Part 3 — CHAPTER XXXI THE SEPARATION OF SPACE-TIME INTO SPACE AND TIME IN A GRAVITATIONAL FIELD

WE must remember that the one fundamental continuum of the world is space-time; space and time considered separately, varying as they do with the observer’s motion, have no absolute significance. The law of curvature, or of gravitation, which we have discussed is one of space-time curvature. It is the law giving the distribution of the four-dimensional metrical field of space-time in the interior of matter and around matter to infinity. It is true that we may be able to effect some sort of separation of space-time into the space and time of some particular observer; and this may enable us to represent the law of curvature of space-time as two separate laws of curvature, one affecting space and the other affecting time. But for the reasons we have just mentioned, no absolute significance could be attached to the curvatures we should obtain. Thus, in interstellar space, an observer situated in a Galilean frame would contend that neither space nor time manifested any trace of curvature, whereas an observer on the rim of a rotating disk would recognise that both space and time were curved. On the other hand, regardless of his frame of reference, either observer would always recognise that space-time possessed no trace of curvature.

It is because space-time transcends the observer that its curvature or flatness, as the case may be, can be considered absolute; whereas a curvature of space or of time alone can never represent anything but a mere relationship between the intrinsic condition of space-time and the observer’s motion or frame of reference.

Of course, if we wish to split up space-time into the particular space and time directions of some particular observer we may always attempt to do so. But in the majority of cases this procedure will be impossible.

If these considerations are properly understood, there is no harm done in trying to separate the curved space-time which endures around the sun into the space and time of an observer at rest with respect to the sun and viewing the solar system as a whole. This separation, however, is possible only in the event that the masses of the moving planets can be ignored, so that the only field we have to consider is that of the sun, remaining unchanged as time passes. It is owing to this permanent condition of the gravitational field that the separation into space and time is possible. Fields of this type are called stationary. For non-stationary fields, such as those where the distribution of the field changes with time, the separation into space and time becomes impracticable.

We will assume, therefore, that the gravitational field of the sun is stationary; that is, that the masses and motions of the planets do not modify the field to any appreciable extent. We may then proceed to split up curved space-time into the space and time of an observer at rest with respect to the sun and non-rotating with respect to the stars. By so doing we shall be splitting up into two separate fields the metrical field of space-time, that four-dimensional field which pertains neither to space nor to time. In this way we shall obtain, first, a purely spatial metrical field, which will be responsible for the measurements of our rods when these are maintained at rest in our frame; and, secondly, a purely temporal metrical field, which will determine the behaviour of our clocks or vibrating atoms while these are at rest in various regions of the space around the sun.

When Levi-Civita and Schwarzschild had performed this separation, it was found that the curvature of the observer’s space along a plane section through the centre of the sun would be identical with that of the surface of a paraboloid of revolution disposed around the sun; and that the observer’s time direction would also be curved in a definite way. Both curvatures would die down gradually for regions of space farther and farther removed, to infinity. We must not forget, however, that this separation and the curvatures we have mentioned hold solely for the observer at rest with respect to the sun when surveying the solar system as a whole.

The advantage of this separation of space-time into space and time is that it allows us to express the purely formal results deduced from Einstein’s space-time gravitational equations in terms of our familiar notions of space and time. It thus becomes possible to describe the world lines of the planets under Einstein’s law by representing them as spatial orbits and motions along these orbits; hence we are enabled to check up Einstein’s previsions with astronomical observation. Furthermore, this splitting up of space-time enables us to study separately those effects which must be attributed to the respective curvatures of space and of time, considered by themselves and independently of one another.

We will first consider the curvature of time alone. Let us assume that for some reason or other the curvature of space were to disappear, so that all we should be concerned with would be a more or less curved time direction. In this case, it is found that the motions of the planets would be in perfect accord with Newton’s law. The precessional advance of Mercury’s perihelion would be non-existent, and the bending of a ray of light grazing the sun’s limb would be 0".87, as required by Newton’s law, and no longer twice that amount, as required by Einstein’s law. From this we see that it is the additional curvature of space around the sun (and around matter generally) that is the distinguishing feature of Einstein’s law of gravitation.

