Part 3 — CHAPTER XXVIII THE DISCOVERY OF THE EINSTEINIAN LAW OF GRAVITATION

IN the two preceding chapters we have discussed in a very general way the subject of tensors and of the laws of nature. It now remains to be seen how Einstein, guided by the requirements that accompanied all general laws, succeeded in formulating with mathematical precision the true law of gravitation.

In order to understand what is to follow we must revert to Newton’s law. This law defines the intensity and distribution of the field of gravitational force in the empty space which surrounds matter. Also, it enables us in turn to ascertain how the field of force would be distributed in the interior of matter, in the interior of the earth, for instance. Thus, in its most general aspect, Newton’s law of gravitation defines the intensity and distribution of the gravitational field both inside and outside of matter.

Now we have seen that the presence of a field of force connotes the existence of an uneven distribution of potentials in the space where the field exists,[92] since any perfectly even and constant distribution of potentials would correspond to the complete vanishing of the field. Hence, Newton’s law of gravitation can also be written as a law of potential distribution inside and outside of matter. Expressed in this form, it is known as Poisson’s equation and is written In this celebrated equation, represents the potential at each point; a complicated mathematical expression linking the potential at a point and its value at all neighbouring points; expresses the density at each point of the matter producing the field, and is a constant of proportionality known as the constant of gravitation. And since the equation gives us the law of potential distribution at each point of space where the density of matter is , in regions outside matter we obviously have since outside matter , is of course zero. Expressed in this latter form (), in which it gives us the potential distribution outside matter, the Newtonian law of gravitation is known as Laplace’s equation.

Thus, we may say that Newton’s law of gravitation inside and outside of matter is given by , in which is zero or has some definite non-vanishing value according to whether we are considering a region outside matter or in its interior. Now, this potential distribution in Newton’s law refers to the distribution through space of one same potential, an invariant in space, known as the Newtonian potential. But we know that if Einstein’s ideas are correct, if gravitation is to be attributed to space-time curvature, Newton’s single potential (an invariant) will have to be replaced at every point by the ten separate ’s of space-time. These, as we know, are no longer invariants, they are second-order tensors;[93] and Newton’s invariant potential turns out to be none other than one of the ’s, namely, the time one, .[94]

According to Einstein’s views, therefore, the expression of the gravitational potential distribution in the interior of matter will be given by an equation between some complicated expression connecting the ten ’s at every point and at neighbouring points, and some quantity representing the characteristics of matter at every point. This equation will then constitute Einstein’s law of gravitation in the interior of matter, and will represent the Einsteinian substitute for Poisson’s classical equation.

Now Poisson’s equation was an equation between two invariants in space. Einstein’s law, on the other hand, in accordance with the requirements that must be satisfied by all general laws in the space-time theory, will have to be an equation between invariants or tensors in space-time. By analogy with Poisson’s equation we see that the left-hand side of Einstein’s law will be a space-time invariant or tensor which will represent the potential distribution, or -distribution, from point to point, while the second side will be given by an invariant or a tensor representing matter.

A tensor representative of matter and describing it fully had already been discovered by Einstein. It was a second-order symmetrical tensor generally written and its ten separate components, as referred to any definite space-time mesh-system, defined such physical magnitudes as the density of matter at a point, its internal stresses, vis viva, and momentum in that mesh-system. Matter, as a source of gravitation, was obviously described much more completely by this tensor than was the case in classical science, where the mere mention of its density of mass was considered sufficient to determine its gravitational field. Thus, Einstein was in possession of the right-hand side of the equation of gravitation in the interior matter, that corresponding to in Poisson’s equation (neglecting constants).

It remained to discover the left-hand side of the equation. Owing to the condition of covariance, this left-hand side had to be represented by a tensor of the same order and nature as the right-hand side, hence, in the present case, by a symmetrical tensor of the second order. Further, it would have to be built up of the ten ’s or potentials, expressing a relationship between these ten ’s at every point and at the neighbouring points. It is here that Riemann’s discoveries fit in with Einstein’s requirements; for Riemann and his successors had discovered tensors built up from the fundamental structural tensors of a continuum, namely, from the ’s. All these tensors served to represent various possible types of structure, geometry or curvature.

In a previous chapter we mentioned these tensors. First, there was the Riemann-Christoffel tensor . It was a tensor of the fourth order, as is indicated by the four indices underlying the letter . Then there was a second-order symmetrical tensor . Lastly, there was an invariant of curvature , no longer a tensor in the ordinary sense of the word. By differentiating and combining these magnitudes, we could of course obtain others, but the ones we have mentioned are the simplest.

