CHAPTER XV CONSEQUENCES OF THE NEW SPACE AND TIME MEASUREMENTS—SIMULTANEITY
IN the preceding chapters we endeavoured to show how Einstein’s fundamental premises—the relativity of Galilean motion through space or the ether and the postulate of the invariant velocity of light—had been posited as a result of induction from ultra-precise experiment. Once these postulates are accepted, all the strange consequences of the special theory are obtained deductively by elementary mathematical reasoning.
In view of the extreme importance of the problem let us recall that the postulate of the invariant velocity of light assumes that a ray of light, in vacuo, will pass through any Galilean frame whatever with the same invariant speed in all directions, provided this speed be measured by the observer at rest in the frame, and not by some other observer in some other frame, whether Galilean or accelerated.[52] And here we must recall that classical science, believing in the absoluteness of time, assumed that events which occurred “now” anywhere throughout space would continue to occur in the same simultaneous fashion regardless of the motion of the observer. This belief was equivalent to assuming that a propagation moving with infinite speed with respect to one observer would also advance with infinite speed when measured by any other observer. In other words, an infinite speed was an absolute; it constituted the invariant speed of classical science. No such physical propagation was known to science, though it was suspected that gravitation might be of this type. At any rate, classical science accepted the existence of this infinite invariant velocity in a conceptual way. This implied the existence of the Galilean transformations, hence of the ordinary laws for the addition of velocities. Inasmuch as all measurements with rigid rods, and with clocks such as vibrating atoms, appeared to bear out the anticipations of classical science, at least so far as crude observation could detect, the classical stand appeared to be vindicated a posteriori even though the hypothesis of absolute time presented no a priori justification.
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Now, in the theory of relativity, Einstein asks us to agree that a certain finite invariant velocity which turns out to be that of light in vacuo must be considered invariant for all Galilean observers.[53] We are thus led, as explained previously, to the Lorentz-Einstein transformations, an immediate consequence of which is that molar matter can never move with a velocity greater than or even equal to that of light. This in itself is sufficient to exclude our taking into consideration observers moving with any such speeds.
But suppose that for argument’s sake we wish to consider, even though it be in a purely conceptual way, a velocity greater than light—in particular, a velocity which would constitute infinite speed for any definite Galilean observer. How would a velocity of this sort manifest itself to some other Galilean observer in motion with respect to the first?—The sole point it will be necessary to stress is that this erstwhile infinite velocity could never be invariant without coming into conflict with the invariance of the velocity of 186,000 miles per second. This becomes obvious when we consider that, according to classical science, if we added any speed—say, the speed of light —to the infinite speed, we should obtain the infinite speed; whereas, in Einstein’s theory, we should obtain the speed of light. From this it follows that the premises of classical science and of relativity are incompatible. Inasmuch as it is Einstein’s theory that we propose to discuss, we shall abandon the classical belief in the invariance of the infinite velocity.
Henceforth the invariant velocity of the universe is to be , or 186,000 miles per second. As for an infinite velocity, this will now become a relative, infinite for one observer, finite for another. Now it is obvious at first sight that if our space and time measurements were such as classical science believed them to be, it would be impossible for a ray of light to pass us with the same speed regardless of whether we were rushing towards it or fleeing away from it. A simple mathematical calculation shows us, however, that we can make our results of measurement compatible with the postulate of invariance provided we recognise that our space and time measurements are slightly different from what classical science had assumed. This is purely a mathematical problem and can be solved by mathematical means. It leads us, of course, to the Lorentz-Einstein transformations; and from these transformations it is easy to see that rods in relative motion must be shortened, durations of phenomena extended, and the simultaneity of spatially separated events disrupted.
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Einstein’s premises being consistent (as can be proved mathematically), no question can arise as to the consistency of Einstein’s conclusions; premises and conclusions must all stand or fall together. The major problem, then, is to analyse the legitimacy of the theory from the standpoint of its ability to portray the real world of physical phenomena. Some lay critics, such as Bergson, alarmed by the revolutionary nature of Einstein’s discoveries and animated by a desire to retain their classical belief in the absolute nature of simultaneity remaining the same for all observers, have endeavoured to uphold this intermediary view. They have maintained that from a purely mathematical standpoint the systems of measurement of relativity might confer greater simplicity on the purely mathematical treatment of the equations of electrodynamics; but that in no case was it permissible to assume that the anticipations of the theory could ever correspond to what we should actually measure with rigid rods and ordinary clocks. We may dismiss this view summarily. Were it correct, none of the physical anticipations of Einstein’s theory would have been verified by physical experiments.
Furthermore, a theory such as Einstein’s is one of mathematical physics, not one of pure mathematics. It is not merely necessary that the deductions of the theory be mathematically consistent with the premises. They must also lead us to anticipations which will be verified by experiment. If they failed in this respect, they would still portray the workings of a possible rational world; but the theory itself, yielding erroneous physical previsions, could never allow us to foresee and to foretell, and no scientist would ever dream of upholding it. In short, we must recognise that if Einstein’s premises are accepted, we must assume that measurements conducted with natural rods and natural clocks would yield the previsions of the theory. We must also remember that by natural clocks we do not mean clocks arbitrarily adjusted. We mean clocks such as vibrating atoms, or, more generally, all periodic physical processes evolving in a state of isolation. Likewise, by natural rods we mean our ordinary material rods maintained in the same conditions of temperature and pressure.
The type of measuring instruments is thus exactly the same as in classical science; and all that relativity tells us is that if our classical measurements had been pursued with greater precision, we should have realised long ago that the classical anticipations were belied by experiment. There is no use in pursuing the study of the theory any farther unless we succeed in understanding its physical significance. So far as the mathematician is concerned, with the Lorentz-Einstein transformations issuing as they do from ultra-precise experiment, their mathematical consequences must be deemed to correspond to the workings of the real world; and these consequences indicate clearly that length, duration and simultaneity must be relatives in a real physical sense. However, in view of the importance of the new ideas, we may consider the problem in an indirect way and thereby throw additional light on the concrete significance of these transformations.
Accordingly we shall have to discuss what the physicist means by an “event.” An event is exemplified by a something happening in a certain region of space at a certain moment of time. It matters not what actually happens. It may be a thought flashing through our minds, a bird bursting into song, or two billiard balls colliding; what is essential is a specification of position in space and in time.
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When we try to think of an event, we are compelled to credit it with more or less extension in space, and more or less duration in time. But in all problems where accuracy of localisation is demanded, we must conceive of an event as ideally restricted to an instantaneous point in space; or, if we prefer, to a mathematical point in space at a mathematical instant in time. We then get what has been called by Minkowski a Point-Event. This procedure of restricting events ideally is of course purely a mathematical abstraction and is in every way similar to the procedure followed by the geometrician when he speaks of points without dimensions, lines without width, or surfaces without thickness.
Even in ordinary life we are obliged to follow this same method of restricting dimensions ideally, in order to render a precise localisation possible. For instance, when we say we are so many miles from New York our assertion is extremely vague. What part of New York are we referring to? Even when we specify a definite building in the city, it could always be asked: “What part of the building?” We see, then, that to obtain any degree of precision we must select a definite point, and our statement that we are fifty miles from New York is equivalent to regarding New York as reduced to a point. Of course we know full well that a city is not reduced to a point; but arguments of this sort are irrelevant to the problem of localisation with which we are faced.
It is the same for time. An event lasts, it manifests duration; but in order to obtain precision we must conceive of it as ideally restricted. We thus obtain an instantaneous event occurring at a definite point of our space.
We may say, then, that a billiard ball, subsisting motionless at a point of the space of our frame, constitutes a continuous succession of point-events all located at the same point of our space (where the point-event itself would be represented by the instantaneous existence of the ball considered as reduced in size to a spatial point). Likewise, a billiard ball moving through the space of our frame would be represented by a succession of points in space at a succession of instants in time; hence by a chain of point-events.
It is not absolutely necessary to conceive of an event as something objective, actually occurring or existing. We may conceive of events and point-events as empty. Thus, a point of our space a million miles away in a definite direction at six o’clock constitutes a point-event. In the same way it is not absolutely necessary to designate a point on our sheet of paper by a pencil mark; for we can define its position by numbers. In physics, however, we are more generally interested in discussing events that are observable; so although in theory a point-event is merely the localisation of an abstract entity at a point of space at an instant of time, in practice the majority of events to which we shall refer will serve to localise observable existents. For similar reasons we usually tint the colourless liquid in the thermometer so as to render it observable; though by so doing, nothing is changed, of course, in the reading of the instrument.
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Let us now pass to the problem of determining the simultaneity of events. As a fact of common experience it is possible to localise our various thoughts and sense impressions in the time-stream of our individual consciousness; and the judgment that two sensations are simultaneous or follow in succession is essentially a priori, in that it cannot be analysed further. Science, however, could never have existed had we restricted ourselves to registering sense impressions without attempting to co-ordinate them and account for them. To these attempts of co-ordination we are indebted for the conception of the outside universe of space and time (or space-time according to the modern view)—the common objective world of science in which events occur. It is objective in that we are enabled to conceive of it as presenting the same relationships of structure for all men, hence as pre-existing as a concrete existent to the mind that discovers it and explores it bit by bit. Whether this objective universe represents a reality in the metaphysical sense (a reality which has merely been discovered by us) or whether, as the majority of scientists would assert, it reduces in the last analysis to a mere mental construct, the product of a synthesis of the mind, is a problem that need not detain us in the present chapter. In either case the physicist may speculate on this objective universe as though it were a reality pre-existing to the observer who discovers it.
Commonplace reason, or rudimentary science, was thus led to conceive of a common external objective universe in which external events occurred, occupying definite positions in space and in time. These external events, such as the explosion of a barrel of gunpowder, were assumed to be the causes responsible for our immediate sense perceptions of light and of sound. Until some three hundred years ago it was thought permissible to identify the instant of time at which an external event occurred with the precise psychological instant at which the event impressed itself on our consciousness in the form of a sound or of a visual sensation. But this very crude view led to inconsistencies when it was realised that an explosion which occurred at our sides and which appeared to us as a simultaneous occurrence of light and sound impressions would have appeared as a sequence of two separate events, a luminous one followed by a sonorous one, had we been situated elsewhere.
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If we were to retain our belief in the possibility of conceiving of a common objective universe in which a single event remained a single event regardless of the position of the observer, it became imperative to account for these discrepancies by assuming that sound transmission from place to place was not instantaneous. It was still thought, however, that light propagation was instantaneous, so that at least the instants of our visual perception of external events could be identified with the instants of their occurrence in the objective world. For example, it was usual to claim three hundred years ago that when we beheld a sudden red brightness in a star we were witnessing a stellar catastrophe occurring there-now. It appeared, then, as though men were gifted with the same a priori intuitional understanding of the simultaneity or succession of events throughout space which experience had shown to exist in the case of sense impressions located in the individual streams of consciousness. But, as Eddington points out, “This crude belief was disproved in 1675 by Römer’s celebrated discussion of the eclipses of Jupiter’s satellites; and we are no longer permitted to locate external events in the instant of our visual perception of them.” Following Römer’s discovery, scientists were compelled to discriminate between the instant an external event occurred and the instant we became cognisant of its occurrence. This discovery led them to recognise that pure intuition could yield us no reliable information on the order of succession and on the simultaneity of external events throughout space. For instance, when we suddenly perceive some stellar conflagration through the telescope, we have no means of deciding by direct intuition whether the stellar catastrophe was simultaneous with the signing of the Armistice, the battle of Waterloo or the building of the Pyramids. In the same way, were we to perceive two stellar catastrophes, it would be impossible for us to decide a priori which had preceded the other.
Now it is the problem of locating external events in a common space and time, or a common space-time according to the modern view, that is the one of paramount importance for the student of nature. Direct intuition failing him, if duration and simultaneity as between external events in the outside world were to have any meaning at all, they could be defined only as a result of some convention similar to that selected for the measurement of space with rigid material bodies. The aim of physics was therefore to decide on some conventional means of determining the simultaneity and succession of external events in a manner compatible with the existence of a common objective universe.
The most obvious method of arriving at some consistent determination of the simultaneity of two external events was to appeal to physical propagations, such as those of light or sound waves, by taking into consideration the times these propagations required to reach us.
Of course, in everyday life, the distinctions the scientist is compelled to consider are very often of scant importance. When, for instance, we bark both our shins and are conscious of a psychological simultaneity of painful impressions, we are perfectly justified in practice in assuming that we have barked our shins simultaneously. It is only when we subject an example of this sort to the rigorous logic of scientific thought that the crude view becomes untenable in theory though of course still permissible as an approximation.
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The fact is that the barking of each of our two shins constitutes an external event,[54] and our awareness of each of the two pains that ensue constitutes in turn a psychological event. Thus we have to take into consideration two external events and two psychological events. We have no more right to muddle up these two species of events than we have to confuse the firing of a shot with the hitting of the target. In any case, a certain period of time, however short, must elapse between the instant we bark one of our shins and the instant we become aware of the pain. Physiologists who have measured the speed of nerve transmissions could tell us exactly what the lapse of time would be, as measured by a clock. Unless we have reason to assert that the nerve transmissions proceed at the same speed along both legs, and unless the bruised spots on our two shins are situated at the same distance from the seat of our consciousness, our intuitional sense of the psychological simultaneity of the two sensations of pain does not permit us to decide on the simultaneity of the two external events (the barkings of our shins against some object). In practice, however, as we have mentioned, we may neglect all these subtleties; for as our psychological awareness of simultaneity is after all extremely crude, there would be no advantage in submitting our results to corrections of comparatively insignificant importance.
Nevertheless, although the differentiation we have established is of no practical interest in the case we have discussed, it possesses a vast theoretical importance, as will be seen from the following illustration: If we were to confuse the external events with our awareness of them, we might argue that a human giant with legs extending to the stars and touching, for an instant, the sun with his left knee and Sirius with his right foot, would be aware by direct intuition of the simultaneity of these events; since all he would have to do would be to register the simultaneity of his two conscious impressions—the two burns. But the argument would be faulty; for in a case of this sort, where the external events occurred at such vast distances from the seat of his consciousness, the giant could no longer afford to neglect possible variations in speed of the two nerve propagations and the difference in distance of the two points on his legs at which the external events had occurred. The only reason the popular mind retains a lingering belief in some vague intuitional understanding of the simultaneity of external events, however distant, is because when the events are near us their simultaneous perception justifies our belief in their simultaneous occurrence. We then extrapolate these results unconsciously from place to place, and fall into the erroneous belief that we possess this same intuitional understanding of the simultaneity of external events throughout space.
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The significance of the preceding discoveries can be summarised as follows: When considering the simultaneity or order of succession of events, we must differentiate sharply between events which reduce to our awareness of sense impressions, and those which refer to external occurrences. Our recognition of the simultaneity or order of succession of sense impressions or of thoughts flashing through our mind is fundamental in that it cannot be analysed further. For this reason a physical theory cannot be called upon to give a physical definition of judgments which it recognises as a priori. In contradistinction to the temporal position of sense impressions, we have to consider the simultaneity or order of succession of external events. In this case direct recognition fails us. All we can do is to note the instants at which we became conscious of these external events, and then by appealing to physical propagations deduce therefrom the instants at which the events occurred in the outside world. In order to avoid any confusion we shall therefore refer to the simultaneity of external events throughout space as physical simultaneity, reserving the name psychological simultaneity for our awareness of the simultaneity of two sense impressions.
Now all the conclusions we have reached up to this point are quite irrelevant to Einstein’s theory proper. They embody the discoveries of classical science and, while they still hold in the relativity theory, we shall see that Einstein’s contributions to the problem of simultaneity lie in another direction entirely. If these points are understood we may pass to a definite example and investigate how the physicist will proceed when he wishes to determine whether or not two external events occurring in different places are simultaneous in the physical sense.
Suppose that two instantaneous flashes of light occur at two points, and , on the road; and that we wish to discover whether or not the two flashes were produced simultaneously (always in the physical sense). The mere fact that we might perceive the two flashes simultaneously or in succession would not of itself permit us to settle the question, for the order of our perceptions would be found to vary with our position (of course only to a very small degree in the case of visual perceptions). But suppose now that we stood at the midpoint between and , as measured with our rigid rods. If we perceived the two flashes simultaneously (in the psychological sense), should we be justified in asserting that the two flashes had occurred simultaneously in the physical sense? Classical science answered no; for we were not at all certain that the wave of light was propagated from to with the same speed as the wave propagated in the opposite direction from to . Classical science assumed that the unknown velocity of the ether drift caused by the earth’s absolute velocity would interfere with our determinations, so that nothing definite could be said.
But the entire situation changes as soon as we accept Einstein’s principle of relativity and his postulate of the invariant velocity of light. We are then in a position to assert that the ether drift, whether it exists or not, can have no physical significance; and that we are therefore justified in assuming that the rays of light are propagated with the same speed in all directions. If this is the case, we have a right to conclude that the simultaneity of our perceptions of the two light flashes (in the psychological sense) ensures the simultaneous occurrence (in the physical sense) of the two external events, namely, the two flashes on the road.
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Now, in the particular example we have considered, the observer was able to station himself at a midpoint between the two points and . But situations may arise where a procedure of this sort would be impossible. Suppose, for instance, that we wish to ascertain whether two events occurring one on the sun and the other on the earth by our sides are simultaneous or not. How shall we proceed?
We will note on our watch the precise instants at which we perceive both events. Then we will take into consideration the time required for the light message from the solar event to reach us; we will deduct this duration from the instant at which the solar event was perceived, and we thus obtain the instant at which the event actually occurred on the sun. If this instant of time is identical with the psychological instant at which the earth event at our sides was perceived, we will assert that the two events occurred simultaneously. And now let us notice that by subtracting the time that light has taken to reach us from the solar event and by noting the instant thus defined, we are in effect re-establishing matters as they would have presented themselves had the news of the solar event been propagated to us through our frame instantaneously with infinite speed, so that we may present our definition of simultaneity in the following conceptual way:
We will say that if a perturbation (say, a telepathic one) is propagated with infinite speed from the surface of the sun, coincidently with the occurrence of the solar event and carrying the news of the solar event, and if this perturbation reaches us on earth at the precise psychological instant at which the earth event is perceived at our sides, then the two events will be simultaneous. Likewise, always considering the hypothetical case of instantaneous transmission, if the solar perturbation reaches us prior or subsequent to our perception of the earth event occurring at our sides, we will conclude that the solar event has preceded or followed the earth event. Of course this definition by means of a propagation moving through our frame with infinite speed is purely conceptual, since no such type of propagation is known to physicists; but as it would lead to exactly the same determination of simultaneity, we may retain it provisionally, as it will facilitate our understanding of the nature of Einstein’s discoveries.
Now, according to classical science, simultaneity having been defined in this conceptual way, any other observer, regardless of his motion, would have agreed that the perturbation had travelled from sun to earth with infinite speed, i.e., in zero time; so that in all cases the two events could be regarded as simultaneous in an absolute sense. By this we mean that they would be simultaneous not merely from the local point of view of the observer on earth, but also for all observers. In other words, physical simultaneity would be absolute; likewise, the stretch of physical time separating two successive pairs of simultaneous events occurring in different regions of space would be absolute; in other words, physical time itself would be absolute. This indeed was the belief of classical science.
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But if now we view the same problem in the light of Einstein’s theory, and if we remember that an infinite speed is a relative depending on the frame of reference or relative motion of the observer, we find that these opinions are no longer tenable. As before, the observer on the earth will be perfectly justified in believing that the two events (the solar one and the one occurring at his side) are simultaneous; but, on the other hand, another observer in relative motion would discover that the earth observer was wrong, for as referred to his own frame, the perturbation would not have spanned the sun-earth distance in zero time with infinite speed. According to him, therefore, the two events would not be physically simultaneous.
In short, physical simultaneity and duration, which were considered absolutes by classical science, must now be considered relatives having no absolute, universal significance, since they vary when computed in frames moving at various relative speeds.
