, and not the Cartesian co-ordinates of Euclidean geometry, all the other physical laws must also receive a general form that is independent of the special choice of co-ordinates. The mathematical instrument for remoulding these formulæ is given by the general calculus of differentials.
This theory, which is built up from the most general assumptions, leads, for a first approximation, to Newton's laws of motion. Wherever deviations from the theory hitherto accepted reveal themselves, we have possibilities of testing the new theory experimentally. Before we turn to this question, let us look back, and become clear as to the attitude which the general theory of relativity compels us to adopt towards the various questions of principle we have touched upon in the course of this essay.
(b) RETROSPECT
1. The conceptions "inertial" and "gravitational" (heavy) mass no longer have the absolute meaning which was assigned to them in Newton's mechanics. The "mass" of a body depends, on the contrary, exclusively upon the presence and relative position of the remaining bodies in the universe. The equality of inertial and gravitational mass is put at the head of the theory as a rigorously valid principle. The hypothesis of equivalence herein supplements the deduction of the special theory of relativity, that all energy possesses inertia, by investing all energy with a corresponding gravitation. It becomes possible—on the basis (be it said) of certain special assumptions into which we cannot enter here—to regard rotations unrestrictedly as relative motions too, so that the centrifugal field around a rotating body can be interpreted as a gravitational field, produced by the revolution of all the masses in the universe about the non-rotating body in question. In this manner mechanics becomes a perfectly general theory of relative motions. As our statements are concerned only with observations of relative motions, the new mechanics fulfils the postulate that in physical laws observable things only are to be brought into causal connection with one another. It also fulfils the postulate of continuity; since the new fundamental laws of mechanics are differential laws, which contain only the line-element
and no finite distances between bodies.
2. The principle of the constancy of the velocity of light in vacuo, which was of particular importance in the special theory of relativity, is no longer valid in the general theory of relativity. It preserves its validity only in regions in which the gravitational potentials are constant, finite portions of which we can never meet with in reality. The gravitational field upon the earth's surface is certainly so far constant that the velocity of light, within the limits of accuracy of our measurements, had to appear to be a universal constant in the results of Michelson's experiments. In a gravitational field, however, in which the gravitational potentials vary from place to place, the velocity of light is not constant; the geodetic lines, along which light propagates itself, will thus in general be curved. The proof of the curvature of a ray of light, which passes by in close proximity to the sun, offers us one of the most important possibilities of confirming the new theory.
3. The greatest change has been brought about by the general theory of relativity in our conceptions of space and time.[13]
[13]This aspect of the problem has been treated with particular clearness and detail in the book "Raum und Zeit in der gegenwärtigen Physik," by Moritz Schlick, published by Jul. Springer, Berlin. The Clarendon Press has published an English rendering under the title: "Space and Time in Contemporary Physics."