CHAPTER VITIME
OUR awareness of the passage of time constitutes one of the most fundamental facts of consciousness, and our sensations range themselves automatically in this one-dimensional irreversible temporal series. In this respect we must concede a certain difference between space and time, the former being, as we have seen, a concept due in the main to a less inevitable synthetic co-ordination of sense impressions. Yet, on the other hand, there exists a marked similarity between space and duration, in that they both manifest the characteristics of sensory continuity.
Thus, in the case of time, two sensationsandmay be so close together that it will be impossible for us to determine which of the two was sensed first; the same may happen in the case of two sensationsandand yet we might have no difficulty in ascertaining thatwas prior to.These facts are expressed by the inconsistent relations,,,which, as we have seen, are characteristic of sensory continuity, be it in space, or in time, or in whatever domain we please. In the case of time, as in that of space, only by a process of abstraction can we remedy this inconsistency. We thereby conceive of mathematical duration, a duration in which mathematical continuity is assumed to have replaced sensory continuity.
Now, owing to the ordering relation or path of continuous transfer which appears to be imposed upon us by nature, perceptual time must be regarded as uni-dimensional, and by retaining the natural ordering relation, we will attribute the same uni-dimensionality to mathematical duration. It is, of course, to this mathematical duration that we appeal in all theoretical investigations in mechanics and physics.
Primarily the only time of which we are conscious is an “egoistic” time, the time ofourstream of consciousness. But as a result of our perception of change and motion in the exterior universe of space, it becomes necessary to extend the concept of duration throughout space, so that all external events are thought of as having occurred at a definite instant in time as well as at a definite point in space.[22]
Now, when discussing space, we saw that the concept of the same point in space, considered at different times, was ambiguous; it was necessary to specify the frame of reference by which space was defined, and according to the frame selected, an object would continue to occupy the same point or else manifest motion. The analogous problem in the case of time would be to conceive of the same instant in time at twodifferent places,i.e., of the simultaneity of two spatially separated events.
If we allowed ourselves to be guided by our analysis of position in space, we should be inclined to say that a determination of the simultaneity of spatially separated events must necessitate the choice of a frame of reference. The observer who was to pass an opinion on the simultaneity of the two events would then select that particular frame in which he stood at rest. But from this it would follow that a change in the motion of the observer, hence a change of frame, might bring about a modification in his understanding of simultaneity. Simultaneity would thus manifest itself as relative, just like position.
Although these possibilities cannot be discardeda priorion any rational grounds, classical science and common-sense philosophy refused to entertain them. The fact is that this thesis would have tended to surround with ambiguity the concept of time in physics, since the duration of the same event would vary with the motion of our frame of reference. It had always been felt that although, through an act of our will, we might change our motion through space, yet on the other hand the flowing of time transcended our action. These views were confirmed by the behaviour of physical phenomena known to classical science; hence the uniqueness of time and the absolute character of simultaneity had been accepted. In other words, the simultaneity and the order of succession of two spatially separated events were assumed to constitute facts transcending our choice of a system of reference.
This stand was fully justified in Newton’s day; for inasmuch asa prioriarguments could not decide the question, so long as experiment did not compel us to recognise the relativity of simultaneity and the ambiguity of duration there appeared to be no good reason to hamper science with an unnecessary complication. But the point we wish to stress is that if perchance experiment were ever to suggest that simultaneity was not absolute, no rational argument could be advanced to prove the absurdity of this opinion. As we shall see, Einstein’s interpretation of certain refined electromagnetic experiments consists precisely in recognising the relativity of simultaneity and the ambiguity of duration.
Let us now pass to the problem of congruence for time. As is the case with space, mathematical time is an amorphous continuum presenting no definite metrics. The concept of congruence or of the equality of successive durations has no precise meaning; and a definition of time-congruence can be obtained only after we have imposed some conventional measuring standard on an indifferent duration. We might define as equal the durations separating the rising and the setting of the sun, or, again, the durations separating successive rainstorms, etc. Theoretically, all such definitions would be legitimate, since in the final analysis duration, just like space, presents no intrinsic metrics.
In common practice, however, we find that just as in the case of space, a definite species of congruence appears to be imposed on human beings and animals alike. For instance, so long as we are in a conscious state, we all agree that the successive tickings of theclock correspond to what our psychological sense of duration would qualify as congruent intervals. In other words, we appear to be gifted with a species of psychological time-keeping device which allows us to differentiate between durations which we have come to call “seconds” and those which we have come to call “hours.” What is commonly called the sense of rhythm, which manifests itself in the primitive music of drum beats and in the dances of savages, is nothing but a consequence of this intuitive understanding of time-congruence.