But it is most important to realise that this relativity of simultaneity holds only for events occurring in different places and does not apply to events occurring in the same place. The reason for this distinction is easily understood. Suppose that two instantaneous events occur at the same point and at the same time in our frame of reference. The events, being simultaneous and copunctual in our frame, are said to constitute a coincidence of events. If now we change our frame of reference, will the two events as referred to this new frame also be simultaneous and copunctual? Obviously yes. The principle of sufficient reason itself should satisfy us on this score, since whatever disruption in copunctuality or simultaneity a change of frame might bring about, as there is nothing to distinguish one event from the other, either in position or in time, there is no reason why one event should precede rather than follow the other, either in position or in time, when the frame is changed. A concrete illustration would be given by a collision between two automobiles. Here we have two bodies situated at the same point of the road (or thereabouts) at the same instant of time; hence the observer on the road is witnessing a coincidence of events in the road-frame. Obviously, were he to view these happenings from some passing train, the collision would occur just the same; hence, so far as the train-reference-system was concerned, the two events would continue to constitute a coincidence. From this we see that, in contradistinction to the simultaneity of events occurring in different places, which is essentially a relative depending on the choice of our frame of reference, a coincidence of events is an absolute, that is, remains a coincidence or a simultaneity in all frames of reference.
We may note that in a world in which coincidences were relative, that is, in a world in which two cars would appear to collide and all the passengers killed when viewed from the road whereas no such collision would take place when viewed from the train, science would be quite impossible; for the opinions of the various observers would be relative to the point of being inconsistent. In the squirming world of relativity something at least must be absolute, and one of these absolutes is found to reside in the coincidences of events.
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We may contrast the teachings of classical science and of relativity in the following way: Just as classical science recognised that there was no physical significance in speaking of the same point of space at different times until, by selecting a frame of reference, we had objectivised, as it were, the space we were discussing, so now relativity compels us to add: “There is no meaning in speaking of the same instant of time in different places until we have objectivised time, as it were, by specifying our frame of reference.” In both sciences, however, the classical and the relativistic, the coincidence of events remains an absolute, transcending the choice of a frame of reference. From a philosophical point of view, this discovery of the relativity of simultaneity marks a date of the same momentous importance as did the discovery of the Copernican system in astronomy.[55]
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Having established the relative nature of physical simultaneity, we find it an easy matter to rediscover that other consequence of the theory, namely, the contraction of length. Consider, for example, an observer at rest on a track observing a train also at rest. What is the length of the train as referred to the observer’s frame, namely, to the earth? Obviously it is the difference in his distance from the engine and from the rear car. But if now we consider the more general case where the train or the observer with his reference frame is in relative motion (either choice comes to the same thing, so long as the motions are Galilean), it becomes imperative to state that the measurements must be performed at the same instant of time. It would be absurd to measure our distance from the rear car at one o’clock, then our distance from the engine at two o’clock; for we might find that the train had a length of over sixty miles owing to its displacement during the interval. But whereas in classical science the significance of the same time in two different places was absolute, the same for all observers, in relativity it becomes indeterminate. According to the relative motion of the observer, different simultaneity determinations will be obtained, and as a result the length of the same train will vary in value. This is what is meant by the relativity of length. Calculation proves that the greater relative velocity, the shorter will the train measure out.
In short, classical science recognised that apart from our intuitional understanding of the succession or simultaneity of the awareness of two sense impressions, such as two impressions of pain for instance (i.e., the here-now type of simultaneity), the succession or simultaneity of events could be determined only indirectly by physical means. And even so, there always remained a certain indeterminateness owing to our ignorance of the velocity of the ether drift. Assuming, however, a spread of simultaneous events to have been determined by any given observer, it was thought that these same events would remain simultaneous for any other observer. In other words, time and simultaneity were absolute.
Einstein’s theory compels us to revise the classical views. It still permits us to retain our belief in the absolute nature of a coincidence of events, that is, of events which are copunctual and which occur at the same time in any given frame. But it denies the absoluteness of the simultaneity of spatially separated events. Thus, if two observers pass each other, the events which are simultaneous with one another and with this passage according to one of the observers would be recognised as unfolding themselves in succession for the other observer. It follows that a simultaneity of events, or a sameness of time throughout space, becomes an indeterminate concept until an observer or a frame of reference has been specified; in much the same way, the weight of a given brick situated anywhere in space will be indeterminate until such time as the distribution of the surrounding masses has been decided upon. So far as the special theory is concerned, this is the sole difference between classical science and relativity.
We may say that in the general theory, where large masses of matter are in relative motion, the entire concept of the simultaneity of external events becomes obscured, for space and time determinations over extended areas become impossible. In the general theory, therefore, relativity breaks away completely from classical science. And here we may mention a number of criticisms directed against Einstein’s selection of optical signals for the purpose of defining simultaneity.
Thus, Dr. Whitehead writes: “The very meaning of simultaneity is made to depend on light signals. There are blind people and dark cloudy nights and neither blind people nor people in the dark are deficient in a sense of simultaneity. They know quite well what it means to bark both their shins at the same instant.”[56]
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In this illustration, Dr. Whitehead, by referring somewhat loosely to a sense of simultaneity, is confusing the two species of simultaneity which physicists have found it imperative to treat separately. We may recall that these two types were exemplified by a simultaneity in our awareness of two impressions (psychological simultaneity) and by a simultaneity in the occurrence of two external events (physical simultaneity). Of course, in the illustration selected by Whitehead, all the events considered, namely, our awareness of the two pains and the actual coming into contact of our shins with the rail, are so close together both in space and in time that for all practical purposes the confusion would be legitimate. Nevertheless, inasmuch as a confused and obscure illustration is liable to generate confused conclusions, it is necessary to clear up the matter from the start and specify the type of simultaneity we propose to discuss.
First let us assume, for argument’s sake, that it is the psychological type of simultaneity which is at stake. The criticism would then amount to this: Einstein has sought to render our immediate awareness of the simultaneity of two sense impressions contingent on the results of optical measurements executed a posteriori. In other words, if we should ask our friend whether he had sensed two sounds simultaneously or in sequence, he would answer: “I cannot tell you before I have looked up Einstein’s rules for simultaneity determinations and performed certain optical measurements.” It would follow that were no light rays obtainable or were our friend blind, he would never be able to give us an answer. Einstein would thus have made our immediate recognition of a simultaneity or of a sequence of sensations contingent on the accidental circumstance that we were gifted with eyesight. This seems to be Dr. Whitehead’s contention, as would appear from his reference to “blind people” in his criticism.
It seems scarcely necessary to state that neither Einstein nor any other scientist has ever attempted to defend so paradoxical an assertion. Quite the reverse. Both classical and relativistic science are fully agreed that our ability to recognise the simultaneity of two impressions (be they visual, tactual or auditory) is in the nature of a fundamental recognition which cannot be analysed further. We know that we feel two pains at the same instant, or that two thoughts flash through our mind simultaneously, and that is all there is to be said about it. Any discussion as to why we know or how we know does not concern the physicist. It is a problem which must be left to the brain specialist or psychologist. So far as the physicist is concerned, it constitutes a form of a priori knowledge over which it is useless to argue. On no account, then, can it be claimed that the theory of relativity professes to interpret this psychological species of simultaneity in terms of optical measurements or, for that matter, of physical measurements of any kind.
But there is another way to view the problem. We might claim that though Einstein has never sought to render psychological simultaneity contingent on the performance of optical measurements, yet, on the other hand, his theory fails to take this fundamental type of simultaneity into account.
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But any one at all conversant with physical science would perceive the fallacy of this contention at first blush. He would realise that physical science would be quite impossible were we to disregard the data furnished us by our fundamental recognitions. Consider, for instance, the manner in which Einstein defines the simultaneity of external events, e.g., of two flashes occurring on the embankment. He tells us that this simultaneity will be realised if the observer standing at a midpoint between the two flashes becomes aware of the two luminous impressions simultaneously. Obviously, for this definition to convey any meaning, it is essential that we credit the observer with the ability to recognise the simultaneity of these two luminous impressions. Had this fact been ignored or denied by the theory, Einstein would never have suggested his definition.
Summarising, we may say that in relativity, as in classical science, the sophisticated rules given for the determination of a simultaneity of external events presuppose that we possess the innate ability to differentiate between a simultaneity and a succession of sense impressions. These fundamental recognitions then serve as a basis for future discovery; and the sophisticated rules are used merely to allow us to pass from the recognition of a simultaneity which is a priori to those which cannot be determined a priori. As for the rules themselves, those suggested by relativity constitute but a refinement of the ones entertained by classical science. Aside from this purely quantitative difference, no change in method is involved.
It is therefore obvious that Dr. Whitehead’s criticism, when interpreted as referring to psychological simultaneity, appears to be totally unwarranted. Let us see whether it would possess any greater merit when interpreted as referring to a simultaneity of external events. The criticism would then have the following significance: Our recognition of the simultaneity of external events is fundamental. We know that two events, one on the sun and the other on the earth, are simultaneous, when we perceive them simultaneously. Physical measurements are irrelevant to the problem. Einstein’s theory fails to recognise this fact.
Now there is no doubt that the theory of relativity does fail to recognise the type of simultaneity presented in this last interpretation of Dr. Whitehead’s criticism. Hence the man who might have this understanding in mind would be perfectly justified in dissenting from a theory which would characterise his conception of simultaneity as altogether indefensible. But, for that matter, so would classical science. Both sciences refuse to agree that we possess any a priori means of determining a simultaneity of external events. Both sciences claim that in all such cases we are compelled to appeal to physical measurements of one kind or another. In other words, both sciences refuse to accept that bizarre doctrine known as “neo-realism” which consists in confusing the event that, occurs with the event that we sense.
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According to neo-realism as championed by Whitehead, and by a number of other philosophers, vision is a matter of direct apprehension[57] Events would thus be simultaneous with the instants of their perception. In the same spirit, the neo-realist speaks of events occurring behind a mirror, or of the space he sees behind the mirror, or of the convergence of the rails on the embankment. No distinction in terminology is made between the status of such perspectives and that of the physical entities existing in the objective world of science. In ancient times Æsop warned his countrymen against the dangers of neo-realism when he composed his fable of the dog mistaking the bone for its reflection in the water. Indeed, for the plain man there appears to be no difference between the attitude of the neo-realist and that of the unsophisticated infant or primitive savage who views a mirror for the first time.
In order to extricate himself from the paradoxes in which he finds himself involved, the neo-realist is finally compelled to agree that the objects behind the mirror, etc., are not quite the same things as physical objects. So he calls them sensa or sense-objects. When requested to justify his belief in these mysterious, shadowy existents, he tells us that sensa and sense-objects certainly exist, for otherwise how could a round coin ever be seen as an ellipse? Obviously (so he maintains) in addition to the round physical object, there must be an “elliptical sensum” present, and it is this that we are sensing.
It is scarcely necessary to say that no scientist could be in sympathy with such loose arguments, and accept as sound this arbitrary materialisation of qualities. Forms and colour sensations are all that we are conscious of visually. It is only by inference, as a result of a co-ordination of sensations, that we are finally led to believe in the existence of a round coin in the objective world. After we have reached this stage of knowledge it is a simple matter to account for the elliptical impression we receive of the coin, since an image of this shape can be proved to have been formed on the retina. An application of the laws of geometrical optics, and a slight knowledge of the eye’s structure, suffice; and the “sensa” and “sense-objects” of the neo-realist can be discarded as perfectly useless hypotheses.
A more detailed study, which we shall spare the reader, would convince us that neo-realism, when taken literally, rests on a confusion between inferences and direct recognitions, and leads to contradictions when an attempt is made to conceive of an objective world transcending our private views. On the other hand, if taken with a grain of salt, mitigated, as occasion demands, by such palliatives as “pseudo” or “peculiar sort of” or “not quite the same as,” it resolves itself into a mere play on words, into a giving of the same name to things which even the neo-realist would concede to be essentially different.
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Physical science, it must be remembered, deals with a common objective world. This world is not apprehended directly, but issues from a long, laborious, ever-widening synthesis of private views. Primarily, all we are aware of is a complex of sensations. Even the crude knowledge of the savage is therefore itself the result of a highly complex synthesis. But science must pursue the synthesis still farther, taking the results of previous syntheses as the elements of its more sophisticated ones. And so what was formerly the synthetic view becomes the private one when we rise to a higher plane. Thus, if we consider the aspect of the earth’s surface, viewed from any one spot, our private view would suggest that the earth was flat. Only a synthesis of a number of private views and experiences can lead us to assert that the earth is round; and science accordingly speaks of the rotundity of our planet and ignores its flatness. To put it in more technical language, the search for the objective world is similar to that of the search for the invariants or invariant equations of a group of transformations. In this illustration, each transformation corresponds to the passage from one private perspective to another, whereas the invariants correspond to the entities of the objective world with which science deals. The discovery of the space-time universe, to be discussed in a future chapter, illustrates the procedure.
To be sure, in many instances the private view may be of greater interest than the collective one. This will be the case if an artist wishes to paint a landscape on his canvas. He will, of course, draw distant objects on a smaller scale and make the rails converge. But, in contradistinction to the artist, the scientist is not particularly interested in the private view except as a means to an end. His ultimate goal is the common objective world.
The only charge that might be directed against the scientific attitude (which in this case is also that of the man in the street) is that this objective world resolves itself into a series of inferences, a mere mental construct, hence cannot possess the same measure of reality as the impressions we experience directly. The majority of scientists would grant the justice of this charge if by “what we experience directly” we mean our awareness of sensations. But they would strenuously oppose the claims of the neo-realist who endeavours to extend the meaning of these words so as to embrace the existence of his sensa and sense-objects; these represent nothing but inferences which he has drawn from his awareness of sensations. And this is precisely the error into which the neo-realist falls when, on perceiving a red flash in the sky, he talks of a star bursting into prominence “there-now.” If he wishes to avoid inferences, he should confine himself to saying: “I experience ‘here-now’ a luminous impression on a dark background.” For all he knows, the luminous impression may be due to an aviator’s light, or to a firefly, or to a purely subjective hallucination.
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At any rate, the point we wish to stress is that while it is correct to say that the objective world is a synthesis of private views or perceptions, we should not forget to add that the private views are themselves the product of a synthesis of sensations. But inasmuch as science must deal with the general, and not merely with the particular, and inasmuch as it is the common objective world that renders this general knowledge possible, it will be this world that the scientist will identify with the world of reality. Henceforth the private views, though just as real, will be treated as its perspectives. We must recognise, therefore, that the common objective world, whether such a thing exists or is a mere convenient fiction, is indispensable to science, if in no other capacity than as a working hypothesis. And neo-realism, as we have said, when pursued consistently, renders a collective view impossible. Furthermore, from the standpoint of science, whether it be classical physics, relativity, physiology or psychology, neo-realism is untenable, for under no circumstances can vision and our knowledge of nature in general be considered a matter of direct apprehension. In much the same way, we know that 7 times 8 is 56, and we obtain this result immediately, without reflection and without having to count on our fingers. Were it not for the fact that the neo-realist recollects the difficulty he experienced in learning the multiplication table, he would presumably be telling us that our knowledge of the properties of numbers was also a matter of direct recognition. From the standpoint of science, there is very little difference between the alleged immediacy of perception and our knowledge of the product of 7 times 8. Neither of these two forms of knowledge can be viewed in the light of direct recognitions. At any rate, regardless of our attitude towards neo-realism, in scientific discussions we must define the meaning of our words clearly and cease to apply the same word, “simultaneity” to a simultaneity in the scientific sense, as holding for the objective world, and to a simultaneity in the neo-realistic sense, as assumed to be apprehended directly. It is remarkable that Dr. Whitehead, who has coined such a profusion of new words when the necessity was scarcely apparent, should have missed this splendid opportunity when it happened to be imperative.[58]
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However, it is not our intention to discuss neo-realism further, for the doctrine is of no interest to science. All we wish to point out is that in any scientific theory, whether it be due to a Newton, a Maxwell or an Einstein, we can make up our minds from the start that the author will deal with the objective world of science, treating the private views as mere perspectives, and will frame his concepts and definitions accordingly. It would appear, then, that if Dr. Whitehead wishes to defend a doctrine which stands in conflict with the views of every scientist from Newton to Einstein, it would have been conducive to greater clarity had he restricted his attacks to classical science. For when we attack a new theory such as Einstein’s, which has introduced such a wealth of revolutionary ideas into our understanding of nature, it is to be presumed that the reasons for our dissent arise from those ideas and not from the classical ones which the theory has left intact. Thus, if it were to be proved that matter was continuous and not atomic, we should not hold Einstein’s theory responsible for our past belief in the atomicity of matter. Rather would we challenge the experiments and theories of those classical physicists who thought they had established atomicity. On the other hand, if we objected to the view that the simultaneity of external events was relative to the observer’s motion, claiming it to be absolute, then of course we should be justified in criticising Einstein’s theory proper, since the relativity of simultaneity is peculiar to it.
To be sure, Dr. Whitehead appears to be under the impression that he is merely attacking the new revolutionary ideas introduced by Einstein. Thus, he tells us that he is merely defending “the old-fashioned belief in the fundamental character of simultaneity,” but one may well wonder to what period of human thought he is referring. Classical science, some three hundred years ago, had recognised the error of this confusion between external events and the instants at which they were perceived, precisely because this attitude was found to lead to results in utter conflict with experience.
As Weyl expresses it, referring to Römer’s discovery in 1675: “The discovery that light is propagated with a finite velocity gave the deathblow to the natural view that things exist simultaneously with their perception.” Needless to say, that which applies to luminous transmissions applies in exactly the same measure to the propagation of nerve excitation through the body and to all other species of transmissions.
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The general conclusions we have reached from a discussion of simultaneity in different places apply without great change when we consider a number of other concepts which in a dim way we half feel and half comprehend. Thus, consider the case of temperature. Why does the scientist have recourse to thermometers rather than trust to the immediate disclosures of his senses? Merely because the thermometer furnishes an objective definition of the equality of two temperatures, and because it can discern differences which our senses would be unable to detect. The result is that when his thermometer indicates differences in temperature of several hundred degrees centigrade—say, temperatures of -200° and +400°—he maintains that such a difference exists, even though to our senses a scalding sensation would result in either case. But, however imprecise and unreliable our direct appreciation of temperature may be, our direct recognition of a simultaneity of spatially separated events is vaguer still. For whereas men may have no great difficulty in agreeing that one room is warmer than another, even without the aid of a thermometer, they will express the most conflicting opinions when asked which of two explosions occurred first. Their answers, unless governed by the results of physical measurements, will vary with their relative positions.
Now, because the scientist measures temperature by the thermometer, no one would accuse him of wishing to imply that the concept of temperature would cease to exist were his thermometer to fall to the ground and be broken. Why, then, should the fact that we determine simultaneity in different places by means of light signals imply that we are making the concept of simultaneity dependent on the behaviour of light, and that simultaneity could have no meaning for blind people?
The sole purpose of all these measurements is to render precise and to clarify concepts which we understand only in a dim, indefinite way. In the majority of cases these concepts have arisen from crude experience. There are instances, however, where they have arisen in a more sophisticated manner. Suppose, for example, that our bodies were insensible to heat or cold. The behaviour of our thermometers would still suggest an abstract concept which we might call “temperature.” In fact, even were no thermometers at our disposal, the concept of temperature would be introduced eventually into physics for theoretical reasons. “Entropy” affords us an illustration of a concept introduced in this way. Our senses cannot detect it and our instruments do not record it directly. Nevertheless, it has imposed itself on science. A difference in entropy can be measured indirectly, just as temperature, duration and space can be measured, but always by physical means; and this is indeed the only way we can increase our understanding of these concepts, and agree on what we are talking about. Incidentally, we may recall that it was through the medium of physical measurements that Einstein was able to demonstrate the relativity of simultaneity, which our so-called direct apprehension of nature had never even led us to suspect. Hence, to accept the relativity of simultaneity, which issues from highly sophisticated physical measurements, and then to cling to the theory of a direct apprehension of nature, is a course which, to any logical mind, must appear to be the height of inconsistency.
We have now to consider a second criticism also advanced by Dr. Whitehead. In it he appears to have abandoned momentarily the doctrine of direct apprehension, recognising (presumably for the sake of argument) that a determination of simultaneity at spatially separated points may require the introduction of physical propagations, but he objects to Einstein’s exclusive appeal to optical transmissions for this purpose. Thus, he tells us:
“Also there are other physical messages from place to place; there is the transmission of material bodies, the transmission of sound, the transmission of waves and ripples on the surface of water, the transmission of nerve excitation through the body, and innumerable other forms which enter into habitual experience. The transmission of light is only one from among many.”
