’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.