Part 3 — CHAPTER XXV THE INCLUSION OF GRAVITATION IN THE MODEL OF SPACE-TIME

IN the preceding chapter the following results were established:

1. No essential difference exists between a field of gravitation produced by matter and a field of inertia produced by the acceleration of our frame of reference (postulate of equivalence).

2. Although these two types of fields of force are essentially of the same nature, yet the actual spread of the gravitational field around matter is never quite the same as that of the field generated by the acceleration of a frame in free space far from matter. As a result of this feature, whereas it is always possible to cancel a field of force entirely by changing the motion of our frame in a suitable way when the field contemplated exists in regions remote from matter, on the other hand, in the neighbourhood of matter, the field of force can be modified, and even cancelled in a small region, but never cancelled over any wide extension. It is like a protuberance which can be flattened out in one place, only to reappear with increased intensity elsewhere.

These are the results which we must weave into the general model of space-time, unless we reject Einstein’s postulate of equivalence, and maintain that fields of gravitation produced by matter have no more in common with fields of inertia than they have with electromagnetic fields, for instance. The postulate of equivalence, as we have seen, appears to be justified owing to the empirical fact that the two types of masses are identical. But we now propose to show that quite independently of this empirical fact, even if the postulate of equivalence had never been stated in a specific manner, a natural mathematical generalisation of the geometry of space-time would have led us indirectly to exactly the same identification of forces of inertia and gravitation. Accordingly, let us recall briefly the results we arrived at in Chapter XXIII when we examined the origin of forces of inertia in our space-time model.

We remember that space-time was assumed to be perfectly flat, so that its geodesics were Euclidean straight lines when referred to Cartesian mesh-systems of equal four-dimensional cubes. These Cartesian mesh-systems, splitting up space-time into mutually perpendicular rectilinear directions, constituted the geometrical representations of our methods of breaking up space-time into space and time for purposes of referring our measurements to Galilean systems of reference. In these Cartesian mesh-systems the ’s of space-time maintained constant values throughout, and the numerical values of these ’s represented the potentials of the field of inertial force. These potentials being constants, remaining unchanged in value from point to point in our Cartesian systems, no field of force could be present; and this was in full accord with what we know of Galilean systems, since these systems are characterised by the total absence of all such fields. Furthermore, as all free bodies and light waves followed geodesics in space-time according to the requirements of Least Action, and as these geodesics when referred to Cartesian systems were straight lines, free bodies and light waves would always appear to follow straight lines with constant speeds when observed from Galilean frames. This was in accord with Newton’s law of inertia.

Subsequently we saw that accelerated frames moving in interstellar space would be represented by curvilinear or Gaussian mesh-systems. In these systems the values of the ’s of space-time were no longer constant, but varied from place to place. This variation in the values of the ’s connoted the variation in value of the potentials; and as a result the mathematical expression of a force (given as it was by relations between these variations) no longer vanished. These curvilinear mesh-systems represented, therefore, systems in which fields of force would be present. This again was in perfect accord with what we knew of accelerated or non-Galilean frames. Further, as referred to these curvilinear mesh-systems, the geodesics of space-time would appear to be curved. The physical significance of this fact would be to imply that the motion of free bodies and light waves would appear to be accelerated when viewed from accelerated frames. Once more the complete accord with experiment is apparent.

Now let us proceed to the natural generalisation of these results. So far, we have assumed space-time to be flat. Indeed, we could not escape this assumption if we were to admit that Galilean frames, free from all fields of force, could exist in space far from matter, and that as observed from these frames the geodesics of space-time would be Euclidean straight lines. Only thus would Newton’s law of the motion of free bodies in interstellar space be verified. But suppose now that, for some reason or other, space-time were to develop non-Euclideanism or curvature, and so depart from its flatness. How would such a condition manifest itself physically?

In the first place, it would be impossible to split up this curved space-time into the straight directions which define a Cartesian mesh-system. Physically speaking, this would imply that in those regions where space-time was curved no Galilean systems could exist. We might, of course, conceive of mesh-systems less curved than others; but even the straightest of these systems would at best be only approximately Cartesian within a restricted region. As we considered parts of the mesh-system farther and farther removed from this special region, the meshes would become more and more widely spaced or more and more closely gathered together.

An elementary illustration of this would be given by considering a curved surface of two dimensions, such as that of a sphere. On a sphere we can trace an infinite variety of mesh-systems, but none of these mesh-systems is made up of equal Euclidean squares. The nearest approach to a network of squares would be afforded by a mesh-system of meridians and parallels. But even a mesh-system of this sort could be likened to a network of equal squares only in the immediate vicinity of the equator. As we examined our mesh-system at regions farther and farther removed from the equator, we should notice that the meridians had a tendency to run closer and closer together, so that the quadrilaterals bounded by the meridians and parallels would depart more and more in shape from those of Euclidean squares. In the general case, where we consider an arbitrarily curved surface and no longer a uniformly curved one, such as a sphere, it could be shown that the straightest type of mesh-system compatible with the surface was assimilable to one of squares only around a point, and not, as in the case of a sphere, along a line (such as the equator). The point could be selected arbitrarily, and our pseudo-straight mesh-system drawn around this point accordingly.

