Chapter VI: completes the preliminary course in the fundamentals of (3)

Here is where we begin to get a grip on the problem of the earth and the ether. In any problem involving the return-trip principle, there will enter two velocities--that of the swimmer and that of the medium; and the time of retardation. If we know any two of these items we can calculate the third. When the swimmer is a ray of light and the velocity of the medium is that of the ether as it flows past the earth, we know the first of these two; we hope to observe the retardation so that we may calculate the second velocity. The apparatus for the experiment is ingenious and demands description.

THE MICHELSON-MORLEY EXPERIMENT

The machine is of structural steel, weighing 1,900 pounds. It has two arms which form a Greek cross. Each arm is 14 feet in length. The whole apparatus is floated in a trough containing 800 pounds of mercury.

Four mirrors are arranged on the end of each arm, sixteen in all, with a seventeenth mirror, M, set at one of the inside corners of the cross, as diagrammed. A source of light (in this case a calcium flame) is provided, and its rays directed by a lens toward the mirror M. Part of the light is allowed to pass straight through M to the opposite arm of the cross, where it strikes mirror 1. It is reflected back across the arm to mirror 2, thence to 3, and so on until it reaches mirror 8. Thence it is reflected back to mirror 7, to 6, and so on, retracing its former path, and finally is caught by the reverse side of the mirror M and is sent to an observer at O. In retracing its path the light sets up an interference phenomenon (see below) and the interference bands are visible to the observer, who is provided with a telescope to magnify the results.

A second part of the original light-beam is reflected off at right angles by the mirror M, and is passed to and fro on the adjacent arms of the machine, in exactly the same manner and over a similar path, by means of the mirrors I, II, III, ... VIII. This light finally reaches the observer at the telescope, setting up a second set of interference bands, parallel to the first.

A word now about this business of light interference. Light is a wave motion. The length of a wave is but a few millionths of an inch, and the amplitude is correspondingly minute; but none the less, these waves behave in a thoroughly wave-like manner. In particular, if the crests of two waves are superposed, there is a double effect; while if a crest of one wave falls with a trough of another, there is a killing-off or "interference".

Under ordinary circumstances interference of light waves does not occur. This is simply because under ordinary circumstances light waves are not piled up on one another. But sometimes this piling up occurs; and then, just so sure as the piled-up waves are in the same phase they reinforce one another, while if they are in opposite phase they interfere. And the conditions which we have outlined above, with the telescope and the mirrors and the ray of light retracing the path over which it went out, are conditions under which interference does occur. If the returning wave is in exact phase with the outgoing one, the effect is that of uniform double illumination; if it is in exactly opposite phase the effect is that of complete extinguishing of the light, the reversed wave exactly cancelling out the original one. If the two rays are partly in phase, there is partial reinforcement or partial cancelling out, according to whether they are nearly in phase or nearly out of phase. Finally, if the mirrors are not set absolutely parallel--as must in practice be the case when we attempt to measure their parallelism in terms of the wave-length of light--adjacent parts of the light ray will vary in the extent to which they are out of phase, since they will have travelled a fraction of a wave-length further to get to and from this, that or the other mirror. There will then appear in the telescope alternate bands of illumination and darkness, whose width and spacing depend upon all the factors entering into the problem.

If it were possible for us to make the apparatus with such a degree of refinement that the path from mirror M via mirrors 1, 2, 3, etc., back through M and into the telescope, were exactly the same length as that from flame to telescope by way of the mirrors I, II, III, etc.--exactly the same to a margin of error materially less than a single wave-length of light--why, then, the two sets of interference fringes would come out exactly superposed provided the motion of the earth through the "ether" turn out to have no influence upon the velocity of light; or, if such influence exist, these fringes would be displaced from one another to an extent measuring the influence in question. But our ability to set up this complicated pattern of mirrors at predetermined distances falls far short of the wave-length as a measure of error. So in practice all that we can say is that having once set the instrument up, and passed a beam of light through it, there will be produced two sets of parallel interference fringes. These sets will fail of superposition--each fringe of one set will be removed from the corresponding fringe of the other set--by some definite distance. Then, any subsequent variation in the speed of light along the two arms will at once be detected by a shifting of the interference bands through a distance which we shall be able to measure.