Here it might be asked: “Since a peculiar type of space-time curvature (curved time and flat space) would lead us to Newton’s law, why should Einstein have rejected the classical law a priori and urged his own more complicated type of space-time curvature in its place? It could not have been for the purpose of accounting for the double bending of a ray of light, since this effect was unknown at the time. Was it for the sole purpose of explaining the motion of the planet Mercury?”

The answer must be decidedly in the negative. When Einstein was led to his law of space-time curvature , he never could have recognised in advance that this law would account for the motion of Mercury. He may have suspected the fact, convinced as he was that he was on the right track; but in order to establish the nature of the planetary motions when governed by this law, it was necessary to submit the law to a mathematical integration. It was only after the integration had been performed that the motion of Mercury, as required by the law, was found to be in agreement with previous astronomical observations. Hence we must discard all ideas of a law patched up for the sole purpose of harmonising with the planetary motions.

In order to understand Einstein’s procedure more fully, we must recall that the general principle of covariance requires that all natural laws be expressed in the form of space-time tensor equations. This criterion alone, however, does not permit us to decide a priori on any particular law of space-time curvature. But Einstein, as we remember, introduces a second restriction, according to which, in the empty space surrounding matter, none but the structural tensors of space-time, the ’s, should enter into the law of curvature.[98] This second restriction is a natural one to set, for were foreign tensors to be introduced at each point of space-time, the space could no longer be empty.

When these two restrictions are taken into consideration, we find that the law is practically the only one that can be constructed; at all events, it is the simplest. Theoretically, could also be countenanced; but this law, indicating perfect flatness, would connote the absence of gravitation. Then again, we might suggest , and this would lead to de Sitter’s finite universe, a possibility which has not yet been discarded.

But in any case, with the restrictions imposed, Newton’s law is barred, so that unless we wish to introduce foreign tensors, acting in the capacity of hypotheses ad hoc, it is hard to see how Einstein’s law can be escaped.[99] If we convince ourselves on this score, we can see that it is quite impossible to think that we may adjust the curvatures of space and of time to suit ourselves. These curvatures are determined in advance by the law of space-time curvature, itself imposed by considerations of a rational order.

Let us revert to our separation of curved space-time into curved space and curved time. We have seen that if the curvature of space were non-existent, the curvature of time alone remaining, the motions of planets and free bodies would be in accord with Newton’s law. We may understand the reason for this by following a different line of thought.

If space is flat, the ’s which refer to space remain constant (in a Cartesian mesh-system at least).[100] The only , or potential, that can vary is therefore the time one . This last potential is therefore the only one that can be responsible for a field of force. It follows that by considering space to be flat, we are preserving only one of the ten potentials of space-time; and we find ourselves in exactly the same position as under Newton’s law, where only one potential was present, i.e., the Newtonian potential. This allows us to identify the time-potential with the Newtonian potential , and the proof of the statement we made in a previous chapter is given thereby.[101]

It now remains to be seen how this curvature of time may be detected directly. Here we must recall that the curvature of time is expressive of the repartition of the temporal metrical field obtained by splitting up the metrical field of space-time into one of space and one of time. The repartition of this temporal metrical field in the space and time separation we are discussing shows that its curvature will increase as we consider regions of space nearer and nearer the sun, indicating thereby a slowing down of time as we approach the sun. Now it is this temporal metrical field which controls the rate of evolution of any isolated periodic phenomenon from place to place, hence which defines practical time-congruence as given by perfect clocks. In the present case, therefore, the action of this field would be to slow down the temporal evolution of all phenomena when these have their spatial locus nearer and nearer the sun. This effect was anticipated by Einstein and is known as the Einstein shift-effect.

Of course, if we, as conscious beings, were to be situated at any particular point of the sun’s gravitational field, we should never appreciate this slowing down of time and of the evolution of all things in our immediate neighbourhood; for our consciousness would participate in the same slowing down that was active around us. It would only be when we viewed distant phenomena occurring in other parts of the field of gravitation that discrepancies in their speeds of evolution would become apparent. Thus, although we should not be directly conscious of aging at various rates in the different parts of a gravitational field, we should discover eventually, either by direct vision or by meeting again, that our friend who had lived in the proximity of the sun had aged less than ourselves, who had lived farther away from its mass. Gravitation would thus preserve youth, and, more generally, two men living on two different stars of unequal mass would not age at the same rate.

