Part 3 — CHAPTER XXVII THE PRINCIPLE OF GENERAL COVARIANCE, OR THE GENERAL PRINCIPLE OF RELATIVITY

WE have endeavoured to show that the great change in our views of the world brought about by the theory of relativity arises from the discovery that four-dimensional space-time and not the three-dimensional space and the independent time of classical science correspond to reality. It might seem but a natural generalisation to extend to space-time the conclusions we arrived at in the last chapter when discussing the covariance of the natural laws to a change of mesh-system in space. We should then expect the laws of nature to express relationships between vectors and tensors in space-time, hence to remain covariant to a change in our space-time mesh-system.

Although this generalisation would turn out to be perfectly legitimate, its validity is by no means self-evident a priori. We can accept it only after we have convinced ourselves that the laws of physics, such as experiment has revealed them to be, can indeed be regarded as expressing relationships between entities (vectors and tensors) in space-time. Now, from a purely formal standpoint, it is conceivable that such might not be the case. Suppose, for instance, that the four-dimensional spatio-temporal background were characterised by some finite invariant velocity greater than Maxwell’s constant , which is given, as we know, by the velocity of light in vacuo. In an extended sense we could still regard our universe as a four-dimensional space-time, since, the invariant velocity being finite, and not infinite, the fusion of space and time would hold, just as it did in Minkowski’s space-time.

But, on the other hand, owing to the discrepancy between and , the laws of electromagnetics would cease to be covariant to a change of Galilean frame, hence to a change of space-time mesh-system. And so we could not view these laws as expressing relationships between the absolute entities of the fundamental space-time continuum.

The situation changes when we find, as a result of experiment, that the laws of electromagnetics remain covariant to a change of Galilean frame. This discovery results, of course, from the negative experiments in electromagnetics and from others of a similar nature. The first consequence of this discovery is to prove that , the invariant velocity of space-time, and , Maxwell’s constant, must be one and the same, since only on this condition will the laws of electromagnetics remain covariant.

Einstein then extended this property of covariance to all physical laws. We thus obtain the special principle of relativity, according to which the laws of nature remain covariant when we refer them to one Galilean frame or another. In short, the special theory deals solely with Galilean frames.

From the standpoint of mesh-systems, let us recall that Galilean frames are given by Cartesian mesh-systems in space-time, the differences in their orientations representing the relative velocities existing between the frames. When we pass from measurements performed in one of these Cartesian mesh-systems to those executed in some other one of these frames, we must apply the transformations corresponding to a rotation in space-time; and these are the Lorentz-Einstein transformations. We may say, therefore, that in the special theory the principle of covariance entails the covariance of all natural laws when submitted to the Lorentz-Einstein transformations.

Now the fact that the laws of nature maintain the same form regardless of the orientation of our Cartesian space-time mesh-system, or frame of reference, suggests that they must be the expressions of relationships between such entities as vectors and tensors in space-time. If this be the case, the laws should remain covariant not merely to a change of Cartesian mesh-system, but also to a change to any species of mesh-system, whether Cartesian or curvilinear. In other words, they should be expressible as vector or tensor equations in flat space-time.

This discovery permits us to advance a step farther. For let us suppose that this erstwhile flat continuum manifests curvature in certain regions. Einstein assumed that in this case its curvature would curve along with it the entities embedded in its substance. Under the circumstances, the covariance of the laws of nature should continue to hold regardless of the flatness or curvature of space-time, hence also of the species of mesh-system we might appeal to. It is this generalisation of Einstein’s which constitutes the principle of general covariance, or the general principle of relativity. We see that it represents an extension of the special principle of covariance, or of relativity, extending as it does to all regions of space-time, whether flat or curved, and to all mesh-systems, whether Cartesian or curvilinear, the covariance established originally only in a special case. Henceforth it becomes permissible to say that all the laws of nature express relationships between entities of space-time (vectors and tensors); and as a result we are able to extend to space-time the conclusions we arrived at when discussing space in the previous chapter. After this preliminary discussion, we may investigate the objectivity of our physical magnitudes.