A superficial analysis of the situation might lead us to assume that, contrary to our previous assertions, a definite species of congruence was intrinsic to duration. This opinion, however, is quite untenable unless supplemented by additional explanations. The problem with which we are confronted is in all respects similar to the one we mentioned when discussing space. Of itself, space, or at least mathematical space, is amorphous. In spite of this fact, we have no difficulty in agreeing that a coin is round while an egg is oval. But we saw elsewhere that our unreasoned predilection for a definite type of congruence in space was imposed by the behaviour of material bodies when displaced. Visually these bodies could be made to present us with the same appearances, whether situated here or there, and for that reason they imposed themselves as defining congruent space intervals. Furthermore, our own human limbs, the steps we take, the ground on which we walk, all suggest the same definition of congruence.
Likewise in the case of time-congruence. The human organism is replete with rhythmical motions such as the beating of the heart, the periodic contractions of the arteries, or the respiratory motions of the lungs. Again, an infant feels the pangs of hunger at periodic intervals. Quite independently of these internal processes and changes, certain external mechanical motions affecting our bodies must be considered. Thus, when we walk with our normal gait, our successive steps follow one another in time in a certain definite way, and the same applies to the swinging of our arms. By an effort of our will we might vary the rhythm of our steps, alternating short steps with long ones, now tarrying, now hastening. But we should soon be aware that something was wrong, for we should find that our irregular motions demanded additional effort, so that on relaxing our will we should relapse automatically into our normal gait. It may be that in addition to these periodic motions we have mentioned, cyclic processes occur in our brains, conferring on our consciousness direct norms of time-congruence. This, however, is a point for physiologists to decide. At any rate, when it is realised that these various methods, whether they appeal to our respiratory motions or to our successive steps when walking normally, all issue in the same definition for time-congruence, there is no cause to be surprised at a definite sense of time-congruence imposing itself naturally upon us, and this in spite of the fact that pure duration of itself possesses no metrics.
Thus far we have been considering time-congruence from the standpointof the savage and the animal. For the most rudimentary purposes, a definition of equal durations obtained in this crude way would suffice; yet it is obvious that no great precision could thus be arrived at. Our crude psychological clock is subject to variations according to our mental and physical condition; furthermore, though it can differentiate between what we have come to call “one second” and “one minute,” it would be incapable of differentiating between fifty-nine minutes and one hour. In the case of animals this imprecision in their sense of duration is manifested when shepherd dogs lead the flocks back during an eclipse or hens lay eggs under the influence of artificial light, believing that day has come. However, as these last illustrations permit of other interpretations, we need not dwell on them and may confine ourselves to the well-known fact that our human sense of duration varies between one individual and another, and between one psychological state and another.
In addition to these difficulties, our understanding of equal durations is entirely subjective and can give rise to no common knowledge. We must needs objectivise it, hence appeal to some objective phenomenon which all men can observe. Let us assume, for the sake of argument, that the monotonous dripping of water drops falling on a pan may have served as a primitive clock. Still, something would be lacking. We should not be completely satisfied, for we might sense now and then an irregularity in the successive thuds of the drops falling on the pan. Furthermore, if we were to establish two such water clocks we should observe that the drops sometimes fell in unison and sometimes in succession. We should wonder then which of the clocks was most reliable and be in a quandary to know to which to appeal. Just as in the case of space, where we substituted wooden rods for the lengths of our steps, then metallic rods maintained at constant pressure and temperature, so now, in the case of time-keeping devices, we must find some way of selecting clocks which will satisfy all our requirements.
We thus arrive at more scientific methods of time-determination. The psychological clock, water clock, sand clock and burning of candles, which had served men in the earlier days of civilisation, have now been discarded in favour of more precise mechanisms. These, as we shall see, will not differ perceptibly in their indications from our psychological clock, but their measurements will be given with increased refinement. We are thus led to examine what are the conditions that must be satisfied for a time-keeping device to be considered perfect.
In a general way we obtain a definition of time-congruence by appealing to the principle of causality. The following passage, quoted from Weyl, expresses this fact:
“If an absolutely isolated physical system (i.e., one not subject to external influences) reverts once again to exactly the same state as that in which it was at some earlier instant, then the same succession of states will be repeated in time and the whole series of events will constitute a cycle. In general, such a system is called aclock. Each period of the cycle lastsequally long.”