THE VERDICT

Under the theories and assumptions governing at the time of the original performance of this experiment, it will be readily seen that if this machine be set up in an "ether stream" with one arm parallel to the direction of the stream and the other at right angles thereto, there will be a difference in the speed of the light along the two arms. Then if the apparatus be shifted to a position oblique to the ether stream, the excess velocity of the light in the one arm would be diminished, and gradually come to zero at the 45-degree angle, after which the light traveling along the other arm would assume the greater speed. In making observations, therefore, the entire apparatus was slowly rotated, the observers walking with it, so that changes of the sort anticipated would be observed.

The investigators were, however, ignorant of the position in which the apparatus ought to be set to insure that one of the arms lie across the ether drift; and they were ignorant of the time of year at which the earth's maximum velocity through the ether was to be looked for. In particular, it is plain that if the solar system as a whole is moving through the ether at a rate less than the earth's orbital velocity, there is a point in our orbit where our velocity through the ether and that around the sun just cancel out and leave us temporarily in a state of "absolute rest." So it was anticipated that the experiment might have to be repeated in many orientations of the machine and at many seasons of the year in order to give a series of readings from which the true motion of the earth through the ether might be deduced.

For those who have a little algebra the demonstration which Dr. Russell gives on a subsequent page will be interesting as showing the situation in perfectly general terms. It will be realized that the more complicated arrangement of mirrors in the experiment as just described is simply an eightfold repetition of the simple experiment as outlined by Dr. Russell, and that it was done so for the mere sake of multiplying by eight the distances travelled and hence the difference in time and in phase.

And now for the grand climax. The experiment was repeated many times, with the original and with other apparatus, indoors and outdoors, at all seasons of the year, with variation of every condition that could imaginably affect the result. The apparatus was ordinarily such that a shift in the fringes of anywhere from one-tenth to one one-hundredth of that which would have followed from any reasonable value for the earth's motion through the ether would have been systematically apparent. The result was uniformly negative. At all times and in all directions the velocity of light past the earth-bound observer was the same. The earth has no motion with reference to the ether!

[The amazing character of this result is not by any possibility to be exaggerated.]* [According to one experiment the ether was carried along by a rapidly moving body and according to another equally well-planned and well-executed experiment a rapidly moving body did not disturb the ether at all. This was the blind alley into which science had been led.]232

THE "CONTRACTION" HYPOTHESIS

[Numerous efforts were made to explain the contradiction.]* [It is indeed a very puzzling one, and it gave physicists no end of trouble. However Lorentz and Fitzgerald finally put forward an ingenious explanation, to the effect that the actual motion of the earth through the ether is balanced, as far as the ability of our measuring instruments is concerned, by a contraction of these same instruments in the direction of their motion. This contraction obviously cannot be observed directly because all bodies, including the measuring instruments themselves (which after all are only arbitrary guides), will suffer the contraction equally. According to this theory, called the Lorentz-Fitzgerald contraction theory,]272 [all bodies in motion suffer such contraction of their length in the direction of their motion;]283 [the contraction being made evident by our inability to observe the absolute motion of the earth, which it is assumed must exist.]272 [This would suffice to show why the Michelson-Morley experiment gave a negative result, and would preserve the concept of absolute motion with reference to the ether.]283

[This proposal of Lorentz and Fitzgerald loses its startling aspect when we consider that all matter appears to be an electrical structure, and that the dimensions of the electric and magnetic fields which accompany the electrons of which it is constituted change with the velocity of motion.]267 [The forces of cohesion which determine the form of a rigid body are held to be electromagnetic in nature; the contraction may be regarded as due to a change in the electromagnetic forces between the molecules.]10 [As one writer has put it, the orientation, in the electromagnetic medium, of a body depending for its very existence upon electromagnetic forces is not necessarily a matter of indifference.]*