In classical science these magnitudes were required to be invariants, vectors or tensors in space. In the light of the new discoveries, we must now substitute space-time for space. The result is that both gravitational force and potential energy turn out to be deprived of all objective significance, for by merely changing our mesh-system in space-time we can annihilate or bring into existence the components of these magnitudes at a point. Potential energy entering as it does into the expression of the law of conservation, it became apparent that this law of classical science was obviously of an artificial nature. We shall discuss the principles of conservation in greater detail in a later chapter.

Let us now examine the subject of the general laws of nature. In classical science these laws had to remain covariant to a change in the shape and orientation of our mesh-system in three-dimensional space. The new requirement is that these laws shall remain covariant to a change of shape and orientation of our mesh-system in four-dimensional space-time. This new condition will obviously exert an additional restrictive influence on the formulation of our laws, for it comprises as a special case all the classical restrictions as to covariance in space, supplementing them with the added restriction of covariance in space-time. When, therefore, we consider the laws of classical science, we shall be able to accept them without modification only provided they happen to satisfy the restrictive condition imposed by the space-time theory.

It must be kept in mind, however, that the space-time tensor test cannot aspire to tell us with certainty what the laws of the real world will be. Its action is merely one of elimination; it tells us what these laws can never be. When it comes to deciding which one of the many possible laws is to be accepted as the true law, experiment must be our guide. We can also foresee that the classical laws, though they may appear impossible in the light of the new test, must nevertheless be approximately correct since classical science had considered them to be borne out by experiment. We must not be surprised to find, therefore, that from a practical point of view very slight differences exist between the new laws and the old ones when crude experiments at least are considered. But from a theoretical point of view the difference is of momentous importance. The new laws pass the test; some of the classical ones do not.

It was soon noted that the classical equations of electrodynamics satisfied the new theoretical requirements, for they could be expressed as tensor laws in space-time. Inasmuch as they were in perfect harmony with experiment, they were left undisturbed. On the other hand, the classical equations of mechanics and, with them, Newton’s law of gravitation were certainly incorrect, for they did not remain covariant to a change in our space-time mesh-system.

Accordingly, Einstein had no difficulty in writing out the correct laws for mechanics by modifying ever so slightly the classical ones so as to make them conform to the restrictive space-time condition. We have stated elsewhere that in this way mass was proved to be a relative, no longer an invariant, its value varying according to our space-time mesh-system. We have also mentioned that the most exacting experimental tests have proved the correctness of the new laws and the fallacy of the classical ones.

In the preceding chapter we stated that classical science never paid much heed to the tensor notation, because it so happened that three-dimensional space was assumed to be permanently Euclidean and flat, and hence a standard mesh-system of the Cartesian variety was applicable everywhere throughout space. In the special theory, space-time is also considered flat, so here again the tensor symbolism can be dispensed with. We have seen, indeed, that the special theory was concerned only with Galilean observers in space far from matter, and we know that observers of this type always split up space-time into space and time through the medium of Cartesian mesh-systems. These mesh-systems differ from one another merely by their various orientations in space-time; and we pass from one mesh-system to another by means of the Lorentz-Einstein transformations.

On the other hand, when we come to consider accelerated observers, we are dealing with observers who split up space-time with curved mesh-systems. Again, when masses of attracting matter are present, space-time assumes an intrinsic curvature which renders the utilisation of Cartesian mesh-systems impossible. Certain particular observers, such as those who are falling freely in a gravitational field, adopt mesh-systems which are approximately Cartesian in the immediate neighbourhood of the observer; but were we to prolong these mesh-systems sufficiently, we should soon discover that they became curvilinear, hence varied in shape from place to place, owing to the intrinsic curvature of space-time. In much the same way, a network of parallels and meridians on the surface of a sphere can be assimilated to a Cartesian one of equal squares only in the neighbourhood of the equator, since the meshes taper together as we near the poles. Thus, in a general way, we see why it is that when we leave the special theory, where space-time is flat and where the observer, being Galilean, abides by Cartesian mesh-systems, and when we investigate the general theory, where the observer is accelerated or space-time is curved, the Lorentz-Einstein transformations must give place to a more general type. Under these circumstances the tensor calculus becomes a mathematical instrument of great power.