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Inasmuch as these arguments are advanced in a spirit of criticism, we must presume that according to Dr. Whitehead, had transmissions other than optical ones been considered, determinations differing from Einstein’s would have ensued.[59] But a contention of this sort would be fundamentally untrue. Einstein’s determinations would have yielded exactly the same results had sound transmissions been substituted for optical ones; and the only reason optical ones are continually referred to is because they permit observations of greater accuracy. We shall now consider the reasons for these statements.
The problem of time determination may be divided into two parts: First, we wish to define equal successive durations at the same point of space, for instance, at the point where we happen to be standing. Secondly, we wish to co-ordinate our time reckonings with those computed by other observers situated at different points of space.
As we have already considered the problem of time-congruence at a point, we will not dwell on it unduly. Suffice it to recall that our direct intuition of time-congruence is far too vague and uncertain to be of any use in scientific investigation. Accordingly, throughout the course of history we find men relying on physical processes of one sort or another, the burning of candles, sand clocks, mechanical clocks, rotation of the earth, vibrations of atoms, etc. Newton gave a theoretical definition of time-congruence when he formulated the law of inertia, according to which a perfectly free body described equal distances in equal times. According to this definition, by measuring equal distances along the body’s path, we were enabled to mark out successive equal durations. But in common practice it was more convenient to appeal to chronometers regulated ultimately by the earth’s rotation. We may now pass to the problem of synchronisation in classical science.
Let us assume that we are in possession of a number of chronometers which, when placed side by side, beat out their hours, minutes and seconds in perfect unison, hence advance at the same rate. Suppose now that having drawn a circle of, say, ten miles’ radius, we maintain one chronometer at the centre and carry the remaining chronometers to various points distributed on the circumference of the circle. If some one were to play a prank and displace the hands of the chronometers located on the circumference, we should be faced with the problem of re-establishing synchronism. Classical science suggested various methods for obtaining this result.
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First, we might transport our centre chronometer to each of the circumference chronometers in turn, and thus re-establish synchronism. But this method was open to the objection that by displacing a chronometer back and forth we might in some way disturb its working.
A second method was to appeal to the earth’s rotation on its axis. By ascribing a value of 24 hours to the complete rotation of our planet, we could divide its surface with equally distanced meridians, and then assert that the passage of the same star from one meridian to the next would always take the same time. Suppose, then, for argument’s sake, that the earth’s surface had been divided by 24 half meridians, all equally spaced. Then if a star passed a certain meridian at at a certain time , it would pass the contiguous meridian one hour later. In order to synchronise two clocks situated on these two meridians and , all that would be necessary would be to set the first clock at “zero” hour as the star passed its meridian , then set the second clock at the hour “one” as the star passed over the second meridian . The clocks would then be synchronised, so that an event occurring by the first clock as its hands marked a time , would be simultaneous with an event occurring by the second clock as its hands also marked . But this method again was open to certain objections which it is unnecessary to recall since we have discussed them in a previous chapter. They deal with the retarding influence of the tides and with the earth’s “breathing.” To be sure, these objections are of a hair-splitting nature, but they are theoretically valid just the same since they do not reduce to mere errors of observation.
A third method of synchronisation considered by classical science was obtained by appealing to propagations, which would serve to establish communications between the spatially separated clocks. The propagation of sound waves suggested itself in this capacity. According to this method we should operate as follows: At noon, sharp, as marked by our chronometer at the centre, we should fire a gun. All the observers standing by their clocks on the circumference would then be instructed to set their clocks, the instant they heard the report, at the hour “noon plus the time required by the sound to progress from centre to circumference.” Given the velocity of sound in air, and given the distance from centre to circumference, this lapse of time could be computed, and the clocks synchronised. But here again we should have to make sure that no wind was blowing; also, even in the case of a stagnant atmosphere, we should have to assume that the speed of sound was constant and remained the same in all directions, i.e., that its propagation was isotropic. We knew that this was the case to a first approximation, but no highly refined experiment had ever established the fact.
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Finally, for sound waves we might substitute light signals or radio signals. This was the method most generally employed by classical science; it permitted greater accuracy than sound signals and was of wider applicability than determinations in terms of the earth’s rotation. When, for instance, the astronomer wished to determine the present position of a comet, the optical method was the only one possible. Accordingly, he was compelled to take into consideration the time required for the comet’s light to reach him. In other words, simultaneity was determined by optical signals. In much the same way, the Eiffel Tower in Paris sends out a radio message at noon every day, and ships at sea can regulate their clocks accordingly. (Needless to say, all this has nothing to do with the Einstein theory.) Nevertheless, while recognising that a determination of simultaneity via optical signals was the most precise method possible, classical science had certain misgivings as to its theoretical accuracy. The effect of the ether drift, generated by the earth’s motion, was assumed to produce an anisotropy in the propagation of light, much as the wind would do in the case of sound, causing it to proceed more rapidly in one direction than in another. Inasmuch as the velocity of this ether drift was unknown, there was no means of introducing the necessary corrections.
Now, what Einstein urges us to recognise, as a result of the negative experiments in electromagnetics, is that these misgivings of classical science were unfounded. Whether it exist at all or whether it be a myth, the ether drift exerts no effect on the propagation of light. The space of a Galilean frame is isotropic so far as the propagation of light is concerned; that is to say, light proceeds with the same speed in every direction. These results having been established by the most accurate experiments known to science, there is every reason to accept them until such time at least as their validity is disproved by further experiments still more precise in nature. Under the circumstances, the optical method of fixing simultaneity, which was already the one followed prior to the theory of relativity, becomes, in view of Einstein’s discovery, the most perfect method at the command of science.
From this rather long discussion we see that in the determination of simultaneity Einstein’s theory does not suggest any essentially new method. With either science, the measure of time and the determination of simultaneity in different places can be arrived at only by an appeal to physical processes of one sort or another. If we wish to discard physical determinations, and define things in the abstract, the concept of simultaneity in different places degenerates into a mere verbal expression. The following passage quoted from Einstein, in his Princeton lectures, may clarify the matter:
“The theory of relativity is often criticised for giving, without justification, a central theoretical rôle to the propagation of light, in that it founds the concept of time upon the law of propagation of light. The situation, however, is somewhat as follows: In order to give physical significance to the concept of time, processes of some kind are required which enable relations to be established between different places. It is immaterial what kind of processes one chooses for such a definition of time.[60]
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“It is advantageous, however, for the theory, to choose only those processes concerning which we know something certain. This holds for the propagation of light in vacuo in a higher degree than for any other process which could be considered, thanks to the investigations of Maxwell and H. A. Lorentz.
“From all these considerations, space and time data have a physically real, and not a mere fictitious significance.”
Let us return to Whitehead’s criticisms. It must be admitted that from the standpoint of experiment and observation, optical determinations are amenable to greater accuracy than all other methods of observation based on the propagation of sound, etc. Not only are they easier to observe, but they are also less contingent on variations in surrounding conditions (e.g., velocity of wind). Now, as we have remarked previously, when the critic attacks the theory of relativity on the ground that it defines simultaneity by optical methods, it must be presumed that he attributes the strangeness of Einstein’s theory to its exclusive use of light signals. Were this opinion correct, it would certainly appear rather arbitrary to found a theory of the universe on one particular choice of time determinations, when other methods would have yielded our classical results. Why, indeed, define time in terms of light rather than of sound or moving bodies? But, as mentioned elsewhere, the opinion is not correct, for regardless of the type of propagation to which we appeal, exactly the same determinations of simultaneity would ensue under ideal conditions of observation. Every experimental test which has verified the anticipations of the theory proves this last statement indirectly.
In order to make this fundamental point clearer, let us consider the following concrete example: Suppose we stand at a midpoint between two guns. According to Einstein’s optical criterion, the two guns would have been fired simultaneously were we to perceive the two flashes at the same instant of time. Classical science would have dissented from this view on the ground that the ether drift would have vitiated our results in a minute degree; but even classical science would have granted that had the physical effect of the ether drift been proved inexistent, there was nothing further to dissent from in Einstein’s definition. If, then, with Einstein, we assume that the negative experiments have proved the irrelevance of the ether drift, both classical science and relativity would recognise that a simultaneity in the perception of the two flashes would have entailed a simultaneity in the hearing of the two reports (assuming no wind was blowing). Also, both bullets would pass us at the same instant (assuming the guns, charges and bullets were identical).
It follows from this that Einstein’s theory, since it has started out by claiming the effect of the ether drift to be nil, must inevitably lead to the conclusion that however we elect to define simultaneity (whether by the optical, acoustic or bullet method), exactly the same simultaneity determinations will ensue. Dark, cloudy nights or the accidental circumstance that we are gifted with eyesight have therefore nothing to do with the matter; and blind men would not be debarred, since they could always synchronise their clocks by means of sound signals.
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We see, then, that the strangeness of Einstein’s theory arises not from his use of optical signals, but from his fundamental premise that the ether drift is incapable of affecting the course of physical phenomena. Those who wish to dissent from the theory should therefore concentrate their attention on the ether-drift problem. Einstein’s preference for optical signals is a mere side issue which could perfectly well be discarded in favour of sound transmission were it not for reasons of practical convenience and accuracy. No change in the theory would ensue on this score. We have established this point in the case of simultaneity, but in view of its importance we may consider the problem in a more general way.
As we have noted, once the irrelevance of the ether drift is admitted and the equations of electromagnetics accepted as accurate, we are led to those space and time transformations known as the Einstein-Lorentz transformations. When we assert that duration might just as well have been defined in terms of moving bodies or sound transmissions (under ideal conditions of observation), our statement is equivalent to asserting that the Lorentz-Einstein transformations, though obtained from a study of electromagnetic and optical phenomena, are yet of universal applicability to all manner of phenomena. Here the critic may object that this is a gratuitous assumption. Maybe it is, but we must understand that were its validity to be contested, were the classical transformations to hold for mechanics and acoustics, while the Lorentz-Einstein ones were valid for electromagnetic processes, we should be faced with insufferable contradictions. Thus, according to the Lorentz-Einstein transformations, as applied to electromagnetics, a ray of light proceeding in vacuo will pass any two different Galilean observers with the same speed . But according to the classical transformations, as applied to mechanics, a body which is moving with the velocity with respect to one observer will be moving with a different velocity relatively to the second observer. If, then, a body were moving along with the crest of a light wave in the opinion of one observer, it would either outstrip the wave or fall behind it in the opinion of the second observer. A coincidence for one would no longer be a coincidence for the other. Coincidence would thus cease to be absolute, and it is doubtful whether any science could exist, since all unity in nature would be lacking. Furthermore, the coexistence of two conflicting species of space and time transformations appears very improbable when we remember that it is impossible to establish a clear-cut distinction between mechanics and electrodynamics, since matter seems to be in large part an electromagnetic phenomenon.
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Hence it is only natural that the scientist should assume that the Einstein-Lorentz transformations, though deduced from electromagnetic or optical processes, should yet hold in all domains of physical science, even in the interior of the atom. It is true that these transformations were obtained from a study of optical and electromagnetic phenomena, not from acoustics or mechanics; but this is merely because experiments as sensitive as those of Michelson and Fizeau could scarcely have been performed with sufficient accuracy had sound waves, nerve excitations or ripples on the water been contemplated. And yet, had Fizeau’s experiment been performed with sound waves, there is every reason to suppose that the curious composition of velocities would have been observed.
The arguments thus far presented show that the space and time determinations derived from electromagnetics and optics are in all probability also those holding for the entire physical universe. Such is indeed Einstein’s attitude when he unhesitatingly applies the relativistic transformations to all manner of phenomena. But it is precisely because as a result of this application he has anticipated phenomena at variance with those deduced from the classical transformations, and it is precisely because his anticipations have been verified by highly accurate experiments, that his assumption of the universal validity of the relativistic transformations must be considered justified.
On no account, therefore, may we regard time and simultaneity, when defined by optical signals, as constituting a special kind of optical or electromagnetic time and simultaneity, at variance with equally justified determinations issuing from mechanical processes or sound transmissions.
In short, if we wish to co-ordinate natural phenomena with simplicity, we must thus conceive the various time-flows and simultaneities defined by Einstein, to represent the times imposed upon us, not by the caprice of a mathematician, not by optical transmissions alone, but by the entire physical world. The times of relativity are therefore those which will govern all physical processes, not merely optical ones; and for this reason we must regard them as representing the real time-flows of the universe.
An understanding of this point is essential. For instance, in the general theory we shall see that Einstein’s law of gravitation necessitates a slowing down of time in the proximity of large masses of matter. If we accept the general significance of this fact as affecting all physical phenomena, it will entail the slowing down in the rate of vibration of an atom located in a gravitational field. This is the celebrated Einstein shift-effect, which has since been verified. Before it was detected, however, and while it was still considered doubtful, some thinkers had argued that it might perfectly well prove to be non-existent on the ground that the atom might not act like a perfect clock and might fail to adjust itself to the local rate of time-flow. Einstein, however, refused to accept this means of escape and stated that the entire theory would have to be rejected if the effect were not verified, for the times of relativity should be of universal significance or else worthless.
We may conclude by saying that it appears unjustified to refer to Einstein’s time and simultaneity determinations as depending on the behaviour of light. They depend on the processes of nature of which the behaviour of light is but a particular illustration. We must therefore amend Dr. Whitehead’s criticism of relativity, as making the very meaning of simultaneity depend on light signals, by replacing the words “light signals” by the words “physical processes.” But when amended [Pg 186] in this way, the criticism loses all its force. For we may speak of simultaneity throughout space, of an instantaneous space, as much as we please, but until we know enough about these evasive concepts to be able to distinguish events that are simultaneous from those which are not, we cannot claim to have any definite idea of what we are talking about. On this score, both classical science and relativity are in perfect agreement.
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CHAPTER XVI PRACTICAL CONGRUENCE IN RELATIVITY
IN a preceding chapter we mentioned certain of the most important aspects of the problem of physical space. We saw that the concept of spatial equality or congruence was deemed to have arisen from the facts of experience. Certain objects appeared to maintain the same visual aspect wherever displaced, provided we modified our own positions as observers in an appropriate way. But such fundamental recognitions were too vague to be of any use to science; hence congruent bodies were defined as those which, when maintained at constant temperature and pressure, coincided when placed side by side.
Congruence, as thus defined, involved physical measurements with material bodies; and, as Poincaré remarked, all we could ever discover in this way would reduce to the laws of configuration of solid bodies, space itself transcending our experiments, since the bodies might behave one way or another in the same space. Poincaré’s attitude drives us to complete agnosticism so far as the geometry of space is concerned. If physical measurements are denied us, there is no means of solving the problem of space, for we have no a priori means of deciding that the structure of space is this or that. Logical arguments are of no avail, for they do not lead us to any definite solution, only to a variety of possibilities. We are thus thrown back on Poincaré’s main contention, i.e., “space is amorphous.” In it we can define congruence in any way we please (theoretical congruence), although for reasons of practical convenience it is necessary to be guided by the properties of so-called rigid bodies. Thus we obtain a physical definition of practical congruence, which permits us to determine the geometry that for all practical purposes is to be called the geometry of real physical space.
When we consider Poincaré’s arguments, there is very little to be said against them. Nevertheless, the physicist must concentrate his attention on the space with which he can cope, with the one that can be explored by physical methods. This attitude defended by Einstein (which was also the attitude of Gauss and Riemann) leads us to the space whose geometry is defined by the paths of light rays, by the laws of nature when expressed in their simplest form, and again by the numerical results of measurement obtained with material rods maintained under constant conditions of pressure and temperature. It is with this space that we shall be concerned.
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But before proceeding, let us make it quite plain that according to the present view the office of our rods is to reveal the pre-existing structure of space much as a thermometer reveals the pre-existing differences of temperature throughout the house. On no account, therefore, may it be claimed that the rods are assumed to create space any more so than the thermometer creates temperature or the weather glass creates the weather. The rods permit us to explore space because they adjust themselves or mould themselves to its structure, or metrical field. The origin of this structure is still very mysterious. For Riemann it was created by matter; but by matter he did not mean the matter of our rod: he meant the total aggregate matter of the entire universe. Einstein’s cylindrical universe confirms Riemann’s views. At all events, regardless of the mysterious problem of the ultimate origin of the metrical field, regardless of whether this structure of space exist per se as a characteristic of physical space or whether it be generated by the masses of the cosmos, in either case it pre-exists to our local exploration with rods, whose masses are far too insignificant to modify it to any perceptible degree.
Also let us recall that when we discuss space, we must specify it by referring it to a frame of reference, in particular to a Galilean frame (one in which no forces of inertia are experienced). We then proceed to consider the laws of disposition of our congruent bodies in this frame, and the nature of the geometrical results obtained will define the geometry of space. As we know, Euclidean geometry is the outcome (at least to a first approximation).
Now classical science had never thought it necessary to stipulate that our rods should stand at rest in our frame, but in view of the disclosures of relativity we see that it is essential to specify this condition. For two rods whose extremities would coincide when at relative rest would cease to coincide when in relative motion, so that a square traced on the floor of our frame would turn out to be an elongated rectangle when measured by a rod in relative motion. To be sure, similar results would have been expected even in classical science, had it been assumed that the moving rod was compressed or distorted owing to its motion through the ether. In a case of this sort, the modified readings of the rod would not have involved space and its geometry, any more than when by heating or crushing a rod we alter its readings. The blame for the discrepancy would have been placed solely on the disturbing conditions affecting the rod.
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Under Lorentz’s theory this interpretation could be entertained, since the FitzGerald contraction would have been explained in terms of the pressure generated by the rod’s motion through the stagnant ether. But under the theory of relativity this interpretation can no longer be defended. For when relative velocity exists between and (not relative acceleration), there is no sense in enquiring whether it is or which is in motion. This, indeed, constitutes the essence of the special principle of relativity. Hence, when we consider the moving rod, we can ascribe no absolute significance to its motion. Whatever motion exists between frame and rod might with equal justification be attributed to the motion of the frame gliding beneath a motionless rod, there being no absolute term of comparison like the stagnant Lorentzian ether to decide the issue. And so we cannot claim that the so-called moving rod has suffered a physical change by reason of its motion, a change rendering it unfit for measurement purposes.
And yet, if we retain all our rods as equally valid, whether in motion or at rest, we obtain conflicting results which are such as to deprive the geometry of space of all significance. When we analyse this anomalous situation we find that the following facts stand out clearly: Rods remain unmodified whether in motion or at rest, and yet their length varies. The truth is that length is not a definite characteristic of a body; it appears as a shadow of a something else projected into the space of our frame. When the rod is at rest in our frame, its length, or the shadow, presents a maximum; it is as though the something and its shadow coincided. But when relative motion is present, the something becomes tilted—tilted along a fourth dimension out of the space of our frame; and its shadow is shortened in proportion. Yet this rod which is moving in our frame is at rest in some other frame, and in this other frame, therefore, the something again coincides with its shadow. And so we must assume that the various Galilean frames and the spaces they serve to define are variously tilted, rotated in a fourth dimension with respect to one another.
There is then no longer one all-embracing Euclidean space which may be defined with respect to one Galilean frame or another. In its place we must conceive of an indefinite member of Euclidean spaces variously tilted, yet fused together.[61] To each separate Galilean frame one of these spaces will correspond, so that when we change our motion we are also changing our space.
The situation is analogous to that which arises when we consider various verticals drawn to the earth’s surface.
John, in London, will assert that his vertical is truly vertical and that Peter’s at New York is slanting, and Peter will return the compliment. Owing to these discrepancies, we cannot regard the two verticals as defining one same direction, though they both may be truly vertical for John and Peter respectively. Rather must we say that there exist an indefinite number of different vertical directions, no one of which is more truly vertical in any absolute sense than any other. Now replace the concept of vertical by that of three-dimensional space, and we have a picture of the spaces of relativity.
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Very similar conclusions apply to time. Thus, consider two observers and in their respectively moving frames. Each observer holds a watch in his hand, and these watches, when at rest with respect to each other, beat the same time; they define congruent stretches of time. The classical idea of practical time-congruence, founded on approximate observations, led us to maintain that these conditions would still endure regardless of the positions of the clocks in space and regardless of their relative motions.
But it now appears that the time-stretches defined by the ticking of a clock at rest in the frame of reference would appear to an observer in relative motion to be longer than they would appear to . The result is quite general and applies to all processes. Thus, the duration of the revolution of a top will be possessed of varying values according to the relative motion of the observer. Arguments similar to those we elaborated in the case of space go to show that time is no longer unique. A multiplicity of different time directions, variously tilted, must coexist, each one referring to some particular frame of reference, hence to some particular observer. As a result, the simultaneity of two events occurring at different points of space can no longer be absolute, since simultaneity depends essentially on the time which we select, and therefore on our frame of reference.
