A difficulty of such a fundamental character could come to light only owing to the theorem of the equality of inertial and gravitational mass not being sufficiently interwoven with the underlying principles of mechanics, and because, in the foundations of Newtonian mechanics, the same importance had not been accorded to gravitational phenomena as to inertial phenomena, which, judged from the standpoint of experience, must be claimed. Gravitation, as a force acting at a distance, is, on the contrary, introduced only as a special force for a limited range of phenomena: and the surprising fact of the equality of inertial and gravitational mass, valid at all times and in all places, receives no further attention. One must, therefore, substitute for the law of inertia a fundamental law which comprises inertial and gravitational phenomena. This can be brought about by a consistent application of the principle of the relativity of all motions, as Einstein has recognized. This is, therefore, the circumstance chosen by Einstein as a nucleus about which to weave his developments.
The theorem of the equality of inertial and gravitational mass, which reflects the intimate connection between inertial and gravitational phenomena, may be illuminated from another point of view, and thereby discloses its close relationship (vide [page 55]) to the general principle of relativity.
However much the notion of "absolute space" repelled Newton, he nevertheless believed he had a strong argument, in support of the existence of absolute space, in the phenomenon of centrifugal forces. When a body rotates, centrifugal forces make their appearance. Their presence in a body alone, without any other visible body being present, enables one to demonstrate the fact that it is in rotation. Even if the earth were perpetually enveloped in an opaque sheet of cloud, one would be able to establish its daily rotation about its axis by means of Foucault's pendulum-experiment. This peculiarity of rotations led Newton to conclude that absolute motions exist. From the purely kinematical point of view, however, the rotation of the earth is not to be distinguished in any way from a translation; in this case, too, we observe only the relative motions of bodies, and might just as well imagine that all bodies in the universe revolve around the earth. E. Mach has, in fact, affirmed that both events are equivalent, not only kinematically, but also dynamically: it must, however, then be assumed that the centrifugal forces, which are observed at the surface of the earth, would also arise, equal in quantity and similar in their manifestations, from the gravitational effect of all bodies in their entirety, if these revolved around the supposedly fixed earth (vide [Note 19]).
The justification for this view, which in the first place arises out of the kinematical standpoint, is, in the main, to be sought in the fact, derived from experience, that inertial and gravitational mass are equal. According to the conceptions, which have hitherto prevailed, the centrifugal forces axe called into play by the inertia of the rotating body (or rather by the inertia of the separate points of mass, which continually strive to follow the bent of their inertia, and, therefore, express the tendency to fly off at a tangent to the path in which they are constrained to move). The field of centrifugal forces is, therefore, an inertial field (vide [Note 20]). The possibility of regarding it equally well as a gravitational field—and we do that, as soon as we also assert the relativity of rotations dynamically: for we must then assume that the whole of the masses describing paths about the (supposed) fixed body induce the so-called centrifugal forces by means of their gravitational action—is founded on the equality of inertial and gravitational mass, a fact which Eötvös has established with extraordinary precision by making use of the centrifugal forces of the rotating earth (vide [Note 21]). From these considerations one realizes how a general principle of the relativity of all motions simultaneously implies a theory of gravitational fields.
From these remarks one inevitably gains the impression that a construction of mechanics upon an entirely new basis is an absolute necessity. There is no hope of a satisfactory formulation of the law of inertia without taking into account the relativity of all motions, and hence just as little hope of banishing the unwelcome conception of absolute motion out of mechanics; moreover, the discovery of the inertia of energy has taught us facts which refuse to fit into the existing system, and necessitate a revision of the foundations of mechanics. The condition which must be imposed at the very outset (cf. [page 20]) is: Elimination of all actions which are supposed to take place "at a distance" and of all quantities which are not capable of direct observation, out of the fundamental laws; i.e. the setting-up of a differential equation which comprises the motion of a body under the influence of both inertia and gravity and symbolically expresses the relativity of all motions. This condition is completely satisfied by Einstein's theory of gravitation and the general theory of relativity. The sacrifice, which we have to make in accepting them, is to renounce the hypothesis, which is certainly deeply rooted, that all physical events take place in space whose measure-relations (geometry) are given to us a priori, independently of all physical knowledge. As we shall see in the [following section], the general theory of relativity leads us, rather, to the view that we may regard the metrical conditions in the neighbourhood of bodies as being conditioned by their gravitation. In this way the geometry of the measuring physicist becomes intimately welded with the other branches of physics.
If we compress into a short statement what we have so far deduced out of the fundamental postulates formulated at the beginning, we may say: The postulate of general relativity demands that the fundamental laws be independent of the particular choice of the co-ordinates of reference. But the Euclidean line-element does not preserve its form after any arbitrary change of the co-ordinates of reference. We have, therefore, to substitute in its place the general line-element:
Whereas, then, the postulate of continuity (cf. [page 20]) seemed to render it only advisable not to introduce the narrowing assumptions of the Euclidean determination of measure, the principle of general relativity no longer leaves us any choice.
The reason for so emphasizing the latter principle—as, indeed, also the postulate that only observable quantities are to occur in physical laws—is not to be sought in any requirement of a merely formal nature, but rather in an endeavour to invest the principle of causality with the authority of a law which holds good in the world of actual physical experience. The postulate of the relativity of all motions receives its true value only in the light of the theory of knowledge ([Note 22]). One must, above all, avoid introducing into physical laws, side by side with observable quantities, hypotheses which are purely fictitious in character, as e.g. the space of Newton's mechanics. Otherwise the principle of causality would not give us any real information about causes and effects, i.e. the causal relations of the contents of direct experience; which is, presumably, the aim of every physical description of natural phenomena.