We shall obtain a more profound understanding of the nature of the problem by considering the various methods of time-determination that have been contemplated by physicists.
We remember that Newton refers to absolute time as “flowing uniformly.” But of course this allusion to time does not lead us very far, for a rate of flow can be recognised as uniform only when measured against some other rate of flow taken as standard. Obviously some further definition will have to be forthcoming. Now both Galileo and Newton recognised as a result of clock measurements that approximately free bodies moving in an approximately Galilean frame described approximately straight lines with approximately constant speeds. Newton then elevates this approximate empirical discovery to the position of a rigorous principle, theprinciple of inertia, and states that absolutely free bodies will move with absolutely constant speeds along perfectly straight lines, hence will cover equal distances in equal times. When expressed in this way as a rigorous principle, the space and time referred to are the absolute space and time of Newtonian science. In other words, it is the principle of inertia coupled with an understanding of spatial congruence that yields us a definition of congruent stretches of absolute time.
In practice, a definition of this sort entails measurements with rods; that is to say, we measure off equal distances along the straight path of a body and then define as congruent, or equal, the times required by the body to describe these congruent spatial distances. It is obvious that, however perfect our measurements, errors of observation will always creep in. Furthermore, a body moving under ideal conditions of observation can never be contemplated; hence all we can hope to obtain is the greatest possible approximation. But although physical measurements become inevitable as soon as we wish to obtain a concrete definition, an objective criterion of equal durations, we are able to proceed in our mechanical deductions in a purely mathematical way without further appeal to experiment. The principle of inertia, together with the other fundamental principles of mechanics, enables us, therefore, to place mechanics on a rigorous mathematical basis, and rational mechanics is the result.
It will be observed that science, in the case of mechanics, has followed the same course as in geometry. Initially our information is empirical and suffers from all the inaccuracies of human observation and all the various contingencies that characterise physical phenomena. But this empirical information is idealised, then crystallised into axioms, postulates or principles susceptible of direct mathematical treatment. To be sure, as we proceed in our purely theoretical deductions, unless we are to lose all contact with reality, we must check our results by physical experiment; and this necessity is still more apparent in rational mechanics than in geometry. If peradventure further experiment were to prove that our mathematical deductions in mechanics were not borne out in the world of reality, we should have to modify our initial principles and postulates or else agree that nature was irrational. With mechanics, the necessity of modifying the fundamental principles became imperative when it was recognised thatthe mass of a body was not the constant magnitude we had thought it to be; hence it was experiment that brought about the revolution. On the other hand, in the case of geometry, it was the mathematicians themselves who foresaw the possibility of various non-Euclidean doctrines, prior to any suggestion of this sort being demanded by experiment.
And now let us revert once again to the problem of time. Theoretically, the law of inertia should permit us to obtain an accurate determination of congruent intervals, but as it was quite impossible to observe freely-moving bodies owing to frictional resistances as also to the gravitational attraction of earth and sun, it was advantageous to discover some other physical method of determination. This was soon obtained. The principles of mechanics enable us to anticipate that if a perfectly rigid sphere, submitted to no external forces, is rotating without friction on an axis fixed in a Galilean frame, its angular rate of rotation will be uniform or constant, as measured in the frame. By “constant,” we meanconstant as measured in terms of the standards of time-congruence defined by a free body moving under ideal conditions according to the principle of inertia. Hence we are in possession of a method of measuring time, more convenient than that afforded by the motions of free bodies along straight courses. It may be noted that the definition of congruent durations as given by the rotating sphere is in perfect accord with the definition in terms of causality as formulated in the passage quoted from Weyl.
Unfortunately, here again, ideal conditions can never be established, for we can never obtain perfect isolation, perfect rigidity and complete absence of friction. But a near approach to ideal conditions would appear to be given by the phenomenon of the earth’s rotation. And so the rotation of the earth, defining astronomical time, was regarded as furnishing science with the most reliable objective criterion of congruent time-intervals that it was possible to obtain. Nevertheless, it was well known that a definition of this sort was still far from perfect, since the rate of rotation of the earth could not be absolutely uniform, when “uniform” is defined by the standards of the principle of inertia. This realisation was brought home when the frictional resistance generated by the tides was taken into consideration.[23]The tidal friction would have for effect the slowing down of the earth’s rate of rotation (when measured against the accurate mathematical definition given by a perfectly rigid and isolated body rotating without friction). In addition to this first perturbating influence, our planet is suspected of varying in size, expanding and contracting periodically, a phenomenon known asbreathing. According to the laws of mechanics, a change in shape of this sort would entail periodic fluctuations, successive retardations and accelerations in the rate of the earth’s rotation.