Just as each frame had its own space, so also would it now have its own time and its own definition of simultaneity. We may summarise these various discoveries by stating that there can exist no universal definition of practical congruence either for space or for time; there can exist, therefore, no geometry for a universal space or for a universal time.
The philosophical consequence of these new points of view is to deprive universal space and time of the objective significance with which they were formerly credited. Henceforth, in the words of Minkowski, “space and time, by themselves, sink to the position of mere shadows.”
Thus, consider a yardstick. If this same yardstick is measured by various observers passing it with different velocities, it will be possessed of different lengths. The space this rod occupies can then be credited with no particular magnitude, since this magnitude itself depends as much on the point of view of the observer as on the characteristics of the rod. This is what is meant by the relativity of distance. We must not confuse this discovery of Einstein’s with the fundamental relativity of all distance in conceptual space. For while classical science recognised that there was no such thing as absolute distance in conceptual space, yet it believed that in the real space of the physicist the distance between two given points in a frame of reference could be defined unambiguously for all observers, by means of rigid rods. It is this last opinion which is shattered by Einstein’s discoveries.
It is the same for duration when we remember that the duration of any phenomenon will be possessed of varying values according to the relative motion of the observer; and it is this fact which is expressed by the statement that time or duration is relative.
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Thus, neither space nor time by itself can exist in the real phenomenal world which the physicist explores. Space and time appear as mere modes of perception, mere relations, varying and changing according to the conditions of relative motion existing between the observer and the observed. With the disappearance of the objectivity of space and time regarded as absolute and universal forms of our perception, the objectivity of the entire physical world seemed to sink into a mist. If, then, any kind of objectivity was to be retained, it was necessary to effect a synthesis of all the individual points of view of all the different observers, to mould into one sole representation the multiplicity of individual spaces and the multiplicity of individual durations.
Could this result be accomplished, the relativity of practical congruence for both space and time would give way to some universal definition of practical congruence holding for all observers regardless of their relative motion, and connoting, therefore, the existence of some absolute continuum no longer a mere shadow. In place of the individual point of view varying with the relative motions of the observers, we should obtain an impersonal and hence common objective understanding of nature.
The achievement of this supreme synthesis of the points of view of all observers was accomplished by Minkowski in 1908. He succeeded in obtaining an invariant definition of congruence by combining any given observer’s definitions of practical congruence for space and for time, and by showing that this combination possessed an invariant value holding equally for all other observers. Of course, the type of congruence obtained was one neither of space nor of time. It was a combination of both. But its existence showed us that, transcending space and time, there existed an impersonal and fundamental four-dimensional metrical continuum which Minkowski called space-time. The definition of congruence obtained by him for this mysterious continuum proved it to be Euclidean (more properly, semi-Euclidean).
The elusiveness of space and time, or rather the ambiguity of these concepts, depending as they do on the observer who measures and senses them, is replaced by the common objectivity of this fundamental continuum, which is the same for all and transcends the particular conditions of motion of the observer. This continuum, though of itself neither space nor time, yet pertains to both in that the observer is able to carve it up into that particular space and that particular time which are characteristic of the frame of reference in which he is stationed, and which constitute the space and time in which he lives and experiments.
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From now on, the real extension of the world has a fourfold order, and the geometry of relativity is necessarily four-dimensional. What we call the space of our Galilean frame at an instant, is but a cross-section of this four-dimensional space-time world; and what we call the time of our frame lies along a perpendicular to this section. As we pass from one Galilean frame to another, we change our space and we change our time. Our new instantaneous space, though still a three-dimensional cross-section of the four-dimensional space-time world, is tilted with respect to our former cross-section; and the same tilting ensues for the perpendicular direction called time. In fact, the situation presents a striking analogy with the various directions of the verticals to the earth’s surface. They, also, manifest different directions because the various two-dimensional portions of the earth’s surface are variously tilted in three-dimensional space. The great change in our understanding of the spatio-temporal background, the great difference between space-time and the separate space and time of classical science, resides, therefore, in this tilting process which accompanies a change of relative motion.
In order to make these points clearer, let us understand that when we speak of the instantaneous spaces of two observers in relative motion as being tilted one with respect to the other, hence as lying outside of each other (except for their common two-dimensional cross-section), we do not mean that the space of one is some incomprehensible entity in the opinion of the other. The tilted instantaneous spaces will enter into our perception, but not as spaces alone: they will appear as a succession of two-dimensional surfaces moving with uniform speed.
All these spaces are equally justified. No one of them stands out more prominently than any other on the ground of symmetry with respect to the four-dimensional world. Only when we specify the frame of the observer will one particular space and one particular duration be allotted. Hence we may say that practical congruence exists for space and for time, as in classical science, provided the Galilean observer effects his measurements with rods and periodic mechanisms, such as clocks, which do not move about in his Galilean frame while performing measurements. There is thus a perfectly definite physical meaning in stating that the distances between two point-pairs at rest in the frame are congruent or unequal; but we must specify that this distance is measured according to the standards of the frame; and the same applies to time-stretches. Likewise, as in classical science, the Galilean observer will discover upon measurement with his congruent rods that his space is Euclidean. It is only when the rods and clocks are in relative motion or when, while at rest in our frame, the frame happens to be accelerated or submitted to gravitational action, that they cease to measure congruent stretches. It is only, therefore, when we reason in an impersonal way, without specifying any particular frame, only when we reason from the standpoint of all possible observers, whatever be their motion, that space and time fade away into shadows; and it is only then that we are compelled, whether we like it or not, to reason in terms of the common objective world of relativity, that is, in terms of four-dimensional space-time.
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CHAPTER XVII THE MATHEMATICAL EXPRESSION OF EINSTEIN’S FUNDAMENTAL PREMISES
CONSIDER a Galilean system, and two points and in this system. If a ray of light is propagated from to , the postulate of the invariant velocity of light demands that the distance measured by us in the frame, divided by the duration required (according to the time of our frame) for a light wave to cover this distance, be always equal to , the velocity of light. If, now, we had viewed the same phenomenon from some other Galilean system, the points and of the first system at the instants when the ray of light passed them would have corresponded to some other points and of our second system. But, just as in the first case, we should have been able to measure the velocity of light between the two points and of our second system. The principle of the invariant velocity of light states that in whatever Galilean system we might have operated, the measured velocity of light in vacuo would always be the same.
Of course this velocity may be positive or negative, according to whether the light ray is directed to the right or to the left; but we can obviate this ambiguity of sign by considering squared values.
The mathematical translation of this principle of physics yields us the following equation, which must remain invariably zero in value for all Galilean frames: When we pass from one Galilean frame to another, , , , may vary in value; but the variations must be so connected that the sum total of the above mathematical expression remains invariably zero. Such is the mathematical condition which expresses the principle of the invariant velocity of light.
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Now it is our object to determine exactly in what measure these different magnitudes , , , must vary in value when we pass from one Galilean frame to another, moving with a constant velocity with respect to the first, if the previously written mathematical condition of invariance is to remain satisfied. From a purely mathematical standpoint problems of this type form part of a branch of mathematics known as the theory of invariants. Such problems had been studied many years before, and it was known that the relations between the variables , , , , or, what comes to the same thing, the transformations to which it was necessary to subject these variables (in order to satisfy the condition of invariance set forth above), were given by a wide group of transformations known as conformal transformations.[62]
But when, in addition, the relativity of velocity is taken into consideration it is seen that conformal transformations are far too general. We must restrict them; and when the required restrictions are imposed we find that the rules of transformation according to which the space and time co-ordinates of one Galilean observer are connected with those of another depend in a very simple way on the relative velocity existing between the two systems. These rules of transformation are given by the Einstein-Lorentz transformations.[63]
Now these transformations are, as we have said, more restricted than the conformal transformations; and this lesser generality of the Einstein-Lorentz transformations has, as a consequence, the further restriction of the conditions of invariance of the mathematical expression mentioned previously. Not only will this expression have a zero value for all Galilean frames when it has a zero value for one particular Galilean frame, but in addition, if it does not happen to have a zero value in one frame but has some definite non-vanishing numerical value, it will still maintain this same definite non-vanishing value in all other Galilean frames. In other words, Einstein’s premises are represented mathematically by the invariance of the total value of for all Galilean frames, regardless of whether this value happens to be zero or non-vanishing.
The deep significance of this condition of invariance was first noted by Minkowski, and it led, as we shall explain in the next chapter, to the discovery of four-dimensional space-time.
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CHAPTER XVIII THE DISCOVERY OF SPACE-TIME
THE discovery of this invariant whose value we shall designate by , marks a date of immense importance in the history of natural philosophy. The fact is that with Einstein’s discoveries such familiar absolutes as lengths, durations and simultaneities were all found to squirm and vary in magnitude when we passed from one Galilean system to another; that is, when we changed the constant magnitude of the relative velocity existing between ourselves as observers and the events observed. On the other hand, here at last was an invariant magnitude , representing the square of the spatial distance covered by a body in any Galilean frame, minus times the square of the duration required for this performance (the duration being measured, of course, by the standard of time of the same frame). It mattered not whether we were situated in this frame or in that one; in every case, if had a definite value when referred to one frame, it still maintained the same value when referred to any other frame.
Obviously, we were in the presence of something which, contrary to a distance in space or a duration in time, transcended the idiosyncrasies of our variable points of view. This was the first inkling we had in Einstein’s theory of the existence of a common absolute world underlying the relativity of physical space and time.
Minkowski immediately recognised in the mathematical form of this invariant the expression of the square of a distance in a four-dimensional continuum. This distance was termed the Einsteinian interval, or, more simply, the interval. The invariance of all such distances implied the absolute character of the metric relations of this four-dimensional continuum, regardless of our motion, and thereby implied the absolute nature of the continuum itself. The continuum was neither space nor time, but it pertained to both, since a distance between two of its points could be split up into space and time distances in various ways, just as a distance in ordinary space can be split up into length, breadth and height, also in various ways. For these reasons it was called Space-Time, and the interval thus became synonymous with the distance between two points in space-time.
[Pg 196]
And here certain aspects of the problem must be noted. In the first place, it may appear strange that measurements with clocks can be co-ordinated with measurements with rods or scales. This difficulty, however, need not arrest us; for although is a time which can only be measured with a clock, yet , being the product of a velocity by a time, is a spatial length since it represents the spatial distance covered by light in the time . For this reason we may consider our four-dimensional continuum to possess the qualifications of an extensional space.
Our next problem is to determine the nature of the geometry of this mysterious continuum. The mere fact that it has been possible to write the expression of this square of the interval with squared magnitudes, such as , , etc., without having recourse to terms such as , , etc., would of itself imply that the continuum was flat and free from all trace of non-Euclideanism or curvature. Furthermore, it implies that the mesh-system to which we refer this distance is one of straight lines, and is not curvilinear (see Chapter VII).
The only flat continuum we have studied so far is the Euclidean one, but in the present case we see that though flat, our continuum is not strictly Euclidean owing to the minus sign preceding ; or, again, owing to the fact that , instead of , as in a four-dimensional Euclidean space. Yet, because of the constancy in the values of the ’s throughout the mesh-system, hence because of the flatness or absence of non-Euclideanism or curvature of the continuum, the analogy with a Euclidean continuum is very great, and for this reason it is called semi-Euclidean.[64]
We say, therefore, that space-time is a four-dimensional semi-Euclidean continuum, and that it is differentiated from a truly Euclidean four-dimensional one solely because it has one imaginary dimension and three positive ones in place of four positive ones. In this continuum, time represents the so-called imaginary dimension, and the three dimensions of space represent the three positive ones; though it must be remembered that all we wish to imply by this statement is that there exists a difference between the dimensions of space and that of time. We should be equally justified in calling time a positive dimension, and the three dimensions of space imaginary dimensions.
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Now we have also seen that the mathematical form of the expression which gives the square of the distance depends not only on the geometry of the continuum, but also on the system of co-ordinates we may have adopted. In the present case the form of the mathematical expression tells us that our co-ordinate systems or mesh-systems are necessarily Cartesian, that is, are constituted by mesh-systems of four-dimensional cubes. Hence we see that all our different Galilean mesh-systems, when taken as frames of reference, correspond to differently orientated Cartesian mesh-systems in four-dimensional space-time.[65]
In short, space-time must be regarded as the fundamental continuum of the universe. Of itself it is neither space nor time. Any Galilean observer splits it up into four directions by means of his particular Cartesian mesh-system; one of these directions corresponds to his time measurements, while the three others correspond to his space measurements.
Henceforth an instantaneous event occurring at a certain point of our Galilean system and at a definite instant of our time will be represented by one definite point in space-time, a point in four-dimensional space-time being called a Point-Event. Space-time itself appears then as a continuum of point-events, just as ordinary space was considered to be a continuum of points, and time a continuum of instants. Of course point-events, like points and instants, are mere abstractions; for a point without extension and an instant without duration are mathematical fictions introduced for purposes of definiteness; they are mere abstractions from experience.
Now just as an instantaneous event is represented by a point-event, so a prolonged event, such as a body having a prolonged existence and occupying therefore the same point or successive points in the space of our frame at successive instants of time, will be represented by a continuous line called a World-Line. If this line is straight, the body is at rest in our Galilean system, or else is animated with some constant motion along a straight line. If the world-line is curved, the motion of the body is accelerated or non-rectilinear in our Galilean system; hence it is accelerated or curvilinear in space (owing to the absolute character of acceleration). In this way all phenomena reducible to motions of particles are represented by absolute drawings in space-time.
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According to the Galilean system of reference we may adopt, we divide space-time into space and time in one way or in another. Hence the resolution of the drawings of phenomena into space and time components is effected in various ways. We thus obtain variations in the spatio-temporal appearances of phenomena. Notwithstanding these variations, dependent on our relative motion, the absolute space-time drawings themselves, with their intersections, remain unchanged, and a direct study of the absolute world would consist in a study of these absolute drawings without reference to their varying spatio-temporal appearances.
It should be noted that inasmuch as space-time appears as an absolute continuum, existing independently of our measurements, there is no reason to limit ourselves to Cartesian mesh-systems when we wish to explore it. The mesh-system is, as we know, a mere mathematical device facilitating our investigations; and from a purely mathematical point of view we may always with equal justification split up a given continuum with one type of mesh-system or another. However, it must be remembered that in the case of space-time, when we wish to express the results of our measurements in terms of the space and time proper to our frame of reference, the mesh-system we must adopt is no longer arbitrary.
All Galilean observers, as we have seen, must split up space-time with their particular Cartesian mesh-systems. Certain curvilinear mesh-systems will correspond with more or less precision to the partitioning of space-time by accelerated or rotating observers. But in the most general case, arbitrary curvilinear mesh-systems do not correspond to the space and time partitions of any possible observer, though this fact does not detract from their utility.
So far as the essential characteristics of space-time itself are concerned, these Gaussian mesh-systems are just as legitimate as the Cartesian ones. As a matter of fact, when space-time becomes curved, owing to the presence of gravitation, Cartesian co-ordinates cannot be realised, being incompatible with the curvature of the continuum.
Yet, regardless of the mesh-system we use, certain general characteristics of our space-time drawings remain absolute. Thus, we have seen that the interval between two space-time points has a value which is invariant to a change of mesh-system. While it is true that a world-line which measures out as straight from a Cartesian system may appear curved when referred to a curvilinear or Gaussian one, yet, on the other hand, intersections or non-intersections of world-lines are absolute and are in no wise affected by our choice of a mesh-system. It is for this reason that phenomena such as coincidences are absolute, in contrast to simultaneities of events at spatially separated points. These are relative.
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Thus, if two billiard balls kiss in the observation of one man, they will continue to kiss in the observation of all other men, regardless of the relative motion of these men. The kissing of the balls constitutes a coincidence, an intersection of the world-lines of the two balls; hence it is an absolute. On the other hand, if two billiard balls hit different cushions simultaneously in the observation of one man, they will not in general hit the cushions simultaneously in the observation of a man in motion with respect to the first. The reason is that our second observer will have adopted a new mesh-system, oriented differently from the first; and in this new mesh-system the two space-time point-events represented by the instantaneous impacts of the two balls against the two cushions will no longer lie necessarily at the same distance along the new time direction.
Owing to the common use of curvilinear mesh-systems in the general theory, we must recall that in a curvilinear mesh-system the square of the interval adopts the more complicated form expressed more concisely by
The ’s are no longer constants, as when we had and , with all other ’s vanishing; they now have variable values from place to place, and as for , , , these no longer necessarily represent space and time differences; they are mere mathematical Gaussian number differences. In spite of all this, the numerical value of , the square of the interval, remains the same regardless of the mesh-system selected.
Psychological duration, or aging, can also be represented in space-time. By psychological duration we mean the flow of time which the observer experiences; or, more precisely still, the duration that would be marked out by a clock held in his hand. Psychological duration is measured by the length of the space-time world-line followed by the observer. When we keep this point in mind the paradox of the travelling twin to be discussed in a later chapter loses much of its paradoxical appearance.
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We can also understand how it comes that the discovery of space-time permits us to account for the duality in the nature of motion, relative when translationary and uniform, absolute when accelerated or rotationary. As long as we were endeavouring to account for this duality by attributing an appropriate structure to three-dimensional space, we were met by a peculiar difficulty: If we assumed space to have a flat structure, that is, one of planes and straight lines, it would be possible to understand why a free body, when set in motion with any definite velocity, should be guided along one of these straight lines or grooves in space. It would be following its natural course and there would be complete symmetry all around it; everything would run smoothly and no forces would arise. Then, if we attempted to tear the body away from its straight course, compelling it to follow a curve, we should be forcing it to violate the laws of the space-structure, tearing it away from the space-groove which it would normally follow. It was not impossible to assume that the groove would react and that antagonistic forces would arise. Thus the fact that curved or rotationary motion was accompanied by forces of inertia, hence manifested itself as absolute, did not constitute a difficulty of an insuperable order. But a real difficulty arose when we considered the other species of accelerated motion, i.e., variable motion along a straight line. In this case, the paths of the bodies being straight (hence in harmony with the flat structure of space), no violation of the laws of this spatial structure was evidenced. And yet forces of inertia were again present. It was difficult to interpret their appearance in terms of the space-structure.
When, however, we substitute four-dimensional space-time for separate space and time, the difficulty vanishes. For now we notice that all accelerated motions, regardless of whether they be rectilinear or curvilinear, are represented by curved world-lines in space-time. All these accelerated motions violate, therefore, the flat space-time structure; and for this reason forces of inertia will always be generated by them. On the other hand, when we consider Galilean or uniform or translational motions, we see that, as in the case of three-dimensional space, they will be represented by straight lines. These motions will then stand in perfect harmony with the flat space-time structure, and no forces of inertia will arise.
In this way, thanks to space-time, one of the outstanding difficulties which confronted classical science, namely, the dual nature of space and of motion, is accounted for.
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CHAPTER XIX VARIOUS POSSIBLE WORLDS
WE have seen that the great distinction between Einstein’s theory and classical science arises from the value to be attributed to the invariant velocity of the world. In the belief of classical science, this velocity was infinite, and we were thus led to a world of separate space and time. According to relativity, the value of the invariant velocity is finite and is given by , the constant which enters into Maxwell’s equations; this constant being illustrated physically by the velocity of light in vacuo.
From a purely mathematical standpoint, however, if we disregard the relations of reality, we may consider other purely formal possibilities. Suppose that the invariant velocity, though finite, were some number differing from Maxwell’s constant . Obviously , being a maximum velocity, would have to be greater than or equal to the highest velocity known to physicists, hence would have to be equal to or greater than , velocity of light. Let us therefore suppose that is greater than . As in Einstein’s theory, we should still be faced with a world of four-dimensional space-time, but this world would differ from that of relativity, owing to the discrepancy existing between the invariant velocity and Maxwell’s constant . This would indicate that the equations of electromagnetics would no longer maintain the same form when we changed Galilean frames. As a result, the erstwhile negative experiments in electromagnetics should now yield positive results, and absolute velocity through space would be detected. The theory of relativity would have to be abandoned.