We see, therefore, that if space-time is unevenly curved or unevenly non-Euclidean, the nearest approach to a Cartesian mesh-system around any given point could be approximately Cartesian only in a restricted region around this point. As we moved away from this point in our mesh-system, its curvilinear characteristics would become more and more pronounced. So far as the ’s of space-time are concerned, this would imply that in a region where curvature was present these ’s could maintain constant values only in the more or less immediate neighbourhood of a point. Expressed in terms of fields of force, this assertion is equivalent to stating that in a region of space-time curvature it would be quite impossible to rid ourselves of a field of force throughout space. We might annul it at the point and in its immediate neighbourhood, but the field of force would reappear for more distant regions.

It is easy to see that these physical conditions are precisely those which we have found to exist in a gravitational field produced by matter. The straightest type of mesh-system around a point (namely, a semi-Cartesian mesh-system) would correspond to that of the observer in the falling elevator or in the hollow projectile, since for an observer in a frame of this type the force of gravitation disappears in the immediate interior of his frame, but reappears at more distant points. Further, space-time being curved, its geodesics would no longer be Euclidean straight lines. They would still appear approximately straight when referred to a semi-Cartesian mesh-system in the immediate neighbourhood of the point for which the mesh-system was approximately one of equal four-dimensional cubes; but their curvature would become more and more apparent when we surveyed them from this same mesh-system in more distant regions. The physical interpretation of this geometrical fact would be afforded by the seemingly uniform and rectilinear motion of free bodies and rays of light in the immediate neighbourhood of the falling elevator when viewed by an observer situated in this falling frame, and by the apparent acceleration of these bodies in regions more distant from the falling elevator.

It is therefore obvious that the existence of a non-Euclideanism of space-time around matter would betray itself physically in exactly the same way as a field of gravitation produced by matter.[85] Assuming this generalisation to be correct, we see that no essential difference can exist between a field of gravitation and one of inertia. Both types of field arise from a relationship between the mesh-system adopted by the observer, and the fundamental structure of the space-time continuum. The only difference that exists is that in interstellar space far from matter, space-time being flat, it is always possible to select a Cartesian mesh-system corresponding to a Galilean system in which no field of force is present, whereas in the neighbourhood of matter, owing to the inherent curvature of space-time, the nearest approach to Galilean frames is given by semi-Cartesian or semi-Galilean frames (falling elevator). In such frames the field of force cannot vanish throughout the frame at more or less distant points.

It follows that these same ’s whose values in a given mesh-system represented the potentials of the field of inertia can now with equal justification be considered to yield the potentials of the field of gravitation. The mathematical expression comprising the variations of the ’s from point to point, which was formerly proved to represent the force of inertia, can now with equal justification be deemed to represent the force of gravitation. As was the case with the force of inertia, this mathematical expression of the force, depending as it does on the variations in the values of the ’s from point to point, varies with our choice of a mesh-system; in exactly the same way, the magnitude of a gravitational force in our frame of reference varies with the acceleration of our frame (as was established when we discussed the postulate of equivalence). Furthermore, the value of the mathematical expression of this force will vanish completely throughout a mesh-system when this mesh-system is Cartesian, whereas it will vanish only around a definite point when, owing to the curvature of space-time, a Cartesian mesh-system is impossible, so that we have to be content with a semi-Cartesian one. We see that the analogy is complete in all respects.

Here it must be noticed that, identifying as we are the ’s in a given mesh-system with the gravitational potentials in the frame corresponding to this mesh-system, we find ourselves in the presence of as many different gravitational potentials as there are separate ’s. That is, we have ten different potentials. Newton knew of only one of these, the Newtonian potential, which turns out to be —the one directed along the time-axis of the mesh-system. It is the presence of these other potentials ignored by Newton and discovered by Einstein which is responsible for the major differences distinguishing Newton’s law of planetary motions from that of Einstein. We see, therefore, that this natural generalisation of the geometry of space-time, by compelling us to recognise forces of gravitation and inertia as of the same nature, leads us to exactly the same conclusions as were obtained by Einstein when he started from the postulate of equivalence. So although we laid stress on the postulate of equivalence as a separate discovery in order to abide by the historical sequence of Einstein’s investigations, we see that when we view the problem from a loftier standpoint the postulate does not add anything to what would normally have been discovered, or at least suspected, as a result of the more philosophical generalisation we have discussed.

Of course, if the postulate of equivalence should prove to be unjustified, as it would were experiment to detect a difference between inertial and gravitational mass, the generalisation would also be unjustified and gravitation could never be ascribed to a curvature of space-time. However, as the most refined experiments have invariably disclosed the complete identity of the two types of masses, the space-time generalisation appears to be in order.