It is to be noted that it is not the force of gravitation itself which is responsible for this slowing down of time as we near the sun. It is rather the decrease in the value of the potential; and in the present case we have seen that the only potential which enters into account is , or the Newtonian potential.

All these phenomena predicted by Einstein were entirely new to science, and it was most important to verify their correctness before accepting Einstein’s theory. The method suggested was that of exploring a field of gravitation with vibrating atoms, these being assumed to behave like perfect clocks marking correct time. For instance, if Einstein’s law of space-time curvature was correct, atoms of sodium placed in different parts of the solar field would appear to be vibrating faster or slower than the atom in our immediate neighbourhood, according to whether they were farther removed from or nearer the sun than we happened to be situated. These variations in the rates of vibrations of the atoms would be betrayed physically by variations in the colour of the light they emitted or absorbed. It would follow that while the atom of sodium by our side would always emit a yellow light in our estimation, those atoms nearer the sun would appear redder, and those farther away from the sun would appear greener. In particular, an atom of sodium situated in the sun’s atmosphere would beat more slowly and appear to us redder than an identical atom examined in our laboratory (the earth’s gravitational field being insignificant compared with that of the sun).[102]

For a number of years it was questionable whether this effect really existed, because the difficulty in detecting so minute a displacement of the spectral lines was very great. Quite recently, however, the phenomenon has been observed on a star of enormous density, the companion of Sirius.[103] This is perhaps the most beautiful of all the verifications of Einstein’s theory, since the effect contemplated was totally unsuspected by classical science—even more so than the bending of light in a gravitational field.

Now the postulate of equivalence, by assimilating gravitational fields to fields of inertia, requires that similar phenomena should occur in an accelerated enclosure (allowing, of course, for the slight discrepancies in the actual field-distribution when produced by matter or by acceleration). Thus, consider a train pursuing a Galilean motion, and two identical sodium atoms or clocks placed on the engine and rear car, respectively. So long as the motion of the train remains Galilean, an observer at rest in any point of the train would detect no difference in the rates of vibration of the atoms,[104] hence in the colours which they emitted. On the other hand, were the train to be accelerated, moving faster and faster, a field of inertial force directed from engine to rear car would now be present; and as a result the atom on the rear car would appear to the train observer to be vibrating more slowly than the atom on the engine.

We must realise, therefore, that whereas in the special theory a slowing down of time was always due to relative motion, in the present case, the slowing down of time takes place in spite of the absence of relative motion between the atoms and the train observer. In the same way, there was no relative motion in the example of the Einstein shift-effect in the sun’s gravitational field, and yet we had to accept a slowing down of time. So we see that acceleration, even when unaccompanied by relative motion, will produce a variation in the flow of time; and the same variations would also ensue in the case of distance.

Instead of considering an accelerated enclosure pursuing a rectilinear path, we could of course consider a rotating enclosure, and the argument would be the same. Atoms situated at greater distances from the centre of rotation would appear to vibrate more and more slowly; and a parcel of radium, for instance, placed on the rim of a rotating disk, would be slowed down in its rate of disruption, so that the effect of the inertial field would be to decrease the radioactivity of the atom.

Now that we have discussed the effects due to the curvature of time and have seen that this type of curvature accounts for all the major effects of gravitation, let us consider the additional effects due to the curvature of space. The presence of this curvature of space, so far as the motions of bodies are concerned, will modify the effects due to the curvature of time. It will therefore be responsible for the small discrepancies that differentiate Einstein’s law from Newton’s. However, so long as the motions of bodies in the gravitational field are small compared with that of light, the curvature of space will not affect their motions to any appreciable extent.