[Pg 202]
But we may consider still another case. Suppose that the invariant velocity of the world were some imaginary quantity , where stands, as before, for , and for some arbitrary finite number. The square of this imaginary velocity being a negative number, the square of would now assume the form Thus would be given by a sum of four squares instead of by a sum of three squares and a difference. This would prove that the space-time continuum was Euclidean, having its four dimensions of the same category, instead of semi-Euclidean, as in Einstein’s theory, with an imaginary dimension for time. Time would now fail to be differentiated from space. It is difficult to realise how a world of this sort would manifest itself to us. The only reason we mention this case is in order to show with greater clarity that the classical belief in the separateness of time and space was equivalent to the classical assumption that the invariant velocity was infinite, and not a finite magnitude, whether real or imaginary.
A point that appears to have misled a number of persons relates to the meaning to be given to a four-dimensional space and time continuum. In all the cases mentioned in this chapter, it would be permissible to speak of the world as a four-dimensional continuum of events. But with the separate space and time of classical science the statement was artificial, owing to the absoluteness of time, which stood out by itself as distinct from space; and so the four-dimensional aspect of the world was never stressed. It is different with space-time. For now this aspect can no longer be disregarded, since it becomes impossible to divorce space from time in any absolute way holding for all observers. Thus we see that the four-dimensional nature of the world is a fact which has been disclosed by the theory of relativity. Prior to Einstein’s discoveries, any reference to space and time as one continuum would have been an unjustified extension of the accepted meaning of words.
Now let us return to the various types of worlds we have discussed in the preceding paragraphs. It is possible to give an alternative presentation of the results mentioned. Thus, we start from the mathematical expression of : where represents the invariant velocity without any specification as to its value. (It may be infinite, as in classical science, or equal to Maxwell’s constant , as in Einstein’s theory, or even equal to some imaginary number, as in the case last discussed.) Then it can be shown that by dividing this equation by and equating the result to zero, we obtain an equation which corresponds to relations holding in three-dimensional space. This equation is given by where the ’s are the components of this velocity along the three mutually perpendicular directions of space. When a mathematical expression of this kind is given as invariant or absolute, the theory of groups enables us to determine the nature of the geometrical rules according to which the variables entering into the expression may be added together. In the present case these variables represent velocities, so that we are on the way to discover the rules governing the addition of velocities. From this equation the following results are obtained.
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1°. If , the invariant velocity, is infinite, as in classical science, it is found that the composition of velocities must be Euclidean; so we should be able to combine velocities graphically by the well-known rule of the parallelogram or of the Euclidean triangle. In this case, of course, velocities which lie in the same direction can be added up and subtracted like numbers.
2°. If , the invariant velocity, is no longer infinite but finite, and if it is a real number, as in Einstein’s theory, the rules according to which velocities combine are found to be those of Lobatchewskian geometry. If, therefore, in Einstein’s theory, we wish to compound velocities, we must operate no longer on a Euclidean but on a Lobatchewskian triangle. We might, for example, trace our triangles on the surface of a pseudosphere, since the geometry of this surface is, as we know, Lobatchewskian. In this case velocities lying in the same direction will no longer add up like the numbers of arithmetic.
3°. In a similar way we should find that if the invariant velocity , while finite, happened to be an imaginary number, the composition of velocities would follow the rules of Riemann’s geometry. In this case we saw that four-dimensional space-time would be Euclidean and not semi-Euclidean.[66] We merely mention this third type of universe for motives of symmetry. Obviously it does not correspond to the world we live in, hence we will not refer to it in future, confining ourselves to the classical world and to that of relativity.
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When we are called upon to decide which of the two alternatives is correct (space-time or separate space and time), which one corresponds to reality, a priori speculations are futile, since both solutions are conceivable and satisfy the facts of crude observation. Our only recourse is then to appeal to experiment, and by experiment we mean observations that are more reliable than our crude perceptions unaided by ultra-precise instruments. It is only thanks to such experiments that we may succeed in ascertaining whether or not a finite invariant velocity is demanded by the world-structure. Should our experiments point to the existence of a finite invariant velocity, our problem would be settled in favour of a world of space-time as against one of separate space and time.
Now, experiment suggests that the propagation of light in vacuo furnishes us with a concrete example of this critical velocity. It follows that a measure of the speed of light propagation will yield us the precise numerical value of the critical velocity. For this reason, the propagation of light assumes a position of vast theoretical importance. Needless to say, this importance of light is due solely to its physical behaviour in vacuo, solely to the fact that it moves with the invariant speed, not to the fact that because of some accidental circumstance it happens to be visible to the human eye. For light passing through glass or water is also visible, but its speed, being reduced, is no longer invariant. Hence light propagation through matter can no longer serve to define the invariant velocity directly. Besides we can get away from light as a visible propagation by considering an invisible electromagnetic propagation in its stead.
If we wish, we can do better still and obviate electromagnetic waves entirely. For when account is taken of the negative experiments in electrodynamics, we realise that the invariant velocity is none other than the constant which enters into Maxwell’s equations, whence any experiment capable of measuring would automatically yield us the invariant velocity. In particular, we might measure the ratio of an electric charge when computed in terms of electrostatic, then of electromagnetic units; this again would yield us the value of Maxwell’s constant , hence of the critical invariant velocity. We might also appeal to the necessary consequences of the existence of a critical velocity, for instance, to the peculiar Lobatchewskian law which it would entail for the additions of velocities. We should then check these consequences by experiment, discover whether they were justified by facts, and deduce therefrom the existence and value of the invariant velocity.
[Pg 205]
Fizeau’s experiment permits this verification and proves that the composition of velocities is Lobatchewskian, not Euclidean, and corresponds to the existence of a critical invariant velocity of 186,000 miles per second. It is true that in Fizeau’s experiment we are again dealing with the propagation of light, though no longer in vacuo, where its speed is invariant, but through water. It is precisely because its speed through water is reduced that the velocity of the water will affect the velocity of the light, thereby permitting us to test out the relativistic law of the composition of velocities. But once again the reason why light propagations enter into the majority of these tests is due solely to the fact that optical experiments permit of greater accuracy. Apart from this, there is no need to limit ourselves to experiments dealing with the propagation of light, whether in vacuo or through material bodies such as water. Any speed, be it the speed of sound or of a bullet shot from a passing train, should enable us to detect the same law of Lobatchewskian composition and thus allow us to demonstrate the existence of space-time.
At all events, one point should be clear by now. Whatever method we follow, we can never get away from performing physical measurements of one kind or another. It is therefore meaningless to speak of defining the invariant critical velocity without reference to a measurement of the speed of light in vacuo, or more generally, without reference to physical measurements bearing on the phenomena which the existence of a finite invariant speed in the universe would necessarily entail. From this it follows that without physical measurements of one kind or another we have no means of establishing whether the world is one of separate space and time or one of space-time.
In theory even crude observations should have led us to a decision on this score, for crude observations differ from those of the physicist only in the matter of precision. Theoretically, therefore, prior to Einstein we should have known of the incorrectness of the Euclidean composition of velocities and hence known of space-time. The reason for our failure to make the momentous discovery is easily understood when we realise that owing to the very great magnitude of the invariant finite velocity , the Einsteinian rule can scarcely be differentiated from the classical rule unless very great velocities are involved. From this it is seen that the reason why the effects anticipated by Einstein’s theory are so difficult to detect is because of the very high value of the invariant velocity . Had this finite invariant velocity happened to be small, say a mile a second, the classical conception of a world of separate space and time would have been abandoned years ago; for the curious effects predicted by Einstein would have become perceptible without effort. Again, were we to conceive of this invariant velocity as becoming greater and greater, the distinguishing features of the Einsteinian universe would gradually fade away, until finally, if the invariant velocity became infinite, space-time would give way to the separate space and time of classical science, and all the peculiarities of Einstein’s universe would vanish.
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A direct study of the Lorentz-Einstein transformations would lead to the same conclusions; for if in these transformations we make infinite, we find that they degenerate into the classical Galilean ones. And here a rather delicate point must be noted. The constant —invariant velocity of the universe—has been identified in Einstein’s theory with the velocity of light in vacuo. This identification was inevitable, as we explained when discussing the equations of electromagnetics. But while it is correct to state that Einstein’s theory would degenerate into classical science were , the invariant velocity of the universe, infinite instead of finite, it would be wrong to infer therefrom that were the velocity of light infinite we should be faced with the world such as classical science thought it to be. As we know, in classical science the velocity of light was not infinite; it was only the invariant velocity that was infinite; and these two velocities had nothing in common.
If we wished to conceive of a world in which both the velocity of light and the invariant velocity were infinite, we should have to make infinite in the equations of electromagnetics, since in these equations represents the velocity of light. But then all the terms in these equations in which entered as denominator would disappear. Since these terms are those responsible for induction, in a world in which the velocity of light was infinite, no such phenomenon as induction could exist. There would be no dynamos, no radio; in fact, there would be no light rays, since light is itself a phenomenon of electromagnetic induction. Obviously such a world would not be the one in which we live.
Summarising, we must say that in classical science the velocity of light is finite and that the invariant velocity is infinite. In relativity, , the velocity of light, is finite, as before, but is identical with the invariant velocity, which is therefore also finite. We also understand the deep-seated reason for the curious complicated form of the relativistic space and time transformations when we pass from one Galilean frame to another (the Lorentz-Einstein transformations) as contrasted with the classical transformations. We realise now that this more complicated form is due to the fact that the real world is one of space-time, and not of separate space and time, as was formerly believed.
Before closing this chapter, there is a point which we feel it advisable to discuss more fully. It deals with purely formal definitions in physics, and in particular with that of the critical velocity. We have mentioned that definitions which profess to dispense with the use of the rod, clock and physical measurements in general, are recognised as impossible. Einstein has repeatedly stressed this fact.
Possibly these points will be best understood if we discuss an illustration of a purely formal definition given by Dr. Whitehead. Thus, he writes:
“Experiment shows that this critical maximum velocity is a near approximation to the velocity of light in vacuo, but its definition in no way depends upon any reference to light.”[67]
Now any physicist would readily agree with this statement were it Dr. Whitehead’s desire merely to obviate an appeal to light propagation in the definition of the critical velocity of relativity. For, as we have seen, the critical or maximum velocity can be defined and has been defined by other physical means. Indeed, so far as we know, the existence of the critical velocity is not contingent on that of light. But Whitehead’s idea, it would seem from his writings, is to eliminate physical measurements entirely from the basic definitions of physics; whence his definition, or at least the nearest approach to a definition we have been able to find in his writings:
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“The physical meaning of is also well known; namely, any velocity which in any time-system is of magnitude is of the same magnitude in every other time-system. No assumption of the existence of a velocity with this property or of the electromagnetic invariance has entered into the deduction of the kinematical equations of the hyperbolic type.”
Note that in this definition Dr. Whitehead has failed to identify with either Maxwell’s constant or any other physical magnitude. Thus is some unspecified magnitude; and we may express Dr. Whitehead’s definition in more familiar language as follows:
“The critical or maximum velocity is a velocity of undetermined value which may or may not be demanded by the world-structure. If, however, it should exist, it will be given by the same invariant in every Galilean frame, whatever the value of this invariant may be.”
In this definition no reference is made to any physical phenomena or laws, and this is the aspect which appears to constitute its superiority in Whitehead’s opinion. But the trouble is that if physical measurements or the existence of physical laws established empirically by the use of rod, clock, etc., are to be discarded, the definition turns out to be no definition at all. For what sort of definition is one which fails to apprise us of the magnitude of the velocity it proposes to define? The definition of the velocity of sound, for example, is that of a velocity of so many feet a second under specified conditions of observation. A definition which failed to give us this information would be no definition of the velocity of sound.
Now it might be contended that the precise magnitude of the invariant velocity is a matter of minor importance, and that its property of invariance (mentioned in the definition) is the fundamental point. We may safely say, however, that no physicist could ever subscribe to this view. Our entire outlook on nature depends essentially on the value of this invariant velocity.
If it is finite, and if, in addition, it is equal in value to Maxwell’s constant , hence to the velocity of light in vacuo, we must conceive of a world of space-time in which the laws of nature will be subject to certain rigid conditions. Also, simultaneity, length and duration will be relative.
If the invariant velocity is a finite magnitude greater than the value of Maxwell’s constant , we shall still have a world of space-time in the extended sense, but the relativity of velocity will not hold and we should be able to determine our absolute velocity through space. The rigid restrictions placed on the laws of nature would no longer be present, since the laws of electromagnetics would certainly be in conflict with those restrictions. The entire structure of Einstein’s theory would be overthrown.
Finally, if this invariant velocity were infinite, we should have the separate space and time of classical science entailing the absoluteness of simultaneity, length and duration. There is nothing in Dr. Whitehead’s mathematical deductions to suggest which of these values the invariant velocity is to have; it might even be an imaginary number.
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At all events, we see that the precise magnitude of the invariant velocity is not a matter of any small importance; hence it should inevitably be defined by the definition. Yet this essential piece of information Dr. Whitehead’s definition fails to give, by reason of its being compatible with a whole class of possible invariant velocities ranging from zero to infinity, just as a definition of New York as “a city by the sea” gives us a whole class of possible cities. Definitions of this sort are ambiguous, they are indefinite, and therefore are no definitions at all. The best that can be said of them is that they constitute partial descriptions.[68]
Up to this point we have discussed Whitehead’s definition in the light of a self-supporting mathematical definition that would necessitate no appeal to physical measurements in order to acquire a determinate significance. Such appeared to be his intention. Thus interpreted, the definition is a failure, as we have seen.
It should therefore be obvious that our only means of rendering it determinate must be to establish the existence, then the magnitude of this elusive critical velocity. But this can be done only through the medium of physical measurements. It matters not whether the measurements be accomplished with rods, clocks, light rays, or by electromagnetic means. In every case they will be physical; hence, when finally the value of the invariant velocity is obtained, this magnitude will be fixed, determined or defined by means of physical measurements. Dr. Whitehead’s purely formal definition cannot emancipate us from this restriction. It seems strange, therefore, that he should criticise in Einstein a procedure which he himself is unable to avoid.
At any rate, inasmuch as scientists convinced themselves years ago that in the matter of physical definitions, physical measurements could not be avoided, their aim has always been to select a definition which would define the unknown quantity in terms of magnitudes already known. If this method fails them, they attempt to present their definition in a form which will be of aid to the physicist, directing him towards his goal.
When these requirements are considered, it is seen that Einstein’s definition of the invariant velocity as that of light in vacuo cannot easily be improved upon. To understand this point, we shall contrast the situation of an experimenter to whom Dr. Whitehead’s definition, then Einstein’s, would be given.
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As we remember, Dr. Whitehead’s definition informs the experimenter that he may recognise the critical speed when his measurements have succeeded in detecting a velocity which is invariant. Failing this, he may also fix its value as soon as his measurements succeed in detecting a trace of non-Euclideanism in the law of composition of velocities. But this is obviously a very roundabout way to help him. The experimenter will now be compelled to fire bullets with various speeds and endeavour to detect traces of non-Euclideanism. If his attempts are unsuccessful, he can draw no conclusions, since it might always be supposed that the invariant velocity was too great to exert perceptible effects on the velocities at his disposal. Even were he to detect what might appear to be non-Euclideanism, how would he know that the effect had not arisen from secondary causes? We remember, in this connection, that neither Fresnel nor Lorentz ever thought it necessary to attribute the result of Fizeau’s experiment to a non-Euclidean composition of velocities.[69] Fresnel ascribed this result to interactions between ether and matter, whereas Lorentz attributed it to the electronic constitution of matter. In short, Dr. Whitehead’s definition places the experimenter in the unenviable position of a man looking for a needle in a haystack without having the consolation of knowing that the needle is really there. From this we see that the definition, when interpreted in a purely formal sense, is ambiguous and tells us nothing, and, when interpreted in a physical sense, is of very little use to the physicist.
And now let us consider Einstein’s definition. First we may note that had the bare invariance of the critical or maximum velocity represented a proper definition in his opinion, he would have given it twenty years ago, seeing that this invariance constitutes the A B C of his theory. Yet, as we know, though he often refers to the maximum velocity as the invariant velocity, and vice versa, he considered it necessary to complete this partial description by a proper definition.
For Einstein, the important point was not so much to furnish a physical illustration of this invariant velocity, which might conceivably never be found to be realised in nature as a physical existent. (In the same way, the infinite velocity of classical science was never given by any physical propagation.) The important point was that he should justify his belief that the world-structure demanded the existence of a critical velocity of this sort; and by its existence we mean that were it ever to be realised, it should prove to be an unsurpassable maximum and an invariant.
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We are now in a position to understand the advantage of Einstein’s definition. First, he states that the electromagnetic experiments have established the relativity of velocity. This in turn entails the maintenance of form of all natural laws—in particular, Maxwell’s laws of electromagnetics. Hence it follows that there must exist a critical invariant velocity that is given by the constant c which enters into Maxwell’s equations. Inasmuch as Maxwell had proved that and the velocity of light in vacuo were one and the same, the invariant velocity is defined by this velocity of light. Thus, with Einstein’s definition, we know that the world-structure demands the existence of a finite invariant velocity, and not of an infinite one, as was believed by classical science. And, in addition, we know that this velocity is given by Maxwell’s celebrated constant , a constant whose value had been determined with accuracy long before the advent of the relativity theory. In other words, Einstein defines the unknown in terms of the known, not in terms of the ambiguous.
To return to our illustration of the needle in the haystack, not only has Einstein given us the assurance that the needle is there, but he also tells us exactly where to look for it. For this reason his definition, in contradistinction to Dr. Whitehead’s, satisfies the requirements demanded by physical science.
In these pages we have discussed Whitehead’s definition at some length because it illustrates the danger there is in confusing mathematics and physics. Thus, whereas in mathematics we may postulate anything we please (with certain reservations) and then proceed to reach our conclusions deductively, in physics this procedure is impossible. We must take our cue from experiment and formulate our premises accordingly. The result is that whereas in mathematics we are concerned with formal possibilities, in physics we are limited to an analysis of actual facts. And in every case the transition from the possibilities of mathematics to the actualities of physics necessitates the introduction of physical measurements.
In this connection it is most important to understand the difference between a physical and a mathematical definition. Take a number like n in mathematics. We can define it as the ratio of the length of a circumference to its diameter in Euclidean geometry. Without performing physical measurements, we can deduce from this definition, by purely mathematical means, the precise value of to any order of approximation we please. Thus the definition does not lead to ambiguity, hence is a valid one.
On the other hand, try to give a physical definition by some similar method—say, the definition of the “gram” or of the “dyne.” The purely logical type of definition breaks down, and we are compelled to resort to physical determinations. Accordingly, we define the gram as the mass of a cubic centimetre of distilled water under specified conditions of temperature and pressure.
And why was the purely logical definition possible in mathematics and a failure in physics? Merely because in mathematics—in the case of , for example—the laws of Euclidean geometry were accepted from the start, whereas in physics preliminary information of this sort is denied us. And it is the same with all the magnitudes and constants of physics. It is impossible to predetermine by logical means the value of the invariant velocity, or of the gas constant, or of the size of the earth or of an electron. In every case, we must have recourse to physical measurements of one sort or another, for the constants of physics present no character of rational necessity.
Much of the criticism directed against Einstein’s use of rods, clocks and light propagation arises from a failure to understand this [Pg 211] fundamental point. Purely logical, non-empirical definitions in physics can only lead to ambiguity. As for the theory of relativity, it is one of physics; it does not claim or aspire to be anything else; and a theory of physics which would avoid physical measurements is about as reasonable a conception as a forest without trees.
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CHAPTER XX THE IRREVERSIBILITY OF TIME
WE saw, when discussing the existence of a finite invariant velocity in nature, that the presence of this velocity was going to work havoc with our belief in absolute simultaneity. Thus, if two events take place in different places, we can no longer attribute any universal meaning to the opinion that these two events have taken place at one and the same instant of time or at two different instants. In other words, there exists no absolute clock giving us the correct time at all points of the universe. According to the motion of the observer, two events which are simultaneous for one may follow in sequence for another.
At this point we must guard against possible confusion. It might be argued: “There is nothing very new in this relativity of simultaneity and of the order of succession of our perceptions; for we encounter such examples even in classical science. For instance, if two explosions occur at different spots, by adjusting our position as observers it may be possible for us to perceive the two noises now simultaneously and now in succession.” But such arguments would indicate that the revolutionary notion of the relativity of simultaneity had not been grasped.
