which belongs to it to be distinguished by having simple mathematical properties. Physics, this time as a physics of fields, is again pursuing the object of reducing the totality of natural phenomena to a single physical law: it was believed that this goal was almost within reach once before when Newton’s Principia, founded on the physics of mechanical point-masses, was celebrating its triumphs. But the treasures of knowledge are not like ripe fruits that may be plucked from a tree.”[124]
Whatever may be the ultimate fate of such speculations in revealing to us the mystery of matter, it cannot be denied that they possess a grandeur which is compelling. At all events, in more restricted domains, the principle of action has led to important discoveries.
Hilbert, Klein, Weyl and later Einstein himself, by applying the principle of action to the space-time metrical field, were led to the law of space-time curvature in the interior of matter and outside matter; that is to say, to Einstein’s law of gravitation.
The problem resolves itself into the selection of an appropriate expression for the action; but in the case of the law of gravitation a peculiar difficulty is encountered. We shall best understand the nature of this difficulty by considering the simpler case of the electromagnetic action. Electricity and magnetism present themselves in a substantial form as electrons, magnets and electrons in motion, or currents; and in a more ethereal form as fields, illustrated by the electromagnetic field. Inasmuch as both these manifestations of electricity and magnetism should enter into the laws of electrodynamics, the total electromagnetic action will have to comprise the two corresponding types of action. These can easily be determined, and an application of Hamilton’s principle of stationary action yields the laws of electromagnetics without further ado. But when we consider matter, classical science knew of but one form, the substantial form; the field form was lacking. To be sure, the gravitational forces surrounding matter appeared to present a certain analogy with the electromagnetic fields surrounding electrons and magnets. But, mathematically, at least, the analogy was superficial, for in contradistinction to the electromagnetic field, the gravitational field was not expressed by differential equations indicating continuous action through a medium. Only in name was it a field; and it appeared impossible to determine its action. With Einstein’s discoveries this difficulty was overcome. The gravitational field now appeared as a field analogous to the electromagnetic field. It was the metrical field of space-time, defined, as we know, by the
potential distribution analogous to the potential distribution of the electromagnetic field. It can also be regarded as representative of the structure, or geometry, of space-time. Henceforth, the total material or gravitational action will be given by the action of matter proper plus that of the metrical field; the missing action has been found.
Thus the solution of the problem of gravitation reduces to establishing the mathematical expression of these two separate actions. Now the action of matter in a space-time region is the energy of matter contained in a volume of space-time. According to Einstein’s theory, therefore, it is given by the density of the mass of the matter multiplied by a space-time volume. As for the action of the gravitational field, inasmuch as it is now seen to be the action of the metrical field, it can be discovered without much difficulty. In this connection, we must recall that the relativity theory suggests that the action in a volume of space-time must be an invariant; its numerical value, whatever it may be, must be independent of our choice of mesh-system. In other words, the action of the metrical field must be an invariant built up from the fundamental tensors
, whose distribution defines the metrical field.