It is instructive to understand how this discovery of the earth’s breathing was brought about. It was noticed that the moon’s motion exhibited variations which it seemed impossible to account for underNewton’s law, even when due consideration was given to the perturbating effects of the sun and other planets. Not only did the moon’s motion appear to be gradually increasing, but in addition, superposed on this first uni-directional effect,[24]periodic accelerations and retardations had been observed. The gradual slowing down of the earth’s rate of rotation under the influence of the tides accounted for the apparent acceleration of the moon’s motion, but the other periodic changes in the velocity of our satellite could not be explained away in so simple a manner. Yet everything would be in order were we to assume that the earth’s rate of rotation was subjected to appropriate periodic changes; and the principles of mechanics then showed that a phenomenon of this kind could only be brought about by periodic variations in the earth’s size.
In all these corrections we are dealing with highly complex phenomena involving Newton’s law of gravitation and the laws of mechanics. Our astronomical time-computations have thus been adjusted so as to satisfy the requirements of these laws, regarded as of absolute validity. We may recall that it was this belief in the accuracy of Newton’s law that led Römer in 1675 to the discovery of the finite velocity of light. He noticed that Jupiter’s satellites appeared to emerge from behind the planet’s disk several minutes later than was required by the law of gravitation. Rather than cast doubts on the accuracy of this law, he ascribed a finite velocity to the propagation of light. Needless to say, his previsions have since been justified by direct terrestrial measurements. The calculations leading to the discovery of the planet Neptune were also based on the assumption that Newton’s law was correct.
And now let us suppose that we had elected to define time by entirely different methods. We might, for instance, appeal to the vibrations of the atoms of some element, such as cadmium. Then the intervals of time marked out by the successive beats of the atom at rest in our Galilean frame would be equal or congruent by definition. These successive congruent intervals of time would be measured by the wave length of the radiation, since a wave length is the distance covered by the luminous perturbation during the interval separating two successive vibrations. We might again proceed in still another way in terms of radioactive processes. It is well known that, when measured by ordinary clocks, the rate of disintegration of radium appears to follow an exponential law. And it is a remarkable fact that this rate of disintegration does not appear to be modified in the slightest degree by the surrounding conditions of temperature or of pressure. Thus the phenomenon may be regarded as being in an ideal state of isolation. Curie had therefore suggested defining time in terms of this rate of disintegration.
We see, then, that a great variety of methods for determining time have been considered. Some were mechanical, others astronomical,others optical, while still others appealed to radioactive phenomena. In each case our definitions were such as to confer the maximum of simplicity on our co-ordination of a certain type of phenomena. For instance, if in mechanics and astronomy we had selected at random some arbitrary definition of time, if we had defined as congruent the intervals separating the rising and setting of the sun at all seasons of the year, say for the latitude of New York, our understanding of mechanical phenomena would have been beset with grave difficulties. As measured by these new temporal standards, free bodies would no longer move with constant speeds, but would be subjected to periodic accelerations for which it would appear impossible to ascribe any definite cause, and so on. As a result, the law of inertia would have had to be abandoned, and with it the entire doctrine of classical mechanics, together with Newton’s law. Thus a change in our understanding of congruence would entail far-reaching consequences.
Again, in the case of the vibrating atom, had some arbitrary definition of time been accepted, we should have had to assume that the same atom presented the most capricious frequencies. Once more it would have been difficult to ascribe satisfactory causes to these seemingly haphazard fluctuations in frequency; and a simple understanding of the most fundamental optical phenomena would have been well-nigh impossible.
Since we realise that the definitions of time-congruence we ultimately adopt are governed by our desire to obtain the maximum of simplicity in the co-ordination of some particular group of phenomena, considerable divergences in our definitions might be expected when we considered in turn phenomena of a different nature. Why, indeed, should a determination arrived at in terms of the vibrations of atoms agree in any way with the astronomical definition issuing from the earth’s rotation? Even when restricting our attention to the vibrations of the atoms of the same element, it might have been feared that different atoms of cadmium would have beaten out time at different rates, since, after all, even though identical in many respects, yet their life-histories must have differed greatly. Instead of this we find that all the methods of time-congruence mentioned lead to exactly the same determinations so far as experiment can detect. It would appear as though a common understanding existed between the rates of evolution of the various phenomena mentioned. The situation is similar to that presented in the problem of space. There we saw that rods, whether of platinum or of wood, whether situated here or there, appeared to yield the same geometrical results (provided they were maintained at constant temperature and pressure). Again, it seemed as though some common understanding existed among the various rods.