Among the planets, Mercury alone is animated with a speed (with respect to the inertial frame of the sun) sufficient to render observable the influence due to the curvature of space; and it is this influence which is responsible for the slight precessional advance of Mercury’s perihelion. In the case of light waves, which move with a speed many times greater than Mercury, the influence of the curvature of space will be marked to a still greater degree. It will cause an increased bending of the ray of light equal to that already produced by the curvature of time alone, which, as we know, would correspond to the bending required by Newton’s law. In this way the double bending is accounted for.

In order to exhaust this discussion, we must mention that there exists still another case, in which the curvature of space would exert a perceptible influence, even in the case of slowly moving bodies. This would be in the event of a very intense field of gravitation, many times greater than that of the sun. In such a field, even slowly moving bodies would deviate perceptibly from the course required by Newton’s law of gravitation.

It may appear strange that this curvature of space should seem to be of such minor importance as contrasted with that of time, and produce observable effects only for bodies moving with great velocities with respect to the sun. The solution of this puzzle will be found, however, when we consider some of the peculiarities of the space-time continuum of relativity. In an ordinary continuum like space, where all the dimensions are of exactly the same nature, we have no difficulty in specifying some unit of length which will apply in exactly the same way to measurements computed along the dimensions of height, length and breadth. In space-time, however, the situation is somewhat modified, since the fourth dimension (time) represents something different from the other three. Thus, whereas our measurements along the spatial dimensions will be conducted with material rods, our time measurements will require time-rods, or clocks. The question is, then, how are we to co-ordinate our measurements with clocks and our measurements with rods; or, in other words, how shall we be able to establish any sort of comparison of magnitude between these measurements so different in nature?

The answer to this question is that the variable which represents measurements along the time direction is not t but ct. This means that one second in time corresponds in our formulæ to the distance which light covers during one second, namely, 186,000 miles. It follows, therefore, that if we measure time in seconds we must, in order to co-ordinate results, measure space in terms of unit rods 186,000 miles long.

Obviously there exists a vast discrepancy in actual practice between the magnitudes of our temporal and spatial units; for the great majority of bodies with which we come in contact cover far less than a unit of space (186,000 miles) in a unit of time (one second). Only in the case of bodies approaching the velocity of light in our frame are the two progressions, that in space and that in time, in any way comparable. Thus, consider our own earth. It moves in relation to the sun with a speed of some eighteen miles a second, which is equivalent to approximately one ten-thousandth part of a unit of space per unit of time. This emphasises the fact that to all intents and purposes the motions of bodies with which we are concerned reduce to a motion in time, the displacement in space being insignificant in comparison.

Now when we come to investigate the respective effects of the curvatures of time and space, taken separately, in the gravitational field of the sun, we must remember that for a curvature to manifest itself we must survey relatively large areas on the curved surface (large as compared with the intensity of the curvature). Thus, on the curved surface of the earth, for example, if we content ourselves with surveying restricted areas, no effects of its curvature will be apparent. It is only when we conduct measurements over large areas, say over a few hundred miles, that the direction of the vertical will change perceptibly and that geodesical surveys will render the curvature manifest. It is much the same with the curvatures of space and of time in a gravitational field. Only when we consider planets moving through the sun’s inertial frame with a speed approaching that of light (hence covering large distances in unit time) will the space-curvature produce effects comparable to those produced by the curvature of time. We understand, therefore, how it comes that the curvature of time alone is able to produce the major effect of Newtonian attraction, such as weight, whereas the effects due to the curvature of space remain imperceptible until we consider the motion of Mercury, which is moving fairly fast with respect to the inertial frame of the sun. When we consider the propagation of light rays, the effects due to the curvature of space reassert themselves fully, and as a result the double bending of a ray of light can be attributed in equal proportion to the curvature of time and to the curvature of space.

In short, we now understand why it is that Einstein’s law of gravitation deviates appreciably in its effects from Newton’s only when we consider bodies moving with velocities approximating to that of light. Once again we realise that if this invariant velocity of the universe had happened to be much smaller, the deviations between Einstein’s law of gravitation and that of Newton would have been far easier to detect; and, conversely, had the invariant velocity happened to be much greater than 186,000 miles a second, had it been infinite, for example, there would have been no discrepancies left between the two laws. Thus, once again we reach the same conclusions that we mentioned when discussing space-time. The fundamental difference between Einstein’s theory and classical science arises from the discovery that the invariant velocity of the universe is finite and not infinite.

