The evolution of scientific thought from Newton to Einstein

by A. D'Abro

’s) and which in addition satisfy all our previously mentioned requirements. In order to distinguish these magnitudes from ordinary tensors, which transcend solely our choice of mesh-system, Eddington calls them in-tensors and in-invariants, and indicates them by asterisks. Henceforth it will be to these in-tensors and in-invariants that we shall have to appeal in order to express those magnitudes which (depending neither on mesh-system nor on gauge) are intrinsic to the world itself.[127]

Now, assuming space-time to be a Weylian manifold, a change of gauge-system will modify the expression of the fundamental tensors of Einstein’s theory (except

), hence will modify the aspect of the mechanical laws and of that of gravitation. A natural extension compels us to assume that all the Einsteinian laws should be generalised and expressed in terms of in-tensors and in-invariants, so as to yield in-tensor equations instead of tensor equations. Only thus can we render them indifferent to a change of gauge-system, just as their original tensor form had already rendered them indifferent to a change of mesh-system. This modification in the form of the Einsteinian laws will affect only the mechanical and gravitational laws, since the electromagnetic ones defining the Weylian curvature are already in-tensor equations.

However, we may assume that the Einsteinian tensor laws, in the form in which they were given by Einstein, are in harmony with some particular gauge-system; and this Weyl calls the natural gauge. We shall have to conceive of this natural-gauge system as one imposed upon us by nature and defined by certain intrinsic properties of the universe. But the only determinate magnitude which is offered us by nature at every point appears to be the radius of curvature of the universe at every point. Hence Weyl defined the natural gauge at every point as given by a unit of length equal to the radius of curvature of the universe at that point or to some definite fraction thereof. If these ideas are correct, the universe cannot help but possess a curvature at every point, and hence must be finite, since were it infinite there would be no natural gauge whereby to measure distances.

Weyl’s theory, as outlined so far, represents the theory in its original form; but we shall see that Weyl was led to modify his original views in certain respects. The fact is that empirical results would appear to conflict with his theory in its initial form. The theory compels us to assume that in regions where electromagnetic fields are present, in regions, therefore, where space-time is Weylian, transference of length is an indeterminate process. Of course, when we speak of a length in space-time, we are primarily considering a space-time distance, that is, an Einsteinian interval. But an Einsteinian interval, we remember, can be split up into a spatial distance, or into a duration, or into both, as the case may be. This result is achieved when the observer refers the Einsteinian interval to his space and time mesh-system. It would follow, therefore, that in a Weylian space-time not only would the transference of length of an Einsteinian interval yield indeterminate results, but that the same would apply automatically to the transference of spatial, as also of temporal, magnitudes. Thus, two identical electrons situated, first, at any one point then separated, would in general differ in size and in charge when brought together again. Likewise, the same indeterminateness would apply to duration. Hence, it would be quite in order to suppose that two identical atoms situated at the same point and beating with the same frequency would present different frequencies of vibration after they had been separated and brought together again. It would be as though the size of an atom and its frequency of vibration depended on its previous life history. Incidentally, these occurrences would deprive the Einstein shift-effect of all value, since any slowing down in the rate of vibration of the solar atom could be attributed to a difference in the previous life histories of the solar and terrestrial atoms, and not necessarily to the influence of the sun’s field of gravitation.

Now, all these anticipations of Weyl’s theory in its original form are in conflict with precise observation. So far as can be detected by experiment, the charges and sizes of all electrons are exactly alike regardless of their past history; and this also applies to the frequencies of vibration of atoms of the same chemical element.[128] It would appear, therefore, that if space-time is a Weyl space, the Weylian characteristics are, for some reason or other, completely obliterated; so that, in fine, space-time, though Weylian, manifests only the properties of the more restricted Riemann type of space.

Weyl was thus compelled to draw a distinction between adjustment and persistence, and to assume that bodies adjusted themselves automatically to the natural gauge, that is to say, to the radius of the universe. If bodies followed the tendency of persistence, the Weylian discrepancies would be observed; but with bodies adjusting themselves afresh from place to place and from instant to instant to the form of the universe, and with the frequencies of vibration of atoms acting likewise, these discrepancies would be banished automatically. We may note the similarity between these views and those reached by Eddington in Einstein’s theory proper (see chapter on the finiteness of the universe).

These views also enable us to attain a better understanding of the significance of the natural gauge. Since objects in equilibrium, such as rigid objects, electrons and atoms, all adjust themselves to the radius of curvature of the universe, it is only natural that results of greatest simplicity should be obtained when we select this radius or some definite fraction thereof as a gauge at every point.