Part 2 — 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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]

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]

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.

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.

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.

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.

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]

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.

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.”

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.

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.

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]

“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.

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.

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 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.