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
