; the latter, and hence both, depend closely upon my ideas of simultaneity. Yours depend upon your reading of simultaneity in precisely the same way.]* [Suppose the observer on the track, in the above experiment, wants to measure the length of something on the car, or the observer on the car something on the track. The observer, or his assistant, must be at both ends of the length to be measured at the same time, or get simultaneous reports in some way from these ends; else they will obtain false results. It is plain, then, that with different criteria of what the “same time” is, the observers in the two systems may get different values for the measured lengths in question.]²²⁰

[Who is right? According to the principle of relativity a decision on this question is absolutely impossible. Both parties are right from their own points of view; and we must admit that two events in two different places may be simultaneous for certain observers, and yet not simultaneous for other observers who move with respect to the first ones. There is no contradiction in this statement, although it is not in accordance with common opinion, which believes simultaneousness to be something absolute. But this common opinion lacks foundation. It cannot be proved by direct perception, for simultaneity of events can be perceived directly,]²⁴ [and in a manner involving none of our arbitrary assumptions,]* [only if they happen at the same place; if the events are distant from each other, their simultaneity or succession can be stated only through some method of communicating by signals. There is no logical reason why such a method should not lead to different results for observers who move with regard to one another.

From what we have said, it follows immediately that in the new theory not only the concept of simultaneousness but also that of duration is revealed as dependent on the motion of the observer.]²⁴ [Demonstration of this should be superfluous; it ought to be plain without argument that if two observers cannot agree whether two instants are the same instant or not, they cannot agree on the interval of time between instants. In the very example which we have already examined, one observer says that a certain time-interval is zero, and another gives it a value different from zero. The same thing happens whenever the observers are in relative motion.]* [Two physicists who measure the duration of a physical process will not obtain the same result if they are in relative motion with regard to one another.

They will also find different results for the length of a body. An observer who wants to measure the length of a body which is moving past him must in one way or another hold a measuring rod parallel to its motion and mark those points on his rod with which the ends of the body come into simultaneous coincidence. The distance between the two marks will then indicate the length of the body. But if the two markings are simultaneous for one observer, they will not be so for another one who moves with a different velocity, or who is at rest, with regard to the body under observation. He will have to ascribe a different length to it. And there will be no sense in asking which of them is right: length is a purely relative concept, just as well as duration.]²⁴

[The degree to which distance and time become relative instead of absolute quantities under the Special Theory of Relativity can be stated very definitely. In the first place, we must point out that the relativity of lengths applies with full force only to lengths that lie parallel to the direction of relative motion. Those that lie exactly perpendicular to that direction come out the same for both observers; those that lie obliquely to it show an effect, depending upon the angle, which of course becomes greater and greater as the direction of parallelism is approached.

The magnitude of the effect is easily demonstrated, but with this demonstration we do not need to be concerned here. It turns out that if an observer moving with a system finds that a certain time interval in the system is T seconds and that a certain length in the system is L inches, then an observer moving parallel with L and with a velocity v relative to the system will find for these the respective values upper T division-sign upper K and upper L times upper K, where upper K equals StartRoot 1 minus v squared slash upper C squared EndRoot period C in this expression of course represents the velocity of light. It will be noted that the fraction v squared slash upper C squared is ordinarily very small; that the expression under the radical is therefore less than 1 but by a very slight margin; and that the entire expression K is itself therefore less than 1 but by an even slighter margin. This means, then, that the observer outside the system finds the lengths in the system to be a wee bit shorter and the time intervals a wee bit longer than does the observer in the system. Another way of putting the matter is based, ultimately, upon the fact that in order for the observer in the system to get the larger value for distance and the smaller value for time, his measuring rod must go into the distance under measurement more times than that of the moving observer, while his clock must beat a longer second in order that less of them shall be recorded in a given interval between two events. So it is often said that the measuring rod as observed from without is contracted and the clock runs slow. This does not impress me as a happy statement, either in form or in content.]*