[Granting the plausibility of all this, on the basis of an electromagnetic theory of matter, it leaves us in an unsatisfactory position. We are left with a fixed ether with reference to which absolute motion has a meaning, but that motion remains undetected and apparently undetectable. Further, if we on shore measure the length of a moving ship, using a yard-stick which is stationary on shore, we shall obtain one result. If we take our stick aboard it contracts, and so we obtain a greater length for the ship. Not knowing our "real" motion through the ether, we cannot say which is the "true" length. Is it not, then, more satisfactory to discard all notion of true length as an inherent quality of bodies, and, by regarding length as the measure of a relation between a particular object and a particular observer, to make one length as true as the other?]182 [The opponents of such a viewpoint contend that Michelson's result was due to a fluke; some mysterious counterbalancing influence was for some reason at work, concealing the result which should normally have been expected. Einstein refuses to accept this explanation;]192 [he refuses to believe that all nature is in a contemptible conspiracy to delude us.]*

[The Fitzgerald suggestion is further unsatisfactory because it assumes all substances, of whatever density, to undergo the same contraction; and above all for the reason that it sheds no light upon other phenomena.]194 [It is indeed a very special explanation; that is, it applies only to the particular experiment in question. And indeed it is only one of many possible explanations. Einstein conceived the notion that it might be infinitely more valuable to take the most general explanation possible, and then try to find from this its logical consequences. This "most general explanation" is, of course, simply that it is impossible in any way whatever to measure the absolute motion of a body in space.]272 [Accordingly Einstein enunciated, first the Special Theory of Relativity, and later the General Theory of Relativity. The special theory was so called because it was, limited to uniform rectilinear and non-rotary motions. The general theory, on the other hand, dealt not only with uniform rectilinear motions, but with any arbitrary motion whatever.

TAKING THE BULL BY THE HORNS

The hypothesis of relativity asserts that there can be no such concept as absolute position, absolute motion, absolute time; that space and time are inter-dependent, not independent; that everything is relative to something else. It thus accords with the philosophical notion of the relativity of all knowledge.]283 [Knowledge is based, ultimately, upon measurement; and clearly all measurement is relative, consisting merely in the application of a standard to the magnitude measured. All metric numbers are relative; dividing the unit multiplies the metric number. Moreover, if measure and measured change proportionately, the measuring number is unchanged. Should space with all its contents swell in fixed ratio throughout, no measurement could detect this; nor even should it pulse uniformly throughout. Furthermore, were space and space-contents in any way systematically transformed (as by reflection in curved mirrors) point for point, continuously, without rending, no measurement could reveal this distortion; experience would proceed undisturbed.]263

[Mark Twain said that the street in Damascus "which is called straight," is so called because while it is not as straight as a rainbow it is straighter than a corkscrew. This expresses the basic idea of relativity--the idea of comparison. All our knowledge is relative, not absolute. Things are big or little, long or short, light or heavy, fast or slow, only by comparison. An atom may be as large, compared to an electron, as is a cathedral compared to a fly. The relativity theory of Einstein emphasizes two cases of relative knowledge; our knowledge of time and space, and our knowledge of motion.]216 [And in each case, instead of allowing the notions of relativity to guide us only so far as it pleases us to follow them, there abandoning them for ideas more in accord with what we find it easy to take for granted, Einstein builds his structure on the thesis that relativity must be admitted, must be followed out to the bitter end, in spite of anything that it may do to our preconceived notions. If relativity is to be admitted at all, it must be admitted in toto; no matter what else it contradicts, we have no appeal from its conclusions so long as it refrains from contradicting itself.]*

[The hypothesis of relativity was developed by Einstein through a priori methods, not the more usual a posteriori ones. That is, certain principles were enunciated as probably true, the consequences of these were developed, and these deductions tested by comparison of the predicted and the observed phenomena. It was in no sense attained by the more usual procedure of observing groups of phenomena and formulating a law or formula which would embrace them and correctly describe the routine or sequence of phenomena.

