In the theory of relativity the concept of matter also undergoes important modifications. In classical science, matter was distinguished by its attribute of mass and by its ability to create a gravitational field. It is true that the electromagnetic field was also known to possess momentum and hence inertia (as exemplified by the pressure of light), so that even in classical science the distinction between matter and ether was losing its definiteness. Owing, however, to the fundamental difference that was thought to exist between inertial mass, as representative of inertia, and gravitational mass, as representative of attraction, the electromagnetic field was never suspected of developing a gravitational field. Hence the difference between matter and energy remained fairly clear. Einstein’s theory, by identifying mass and energy and by showing that all forms of energy are capable of producing a gravitational field, tends to render the distinction less legitimate. A further consequence of the theory has been to show that conservation of matter cannot hold. The result is that mass, as a substance, loses its meaning.
Passing to the laws of motion, we remember that there is but one law: All free bodies (when reduced to point-masses) follow time-geodesics in space-time regardless of whether space-time be flat, as it is (at least approximately) in interstellar space, or whether it be curved by the presence of matter. If space-time is flat, the geodesics are straight and the bodies describe straight courses with constant speeds as referred to a Galilean frame. Thus Newton’s law of inertia is seen to express the flatness of space-time. When space-time, and hence its geodesics, are curved by the presence of matter, the courses of free bodies appear to be curved, or else their motion to be accelerated. But whereas, under those conditions, the law of inertia was at fault in classical science, and an additional gravitational influence had to be introduced, in Einstein’s theory the general law of geodesic motion still holds good. Inasmuch as the structure of space-time determines the laws of our geometry, the beatings of natural clocks (atoms) and the motion of free bodies, we see that the theory has brought about a fusion between geometry and physics.
We may also remember that the existence of space-time has solved the old enigma of the dual nature of the structure of space in mechanics, seemingly relative for velocity and absolute for acceleration (see [Chapter XIX]).
Force is another important concept on which the theory of relativity has shed new light; though to avoid any misunderstanding we may state that by force we refer not to muscular pulls, but to the forces of gravitation, of inertia and of the electromagnetic field. Let us recall how the new ideas were obtained. We saw that the fundamental continuum of the world was space-time, and that observers, according to their relative motions, carved it up into space and time in various ways. Now we know that even in flat space-time, far from matter, forces of inertia spring into existence when we place ourselves in a rotating frame or, more generally, in any accelerated frame; translated into terms of space-time, this means when we split up space-time into a curved space and a curved time.
It follows that in an empty region of flat space-time there is no sense in wondering whether forces are present or not. Forces can come into existence only after the splitting up of space-time into separate space and time has been accomplished. Then, according to the method selected forces will appear to be distributed this way or that, or even may not arise at all; the last case would correspond to the choice of a Cartesian mesh-system. But this splitting-up process which may give rise to forces is obviously artificial and corresponds to nothing intrinsic in nature, any more than the splitting up of three-dimensional space into arbitrary directions, which we name above, below, left and right, reveals any intrinsic property of space. So we see that forces are introduced by the observer and that, in the same region of space-time, they may be present for one man and absent for another. We cannot, therefore, attribute to them any absolute existence, one transcending the observer.
Now the forces we have been discussing were those which might arise in flat space-time far from matter. They were the forces of inertia o classical science, and were often referred to as fictitious precisely because their existence depended on our conditions of observation. A distinction was then drawn between these fictitious forces and the force of gravitation, which was assumed to exist around matter regardless of the observer’s frame of reference. But with the relativity theory we see that a discrimination of this sort is no longer permissible. Around matter, space-time manifests curvature, but, just as in the case of flat space-time, it is possible for an observer to carve it up into space and time in such a way that no force will be manifest, at least in his vicinity. This method of partitioning corresponds, as we know, to that of the observer falling freely, hence following a geodesic in space-time. If, then, forces of inertia must be regarded as fictitious, so must forces of gravitation. And it is much the same with electromagnetic forces. Here also, the intensity and nature of the field (whether electric, magnetic, or electromagnetic) depends in large part on the circumstances of observation.
The purport of all these discussions is that if we wish to form an impersonal picture of the universe, there is no place for forces of gravitation or of inertia, since these forces are introduced by the observer. Likewise, in the description of a monument there is no place for shadow, since shadow is introduced only by the sun, and, so far as the monument itself is concerned, the sun and light in general do not constitute essential conditions of its existence. The entire philosophy of the relativity theory is to separate entities which have no significance except as expressing relationships to the frame of the observer, from entities which can be conceived of as enduring regardless of the observer’s presence. We recognise these absolute entities as those which remain unaltered for all observers. Mathematically, they are expressed by invariants and tensors in space-time. Action and the Einsteinian interval are representative of invariants, while the majority of types of space-time curvature, the electromagnetic-field tensors, and the matter tensor are representative of tensors. It is because gravitational force and potential energy are not represented by tensors that we can attribute no absolute significance to them; and as a result, potential energy being discarded as a relative, conservation of energy falls to the ground.
The attitude towards gravitational force which we have outlined is championed more especially by Eddington. Weyl, however, views the problem differently. He argues that the space-time metrical field is related to matter much as the electric field is related to electricity. Just as an electric force is a real force generated by the electric field on a charged body, so now inertial and gravitational forces must be regarded in the same light, arising as they do from the action of the metrical or guiding field on matter. But it should be understood that Weyl’s conception of force is not quite the same as the one commonly accepted. For instance, according to the views of classical science, a body moving through interstellar space in conformity with the law of inertia would be submitted to no forces. It moved as it did in virtue of the principle of inertia, and no question was raised as to why it followed a straight line rather than a curve. Although it was apparent even in classical science that there must be a guiding influence at work emanating from space itself, very little was said of its nature.
It is this “guiding field” whose importance is stressed by Weyl. It is this dynamical property of space or space-time that the theory of relativity has brought into full light. The guiding field, or metrical field, is represented, as we know by the