[The argument that these formulae are contradicted by human experience can be refuted by examining a concrete instance. If a train is 1,000 feet long at rest, how long will it be when running a mile a minute?]²³² [I have quoted this question exactly as it appears in the essay from which it is taken, because it is such a capital example of the objectionable way in which this business is customarily put. For the statement that lengths decrease and time-intervals increase “with velocity” is not true in just this form. The velocity, to have meaning, must be relative to some external system; and it is the observations from that external system that are affected. So long as we confine ourselves to the system in which the alleged modifications of size are stated as having taken place, there is nothing to observe that is any different from what is usual; there is no way to establish that we are enjoying a velocity, and in fact within the intent of the relativity theory we are not enjoying a velocity, for we are moving with the objects which we are observing. It is inter-systemic observations, and these alone, that show the effect. When we travel with the system under observation, we get the same results as any other observer on this system; when we do not so travel, we must conduct our observations from our own system, in relative motion to the other, and refer our results to our system.

Now when no particular observer is specified, we must of course assume an observer connected with the train, or with whatever the body mentioned. To that observer it doesn’t make the slightest difference what the train does; it may stand at rest with respect to some external system or it may move at any velocity whatsoever; its length remains always 1,000 feet. In order for this question to have the significance which its propounder means it to have, I must restate it as follows: A train is 1,000 feet long as measured by an observer travelling with it. If it passes a second observer at 60 miles per hour, what is its length as observed by him? The answer is now easy.]* [According to the formula the length of the moving train as seen from the ground will be 1 comma 000 times StartRoot 1 minus left-parenthesis 88 right-parenthesis squared slash left-parenthesis 186 comma 000 times 5 comma 280 right-parenthesis squared EndRoot equals 999.999 comma 999 comma 999 comma 996 feet, a change entirely too small for detection by the most delicate instruments. Examination of the expression K shows that in so far as terrestrial movements of material objects are concerned it is equal to 1]²³² [within a far smaller margin than we can ever hope to make our observations. Even the diameter of the earth, as many of the essayists point out, will be shortened only 2½ inches for an outside observer past whom it rushes with its orbital speed of 18.5 miles per second. But slight as the difference may be in these familiar cases, its scientific importance remains the same.]*

[A simple computation shows that this effect is exactly the amount suggested by Lorentz and Fitzgerald to explain the Michelson-Morley experiment.]¹⁸⁸ [This ought not to surprise us, since both that explanation and the present one are got up with the same purpose. If they both achieve that purpose they must, numerically, come to the same thing in any numerical case. It is, however, most emphatically to be insisted that the present “shortening” of lengths]* [no longer appears as a “physical” shortening caused by absolute motion through the ether but is simply a result of our methods of measuring space and time.]¹⁸⁸ [Where Fitzgerald and Lorentz had assumed that a body in motion has its dimensions shortened in the direction of its motion,]²²⁰ [this very form of statement ceases to possess significance under the relativity assumption.]* [For if we cannot tell which of two bodies is moving, which one is shortened? The answer is, both—for the other fellow. For each frame of reference there is a scale of length and a scale of time, and these scales for different frames are related in a manner involving both the length and the time.]²²⁰ [But we must not yield to the temptation to say that all this is not real; the confinement of a certain scale of length and of time to a single observation system does not in the least make it unreal.]* [The situation is real—as real as any other physical event.]¹⁶⁵

[The word physical is used in two senses in the above paragraph. It is denied that the observed variability in lengths indicates any “physical” contraction or shrinkage; and on the heels of this it is asserted that this observed variability is of itself an actual “physical” event. It is difficult to express in words the distinction between the two senses in which the term physical is employed in these two statements, but I think this distinction ought to be clear once its existence is emphasized. There is no material contraction; it is not right to say that objects in motion contract or are shorter; they are not shorter to an observer in motion with them. The whole thing is a phenomenon of observation. The definitions which we are obliged to lay down and the assumptions which we are obliged to make in order, first, that we shall be able to measure at all, and second, that we shall be able to escape the inadmissible concept of absolute motion, are such that certain realities which we had supposed ought to be the same for all observers turn out not to be the same for observers who are in relative motion with respect to one another. We have found this out, and we have found out the numerical relation which holds between the reality of the one observer and that of the other. We have found that this relation depends upon nothing save the relative velocity of the two observers. As good a way of emphasizing this as any is to point out that two observers who have the same velocity with respect to the system under examination (and whose mutual relative velocity is therefore zero) will always get the same results when measuring lengths and times on that system. The object does not go through any process of contraction; it is simply shorter because it is observed from a station with respect to which it is moving. Similar remarks might be made about the time effect; but the time-interval is not so easily visualized as a concrete thing and hence does not offer such temptation for loose statement.

