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