-distribution, and according to Einstein, its origin must in all probability be sought in the presence of the universal masses. To quote Weyl:
“Galilei’s principle of inertia shows that there is a sort of ‘forcible guidance’ which compels a body that is projected with a definite velocity to move in a definite way which can be altered only by external forces. This ‘guiding field,’ which is physically real, was called ‘affine relationship’ above. When a body is diverted by external forces the guidance by forces such as centrifugal reaction asserts itself.”
These considerations pertaining to tensors lead us in turn to an understanding of the significance of the laws of nature. The special theory, by demonstrating the identity of the invariant velocity of the space-time world with Maxwell’s constant
, had proved that the laws of nature remained covariant to the Lorentz-Einstein transformations. This was but another way of saying that they expressed relationships between vectors and tensors in flat space-time, at least so far as Galilean observers were concerned. The general theory extends this discovery by assuming it to hold for all observers regardless of their frame of reference and regardless of whether space-time be flat or curved. We thus obtain the general principle of relativity, or of covariance, which expresses the conditions which all laws of nature must satisfy. Those which do not pass this acid test can no longer be regarded as true laws.
We can also approach the problem of the laws of nature in a deductive way by starting from space-time. As soon as we recognise the existence of an absolute universe underlying our spatial and temporal world of immediate perception, and as soon as we realise that variations in our motion correspond to mere differences in our methods of splitting up into space and time the space-time continuum, it becomes evident that all particularities of structure of the absolute universe will subsist regardless of our choice of mesh-system, and hence regardless of our motion. These absolute space-time relations correspond to the laws of nature, which transcend the motion of the observer. From this it follows that the existence of the absolute universe, in which our physical entities are represented by vectors and tensors, and the relativity of all motion so far as the expression of the laws of nature is concerned, are but alternative aspects of one same condition; so that, as we have had occasion to state, the theory of relativity might also be termed the “quest of the absolute.”
And here a rather curious point about Einstein’s theory must be mentioned. Classical science seemed to have established almost beyond the shadow of a doubt that the description of phenomena in terms of the inertial frame of Copernicus had come to stay; since, when we endeavoured to express phenomena in terms of rotating frames, we were compelled to complicate the form of our laws by introducing centrifugal and Coriolis fields of force. But this remained true only so long as we reasoned in terms of separate space and time. With space-time the laws of nature assume invariant forms, so that the mathematical relationships are no longer modified when we pass from the inertial to a rotating frame. It is not that centrifugal and Coriolis forces fail to manifest themselves when we pass from Galilean to rotating frames, but that they no longer have to be introduced as foreign adjuncts, for they are already contained in a latent form in the mathematical expression of the laws. Then the mathematical rules of transformation which correspond to the passage from one frame to another, say from a Galilean to a rotating frame, cause these forces to pass automatically from a latent to an active form.
The heuristic value of this new aspect of the laws of nature is, of course, considerable. In classical science, laws were always tensor equations in separate space and time; but the relativity theory, by compelling us to substitute space-time for space, narrows down our choice of laws in a remarkable manner. Thus, in classical science, the law of gravitation might have been practically anything; it might have been an inverse-cube law, for instance; only the observation of planetary motions could decide between Newton’s law and any other. But with the space-time criterion, a number of conceivable laws, including Newton’s law of the inverse square and those of classical mechanics, are discarded as impossible ab initio.
Here, however, it is necessary to obtain a proper understanding of the limitations which the principle of covariance, or general principle of relativity, imposes upon the laws of nature. The mere fact that all the laws of nature should be expressible in space-time-tensor form is not, of itself, sufficient to determine them unambiguously.[163] By introducing a sufficient number of arbitrary tensors into the laws of gravitation, tensors whose existence we would postulate ad hoc, we might conceivably vary the law ad infinitum while retaining its properties of covariance. Furthermore, even without introducing foreign tensors, the principle of covariance does not restrict the law of gravitation to that simplest form selected by Einstein (