The first principle thus enunciated is that it is impossible to measure or detect absolute translatory motion through space, under any circumstances or by any means. The second is that the velocity of light in free space appears the same to all observers regardless of the relative motion of the source of light and the observer. This velocity is not affected by motion of the source toward or away from the observer,]283 [if we may for the moment use this expression with its implication of absolute motion.]* [But universal relativity insists that motion of the source toward the observer is identical with motion of the observer toward the source.]283

[It will be seen that we are at once on the horns of a dilemma. Either we must give up relativity before we get fairly started on it, or we must overturn the foundations of common sense by admitting that time and space are so constituted that when we go to meet an advancing light-impulse, or when we retreat from it, it still reaches us with the same velocity as though we stood still waiting for it. We shall find when we are through with our investigation that common sense is at fault; that our fixed impression of the absurdity of the state of affairs just outlined springs from a confusion between relativism and absolutism which has heretofore dominated our thought and gone unquestioned. The impression of absurdity will vanish when we have resolved this confusion.]*

QUESTIONS OF COMMON SENSE

[But it is obvious from what has just been said that if we are to adopt Einstein's theory, we must make very radical changes in some of our fundamental notions, changes that seem in violent conflict with common sense. It is unfortunate that many popularizers of relativity have been more concerned to astonish their readers with incredible paradoxes than to give an account such as would appeal to sound judgment. Many of these paradoxes do not belong essentially to the theory at all. There is nothing in the latter that an enlarged and enlightened common sense would not readily endorse. But common sense must be educated up to the necessary level.]141

[There was a time when it was believed, as a result of centuries of experience, that the world was flat. This belief checked up with the known facts, and it could be used as the basis for a system of science which would account for things that had happened and that were to happen. It was entirely sufficient for the time in which it prevailed.

Then one day a man arose to point out that all the known facts were equally accounted for on the theory that the earth was a sphere. It was in order for his contemporaries to admit this, to say that so far as the facts in hand were concerned they could not tell whether the earth was flat or round--that new facts would have to be sought that would contradict one or the other hypothesis. Instead of this the world laughed and insisted that the earth could not be round because it was flat; that it could not be round because then the people would fall off the other side.

But the field of experimentation widened, and men were able to observe facts that had been hidden from them. Presently a man sailed west and arrived east; and it became clear that in spite of previously accepted "facts" to the contrary, the earth was really round. The previously accepted "facts" were then revised to fit the newly discovered truth; and finally a new system of science came into being, which accounted for all the old facts and all the new ones.

At intervals this sort of thing has been repeated. A Galileo shows that preconceived ideas with regard to the heavens are wrong, and must be revised to accord with his newly promulgated principles. A Newton does the same for physics--and people unlearn the "fact" that motion has to be supported by continued application of force, substituting the new idea that it actually requires force to stop a moving body. A Harvey shows that the things which have been "known" for generations about the human body are not so. A Lyell and a Darwin force men to throw overboard the things they have always believed about the way in which the earth and its creatures came into being. Every science we possess has passed through one or more of these periods of readjustment to new facts.

SHIFTING THE MENTAL GEARS

Now we are apt to lose sight of the true significance of this. It is not alone our opinions that are altered; it is our fundamental concepts. We get concepts wholly from our perceptions, making them to fit those perceptions. Whenever a new vista is opened to our perceptions, we find facts that we never could have suspected from the restricted viewpoint. We must then actually alter our concepts to make the new facts fit in with the greatest degree of harmony. And we must not hesitate to undertake this alteration, through any feeling that fundamental concepts are more sacred and less freely to be tampered with than derived facts.]* [We do, to be sure, want fundamental concepts that are easy for a human mind to conceive; but we also want our laws of nature to be simple. If the laws begin to become, intricate, why not reshape, somewhat, the fundamental concepts, in order to simplify the scientific laws? Ultimately it is the simplicity of the scientific system as a whole that is our principal aim.]178

[As a fair example, see what the acceptance of the earth's sphericity did to the idea represented by the word "down." With a flat earth, "down" is a single direction, the same throughout the universe; with a round earth, "down" becomes merely the direction leading toward the center of the particular heavenly body on which we happen to be located. It is so with every concept we have. No matter how intrinsic a part of nature and of our being a certain notion may seem, we can never know that new facts will not develop which will show it to be a mistaken one. Today we are merely confronted by a gigantic example of this sort of thing. Einstein tells us that when velocities are attained which have just now come within the range of our close investigation, extraordinary things happen--things quite irreconcilable with our present concepts of time and space and mass and dimension. We are tempted to laugh at him, to tell him that the phenomena he suggests are absurd because they contradict these concepts. Nothing could be more rash than this.