Just as this same mysterious situation in the case of space suggested that the uniformity of our spatial measurements was due to the presence of an all-embracing physical reality, the metrical field of space, regulating the behaviour of material bodies and light rays, so now, as we shall see in later chapters, the general theory of relativity willcompel us to extend the same ideas to time. We shall find that there exists a space-time metrical field which splits up into a temporal and a spatial field. The temporal metrical field will then be responsible for that apparent understanding which seems to exist among the various spatially separated phenomena.
These new ideas modify the position of the problem of time-congruence. While it is still correct to state that pure duration as a mathematical continuum is amorphous and possesses no intrinsic metrics, yet from the standpoint of the physicist we must concede a certain physical reality to this temporal metrical field which controls the processes of all isolated phenomena.
We may summarise the preceding statements by saying that we define time-congruence so as to introduce the maximum of simplicity in our co-ordinations of physical phenomena, regardless of the particular type of phenomenon we may be considering. That a unique definition of time-congruence should be possible, one holding for mechanics, optics, electromagnetics and astronomy, is a fact that could scarcely have been anticipateda priori; hence we must recognise its undeniable existence as establishing the presence of unity in nature.
And here we may mention a type of objection which is often encountered in people who do not trouble to differentiate between physical time and psychological duration. It has been claimed, for instance, that it is absurd to maintain that our sense of time was originated from a desire to retain harmony between the planetary motions and the requirements of Newton’s law, since men possessed an intuitive understanding of equal durations long before Newton’s law or any other natural law was discovered. We trust that those who have taken the trouble to follow the course of this analysis will realise that the criticism is quite beside the mark. No scientist has ever contended that we have derived our sense of time from astronomical observations, for we know that even savages are not lacking in a sense of rhythm. All that is claimed is that the accurate determination of time-congruence must be based on physical processes or laws, since our crude time-sense is too vague to be of any use in the precise problems with which science is concerned. The fact that our crude appreciation of time agrees in a more or less approximate way with the more refined methods of scientific determination offers no great mystery, for, after all, the laws which govern material processes in the outside world would in all probability also govern that which is material in our organisms. Regardless of our ultimate views as to matter and mind, we cannot banish matter from our anatomy.
At any rate, once the existence of a universal type of time-congruence is conceded, the problem is to select some particular phenomenon which can be studied with a minimum of contingency and a maximum of certainty. Now, when we consider the astronomical definition in terms of the earth’s rotation, we realise that were it not for the fact that we had started by assuming the absolute accuracy of Newton’s law, the corrections we made in order to account for the peculiarities of thelunar motion would have to be discarded or replaced by others. If, therefore, any doubt were to be cast on the accuracy of Newton’s law or on those of mechanics, corresponding modifications would ensue for our definition of time-congruence. But it is obvious that we have no right to assume that Newton’s law or those of mechanics are above all suspicion. The laws of physical science present noa prioriinevitableness; and our sole reason for accepting them is because they appear to be borne out to a high degree of approximation when we compare their rational consequences with the results of observation. Approximation is thus all we can aspire to in physical science; and even classical scientists recognised the possibility that more precise observation might prove Newton’s law and those of mechanics to stand in need of correction. All that could be asserted was that the corrections would certainly be very slight, far too slight to entail any perceptible change in our understanding of time-congruence when applied to the processes of commonplace observation.
We now come to Einstein’s definition. As we shall see, it does not differ in spirit from the definitions of classical science; its sole advantage is that it entails a minimum of assumptions, and is susceptible of being realised in a concrete way permitting a high degree of accuracy in our measurements. Einstein’s definition is, then, as follows: If we consider a ray of light passing through a Galilean frame, its velocity in the frame will be the sameregardless of the relative motion of the luminous source and frame, and regardless of the direction of the ray. It follows that a definition of equal durations in the frame will be given by measuring equal spatial stretches along the path of the ray, and asserting that the wave front will describe these equal stretches in equal times.
As can be seen, the definition is simple enough, but the major question is to decide on its legitimacy. Classical science would have rejected the definition. Why? Because it would have maintained that Einstein’s definition was equivalent to defining time in terms of a rigid body rotating round an axis, butsubmitted to frictional forces. It would have claimed that the velocity of the frame through the stagnant ether would have introduced complications entailing anisotropy, that is to say, a variation of the speed of light according to direction. It would then be necessary to take this perturbating effect into account just as it was necessary to take into consideration the frictional effect of the tides on the earth’s angular rate of rotation.