We may summarise all these results by stating that the major effects of gravitation are produced by the curvature of time, i.e., of the time component of the metrical field of space-time; the curvature of space exerts perceptible effects only when the velocities of the moving bodies are very great or when the gravitational field is of very great intensity. Hence, for weak fields such as that of the sun, and for slow motions, Einstein’s law of space-time curvature leads us to results practically identical with those necessitated by Newton’s law of gravitation.

It now remains to be seen how the curvature of space could be detected by direct measurement, without our having recourse to moving bodies. It can, of course, be detected indirectly, since we have seen that it is responsible for the advance of Mercury’s perihelion and for the double bending of a ray of light passing near the sun. But we now propose to detect it by direct spatial measurements.

We have said that our separation of space-time into space and time was equivalent to splitting up the metrical field of space-time into a metrical field of time and a metrical field of space. It is this metrical field of space with which we are now concerned. To this metrical field of space must be ascribed the structure of our space (curved in the present case). Were the metrical field non-existent, space would, of course, have no structure, no geometry whatsoever. As the geometry of a continuum is revealed by measurement, we see once again that it is the metrical field of space which will control the behaviour of our measuring rods at rest in our frame, hence which will determine practical congruence.

The obvious method of verifying the anticipated lay of this metrical field of space, or, what comes to the same thing, this curvature of space would be to explore space with measuring rods; we should, of course, obtain non-Euclidean results. Calculation shows that the results should be identical with those which would be obtained were we to apply rigid Euclidean rods on an appropriate paraboloid of revolution. The type of curvature, or non-Euclideanism, implied thereby is of the Riemann variety, since a paraboloid of revolution is a surface of positive curvature. Hence, we should expect the ratio of the circumference to its diameter to be smaller when some massive body such as the sun was situated at the centre of the circumference.

But here again we are met with the same difficulty. In order to detect a curvature of this sort, we should have to survey vast areas of space, much greater than that of the solar system; and it is scarcely probable that even in future years our measurements will be sufficiently refined to permit a proper test. The situation would have been different had the earth to which we refer our measurements been moving with great speed through the sun’s gravitational field. The inhabitants of the planet Mercury, for example, would be in a much better position to verify the curvature of space by direct measurement, owing to the greater velocity of their post of observation through the inertial frame of the sun. Once more we see how it comes that the finite invariant velocity of the universe, though most important from a philosophical viewpoint, in practice loses much of its palpable influence by reason of its incredible magnitude. As a result, ultra-precise experimentation must always be resorted to when we wish to test the Einstein theory.

Incidentally, we are now in a position to understand another erroneous assertion that has often been upheld. As soon as Riemann’s discoveries were known, the possibility of interpreting gravitation in terms of a curvature of space had been suggested (chiefly by Clifford). But the great obstacle to this attitude was that the most refined spatial measurements obstinately refused to reveal the slightest trace of non-Euclideanism. Assuming, for the sake of argument, that it had been possible to interpret the curved orbits of the planets and of projectiles in terms of guided motions along curved grooves threading their way through a curved space,[105] even then, in view of the tremendous degree of non-Euclideanism that it would have been necessary to invoke, the scheme would have had to be abandoned; the chief reason being, of course, the incompatibility of this hypothesis with the Euclideanism given by measurements with rigid rods. It was only after Minkowski had discovered space-time that a modified form of Clifford’s premonitions could be vindicated; only then was it possible to reconcile the Euclideanism of our spatial measurements with that non-Euclideanism of space-time which was the source of all gravitational phenomena. In this respect it cannot be urged too strongly that the general theory and the special theory are but two chapters of one same theory. In so far as the general theory cannot work without space-time, any experiment which would conflict with the special theory would thereby conflict with the general theory.

Let us now examine another point. The velocity of light in a gravitational field is no longer an invariant, as it was in the special theory. However, if we were to measure the velocity of light from point to point, displacing ourselves from one point to the next, we should find that wherever we might be placed, this velocity would always appear the same everywhere, owing to the progressive modification in the behaviour of our rods and clocks. It would only be when viewing distant points that we should recognise variations in the velocity of light.