When we consider the results which follow from physical velocities comparable with that of light, we must confess that here are conditions which have never before been carefully investigated. We must be quite as well prepared to have these conditions reveal some epoch-making fact as was Galileo when he turned the first telescope upon the skies. And if this fact requires that we discard present ideas of time and space and mass and dimension, we must be prepared to do so quite as thoroughly as our medieval fathers had to discard their notions of celestial "perfection" which demanded that there be but seven major heavenly bodies and that everything center about the earth as a common universal hub. We must be prepared to revise our concepts of these or any other fundamentals quite as severely as did the first philosopher who realized that "down" in London was not parallel to "down" in Bagdad or on Mars.]*

[In all ordinary terrestrial matters we take the earth as a fixed body, light as instantaneous. This is perfectly proper, for such matters. But we carry our earth-acquired habits with us into the celestial regions. Though we have no longer the earth to stand on, yet we assume, as on the earth, that all measurements and movements must be referred to some fixed body, and are only then valid. We cling to our earth-bound notion that there is an absolute up-and-down, back-and-forth, right-and-left, in space. We may admit that we can never find it, but we still think it is there, and seek to approach it as nearly as possible. And similarly from our earth experiences, which are sufficiently in a single place to make possible this simplifying assumption, we get the idea that there is one universal time, applicable at once to the entire universe.]141 [The difficulty in accepting Einstein is entirely the difficulty in getting away from these earth-bound habits of thought.]*

IV

THE SPECIAL THEORY OF RELATIVITY

What Einstein's Study of Uniform Motion Tells Us About Time and Space and the Nature of the External Reality

BY VARIOUS CONTRIBUTORS AND THE EDITOR

Whatever the explanation adopted for the negative result of the Michelson-Morley experiment, one thing stands out clearly: the attempt to isolate absolute motion has again failed.]* [Einstein generalizes this with all the other and older negative results of similar sort into a negative deduction to the effect that no experiment is possible upon two systems which will determine that one of them is in motion and the other at rest.]121 [He elevates the repeated failure to detect absolute motion through space into the principle that experiment will never reveal anything in the nature of absolute velocities. He postulates that all laws of nature can and should be enunciated in such forms that they are as true in these forms for one observer as for another, even though these observers with their frames of reference be in motion relative to one another.]264

[There are various ways of stating the principle of the relativity of uniform motion which has been thus arrived at, and which forms the basis of the Special Theory of Einstein. If we care to emphasize the rôle of mathematics and the reference frame we may say that]* [any coordinate system having a uniform rectilinear motion with respect to the bodies under observation may be interchangeably used with any other such system in describing their motions;]232 [or that the unaccelerated motion of a system of reference cannot be detected by observations made on this system alone.]194 [Or we can let this aspect of the matter go, and state the relativity postulate in a form more intelligible to the non-mathematician by simply insisting that it is impossible by any means whatever to distinguish any other than the relative motion between two systems that are moving uniformly. As Dr. Russell puts it on a later page, we can assume boldly that the universe is so constituted that uniform straight-ahead motion of an observer and all his apparatus will not produce any difference whatever in the result of any physical process or experiment of any kind.

As we have seen, this is entirely reasonable, on philosophical grounds, until we come to consider the assumptions of the past century with regard to light and its propagation. On the basis of these assumptions we had expected the Michelson-Morley experiment to produce a result negativing the notion of universal relativity. It refused to do this, and we agree with Einstein that the best explanation is to return to the notion of relativity, rather than to invent a forced and special hypothesis to account for the experiment's failure. But we must now investigate the assumptions underlying the theory of light, and remove the one that requires the ether to serve as a universal standard of absolute motion.

LIGHT AND THE ETHER

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.]*

THE MEASUREMENT OF TIME AND SPACE

[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.

THE PROBLEM OF COMMUNICATION

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.

AN EINSTEINIAN EXPERIMENT

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.

WHO IS RIGHT?

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 RELATIVITY OF TIME AND SPACE

[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 $T \div K$ and $L \times K$, where

$$K = \sqrt{1 - v^2/C^2}.$$

$C$ in this expression of course represents the velocity of light. It will be noted that the fraction $v^2/C^2$ 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.