Einstein had still one further clue to guide him. Classical science had assumed that at an infinite distance from attracting matter the law of inertia would hold rigorously; that is to say, a body moving freely would follow a rectilinear course with constant speed when referred to a Galilean frame. Expressed in the language of space-time, this would imply that at an infinite distance from matter, space-time would become rigorously flat, as we had assumed it to be in the special theory. This condition necessitated the vanishing of the Riemann-Christoffel tensor , at infinity. If this condition is maintained, the curvature of space-time around matter will have to be of a type which is capable of dying down to perfect flatness at infinity, degenerating to .

Now the law of curvature was precisely of such a type; hence Einstein, in his original paper, found it natural to select the tensor for the left-side term of his gravitational equations. Accordingly he wrote:

Outside matter, this law would of course reduce to , since, outside matter, , representative of matter, would vanish. It is easy to see that these gravitational equations satisfy all the requirements we have mentioned. First, we are in the presence of covariant equations, since and are tensors of the same nature and order. Secondly, is built up from the ’s or potentials, as it should be. Thirdly, outside matter, our equations reduce to , and the curvature connoted by this relation is compatible with the existence of perfect flatness at infinity, i.e., with at infinity.

Nevertheless, Einstein soon recognised that his equations could not correspond to reality. The fact was that matter appeared to be conserved in nature. By this we mean that matter disappears from here only to reappear there.[95] It followed that the tensor of curvature which would be identified with the presence of matter in the law of gravitation should likewise possess the attributes of conservation. Now the tensor possessed no such attributes; but a very similar tensor, also symmetrical and of the second order, namely, fulfilled all the previously mentioned requirements. Henceforth, the law of potential distribution, or of gravitation inside matter, would have to be (neglecting constant factors) (analogous to Poisson’s equation); and outside matter, as disappeared, there was left which reduces to (analogous to Laplace’s equation). From this we see that Einstein’s initial error affected the law of space-time curvature only in the interior of matter. Outside matter, we had, as before, . [96]

Let us recall that in all examples where tensor equations are discussed, the indices and (in the present case) are to be replaced by all manner of arrangements of the integers from 1 to 4 (since space-time has four dimensions). An equation such as or as represents therefore ten separate equations. As a matter of fact, sixteen equations would be obtained were we to substitute all possible arrangements; but as the tensors are symmetrical, only ten out of sixteen art independent, hence need be retained; the other six are mere repetitions We see, then, that in Einstein’s theory ten separate equations are necessary in order to define the distribution of the gravitational field in the interior and around matter, whereas in classical science one equation, i.e. Poisson’s equation (), was sufficient.

Here we may mention that if we abandon the restriction that the law of inertia holds rigorously at infinity, hence if we abandon the condition of perfect flatness at infinity, we may build up other tensors satisfying all the remaining requirements. This can be done by adding a further tensor (where represents an arbitrary constant) to the left-hand side of our equation. The presence of this additional term will imply that even for very distant regions, space-time would still manifest a trace of residual spherical curvature so that the universe would close round on itself and be finite. In the first development of his theory, Einstein rejected this additional term, but the fact remains that so far as the mathematical requirements of the theory of relativity are concerned there is no reason to exclude it ab initio. At all events, we can assert that if the term exists at all, must be exceedingly minute; for even though finite, the universe is certainly of gigantic proportions, and its curvature therefore exceedingly slight. For the present we shall ignore the term since in any case its influence would be too insignificant to affect perceptibly such phenomena as those of planetary motions. Only when we come to consider the shape of the universe as a whole and its bearing on the Mach-Einstein principle of the relativity of all motion will the presence of assume a vast philosophical importance.

Now the methods whereby Einstein was led to the discovery of his law of gravitation might appear rather piecemeal, much like the fitting together of a jig-saw puzzle in which the structural tensors constituted the pieces. But this, in itself, can scarcely be regarded as a drawback to the theory. It is rather the reverse, since it proves that just as the pieces of a puzzle can only be fitted together in certain ways, so now the universe can scarcely be otherwise than it is. In this respect, there is a marked contrast between the present situation and that which endured in classical science, where practically any law of gravitation might have been expected.

Nevertheless, from the standpoint of mathematical elegance, something more was needed. When, therefore, Einstein had cleared the ground and his ideas were more fully understood, the greatest of the pure mathematicians, such as Klein, Hilbert, Weyl and subsequently Einstein himself, undertook to discover the law of gravitation by more general methods. They found that if we accepted the existence of space-time and if we agreed that the gravitational field around matter was due to a curvature of space-time, the Principle of Stationary Action showed that a large number of laws of gravitation could be entertained. If, however, it was specified that by analogy with Newton’s law, the law of curvature should contain no derivatives of the ’s or potentials to an order higher than the second, Einstein’s law of gravitation with or without the term was the only one possible. Furthermore, all these possible laws of gravitation were found to verify the mathematical condition expressing the conservation of matter; hence, by proceeding in this way through the principle of action, it was possible to obviate the rather artificial condition of conservation which would otherwise have had to be introduced as an additional hypothesis.