But when it was found that contrary to the anticipations of classical science not the slightest trace of anisotropy could be detected even by ultra-precise experiment, the objections which classical science might have presented against Einstein’s definition lost all force. Henceforth it was not necessary to take into consideration the velocity of the frame through the stagnant ether, since this ether drift appeared to exert no influence one way or the other.
Now, it may not be easy to understand how isotropic conditions couldbe demonstrated by experiment, for isotropy signifies that the velocity of light is the same in all directions. And how can we ascertain the equality of a velocity in all directions when we do not yet know how to measure time? Experimenters solved the difficulty by appealing to the observation of coincidences.
i001Fig. I
Fig. I
Fig. I
The following elementary example will make this point clear. Consider a circle along which two bodies are moving in opposite directions. We shall assume that the two bodies leave a pointat the same instant of time, hence in coincidence, and then meet again at a pointhalfway round the circle, hence diametrically opposite.There is no difficulty in ascertaining that the pointis diametrically opposite,since we know how to measure space. All we have to do is to verify the fact that the length of the curveis equal to the length.Neither does the observation of a coincidence, that is, of the simultaneous presence of two objects at the same point of space, entail any knowledge of a measure of duration. Hence, although we know nothing of the measurement of time, we cannot escape the conclusion that the two bodies have taken the same time to pass fromtoby different routes. But these spatial routes are of equal length; hence we must assume that the speeds of the bodies along their respective paths have been the same. In all rigour, we may only claim that the mean speed has been the same, since one body may have slowed down, then spurted on again, making up for lost time. But if we repeat the experiment with circles of different sizes, and if in every case the bodies meet one another at the opposite point, we are justified in asserting that not only the mean speed, but also the instantaneous speed at every instant, is the same for either body. In short, thanks to spatial measurements coupled with the observation of coincidences, we have been able to establish the equality of two velocities, even though we knew nothing of time-measurement.
A more elaborate presentation of the same problem would be given as follows: Waves of light leaving the centre of a sphere simultaneously are found to return to the centrealso in coincidence, after having suffered a reflection against the highly polished inner surface of the sphere. As in the previous example, the light waves have thus covered equal distances in the same time; whence we conclude thattheir speed is the same in all directions.
Inasmuch as this experiment has been performed, yielding the results we have just described, even though the ether drift caused by the earth’s motion should have varied in direction and intensity, the isotropy of space to luminous propagations was thus established. (The experiment described constitutes but a schematic form of Michelson’s.) It is to be noted that in this experiment the observation of coincidences is alone appealed to (even spatial measurements can be eliminated).[25]When it is realised that coincidences constitute the most exact form of observation, we understand why it is that Einstein’s definition is justified.
The optical definition presents a marked superiority over those of classical science. Whereas, with the rotation of the earth, we had to assume the correctness of Newton’s law and of those of mechanics in order to determine the compensations necessitated by the earth’s breathing or by the friction of the tides, in the present case we assume practically nothing, and what little we do assume issues from the most highly refined experiments known to science.
Now, the importance of Einstein’s definition lies not so much in its allowing us to obtain an accurate definition of time in our Galilean frame as in its enabling us to co-ordinate time reckonings in various Galilean frames in relative motion. So long as we restrict our attention to space and time computations in our frame, we may, as before, appeal to vibrating atoms for the measurement of congruent time-intervals and to rigid rods for the purpose of measuring space. It is when we seek to correlate space and time measurements as between various Galilean frames in relative motion that astonishing consequences follow. We discover that the concepts of spatial and temporal congruence of classical science must be modified to a very marked degree. They lose those attributes of universality with which we were wont to credit them. It is then found that congruence can only be defined in a universal way when we consider the extension of four-dimensional space-time.
As we have mentioned, any change in our concepts of space and time congruence is not merely a local affair. Owing to the unity of nature, we shall be led to modify our entire understanding of the relatedness of phenomena, and must not be surprised to learn that a totally new science is the necessary outcome of Einstein’s new definitions. Old laws are cast aside, new ones take their place; and Einstein, by following his mathematical deductions with rigorous logic,without introducing anyhypothesesad hoc, proved that if his new definition of space-time congruence was physically correct, a whole series of hitherto unsuspected natural phenomena should ensue and should be revealed by precise experiment. Wonderful as it may appear, Einstein’s previsions have thus far been verified in every minute detail.