While it is true that in the early days of the special theory Einstein never made use of this form of calculus, with which he was probably unacquainted, most modern treatises on relativity appeal to it from the very start, since it brings out the beautiful unity of the entire theory and shows that the special theory is merely a particular limiting case of the general one. This allows us to treat both forms of the theory simultaneously. We lay stress on this point, because some non-mathematical critics are of the mistaken opinion that the tensor calculus applies only to the general theory. From this belief they derive the equally erroneous opinion that the two theories constitute separate creations, having little in common. We have seen that such is not the case.

In this presentation of Einstein’s theory, we have emphasised the absolute characteristics of space-time, showing how the covariance of the natural laws followed as a necessary consequence as soon as we regarded them as expressing relationships between space-time entities (tensors, vectors) transcending the mesh-system of the observer. Viewed in this way, the theory has sometimes been referred to as “the quest of the absolute.” But we must remember that it is only as a result of the principle of relativity, necessitated by the negative experiments, among others, that space-time and the laws of nature can be viewed in this light. Were it not for the principle, hence for the identity of , invariant velocity, and , Maxwell’s constant, though there might still be an absolute four-dimensional world, the physical laws such as we know them could no longer be conceived of as expressing relations between its tensors and vectors. We might conceive of space-time entities, in this new type of space-time, as entering into laws that would remain covariant to any change of mesh-system, but these entities and laws would not be represented by the physical entities and laws that experiment has revealed. From the standpoint of the physicist, they would be mere fictions, foreign to the world of our experience. It would be as though the entities and laws revealed to us by experience differed from the laws that would normally be expected had the world a unity of structure and of content. In this event the great beauty of Einstein’s space-time theory would be lost, and with it the wonderful unity and harmony it has shown to exist in nature. Thus, although the absolute viewpoint is the one we have stressed in this chapter, we should bear in mind that it was the disclosures of relativity that rendered this absolute viewpoint possible.

Furthermore we may very easily confer a relativistic aspect on the theory if we recall that by selecting curvilinear or Gaussian mesh-systems in space-time, we are merely placing ourselves in the position of an accelerated observer, who refers his measurements to some frame of reference, squirming like an octopus and who times events with clocks running wild. So far as can be gathered, Einstein was guided originally by a desire to prove that all motion was relative; it was thus the relativistic view that preceded the absolute viewpoint in the elaboration of the theory. There is no cause to be surprised at this precedence, seeing that we are referring to days when space-time was as yet unknown. But when space-time was discovered by Minkowski, the theory received an added impetus (though we can realise full well that even before this discovery a theory which aspired to prove that certain permanencies endured regardless of the observer’s motion would obviously lead us to suspect the underlying presence of some absolute world in which these permanencies would be situated). This absolute world could not be one of separate space and time; for in the absolute space of classical science the laws changed in form when our motion was modified. Obviously something more was needed, and space-time was the answer.

It is imperative, however, not to misinterpret the significance of the covariance of natural laws. The mere fact that all observers, whatever their motion, will discover the same laws, does not mean that all observers are alike in every respect. Even in Einstein’s theory, a rotating observer will realise full well, owing to centrifugal forces, that he does not stand in a Galilean frame. The complete relativity of all motion from every point of view is not by any means embodied in the covariance of the laws; and we shall see that this complete relativity of motion, leading to Mach’s mechanics, is far from being accepted by all scientists. The discussion of this subject will have to be left aside for the present, as it is intimately connected with the difficult problem of the shape of the universe, concerning which we shall have something to say in a later chapter.