Thus we see that the principle of the invariant velocity of light is accurately true only in free space far from matter, and, even then, only when computed with reference to our Galilean frame. In an accelerated frame, as in the neighbourhood of matter, the cornerstone of the special theory of relativity no longer applies.

It follows that in the general theory the law of the invariant velocity of light must be replaced by some more general principle, comprising the invariant-velocity principle as a particular case. This more general principle can be expressed by saying that a ray of light moves along a world-line which is a very particular kind of geodesic called a null-line or minimal geodesic. This holds regardless of whether space-time be flat or curved.

Thus far we have studied the curvatures of space and of time outside matter in those particularly simple stationary cases where it is feasible to split up space-time into a separate space and time. Under similar conditions we may investigate these curvatures in the interior of matter. Consider a homogeneous incompressible fluid at rest in our Galilean frame. Conditions being stationary, that is, not varying with time, we can again split up space-time into space and time and study the two types of curvatures separately. We find that the time direction is curved, and this curvature once more is connected with the gravitational forces which exist in the interior of the fluid. We next find that the three-dimensional space inside the fluid possesses a curvature at every point which is proportional to the density of the fluid at every point. Assuming this density to be the same throughout, we see that the curvature of space is constant, and we obtain a spherical space of three dimensions analogous to a cap on the two-dimensional surface of an ordinary sphere in three-dimensional Euclidean space. This curvature of space inside the sphere has nothing to do with the dimensions of our fluid sphere; the curvature is governed solely by the density of the fluid.

Now, in our ordinary three-dimensional Euclidean space, when the curvature or the radius of a sphere is given, its total area is thereby fixed. In a similar way, the density of the fluid predetermines the curvature of space and hence fixes the total volume of the spherical space. Obviously, it would be impossible for us to increase the volume of our fluid indefinitely, for there would come a time when there would be no more room for it to occupy in the spherical space which it had itself created. In the case of a fluid of the density of water, calculation shows that when the sphere of water had attained a radius of about 400 million kilometres, it would be impossible to increase its size further by adding more water to it. When this critical volume was reached, the fluid would fill the spherical space entirely. Moreover, calculation shows that the flow of time would be arrested altogether at the surface of the fluid, so here again we have another reason which would prevent the sphere from growing any larger. Reverting to our two-dimensional analogy, it would be as though the entire surface of the two-dimensional sphere were occupied by the water.

As a matter of fact, this critical volume could never be reached; for even before it could be attained the pressure at the centre of the sphere would have become infinite, and calculation shows that this condition, too, would be accompanied by an arrest in the flow of time. In fine, we discover that the greatest possible volume our fluid sphere could ever occupy would be about of the volume mentioned previously.

It follows that the fluid could never fill completely the spherical volume its density had created in space. Using a two-dimensional analogy, it would be as though on the surface of an ordinary sphere the entire surface could not be covered; an uncovered portion like a cap would always remain. On this account it is sometimes said that space would not close round on itself, since part of the spherical space created by the fluid could never be filled.

Just as a fluid of given density can never increase in volume beyond a certain point, so, conversely, a fluid of given mass could never be compressed into a volume which would cause its density to increase beyond a certain limit. The argument would be the same. The pressure would become infinite, time would stop, and nothing further could happen. A sphere of fluid possessing the same total mass as the earth could never be compressed into a volume less than that of a thimble.

The numerical values mentioned show that there is only a very slim chance of putting these anticipations of the theory to an experimental test. Even in the case of a giant star such as Betelgeuse, its density is so small that the critical volume is well beyond the actual size of the star. Nevertheless, from a theoretical point of view, it is interesting to note that there must exist a definite limit to the size of a star of given density. If these critical limits should ever be reached, time would be arrested, and it is difficult to foresee what the consequences might be. Einstein has called these critical conditions the “catastrophes.” When pressed to give his opinion as to what would happen were they ever to be reached, he is said to have replied that very possibly we should witness the disruption of matter into radiant or electromagnetic energy. However, he preferred not to venture an opinion, as anything he might say would be little better than a guess.