Thus, in the example of the two explosions happening in different places on the earth’s surface, while it is perfectly true that the order of succession in which we hear or see these explosions will depend on our position as observers, yet, on the other hand, if we take into consideration the time which sound or light waves require to cover the distances separating us from the two explosions, we can always decide without ambiguity whether the two explosions occurred simultaneously or in succession. It was never the order or the simultaneity of our perceptions which was considered absolute in classical science; it was solely the simultaneity or order in which these explosions or events had taken place in the outside world.
Now, according to the theory of relativity, not only is the simultaneity of our perceptions relative, but in addition, even when we take into consideration the speed with which the sound waves or light waves advance towards us, it is still impossible to decide whether the two events (regarded as existing in the outside world independently of any observer) are simultaneous or not; for our calculations would show that this simultaneity in the outside world would vary from one observer to another.
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The relativity of simultaneity leads us to the kindred subject of the relativity of the order of succession of two events occurring in different places. Here it is impossible to lay down a rigid rule; for we shall see that the theory compels us to recognise that in certain cases the order of succession is indeterminate or relative to our motion, while in others it is absolute, remaining the same for all observers.
The problem is of sufficient importance to warrant a more detailed explanation. Our awareness of the passage of time constitutes one of the most fundamental facts of consciousness. Not only is time continuously passing, but it is ever flowing in the same direction. To this mystery of the uni-directional passage of time, the theory of relativity contributes no new information, so that we may discuss the problem from the standpoint of classical science. Suppose, then, that in a bottle we place two layers of powder, one white and one black. If we shake the mixture long enough, we know that the final result will be a uniform grey mixture. Now we might have anticipated this result without actually performing the experiment. For if we consider the various ways in which the particles of the powders could be distributed in the bottle, sufficient reason would urge us to assert that all distributions were equally probable. But it so happens that by far the greater number of these possible schemes of distribution would yield the appearance of a uniform grey mixture. Probability would suggest, therefore, that on shaking the bottle long enough, the uniform grey mixture would finally appear. When this homogeneous state had been obtained, only once in æons of time would the black and white separation reappear, and then but for an instant. We may express these facts by saying that the general tendency would be a passage from the heterogeneous (black and white layers) to the homogeneous (uniform grey mixture).
The example we have considered is one of extreme simplicity, but inasmuch as the arguments involved appear to be of universal validity, we may say that natural phenomena present a uni-directional sense of advance, passing from the states of lesser probability to those of greater probability. We may therefore identify the states of lesser probability with the past and those of greater probability with the future. The direction of time’s passage can thus be defined physically in terms of probability considerations, entailing a mere appeal to sufficient reason and to operations of counting.
This was the definition suggested by Boltzmann (the founder of the kinetic theory of gases). According to him the universe was passing from states of lesser probability to that of maximum probability, and the direction of this passage defined that of time. As can be gathered from the preceding explanations, a uni-directional passage in the course of phenomena is to be ascribed to the fact that phenomena are irreversible, i.e., that the various states are not equally probable.
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It is true that there also exist types of phenomena for which all states present the same probability. In this case no privileged direction exists, and the phenomena are called reversible. An adiabatic transformation under ideal conditions, and the rotation of a body in the absence of friction, are illustrations of reversible processes; such phenomena would of course be incapable of defining the direction of time.
But by far the greater number of phenomena in nature are of the irreversible type; with these a definite direction of change is privileged. When we wish to force the phenomenon against its natural trend, work must be furnished. All phenomena entailing friction are of the irreversible sort, for whereas motion generates heat through friction, the heat cannot be used to regenerate the motion. The example of the two powders also presented us with an illustration of the irreversible type of phenomenon, since the natural evolution was from heterogeneity to homogeneity. To be sure, it would be possible to reverse the process, but only through the medium of some intelligent activity sorting out the particles and distributing them according to states of lesser probability. The action of a demon of this sort, Maxwell’s demon, would cause the direction of the irreversible phenomena to be reversed, so that the direction of time would appear to change. Needless to say, however, Maxwell’s demon is but a fiction.
In the illustration of the two powders we can readily understand the reason for the irreversibility, but it was not by this method that the principle involved was first discovered. We must go back to Carnot and to his investigations on the cycle of the steam engine in order to trace the origin of what was to become one of the most fundamental principles of science. Carnot’s celebrated principle relating to the efficiency of the steam engine was shown by Clausius to be a special case of a general physical principle which may be stated thus: “Heat cannot flow unaided from a colder to a hotter body, but tends invariably to seek lower levels of temperature.”
By introducing a new concept called Entropy, defined as the ratio of a quantity of heat to a temperature, Clausius was able to give a more general form to this principle. It then became the Principle of Entropy, according to which, in any irreversible change, the entropy of a system was increased; only in the case of a reversible change would it remain constant. Under no circumstances, however, would the total entropy decrease. Thus all natural processes involve an increase of entropy since none are ideally reversible.
Now when we consider the principle of entropy in its bearing on the concept of energy, we find that it implies a continual loss of potentiality in the energy of the system. The energy is thus constantly dissipated, or, more properly speaking, degraded into that lowest of forms, namely, disorderly molecular agitation, i.e., heat; whence the name, Principle of the Degradation of Energy, under which it is often referred to.
The principle is of such extreme importance that it has been named the Second Principle of Thermodynamics; the appellation First Principle of Thermodynamics being reserved for the principle of the conservation of energy. It is important to note that these two principles, that of degradation and that of conservation, do not conflict. In quantity the energy is conserved; it is solely in quality, or in potentiality, or in its ability to perform work, that it is degraded.
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We now come to Boltzmann’s contributions. Entropy as defined by Clausius was an exceedingly abstruse thermodynamical concept. Boltzmann, by basing his deductions on the kinetic theory of gases, succeeded in giving a more concrete representation of entropy, defining it as the logarithm of a probability. Thus, if we admit the correctness of the kinetic theory, the principle of entropy becomes one of maximum probability, losing thereby its status of absolute validity and assuming a statistical significance. The representation of entropy in terms of probability permits an easy application of the principle to the problems of molecular physics and to atomistic physics in general. We may note that it was by following this method, or rather by combining the two definitions of entropy, that Planck established his quantum radiation formula, and was thus led to his quantum theory. Needless to say, the principle holds only when exceedingly large numbers are considered. In other words, it is a principle governing the chaos, or, again, it is a statistical principle.
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It will be seen that the principle of entropy, by stating that the world is ever passing from states of lesser to states of greater probability, indicates that when the state of maximum probability or entropy is reached, no further change can take place. There will then be silent immobility or death. Under the circumstances it might appear that the principle was inconsistent with the existence of an original state of minimum probability. But here it should be remembered that the principles of science do not aspire to any absolute measure of value. All we can demand is that they be satisfied in the very restricted domain which science can explore. In the case of the principle of entropy, however, there is no reason to fear any conflict with the possibility of rebirth or re-creation. For instance, when shaking the powders, we saw that a uniform grey mixture would be the outcome; but were we to go on shaking the bottle for trillions of centuries, at some time or other the black and white separation would reoccur, and the cycle begin all over again. In short, the entropy would continually increase, but once every trillion years it might suddenly decrease, and then start to increase once more. We may mention that although there is no reason to assume that vital phenomena are in any wise inconsistent with the principle of entropy, it is safer to restrict the principle to the physical world. With biological phenomena, questions of probability are far too complex to be treated with any measure of assurance in the present state of our knowledge. Only in certain special cases, such as those connected with Mendel’s observations, have probability considerations been introduced with any profit.[70]
Now the two points we wish to stress are as follows: First, the uni-directional progress of time is imposed by common experience. Secondly, the principle of entropy is suggested both by experience and by arguments of a rational order. We do not assert thereby that the principle of entropy, even when reduced to the rank of a statistical principle, can be claimed with assurance to be of universal validity. It is far from impossible that in certain phenomena, of which as yet nothing is known, some counterbalancing activity may be at work. Nevertheless, it may be pointed out that a denial of the principle of entropy, at least as regulating those physical phenomena with which science deals, would arrest further progress.
For instance, we can readily understand in what an embarrassing situation we should be placed were a reversal of time to occur. Quite aside from such trivial examples as that of a hen becoming a chick and disappearing into an egg, or from effects becoming causes, a number of more scientific considerations must be mentioned. Phenomena would now pass from states of maximum probability to those of minimum probability; we could not merely interchange the meanings of the words “maximum” and “minimum,” for these are given by purely numerical considerations, which are quite irrelevant to the direction of the time-flow. Hence phenomena would pass from the homogeneous to the heterogeneous. But whereas a homogeneous state is unique, and can be predetermined, a heterogeneous one is more or less arbitrary, and baffles prevision. Thus, in the case of the two powders, the homogeneous state is a uniform grey mixture; whereas a heterogeneous state might be a succession of alternate layers, or only two layers, or a juxtaposition of black and white cubes. In fine, prevision would become well-nigh impossible.
From this we may readily anticipate that should Einstein’s theory render the direction of time’s progress indeterminate, should past and future be interchangeable according to our circumstances of motion, the utmost chaos would ensue. However, we need have no fear on this score, for as we shall see, the theory of relativity leads to no reversal of causality. Let us now consider the space-time theory in greater detail.
[Pg 217]
When we consider the four-dimensional space-time continuum, where space and time are on the same footing, there is nothing to suggest either a flowing of time or a privileged direction for this flow. In order to conform the theory to the facts of experience, it is therefore necessary to postulate that our consciousness rises along the world-line of our body through space-time, discovering the events on its course. Obviously, we might reverse the presentation by assuming that it was space-time that was moving past our consciousness; we might also claim that the very texture of space-time possessed dynamic properties urging our consciousness along time directions. But in view of the vagueness of the subject, not much is to be gained by speculating on this score.
Now when, along the world-lines which each one of us follows, a common direction from past to future has been specified, we find that an absolute past and an absolute future (though not an absolute present) are demanded by the theory of relativity. To be more explicit, we find that whereas, according to the nature of our relative motions, certain events may appear as simultaneous, or as antecedent and consequent, or as consequent and antecedent, others will invariably present themselves in the same order of temporal sequence regardless of our motion.
This division between events whose order of occurrence is absolute and those whose order is relative is obtained as follows: Whenever it is possible for a ray of light to pass from one event to a second event, leaving the first event at the instant it is produced and reaching the second event before or even at the instant at which it is produced, the time sequence of the two events cannot be reversed by relative motion; it is absolute.
On the other hand, if it is impossible for the ray of light to satisfy the preceding requirements, owing to the too great distance or the too small period of time separating the two events, the time sequence of the two events becomes indeterminate. One observer may claim that precedes , whereas the other may claim that precedes , and yet another may assert that and are simultaneous. If we remember that in Einstein’s theory no active influence, no propagation, can be transmitted with a speed greater than light, we see that when a ray of light cannot proceed from to , leaving when is produced and reaching when or before is produced, it is quite impossible for us to suppose that has influenced . No causal connection can exist between these two events. Hence we see that when two events are causally connected, their order of occurrence will remain the same for all observers. Only when no causal connections could possibly exist can the temporal sequence of the events be reversed by an appropriate motion of the observer.
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It is not because a causal connection happens to exist between two events that the order is non-reversible; the non-reversibility is due entirely to the spatio-temporal separation of the events. As for the causal connection, it may or may not exist. The only reason we mention causal connections with reference to this problem is in order to show that there is nothing in the relativity theory to suggest a reversal of causality; hence, though Einstein’s theory entails the relativity of simultaneity and in certain cases that of temporal sequence, there is no danger of a cause appearing as an effect or vice versa. In particular, no relative motion of the observer could ever lead him to believe that the glass we had let fall had in reality leaped up from the floor into our hand. No danger of the black and white powders which we had shaken into a grey mixture, appearing to separate back into black and white, as a result of a change in our relative motion. In short, no danger of the principle of entropy being overthrown.
We may also add that whereas, in classical science, owing to the possible existence of influences transmitted with infinite velocity, any two events happening in space might have been conceived of as causally connected or at least as related, Einstein’s theory, by requiring the velocity of light to be unsurpassable, permits us to restrict the cases of possible causal relationships and thereby to gain a better understanding of the cross influences which may be active in the universe.
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CHAPTER XXI THE REALITY OF THE CONTRACTION OF LENGTHS AND OF THE LENGTHENING OF DURATIONS
LET us now pass to the question of lengths and durations. Consider a rigid rod placed on a table, and suppose that both table and rod are drawn away from us with constant speed along a straight line. The contraction of lengths implies that both rod and table will appear to us to be contracted, owing to their relative motion with respect to us.
Here we must emphasise the fact that this contraction has nothing to do with the motion of the rod through space or the ether. As we know, motion, or at least Galilean motion, through space has no meaning. It is purely a relative; so that exactly the same effects of contraction would ensue were we to conceive of the table and rod as at rest and of the observer as moving away from them. It follows that an observer attached to the rod would perceive no contraction, although his friend in motion with respect to him would assert that a contraction had taken place.
The nature of this contraction is obviously rather mysterious, and the question is to decide as to its reality. Here we are met with the difficulty of defining what we wish to convey by the word “reality”; in some respects the contraction of the rod is a physical reality, in others it is a fiction, but in no case is it an illusion.
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Let us consider the problem of illusions. For instance, we experience an aggregate of visual impressions which we unhesitatingly ascribe to the presence of a table situated “there in the room.” Yet all we have any right to be positive about is that certain visual impressions have been experienced; our belief in the existence of the table situated “there” results from an inference, a judgment formulated in an attempt to ascribe a cause to our visual sensations. By an association of ideas and by recalling past experience, we feel justified in claiming the existence of the table “there” as we see it. In the majority of cases our judgment will be correct, but in a certain number of instances it will be erroneous. Such would be the case were we to be viewing the reflection of a table in the mirror. An illusion is therefore an error of judgment. And what prompts us to state that our belief in the object behind the mirror is an illusion? Obviously, the bare fact that on testing the matter out in a variety of different ways it appears impossible to credit the table with any existence at the spot where we claim to see it in the common objective world of science. In much the same way we see a mirage and interpret our sensations as connoting the presence of water. Yet were a meteor to fall into the water it would not appear to be extinguished by the waters of the lake. Kindred observations of an indirect order would finally compel us to recognise that our initial judgment was at fault, and that no water lay before us.
And now let us consider the FitzGerald contraction. Is it an illusion? To this question our reply must unhesitatingly be in the negative. Not only should we perceive the contraction were our eyes capable of yielding a snap-shot picture of a rapidly moving object, but in addition every conceivable physical experiment performed in our frame would indicate that the contraction was physically present. And yet we have said that the contraction differed essentially from the one contemplated by Lorentz, generated as was the latter by a real compression caused by the motion of the object through the stagnant ether. The explanation of the dilemma is as follows:
In Einstein’s theory, the rod is contracted, but we must add that this contraction holds only in the space of the frame with respect to which the rod is in motion. In the space of the frame accompanying the rod in its motion, no contraction would occur. We see, then, that as there exists no privileged space, only “my space,” “your space,” “his space,” there is no particular length to the rod, only a length “for me,” “for you” and “for him.” The contraction is thus embodied in a two-term relationship extending between the observer and the observed; it is a “relative.” But, once again, for a given observer it is as physically real as a chair or a table. Only when we consider the contraction from an impersonal standpoint, from that of space-time, without mentioning the conditions of observation, is the contraction not an illusion, but downright meaningless. It is, as we see, the concept of spatial magnitude, of primary qualities, which is at stake, in a form entirely different from anything science or philosophy had ever dreamed.
Now although length, size and duration are relatives, the theory of relativity has discovered absolutes which transcend the observer, remaining the same for all. They are to be found in the absolute world of space-time; and it is this thought that Minkowski desired to convey when he said: “Henceforth space and time by themselves are mere shadows.” In the absence of some specified observer, space and time become indeterminate, and with them length and duration; that which endures and transcends the observer is a something in space-time, and this is neither a length nor a duration. Or, again, lengths and durations are the projections of something in absolute space-time on to the space and time directions of the observer. These latter are indeterminate (until the frame of the observer is specified), since no one space and no one time exists above all others. Space and time being indeterminate, the same will apply to those projections which we call length and duration.
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Although we may be repeating ourselves unnecessarily, we cannot help but feel that the easiest way for the layman to grasp these difficult concepts will be to return to the well-known illustration of the relativity of a visual angle. When we perceive an object, say a telegraph pole, we see it under a certain angle, which we may call the visual angle. Is this visual angle real? Of course it is; we can measure it and define it physically as the angle subtended by two Euclidean straight lines extending between our eye and the two extremities of the pole. It is real but it expresses a relationship—a relationship between the positions of the observer and pole. It is certainly not inherent in the pole, since as the observer changes his position the visual angle changes in magnitude, though nothing happens to the pole in the meantime.
Visual angle in classical science is analogous to distance in Einstein’s theory. Our rod has no true length any more than the telegraph pole has any true visual angle. The whole significance of relativity is precisely to deprive length and duration of the absolute characteristics which classical science attributed to them. In short, when we are asked whether the rod has really been contracted and lost part of its length, we might counter by asking whether the telegraph pole has really been modified and lost part of its visual angle. Just as, if we omit to stipulate the position of the observer, the visual angle of the pole has no meaning and can teach us nothing about the pole, so now, in relativity, the length of the rod has no meaning in itself and can teach us nothing about the rod.
All these arguments developed on the score of length apply in similar fashion to duration. Both length and duration are relatives having no absolute significance in the universe.
The critic will generally balk at these statements and be inclined to say: “What nonsense! The length of the rod is immanent in the rod and can in no wise be affected by the observer’s motion or by the frame’s existence. The observer might die, the frame might disappear; what has all this to do with the length of the concrete entity we call the rod? The length of the rod may appear longer or shorter according to its relative motion; but we must not confuse appearance and reality. Thus, a man standing 100 yards away appears to us shorter than when he is close up, but it would never enter our minds to say that he was really reduced in stature. Just as the height of the man is independent of our position and of our existence as observers, so is the length of the rod independent of our relative velocity.”
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Now we might readily concede to the critic the right to maintain that the existence and absolute characteristics of the rod are in no wise dependent on the motion or the existence of an observer. But the critic is begging the question when he assumes without further ado that these absolute characteristics are expressed by the length of the rod. This length enters into existence only when a definite space is specified; till then it is meaningless. And in the absence of an observer there is no such thing as a space, but merely four-dimensional space-time. The absolute characteristics of the rod thus refer to space-time, and this eliminates all primary qualities such as length. The situation is of course not easy to grasp at first blush and is still less easy to explain, but in view of the extreme importance of the matter we shall do our best to clarify it by slightly varying the presentation.
Let us first examine the problem from the standpoint of classical science. Consider an observer who walks away from a point on the surface of the earth. He finally comes to a stop after having walked a distance from the point . He turns round, looks back and sees a black line sticking up from the earth at the point , whence he started. The height of the line appears to him to have a certain magnitude; and in order to register this magnitude accurately, he measures with an instrument the precise visual angle under which he sees the black line. He then continues his journey, stops, turns round again and finds that the black line is now seen under a smaller visual angle, hence appears to him to be reduced in height. His first impulse would be to assume that the black line was due to some squirming mirage and, while a reality in so far as it affected his eyesight, yet could be endowed with no objective existence in any impersonal way. Yet, without even returning to the point and investigating the matter more closely, our observer could soon find out that the black line must be produced by some existent entity. He would make this discovery in the following way: By combining the visual angle under which the black line appeared to him with his distance from the point , he would find that wherever he stood, a certain mathematical expression (i.e., ) representative of a length maintained the same invariant value. Something truly objective would have been discovered since, in contrast to visual angle and distance which varied with his position, here was a length which was an absolute since its value transcended his position. He would immediately attribute this absolute to the objective existence of a real rigid pole of length standing at the point . In short, the test of objectivity is invariance for all observers, and this illustration has enabled us to separate the absolute length of the pole from the relative visual angle and distance . Classical science assumed that this test of objectivity would hold in all cases, regardless of whether our observers were stationary or in motion. The theory of relativity tells us that this is not so. The observer in relative motion would always perceive the pole under a smaller visual angle, so that the mathematical expression purporting to give the length of the pole would no longer be an invariant.[71] Under the circumstances we should again be thrown back on the perplexing problem of whether there could exist any kind of universal objectivity holding for all observers in this universe.