The purely relative aspect of the matter is further brought out if we consider a single example both backwards and forwards. Systems S and S′ are in relative motion. An object in S which to an observer in S is L units long, is shorter for an observer in S′—shorter by an amount indicated through the “correction factor” K. Now if we have, in the first instance, made the objectionable statement that objects are shorter in system S′ than they are in S, it will be quite natural for us to infer from this that objects in S must be longer than those in S′; and from this to assert that when the observer in S measures objects lying in S′, he gets for them greater lengths than does the home observer in S′. But if we have, in the first instance, avoided the objectionable statement referred to, we shall be much better able to realize that the whole business is quite reciprocal; that the phenomena are symmetric with respect to the two systems, to the extent that we can interchange the systems in any of our statements without modifying the statements in any other way.

Objects in S appear shorter and times in S appear longer to the external “moving” observer in S′ than they do to the domestic observer in S. Exactly in the same way, objects in S′ appear shorter to observers in the foreign system S than to the home observer in S′, who remains at rest with respect to them. I think that when we get the right angle upon this situation, it loses the alleged startling character which has been imposed upon it by many writers. The “apparent size” of the astronomer is an analogy in point. Objects on the moon, by virtue of their great distance, look smaller to observers on the earth than to observers on the moon. Do objects on the earth, on this account, look larger to a moon observer than they do to us? They do not; any suggestion that they do we should receive with appropriate scorn. The variation in size introduced by distance is reciprocal, and this reciprocity does not in the least puzzle us. Why, then, should that introduced by relative motion puzzle us?

Our old, accustomed concepts of time and space, which have grown up through countless generations of our ancestors, and been handed down to us in the form in which we are familiar with them, leave no room for a condition where time intervals and space intervals are not universally fixed and invariant. They leave no room for us to say that]* [one cannot know the time until he knows where he is, nor where he is until he knows the time,]²²⁰ [nor either time or place until he knows something about velocity. But in this concise formulation of the difference between what we have always believed and what we have seen to be among the consequences of Einstein’s postulates of the universal relativity of uniform motion, we may at once locate the assumption which, underlying all the old ideas, is the root of all the trouble. The fact is we have always supposed time and space to be absolutely distinct and independent entities.]*

[The concept of time has ever been one of the most absolute of all the categories. It is true that there is much of the mysterious about time; and philosophers have spent much effort trying to clear up the mystery—with unsatisfactory results. However, to most persons it has seemed possible to adopt an arbitrary measure or unit of duration and to say that this is absolute, independent of the state of the body or bodies on which it is used for practical purposes.]²⁷² [Time has thus been regarded as something which of itself flows on regularly and continuously, regardless of physical events concerning matter.]¹⁵⁰ [In other words, according to this view, time is not affected by conditions or motions in space.]²⁷² [We have deliberately chosen to ignore the obvious fact that time can never appear to us, be measured by us, or have the least significance for us, save as a measure of something that is closely tied up with space and with material space-dimensions. Not merely have we supposed that time and space are separated in nature as in our easiest perceptions, but we have supposed that they are of such fundamentally distinct character that they can never be tied up together. In no way whatever, assumes the Euclidean and Newtonian intellect, may space ever depend upon time or time upon space. This is the assumption which we must remove in order to attain universal relativity; and while it may come hard, it will not come so hard as the alternative. For this alternative is nothing other than to abandon universal relativity. This course would leave us with logical contradictions and discrepancies that could not be resolved by any revision of fundamental concepts or by any cleaning out of the Augean stables of old assumptions; whereas the relativity doctrine as built up by Einstein requires only such a cleaning out in order to leave us with a strictly logical and consistent whole. The rôle of Hercules is a very difficult one for us to play. Einstein has played it for the race at large, but each of us must follow him in playing it for himself.

