[95] Later we shall see that this belief of classical science is not rigorously correct but it still remains true under certain special circumstances.
[96] In point of fact, it was when Einstein applied the principle of Action (to be discussed presently) that he first recognised the error in his original law. Also we may note that the law of curvature
, when it does not reduce to
, represents a non-homogeneous type of curvature, and most certainly not a homogeneous spherical curvature as certain lay writers have stated, drawing hasty philosophical conclusions therefrom.
[97] We shall see ([Appendix I]) that the geodesics of space-time are of two major varieties: the so-called time-like and the so-called space-like geodesics. The transition between the two is given by the null-lines or minimal geodesics; these correspond to the paths and motions of light rays. The space-like geodesics would correspond to the paths and motions of bodies moving with a speed greater than that of light. As such motions cannot exist, according to the theory of relativity, we see that free bodies can follow only the time-like geodesics. In future, therefore, when referring to the geodesics of space-time, we shall always have in mind the time-like geodesics. Also we may note that whereas the time-like geodesic defines the longest space-time distance between two points, the null-line or minimal geodesic has always a zero space-time length.
[98] Also constants such as
may enter into the law of curvature in the empty space around matter; but never foreign tensors.