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One thing is certain. We could no longer attribute an invariant length to the pole. This length would no longer be immanent in the pole, since it is impossible to discover an invariant expression of spatial length holding for all observers, hence independent of their existence. We cannot, therefore, escape the conclusion that spatial length by itself can have no more universal existence than the apparent height or visual angle of classical science. Unless we abandon all hope of discovering a something universally objective, we much search for some substitute for that absolute spatial distance in which classical science believed.[72]
Minkowski found that such an absolute could be constructed; but it was made up of the relative length and duration under which every observer perceived any two instantaneous events. In exactly the same way as classical science had constructed the absolute spatial magnitude, so now Minkowski constructed the absolute space-time magnitude, i.e., the Einsteinian interval. Objectivity is at last rediscovered, though this objectivity refers neither to space magnitudes nor to durations, but to a combination of both. In other words, the analogue of the absolute objective length of the pole in classical science is now given by the absolute objective space-time interval. The analogue of the relative visual angle, or apparent height of the pole, of classical science is given by its relativistic length as measured by the observer; and this length is no more an illusion than was the visual angle. By omitting to analyse the meaning of his words, the critic has argued as though the visual angle, or apparent height, in classical science were an illusion. But the visual angle is not an illusion. The illusion arises only from faulty inferences, when, having identified visual angle with apparent height, we then proceed to confuse the apparent height with the real height of the pole, thereby crediting it with too short a length. It is this last confusion that is unjustified, for it amounts to confusing visual angle, or apparent height (which, though real for the individual, has no universal significance, hence which is relative) with real length (which is absolute in classical science).
In the same way the FitzGerald contraction is not an illusion. It can be measured, and manifests itself physically in every conceivable manner. But again it is a relative for the very concept of spatial magnitude is essentially a relative, since, according to our motion, it will have one value or another. That which is absolute, that which endures unchanged, is the Einsteinian interval; and unless we wish to continue to talk about such shadows as space and time, it is in the space-time world and there alone that we shall obtain universal objectivity.
Let us now state the precise nature of those modifications in lengths and durations which will arise as a result of relative motion. The Einstein-Lorentz equations show that a rigid material object, say a yardstick, which, when at rest in our frame, would measure out as one yard in length, would appear to contract more and more in value in the direction of its relative motion (FitzGerald contraction) as this relative velocity increased. Could the relative velocity reach that of light, the rod would become completely flattened down to nothing.
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Likewise, an observer fixed to a clock would note that the durations between the successive second-beats of the clock were truly seconds. But with an increase in the relative motion existing between clock and observer, these beats would measure out as lengthened; and for a relative speed equal to that of light an infinite duration would separate two successive beats.
It must be noted that this lengthening in the durations defined by the tickings of the clock must not be attributed to the fact that the observer is moving farther and farther from the clock and therefore that longer and longer periods of time are necessary for him to take cognisance of these tickings. The results mentioned are those which would be found to hold after all the necessary corrections had been performed. It is solely the relative motion of the observer and the clock which is involved, and not the variations in their mutual distance. Our results would be exactly the same whether we were approaching or receding from the clock, so long as the same magnitude of our relative motion was maintained.
It is customary to designate by the names proper length and proper duration the length and duration obtained by an observer at rest with respect to the rod or clock. Beginners appear to have considerable difficulty in convincing themselves that all these durations, whether proper or elongated, are equally real. We have already explained in what measure they are so, but it may be helpful to give the following additional illustration: Suppose we are suffering from a headache and that this headache lasts an hour when timed by our watch. This will be the proper duration of the headache. Now we might be inclined to claim that inasmuch as we alone are suffering from the headache, we alone are in a position to determine its duration; and when some rapidly moving observer tells us that we have held our watch in our hand for one week and that our headache has therefore lasted a week, we might be tempted to tell him to mind his own business. But it must be realised that although the headache is ours and ours alone, and nobody else can feel it for us, yet the duration occupied by the headache is common property, and any other observer is just as much entitled to live it and to measure it as we ourselves are.
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CHAPTER XXII AND THE TRIP TO THE STAR
THE inevitable rational consequences which follow with mathematical certainty when once the existence of space-time as a model of the universe is conceded, are often puzzling to beginners, who feel that these consequences are too paradoxical to be acceptable. Of course, we must distinguish between a paradox of feeling, and a paradox of logic or an inconsistency. Any inconsistencies would be fatal to a rational construction and would denote either errors of reasoning or the presence of some fundamental contradiction in our basic premises. If, therefore, when analysing any given example in Einstein’s theory, we proceed to follow our deductions by adopting successively two different alternatives in our point of view, both of which would be consistent with the premises of the theory, and if we finally compare the results thus obtained and find them to be contradictory, hence inconsistent, the theory of relativity could never survive. However, it is scarcely probable that any such inconsistencies will ever be discovered by the layman, inasmuch as the theory of relativity is the work of mathematicians, for whom, more than for any one else, accurate and consistent reasoning is the indispensable condition of research.
On the other hand, though no inconsistencies or paradoxes of logic can exist in Einstein’s theory, this does not mean that the conclusions of relativity may not appear strange and lead to paradoxes of feeling. But once more it is important to differentiate between strangeness or paradoxes of feeling, and inconsistency or paradoxes of reason.
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The important point which must be settled, therefore, is whether our habitual notions of time, space and simultaneity are rational necessities, or whether they are merely feasible notions among a number of equally feasible ones. The question we have to decide is analogous to that which Kant proposed to solve when he stated that Euclidean geometry and three-dimensional space were the a priori forms of pure perception. Kant was wrong, as the discovery of non-Euclidean geometries proved without doubt. To-day Einstein has done for space, time and motion what Riemann and Lobatchewski did for the geometry of space alone. He has furnished us with a model just as rational as the model of classical science; and this model has the additional advantage of being in accord with empirical observation to a high order of precision, instead of being a mere approximate construction as was the classical structure. Indirectly, then, Einstein has proved that our innate classical conceptions of space, time, simultaneity and motion were by no means rational necessities.
Many people, however, are of the mistaken opinion that the theory of relativity leads to paradoxes of logic or inconsistencies. But when we analyse their arguments we are always confronted with the same erroneous method of reasoning. In many cases the critics seem to have a very hazy idea of the premises on which the theory is built. Others, while stating that for the sake of argument they are willing to recognise the validity of these premises, appear to forget their good intentions before reaching the end of their discussion. Inadvertently, in the very middle of their demonstrations, they relapse into the premises of classical science and, of course, are led to insufferable contradictions; whereupon they conclude that the theory of relativity leads to inconsistent results.
Let us consider, for example, the relativity of duration. A number of clocks or atoms which beat at the same rate and mark the same time when placed side by side are displaced along different routes to the same terminal point. The relativity theory then demands that when set side by side once again they shall be marking different times. The critic immediately exclaims: “This is inconsistent!” But where is the inconsistency? There would be inconsistency, indeed, had Einstein started out by positing as an axiom that time was absolute. This, in fact, is precisely what the critic has done. But then, when, as a result of this uncalled-for presupposition, he finds that the theory leads him into inconsistencies, he should not blame Einstein; for the theory is inconsistent not with itself as a logical structure, but solely with the critic’s own presuppositions.
Of course, the source of all the trouble lies in the fact that a belief in the absoluteness of duration has become second nature to us; and it requires a certain effort of introspection on our part for us to realise that this belief presents no rational justification. Eventually, however, our ideas become clarified, and we concede that we have no means of figuring out a priori whether time will be absolute or relative. We realise that it is impossible to lay down the law at the start one way or the other, unless we wish to force nature into a preconceived mould—which may not fit.
Our sole means of settling the problem is, then, to appeal to experiment; to ultra-precise experiment in the present case, since the crude variety would not enable us to reach a decision. The experiment of the clocks, could it be performed with sufficient refinement, would constitute one of these. Obviously it would be unwise to assume that its result could be known with certainty in advance; for were men to labour under any such belief, there would be no need for them to experiment. We may, of course, question the likelihood of a certain result occurring, basing our opinion on previous experiments of a similar nature; but in the final analysis our understanding of nature will issue from an accurate observation of facts and not from a-prioristic metaphysics.
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Unfortunately, as we mentioned previously, this experiment with clocks or atoms cannot be performed with sufficient accuracy, so that indirect methods must be considered. But it is precisely because all the indirect experiments thus far attempted have yielded results incompatible with the absoluteness of duration and distance that Einstein was compelled to reject these secular beliefs of mankind. Accordingly, we have every reason to anticipate that the clock experiment would bear out Einstein’s views; in fact, were this not the case, the entire theory would collapse.
Let us abandon these generalities, and consider the clock experiment in a more vivid form. We refer to what is known as the problem of the twins. In this we consider twin brothers one of whom remains on earth, while the other steps on to a magic carpet and visits distant Arcturus. On his return he finds that his brother has grown old and decrepit, while he himself has preserved his youth.
This particular consequence of the theory has been one of the stumbling blocks of practically every lay writer who has devoted his time to criticising the theory of relativity. Some have stated that Einstein never upheld any such absurdity, and that the whole trouble must be ascribed to his too enthusiastic followers. Others have stated that the conclusions violated the rules of logic in that they led to inconsistencies; and that if Einstein’s theory necessitated such results, it should be regarded as a gigantic hoax from beginning to end.
Now, in order to avoid any misconception, we may mention that in our example both twins hold in their hands clocks which, when placed side by side, beat out their seconds and minutes in perfect unison. Einstein’s theory indicates that the clock carried by the travelling twin will be behind time when brought back to earth and compared again with the clock that has been left there, in the keeping of the stationary twin. Some critics, such as Bergson, while accepting the postulate of the invariant velocity of light, deny that this slowing down in the beatings of the moving clock is required by the theory in any real sense. It is a mere appearance, Bergson maintains; a mathematical illusion. Just as the gradual shrinking of a man’s apparent size as he moves away from us would not connote any real decrease in his stature, so now, when the travelling twin returns with his clock to his brother on earth, the illusion would vanish, and both twins would recognise that their respective clocks were still marking the same time in perfect agreement, one with the other.
Now, it cannot be stressed too strongly that this interpretation of the problem is worse than incorrect. Not only is it incorrect from the standpoint of Einstein’s theory, but it denotes on the part of the critic a colossal ignorance of the significance of a theory of mathematical physics. Here it must be recalled that the comparative retardation of the travelling twin’s clock is an inevitable consequence of the Lorentz-Einstein transformations. To deny, therefore, that these transformations correspond to what would actually occur, to what would be detected and measured by the most accurate instruments, would be to deny that the transformations possess any physical basis. If this be the stand we wish to take, well and good; but the logical outcome of this attitude will be to deny the legitimacy of the entire theory; [Pg 228] it would be quite impossible to retain the theory while contesting the real significance of the transformations. Once again a theory of mathematical physics is not one of pure mathematics. Its aim and its raison d’être are not solely to construct the rational scheme of some possible world, but to construct that particular rational scheme of the particular real world in which we live and breathe. It is for this reason that a theory of mathematical physics, in contradistinction to one of pure mathematics, is constantly subjected to the control of experiment. If, therefore, the Lorentz-Einstein transformations failed to yield the results that would be actually measured, Einstein’s theory as a means of physical discovery, as a co-ordination of physical facts, would be not only useless, but entirely incorrect, and we should have to abandon it.
Furthermore, it should also be noted that every precise physical experiment that it has been possible to accomplish has verified the anticipations of the Lorentz-Einstein transformations; it is owing to this fact that the theory was not abandoned years ago. This in itself should warn us against asserting that the transformations possess no physical significance.
Thus far we have been concerned with generalities and have merely attempted to show why Bergson’s solution of the problem must be discarded on first principles. Leaving aside this aspect of the question, it is instructive to consider the formal proof which he presents in support of his contentions.
He appears to realise that the entire matter hinges on the measure of physical reality which should be credited to the Lorentz-Einstein transformations. Accordingly, he attempts to prove that these transformations do not represent reality, but possess a mere fictitious mathematical significance having nothing in common with the physical world. He calls them “phantasmatical,” contending that mathematicians, not being trained in philosophy, are severely handicapped when they attempt to interpret their equations, and are incapable of distinguishing mathematical fictions from physical reality. So Bergson proceeds to demonstrate the fictitious significance of the transformations.
He first serves us with a series of arguments based on the principle of the relativity of motion itself. In the very first paragraph, however, he confuses the relativity of Galilean motion with the complete relativity of all motion, Newtonian and special relativity with visual relativity, velocity with acceleration. All these errors relate to classical science itself; and it is obviously quite impossible to discuss Einstein’s theory intelligently until we have acquired at least an elementary knowledge of classical mechanics. Following this philosophical demonstration, he goes on to give us a mathematical one; but all he succeeds in proving is that he has not the slightest conception of the most elementary properties of mathematical transformations.
Now, without going into mathematical details, let us make an effort to see why it is that the clock of the travelling twin must necessarily have suffered retardation. To understand the reason, we must revert to Einstein’s premises, and in particular to the postulate of the [Pg 229] invariant velocity of light. This postulate, as we have explained on several occasions, states that all Galilean observers situated in their respective Galilean frames will discover, as a result of measurement with rigid rods, that waves of light invariably pass through their frames with the same constant speed of 300,000 kilometres per second. If this postulate be accepted, we must infer that in every Galilean frame, time can be measured by the distance in space covered by a wave of light (the distance being measured with rigid rods at rest in the frame). As a result the postulate allows us to construct a sort of optical clock. We may, for example, consider two parallel mirrors situated at a fixed distance apart of 1.5 kilometres in our Galilean frame, and we may suppose that a wave of light is reflected back and forth from one mirror to the other. These oscillations of the ray of light, covering always the same distance, require equal durations; and in the present case 100,000 such double oscillations would be seen to define a duration of one second for that Galilean frame in which the optical apparatus was situated. We must henceforth consider every Galilean frame as possessing an optical clock of this kind defining the passage of time for its own frame.
Let us now analyse the problem of the two twins; and let us suppose that we as observers are stationed on the earth, say, at the North Pole. To a high degree of approximation we may consider ourselves, therefore, as at rest in a non-rotating or Galilean frame. The moving observer is also situated in a Galilean frame, moving away from us with a speed which we shall assume to be about four-fifths that of light. He visits a distant star which we should recognise as being sixteen light years distant[73] from us, then reverses his velocity and returns to earth.
Both we and the traveller carry optical clocks, of precisely the same dimensions when placed side by side; so that for either one of us, 100,000 oscillations of our respective light rays in our respective optical clocks correspond to one second of our respective times. Let us assume that the optical clock of the travelling observer is placed crosswise to his motion with respect to us, so that his light days oscillate back and forth in a direction perpendicular to his relative motion. So far as the traveller is concerned, the light rays of his optical clock will oscillate along one same straight line 1.5 kilometres in length; and if during his entire trip back and forth he finds that his clock has oscillated times, he will be justified in saying that according to his optical clock his trip has lasted seconds. This, in the precise numerical example we have chosen, would turn out to be twenty years.
Now let us examine the problem from our own point of view; that is to say, let us refer all measurements to our own Galilean system attached to the earth. In this Galilean frame of ours the oscillating wave of light of the traveller will not follow a to-and-fro motion along the [Pg 230] same straight line. It will follow a zigzag line analogous to a series of ’s placed end to end. The total length of this path followed by the light wave must therefore be longer than when referred to the frame of the traveller; in the present numerical example it would be just twice as long. The question then arises: “How long will it take for the traveller’s wave of light to describe this zigzag line back and forth when time is measured according to the standards of our Galilean frame attached to the earth?” The postulate of the invariant velocity of light gives us the answer immediately. Since the zigzag line followed by the light wave in our frame is just twice as long as the succession of superposed up-and-down lines which it follows in the traveller’s frame, and since, whether referred to one frame or the other, the velocity of the light wave along its path must be the same (invariant velocity of light in all Galilean frames), then the immediate conclusion is that the total duration of the oscillations will be twice as long when measured by us as when measured by the traveller.
As these oscillations start and end with the beginning and end of the trip, we must assume that the trip, which lasted twenty years according to the traveller, will have lasted forty years in our own estimation. In other words, any registering device would show when both optical clocks were again placed side by side, that our earth clock had performed oscillations during the trip, whereas the moving clock had performed just half that number, namely, . Obviously, there is no illusion about the matter; for, as we have said, the oscillations of the two clocks could perfectly well be registered graphically, so that we should be dealing with a definite enumeration of registrations which any third party could verify.
It is quite easy to understand why these extraordinary results must take place. The cause is to be ascribed entirely to this invariant velocity credited to any light wave in all Galilean frames. Were it not for this postulate, the mere fact that the total length of the zigzag line was twice that of the up-and-down line would not connote any such difference in duration. Classical science would have assumed that since the path was twice as long, the velocity of the light wave along this path would have measured out twice as great; so that finally both durations would have been the same, in full accord with the classical belief in the uniqueness of time. In short, the comparative retardation of the moving clock follows as an inevitable necessity when once we agree to accept the validity of the postulate of the invariant velocity. On no account is it permissible to accept the postulate and then deny its mathematical consequences. If, therefore, the example of the trip to the star is too great a tax on our credulity, we must reject the postulate of invariant velocity and be faced with all the difficulties which surrounded electrodynamics prior to Einstein’s discoveries.
And now we may discuss another type of criticism, not quite so misplaced as that of Bergson, but still utterly erroneous. It has been argued, for instance, that all we have succeeded in proving is [Pg 231] that if each Galilean observer measures time with the help of his optical clock, it is true that the duration of the trip will measure out differently according to the observer, but that if, in place of optical clocks, each observer had defined duration in terms of the usual mechanical watches, these strange discrepancies would never have arisen. In other words, the so-called slowing down of time in the frame of the travelling twin would be due entirely to the slowing down, not of time, but of the optical clock wherewith he proposed to measure time. Always according to this same criticism, the optical clock measured duration only in virtue of an artificial convention, and its readings could not be claimed to represent true lapses of time. Real time would flow independently of any clock; and to assert that an arbitrarily postulated clock defined true time would be about as reasonable as claiming that we should be dead were we to forget to wind up our watch. In other words, we might adjust our clock as we pleased, accelerate its motion, retard it, and even stop it completely, but we should go on living just the same.
The only way to refute this fallacious argument is to examine once again the premises of the relativity theory. Einstein starts with the assumption that Galilean motion through the ether is relative. From this it follows that no observer attached to a Galilean frame can by any means whatsoever ascertain whether his frame is in a state of uniform rectilinear motion or in a state of rest (special principle of relativity). Then he lays down his famous postulate on the invariant speed of light (300,000 kilometres per second) through any Galilean frame when measured by the observer attached to the frame. This postulate, as we have seen, enables any Galilean observer to construct an optical clock and to measure duration as referred to his clock, hence to his frame.
Now, in order to answer the criticism which charges Einstein’s selection of optical clocks with being artificial, we must proceed to investigate whether the definitions of congruent durations marked out by the optical clocks are justifiable. And, first, what do we mean by a justifiable definition of the equality of two successive durations? Would it have been justifiable to define equal durations by the durations a body falling vertically towards the earth would require to cover equal Euclidean distances in space? Obviously not.
We saw, when discussing congruence generally, that mathematical analysis by itself afforded us no means of exalting above all others one particular definition of congruent stretches either for space or for time; but that when we combined physics with mathematics, certain precise methods of defining congruence imposed themselves immediately in any given Galilean frame of reference. In the case of time-congruence we found it necessary to assume that any periodic phenomenon at rest in our frame, in a complete state of isolation from all external influences, would have to be regarded as passing at regular congruent intervals of time through the same initial state. This assumption was demanded by our belief in causality, and by a [Pg 232] justifiable definition of time-congruence we mean to imply one which would satisfy these consequences of causality.
When, therefore, we stop to consider whether Einstein’s optical clocks yield us a justifiable definition of equal successive durations, the whole question simmers down to deciding whether or not the oscillations of a ray of light in an optical clock constitute an isolated phenomenon, or whether, according to the motion of the Galilean frame in which the clock is situated, certain perturbating influences may not interfere with the state of isolation we credit to the working of the clock.
Until such time as the effect of the ether on an object moving through it had been settled, there was every reason to suppose that in the case of a Galilean frame moving through the ether, the effect of the ether wind would be to disturb the free oscillations of the light rays between the mirrors of the clock. And so the clock could not be considered isolated, and its readings would be valueless, or at least in need of correction. But Einstein’s special principle of relativity has precisely for its object to inform us that the ether behaves as though it were non-existent, so that any perturbative effect which classical science had suspected was ipso facto eliminated. In other words, if we accept the special principle of relativity, the optical clocks may be regarded as perfectly isolated; and their readings must therefore afford us with perfectly reliable measures of congruent time-stretches, as required by the principle of causality.