I need not trespass upon the subject matter of those essays which appear in full by going here into any details with regard to the manner in which time and space are finally found to depend upon one another and to form the parts of a single universal whole. But I may appropriately point out that if time and space are found to be relative, we may surely expect some of the less fundamental concepts that depend upon them to be relative also. In this expectation we are not disappointed. For one thing,]* [mass has always been assumed to be a constant, independent of any motion or energy which it might possess. Just as lengths and times depend upon relative motion, however, it is found that mass, which is the remaining factor in the expression for energy due to motion, also depends upon relative velocities. The dependence is such that if a body takes up an amount of energy E with respect to a certain system, the body behaves, to measurements made from that system, as though its mass had been increased by an amount upper E slash upper C squared, where C is as usual the velocity of light.]¹⁹⁴

[This should not startle us. The key to the situation lies in the italicized words above, which indicate that the answer to the query whether a body has taken up energy or not depends upon the seat of observation. If I take up my location on the system S, and you on the system S′, and if we find that we are in relative motion, we must make some assumption about the energy which was necessary, initially, to get us into this condition. Suppose we are on two passing trains.]* [The chances are that either of us will assume that he is at rest and that it is the other train which moves, although if sufficiently sophisticated one of us may assume that he is moving and that the other train is at rest.]²⁷² [Whatever our assumption, whatever the system, the localization of the energy that is carried in latent form by our systems depends upon this assumption. Indeed, if our systems are of differing mass, our assumptions will even govern our ideas of the amount of energy which is represented by our relative motion; if your system be the more massive, more energy would have to be localized in it than in mine to produce our relative motion. If we did not have the universal principle of relativity to forbid, we might make an arbitrary assumption about our motions and hence about our respective latent energies; in the presence of this veto, the only chance of adjustment lies in our masses, which must differ according to whether you or I observe them.]*

[For most of the velocities with which we are familiar upper E slash upper C squared is, like the difference between K and unity, such an extremely small quantity that the most delicate measurements fail to detect it. But the electrons in a highly evacuated tube and the particles shot out from radioactive materials attain in some cases velocities as high as eight-tenths that of light. When we measure the mass of such particles at different velocities we find that it actually increases with the velocity, and in accordance with the foregoing law.]¹⁹⁴ [This observation, in fact, antedates Einstein’s explanation, which is far more satisfactory than the earlier differentiation between “normal mass” and “electrical mass” which was called upon to account for the increase.]*

[But if the quantity upper E slash upper C squared is to be considered as an actual increase in mass, may it not be possible that all mass is energy? This would lead to the conclusion that the energy stored up in any mass is m upper C squared. The value is very great, since C is so large; but it is in good agreement with the internal energy of the atom as calculated from other considerations. It is obvious that conservation of mass and of momentum cannot both hold good under a theory that translates the one into the other. Mass is then not considered by Einstein as conservative in the ordinary sense, but it is the total quantity of mass plus energy in any closed system that remains constant. Small amounts of energy may be transformed into mass, and vice versa.]¹⁹⁴

[Other features of the theory which are often displayed as consequences are really more in the nature of assumptions. It will be recalled that when we had agreed upon the necessity of employing signals of some sort, we selected as the means of signalling the speediest messenger with which we happened to be acquainted. Our subsequent difficulties were largely due to the impossibility of making a proper allowance for this messenger’s speed, even though we knew its numerical value; and as a consequence, this speed enters into our formulae. Now we have not said in so many words that C is the greatest speed attainable, but we have tacitly assumed that it is. We need not, therefore, be surprised if our formulae give us absurd results for speeds higher than C, and indicate the impossibility of ever attaining these. Whatever we put into a problem the algebra is bound to give us back. If we look at our formula for K, we see that in the event of v equalling C, lengths become zero and times infinite. The light messenger itself, then, has no dimension; and for it time stands still.

If we suppose v to be greater than C, we get even more bizarre results, for then the factor K is the square root of a negative number, or as the mathematician calls it an “imaginary” quantity; and with it, lengths and times become imaginary too.

