(a) THE FUNDAMENTAL LAW OF MOTION AND THE PRINCIPLE OF EQUIVALENCE OF THE NEW THEORY
AFTER the foregoing remarks we shall be able to proceed to a short account of Einstein's theory of gravitation. Within the limits of the mathematics assumed in this book we shall, of course, only be able to sketch the outlines so far that the assumptions and hypotheses characteristic of the theory come into clear view and that their relation to the two fundamental postulates of the [second section] becomes manifest. We start out from the fundamental law of motion in classical mechanics, the law of inertia. Since even in the law of inertia all the weaknesses of the old theory come to light, a new fundamental law of motion becomes an absolute necessity for the new mechanics. It is thus natural that we should start building up the new theory from this point. The new law of motion must be a differential law, which, in the first place, describes the motion of a point-mass under the influence of both inertia and gravity, and which, secondly, always preserves the same form, irrespective of the system of co-ordinates to which it be referred, so that no system of co-ordinates enjoys a preference to any other. The first condition arises from the necessity of ascribing the same importance to gravitational phenomena as to inertial phenomena in the new process of founding mechanics—the law must, therefore, also contain terms which denote the gravitational state of the field from point to point; the second condition is derived from the postulate of the relativity of all motion.
A law of this kind exists in the special theory of relativity in the equation of motion of a single point, not subject to any external influence. According to this equation, the path of a point is the "shortest" or "straightest" line (vide [Note 23])—i.e. the "straight line," if the line-element
is Euclidean. Written as an equation of variation this law is:
If the principle of the shortest path, which is to be followed in actual motions, be elevated in this form to a general differential law for the motion in a gravitational field too, with due regard to the principle of the relativity of all motions, the new fundamental law must run as follows:
For only this form of the line-element remains unaltered (invariant) for arbitrary transformations of the