It follows that if in a Galilean frame we place side by side, at rest, an optical clock, a vibrating atom, a piece of radium gradually losing its mass, and a top revolving without friction, the time determinations defined by all these phenomena will be identical, since all these are perfectly isolated from external influences. To be more precise, if a sodium atom vibrates so many times per second, as measured by the optical clock of its Galilean system, it should still vibrate exactly the same number of times per second (as measured by the clock of the frame) even if the constant velocity of the Galilean frame (with respect to some standard Galilean frame) has been changed to some other constant velocity. In short, all physical phenomena at rest in a frame would evolve at exactly the same rate in terms of the optical clock of the frame, regardless of the magnitude of the Galilean motion of the frame; since, according to the special principle of relativity, this absolute Galilean motion is entirely meaningless.
If observation, by any chance, should prove that the relative rates of vibration of the atom and the clock were to vary according to the Galilean frame in which we set them, we should have a physical means of differentiating one Galilean frame from another, hence of determining the velocity of our frame through the ether; and the special principle of relativity would have to be abandoned. Assuming, therefore, Einstein’s premises to be correct, we must agree that the vibrations of the optical clock constitute as perfect a means of determining congruent intervals of time as it is possible for physicists to obtain. Finally, we see that in any Galilean frame exactly the same [Pg 233] time-determinations would be obtained whether we made use of an optical clock, a top spinning on its axis without friction, a vibrating atom, or a mechanical clock, provided all these bodies were at rest as a whole in our Galilean frame.
Consider, then, two Galilean frames, each containing an optical clock and a top spinning without friction on its axis. If, when the two frames are brought to relative rest, side by side, the two tops are spinning at exactly the same rate, both executing rotations per second when measured in terms of their respective clocks, then exactly the same correlation must exist when the Galilean frames are moving apart. It is therefore a matter of indifference whether the observers in the frames measure time by means of their optical clocks or by means of the rotations of their tops.
Now a planet revolving on its axis in space is nothing but a huge top; and if, for instance, we stand at the non-rotating point of a planet, say the North Pole (so as to be in a Galilean frame), we should be perfectly justified in measuring our unit durations by the successive rotations of our planet with respect to the fixed stars. Consider, then, two planets rotating at exactly the same speed when held side by side, and let us call one of these planets the earth. Regardless of whether these two planets be set flying apart or held together, the observer on either planet will be justified in measuring his unit time-stretches in terms of his own planetary rotations. Classical science assumed that since time was one and universal, both planets would always beat out exactly the same unit time through their rotations, whether they were in relative motion or at rest, so it mattered little which particular planet was selected. Under the circumstances, from motives of convenience, the earth was chosen as standard.
The novel point of Einstein’s theory consists in proving that this attitude is untenable. Assuming the various planets to rotate with the same speed when their axes are at relative rest, each different observer must henceforth abide by the rotations of his own planet and not by those of the other planets when relative motion is present. Only thus can the special principle of relativity be maintained. In particular, if we assume one rotating planet to be transferred to a distant star, then returned near the rotating earth as in the example of the two twins, it would be found that, owing to the solidarity between the optical clocks and the rotations of the planets, the wandering planet would have revolved less often during its trip than would have been the case with the earth. So we see that the slowing down we have mentioned is absolutely general, and must indeed be so if the special principle of relativity is to endure.
Furthermore, we see that the great difference between classical science and relativity consists, not in having substituted an optical clock for the earth’s rotation, since both these determinations of time would be identical, but in having proved that the earth’s rotation or the earth’s optical clock is valid only as a measure of time for an earth observer or for one at rest with respect to him.
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Having established the solidarity of the evolution of all physical phenomena at rest in the same Galilean frame as an inevitable consequence of the relativity of Galilean motion, we must now enquire whether this general solidarity should be assumed to extend to vital phenomena and to our consciousness of the flight of time. Of course we cannot deny that aging is accompanied by a physiological alteration of our bodies; hence, on this ground alone, there can be no doubt that physiological aging will proceed in terms of the general solidarity of all physical phenomena. But there exists another aspect of the passage of time, and that is our consciousness of its stream.
Our awareness of the flight of time is probably the most fundamental fact of consciousness; and it remains an ever-mysterious enigma. If we credit it to certain physiological processes occurring in the brain, it must of course accompany the general evolution of physical processes; but if we wish to regard this flight of time as a vital or spiritual phenomenon different in its essence from anything physical, there can be no a priori reason for assuming that it would accompany the general physical processes. Of course our awareness of the flight of time is an extremely vague feeling; nevertheless, so long as we retain a conscious state, we have no difficulty in differentiating a duration of twelve hours from one of one second.
If, then, in the example of the two twins, we assume that the length of the trip and the velocity of the travelling twin have been so selected that when the twins meet again, the clock of the traveller marks an advance of only one second, whereas that of his stationary brother marks an advance of ten years, our differentiation between physiological time and our consciousness of the time-flow would lead to the following result: The travelling twin would not have aged, physiologically speaking, but he would be fully aware, in his inner consciousness, that a stretch of time far in excess of one second had elapsed during his trip. However, we cannot fail to observe that if such were the case, if there existed two crates of aging for the same man, one physiological and the other corresponding to a psychological sensing of the flow of time, the special principle of relativity, and with it Einstein’s theory, would have to be abandoned.
The reason for this last statement is easy to understand: If the two species of time-flow varied independently, the rate of evolution of all processes would appear to vary according to the frame in which we might be situated—a tortoise would appear to crawl or to be galloping madly through space. Theoretically, therefore, it would be possible to detect a difference as between Galilean frames, hence to discover absolute velocity through the ether or through space. And this would be contrary to the principle of relativity.
When one places in the balance the mass of empirical evidence which lends support to the special principle of relativity, and the total lack of evidence of any sort or kind which would encourage the belief that the time sensed and lived is different from the time of physics and of physiological aging, one cannot be surprised at the total lack [Pg 235] of interest the mathematicians of relativity must take in such vague hypotheses of certain philosophers.
Moreover, there is yet another aspect of the question. The fundamental reason for all the criticisms we have mentioned, is that deep down at the bottom of his heart the critic wishes to preserve, in spite of all, his hereditary belief in the absoluteness of time. Even when convinced that as a result of Einstein’s physical discoveries any such belief is incompatible with the evolution of things physical, he endeavours to assert that absolute time must continue to endure when the sensing of the flow of time and our consciousness of being alive are contemplated. In this way he proceeds to draw a distinction between time actually lived and time as measured by our optical clock. But the critic overlooks the fact that this belief in an absolute time which he refuses to abandon is in no wise a rational necessity, any more than is the hypothesis of the plurality of times. Experiment alone can yield us a clue; and granting that physical experiment may not be capable of throwing light on the spiritual, nevertheless, if we are to be eternally ignorant of things spiritual, there is surely no reason to state a priori that they must be governed by absolute time or that time itself should have any meaning so far as they are concerned. Thus the trouble with the critic who wishes to understand the significance of Einstein’s discoveries is that in spite of himself he remains obsessed by this unwarranted belief in the absoluteness of time. This conviction is undoubtedly hard to eradicate, and only prolonged introspection can gradually rid us of an obsession which is so thoroughly ingrained in the species. Then, and only then, will the so-called paradox of the two twins cease to be a paradox.
For all these reasons, if we accept Einstein’s premises there is no alternative but to recognise that the duration marked out by an optical clock in a Galilean system will give us the same measures of duration as will any other isolated periodic physical phenomenon; and that our aging, our awareness of the flow of time, and the time we are conscious of living accompany the oscillations of the optical clock. In short, therefore, if the optical clock in our Galilean system indicates that one year has elapsed, we must have aged by one year. We shall have gone to sleep and we shall have risen from bed on 365 different occasions, and we shall have had enough time to perform all those duties which usually occupy the space of one year.
Under these circumstances, to what does the example of the two twins finally reduce? Simply to this: The departure of our twin brother and his return to earth constitute two definite events. The duration separating two events being robbed of any definite value by the theory of relativity, there is no cause to be surprised that this duration should manifest different magnitudes to different observers.
Again let us put the paradox in the following alternative form: Our twin brother covers a distance of thirty-two light years (when he goes to the star and then returns), and the duration of his trip is forty years; hence his speed along the distance is four-fifths that [Pg 236] of light. How, then, could he ever be misled into believing that he had required only twenty years to cover this total distance? Would not this belief suggest to him that his speed had been greater than that of light, and would not this fact conflict not only with relativity but also with the assertion which he would voluntarily offer, that his speed with respect to us was only four-fifths that of light?
But here, again, the criticism is occasioned by another erroneous obsession, that of absolute distance. The critic does not grasp the fact that the distance from earth to star, just like any other distance, is indeterminate and has no meaning in itself. While it is true that, so far as we are concerned, this distance is sixteen light years, yet in the opinion of the traveller it is only eight light years; and his opinion expresses exactly the same measure of reality as does our own. We can only repeat once more that absolute duration and distance are incompatible with Einstein’s premises; and if we insist upon preserving our classical obsessions, a study of Einstein’s theory will be little better than a nightmare.
Let us now pass in review a totally different order of criticism. It is contended, for example, that the principle of relativity permits us to interpret any problem of relative motion between two bodies in two different ways: by considering either one body or the other as being at rest. Then, according to the critic, since in the problem of the two twins it is the twin’s motion that is the cause of the slowing down of his time, all we need do is to consider him at rest and his brother in motion, in order to reach the ridiculous conclusion that either twin is younger than his brother.
This argument is radically incorrect for two separate reasons. In the first place, the special principle of relativity does not state that, in the general case, when two bodies are in relative motion we can consider either one of them as being at rest. The special principle emphasises the fact that a reversibility of this kind is possible only when the motions of both bodies are Galilean.
In the present case the state of the twin remaining on earth is truly Galilean (if we neglect the earth’s rotation and the curvature of its orbit); but this Galilean condition no longer applies to the twin who reverses his course when he reaches the star. In this latter case there is acceleration, and the well-known symptoms of accelerated motion, namely, forces of inertia, would make their appearance and would bring about well-defined physical disturbances.
As a matter of fact, we can amend the problem of the two twins so as to eliminate acceleration by assuming that the so-called travelling twin passes both earth and star without ever stopping or reversing his course. In this case the motions of both twins are Galilean; and, true enough, the special principle of relativity allows us to consider either one of the twins as at rest and the other as in motion. But, again, we should see that exactly the same relative agings followed, whether we considered one twin or the other as being at rest; hence the critic’s statement that motion produces a slowing down of time is a [Pg 237] loose one unless explained more fully.
We may make these points clearer by examining the slightly modified example we have suggested. First, suppose that the earth twin, the earth and the star are all at rest; and that the so-called travelling twin travels without a stop past earth and star. We will further suppose that a rod extends from earth to star. To return to the numerical values selected at the beginning of the chapter, calculation would show that the duration of the passage of the travelling twin from earth to star would be twenty years in the opinion of the earth observer and only ten in the opinion of the wanderer. Why this difference?
Because, as we have already mentioned, since a relative motion exists between the so-called wanderer and the rod, the distance from earth to star, or the length of the rod, is contracted to half its length when computed by him. And here we must note that this contraction is not due to the so-called wanderer’s motion (for motion, unless accelerated, has no absolute significance). It is due to the relative motion between the wanderer and the rod, or the earth-star distance. If, therefore, the wanderer were to consider himself at rest and the earth-twin, together with the earth and star, travelling past him, exactly the same conclusions would endure; since once again the rod, or earth-star distance, would be animated with exactly the same relative velocity in respect to the so-called wanderer, and it is relative velocity, and not absolute velocity, that counts. We see, therefore, that no inconsistencies can arise whether we consider one twin or the other as at rest.
Let us now revert to the complete problem where the wanderer returns to earth. In this case, the wanderer has led an entirely different life history from his brother, since he, and he alone, has suffered acceleration.
It would not be legitimate to attempt to apply the special principle of relativity, and to argue as though the wanderer had always been at rest; for by so doing we should be assuming that he had never suffered acceleration, and we should therefore have to credit his brother with having experienced this acceleration. The problem would be completely modified; and, indeed, the relative aging would then be reversed. But no argument against the theory could be invoked, for the simple reason that we should be considering a totally different problem, one in which the circumstances of acceleration were no longer the same.
For the sake of completeness let us assume that it is now the wanderer who remains at rest, while the earth twin, the earth and the star execute a backward-and-forward motion, being of course subjected, in so doing, to acceleration when their velocity is reversed. Retaining the same numerical values, we should find that the erstwhile wanderer had waited twenty years, as before, for the return of his brother, but this earth brother would now conclude that his absence had lasted ten years instead of forty. So even from a purely numerical standpoint conditions would not have been completely reversed. Thus, in a general way, we see that when two observers leave each other, then meet again, one of the [Pg 238] two, if not both, must have been submitted to acceleration; otherwise they could never meet again but would wander farther and farther apart ad infinitum.[74] If we consider twins separating, then meeting again, the younger of the two will invariably be the one who has suffered acceleration.
We need not be surprised at these very real effects produced by acceleration, for we must remember that acceleration is an absolute and not a relative like mere velocity. A body has one velocity or another, according to our point of view or choice of a frame of reference; but when a body is accelerated the choice of our frame of reference is irrelevant for the simple reason that acceleration betrays itself by physical forces and stresses which may exert palpable influences.
We must not allow ourselves to be misled by a further generalisation of Einstein’s, known as the Postulate of Equivalence, according to which even an accelerated observer may be considered at rest. This new postulate modifies to a certain degree the position of the problem of the two twins. Whereas the special principle did not allow us to consider the accelerated wanderer as at rest, the postulate of equivalence permits us to adopt this alternative view. However, those who hope to discover by this means a logical inconsistency in the theory, will again be disappointed; for the postulate states expressly that an accelerated observer may be considered at rest only if we assume that a field of gravitation takes the place of the field of inertial forces to which the accelerated observer was submitted. Then, in virtue of the postulate of equivalence itself, exactly the same relative rejuvenation will take place except that it will now be ascribed to gravitation instead of to acceleration.
It is hoped that the paradox of the two twins has been presented in a sufficient number of ways to enable the conscientious student to master the difficulty. The problem requires a certain effort of introspection; but once we have succeeded in ridding ourselves of our prejudiced ideas regarding space and time, the paradox will vanish of itself.
And now let us mention another type of so-called paradox which appears to cause considerable trouble to beginners. We refer to the Einsteinian addition of velocities. Thus, the critic begins by stating that according to the dictates of common sense , so that it is only natural to assume that a velocity of one mile per second plus another velocity of one mile per second equals a velocity of two miles per second. Einstein’s contention that the resultant velocity is slightly less than two miles per second is therefore an absurdity on the face of it. But if we examine these arguments carefully we shall see that they again are based on a faulty understanding of the problem.
Even in classical science it is not necessarily correct to state that velocities add up like the numbers of arithmetic. Consider, for instance, the following illustration: If, while we are standing on an embankment, a train moves away from us with a speed and if a rifle bullet is shot from the train with a speed , it is not always correct to state that the velocity of the bullet relatively [Pg 239] to us on the embankment is . This statement is true only when the velocities of the train and the bullet lie along the same straight line. If we neglect to specify this restriction, our statement is incorrect. In Einstein’s theory the problem is very similar.
If a Galilean frame moves away from us with a speed and if, with respect to this frame and in the direction of its motion, a bullet speeds with a velocity , the resultant speed of the bullet might be only provided our methods of measuring a velocity, and those of the observer in the Galilean frame, were identical; that is, provided our measurements of space and time were identical. This was always tacitly assumed to be the case by classical science; but we know that in Einstein’s theory the identity of our standards of measurement no longer holds, since such a state of affairs would be incompatible with the invariance of the velocity of light.
We may therefore assimilate the conditions surrounding the various space and time measurements in Einstein’s theory to those which would endure in classical science if by reason of some law of nature it were impossible for the observer in the train to fire a bullet in the precise direction of the train’s motion. In other words, in Einstein’s theory, velocities lying in the same direction add up like algebraical numbers, provided the velocities we propose to add and subtract are computed under exactly the same conditions, that is, by the same Galilean observer. It is when one observer computes one velocity and a second observer, in relative motion, computes the second velocity that the classical law of addition is at fault.
Very similar to this problem of the addition of velocities is that of simultaneity. Thus, Einstein’s theory implies that if two events and occur simultaneously for one observer, John, and if in the opinion of another observer, Peter, in motion with respect to John, the event and another event occur simultaneously, it is not correct to assume that and will be simultaneous in the opinion of either of the two observers. Here the critic assumes, without further ado, that the symbolic expression of these relationships would be as follows: ; ; . Therefrom he infers that the theory of relativity is contrary to logic and is a manifest absurdity. And yet, if we examine these statements carefully, we shall have no difficulty in seeing that the error in logical reasoning lies not with Einstein but with the critic himself.
In order to dispel any doubt on this score, let us ask the critic what he intends to convey by the symbol =. Suppose, for example, that we are considering two objects, and , of the same size. Are we entitled to write ? The answer should be: Yes, if the symbol = stands for “is of the same size as”; or No, if it stands for most other types of identity. In order to avoid confusion let us then represent the symbol by ( standing for size) and let us write .
And now consider a third object, . We will assume that is of the same colour as . Beyond that we know nothing of it. Once again, are we entitled to write ? The answer is: Yes, if the [Pg 240] symbol = stands for “is of the same colour as”; No, if it stands for some other type of equality. In order to avoid confusion let us then write our symbol: ( standing for colour).
Our two equalities now become ; ; and obviously there is no reason for us to infer that should equal either in size or colour, since the symbols and do not convey the same meaning.
It may be assumed that even the most careless of critics would grant the correctness of the preceding argument. But in the case of his criticism of Einstein’s theory he would argue: “All this is very fine, but in the relativistic absurdity which I am pointing out to you I am perfectly entitled to write ; and . For in the present case the symbol = between and , and between and , stands for exactly the same concept, namely, that of identity in time; it does not refer to such different concepts as ‘size’ in the first instance and ‘colour’ in the second.”
Obviously, it is easy to detect the petitio principii unconsciously introduced into this argument. By stating that the two symbols = must mean the same thing since they both connote identity in time, the critic is accepting a priori as self-evident that there can be only one time-stream, or, in other words, that time is absolute. Of course, if he wishes to introduce his own private assumptions, he may do so; but he must not blame the theory if as a result thereof he is faced with a logical contradiction. He should realise that the contradiction arises solely as a result of his own initial unwarranted assumption, and that were he to withdraw this initial assumption the contradiction would automatically disappear. Then, indeed, he would have to recognise that the two symbols = need not have the same meaning; that the first, which we may write , would represent “at the same time according to John,” whereas the second one, , would represent “at the same time according to Peter.” As a result the relations would be written: and just as in the case of colour and size, he would recognise that the first two equalities in no wise compelled the equality of and .
But we may wonder why it is that the critic does not realise that all the trouble that has arisen is of his own making. The answer appears perfectly clear. It would be somewhat as follows:
“At no time does he realise that he has introduced an arbitrary assumption and has thereby begged the entire question. The traditional belief in absolute time so dominates his mind that he thinks in terms of it even when he knows that the theory of relativity is not based on any such a priori belief.”
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Once again, the theory of relativity as a theory of mathematical physics may survive or may succumb. It is for experiment to give the answer. If the anticipations of the theory should not conform to experiment, if, for instance, the recent experiment of Miller were considered conclusive as detecting velocity through the ether, it would be impossible to save the theory, at least in its present form.[75] But we must not confuse the ability of the theory to portray the workings of nature with its legitimacy as a perfectly consistent and rational doctrine of thought. On this last point no doubt exists. The theory has been scrutinised by the greatest of living mathematicians and not the slightest error of mathematical reasoning, nor therefore of logic, has been found in it.
Fig. VIa
As a final illustration of this same problem of the relativity of simultaneity we may present a geometrical analogy. We have only to consider two points and at the same distance above a line , and then to consider the points and at the same distance above another line . We have only to look at the figure to realise that and are at the same distance neither from nor from . Only should the two lines and coincide would they be so.
It is much the same in relativity. The straight lines and correspond to the time directions of the two different observers. Classical science, assuming time to be absolute, considered the two lines coincident. Relativity has proved the error of this belief.
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PART III THE GENERAL THEORY OF RELATIVITY
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