The fact that time stops for it, and the fact that it is the limiting velocity, give to C certain of the attributes of the mathematician’s infinity. Certainly if it can never be exceeded, we must have a new formula for the composition of velocities. Otherwise when my system passes yours at a speed of 100,000 miles per second, while yours passes a third in the same direction at the same velocity, I shall be passing this third framework at the forbidden velocity of 200,000 miles per second—greater than C. In fact Einstein is able to show that an old formula, which had already been found to connect the speed of light in a material medium with the speed of that medium, will now serve universally for the composition of velocities. When we combine the velocities v and u, instead of getting the resultant v plus u as we would have supposed, we get the resultant left-parenthesis v plus u right-parenthesis slash left-parenthesis 1 plus left-parenthesis u v slash upper C squared right-parenthesis right-parenthesis comma or upper C squared left-parenthesis v plus u right-parenthesis slash left-parenthesis upper C squared plus u v right-parenthesis

This need not surprise us either, if we will but reflect that the second velocity effects a second revision of length and time measurements between the systems involved. And now, if we let either v, or u, or even both of them, take the value C, the resultant still is C. In another way we have found C to behave like the mathematician’s infinity, to which, in the words of the blind poet, if we add untold thousands, we effect no real increment.

A good many correspondents who have given the subject sufficient thought to realize that the limiting character of the velocity C is really read into Einstein’s system by assumption have written, in more or less perturbed inquiry, to know whether this does not invalidate the whole structure. The answer, of course, is yes—provided you can show this assumption to be invalid. The same answer may be made of any scientific doctrine whatever, and in reference to any one of the multitudinous assumptions underlying it. If we were to discover, tomorrow, a way of sending signals absolutely instantaneously, Einstein’s whole structure would collapse as soon as we had agreed to use this new method. If we were to discover a signalling agent with finite velocity greater than that of light, relativity would persist with this velocity written in its formulae in the place of C.

It is a mistake to quote Einstein’s theory in support of the statement that such a velocity can never be. An assumption proves its consequences, but never can prove itself; it must remain always an assumption. But in the presence of long human experience supporting Einstein’s assumption that no velocity in excess of C can be found, it is fair to demand that it be disputed not with argument but with demonstration. The one line of argument that would hold out a priori hope of reducing the assumption to an absurdity would be one based on the familiar idea of adding velocities; but Einstein has spiked this argument before it is started by replacing the direct addition of velocities with another method of combining them that fits in with his assumption and as well with the observed facts. The burden of proof is then on the prosecution; anyone who would contradict our assertion that C is the greatest velocity attainable may do so only by showing us a greater one. Until this has been done, the admission that it may properly be attempted can in no way be construed as a confession of weakness on the part of Einstein.

It may be well to point out that in no event may analogy be drawn with sound, as many have tried to do. In the first place sound requires a material medium and its velocity with regard to this rather than relative to the observer we know to be fixed; in the second place, requiring a material medium, sound is not a universal signalling agent; in the third place, we know definitely that its velocity can be exceeded, and are therefore barred from making the assumption necessary to establish the analogy. The very extraordinary behavior of light in presenting a velocity that is the same for all observers, and in refusing to betray the least material evidence of any medium for its transmission, rather fortifies us in believing that Einstein’s assumption regarding the ultimate character of this velocity is in accord with the nature of things.

A great deal can be said in the direction of general comment making the Special Theory and its surprising accompaniments easier of acceptance, and we shall conclude the present discussion by saying some of these things.]* [It has been objected that the various effects catalogued above are only apparent, due to the finite velocity of light—that the real shape and size of a body or the real time of an event cannot be affected by the point of view or the motion of an observer. This argument would be perfectly valid, if there were real times and distances; but there are not. These are earth-bound notions, due to our experience on an apparently motionless platform, with slow-moving bodies. Under these circumstances different observations of the same thing or of the same event agree. But when we no longer have the solid earth to stand on, and are dealing with velocities so high that the relativity effects become appreciable, there is no standard by which to resolve the disagreements. No one of the observations can claim to be nearer reality than any other. To demand the real size of a thing is to demand a stationary observer or an instantaneous means of information. Both are impossible.

