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.