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.