When relativity asks us to give up our earth-bound notions of absolute space and absolute time the sensation, at first, is that we have nothing left to stand on. So must the contemporaries of Columbus have felt when told that the earth rested on—nothing. The remedy too is similar. Just as they had to be taught that falling is a local affair, that the earth is self-contained, and needs no external support—so we must be taught that space and time standards are local affairs. Each moving body carries its own space and time standards with it; it is self-contained. It does not need to reach out for eternal support, for an absolute space and time that can never quite be attained. All we ever need to know is the relation of the other fellow’s space and time standards to our own. This is the first thing relativity teaches us.]¹⁴¹

[The consequences of Einstein’s assumptions have led many to reject the theory of relativity, on the ground that its conclusions are contrary to common sense—as they undoubtedly are. But to the contemporaries of Copernicus and Galileo the theory that the earth rotates on its axis and revolves around the sun was contrary to common sense; yet this theory prevailed. There is nothing sacred about common sense; in the last analysis its judgments are based on the accumulated experience of the human race. From the beginning of the world up to the present generation, no bodies were known whose velocities were not extremely small compared with that of light. The development of modern physics has led to discovery of very much larger velocities, some as high as 165,000 miles per second. It is not to be wondered at that such an enlargement of our experience requires a corresponding enlargement or generalization of the concepts of space and time. Just as the presupposition of primitive man that the earth was flat had to be given up in the light of advancing knowledge, so we are now called upon to give up our presupposition that space and time are absolute and independent in their nature.

The reader must not expect to understand the theory of relativity in the sense of making it fit in with his previous ideas. If the theory be right these ideas are wrong and must be modified, a process apt to be painful.]²²³ [All the reader can do is to become familiar with the new concepts, just as a child gets used to the simple relations and quantities he meets until he “understands” them.]²²¹ [Mr. Francis has said something of the utmost significance when he points out that “understanding” really means nothing in the world except familiarity and accustomedness.]* [The one thing about the relativity doctrine that we can hope thus to understand at once and without pain is the logical process used in arriving at our results.]²²¹ [Particularly is it hard to give a satisfactory explanation of the theory in popular language, because the language itself is based on the old concepts; the only language which is really adequate is that of mathematics.]²²³ [Unless we have, in addition to the terms of our ordinary knowledge, a set of definitions that comes with a wide knowledge of mathematics and a lively sense of the reality of mathematical constructions, we are likely to view the theory of relativity through a fog of familiar terms suddenly become self-contradictory and deceptive. Not that we are unfamiliar with the idea that some of our habitual notions may be wrong; but knowledge of their illusory nature arises and becomes convincing only with time. We may now be ready to grant that the earth, seemingly so solid, is really a whirling globe rushing through space; but we are no more ready immediately to accept the bald assertion that this space is not what it seems than our ancestors were to accept the idea that the earth was round or that it moved.]¹⁵⁶ [What we must have, if we are to comprehend relativity with any degree of thoroughness, is the mathematician’s attitude toward his assumptions, and his complete readiness to swap one set of assumptions for another as a mere part of the day’s work, the spirit of which I have endeavored to convey in the chapter on non-Euclidean geometry.]*

[The ideas of relativity may seem, at first sight, to be giving us a new and metaphysical theory of time and space. New, doubtless; but certainly the theory was meant by its author to be quite the opposite of metaphysical. Our actual perception of space is by measurement, real and imagined, of distances between objects, just as our actual perception of time is by measurement. Is it not less metaphysical to accept space and time as our measurements present them to us, than to invent hypotheses to force our perceptual space into an absolute space that is forever hidden from us?]¹⁸² [In order not to be metaphysical, we must eliminate our preconceived notions of space and time and motion, and focus our attention upon the indications of our instruments of observation, as affording the only objective manifestations of these qualities and therefore the only attributes which we can consider as functions of observed phenomena.]⁴⁷ [Einstein has consistently followed the teachings of experience, and completely freed himself from metaphysics.]¹¹⁴ [That this is not always easy to do is clear, I think, if we will recall the highly metaphysical character often taken by the objections to action-at-a-distance theories and concepts; and if we will remind ourselves that it was on purely metaphysical grounds that Newton refused to countenance Huyghens’ wave theory of light. Whether, as in the one case, it leads us to valid conclusions, or, as in the other, to false ones, metaphysical reasoning is something to avoid. Einstein, I think, has avoided it about as thoroughly as anyone ever did.]*