,
and
will represent the three components of vis viva, or energy of motion, coupled with the stresses in the matter at the point considered, etc.
The equations of gravitation thus signify that whenever we recognise the existence of one of these physical magnitudes it is always accompanied by corresponding curvatures of space-time. It is usual to assume that the curvatures are produced by those concrete somethings which we call mass, momentum, energy, pressure. In this way, we must concede a duality to nature; there would exist both matter and space-time, or, better still, matter and the metrical field of space-time. Einstein, when he elaborated his hypothesis of the cylindrical universe, attempted to remove this duality by proving that it was possible to attribute the entire existence of the metrical field, hence of space-time, to the presence of matter. This attitude led to a matter-moulding conception of the universe, elevating matter over the metrical field of space-time. And, as we recall, only when this attitude was adhered to could Mach’s belief in the relativity of all motion be accepted.
Eddington’s attitude is just the reverse. He prefers to assume that the equations of gravitation are not equations in the ordinary sense of something being equal to something else. In his opinion they are identities. They merely tell us how our senses will recognise the existence of certain curvatures of space-time by interpreting them as matter, motion, and so on. In other words, there is no matter; there is nothing but a variable curvature of space-time. Matter, momentum, vis viva, are the names we give to these curvatures on account of the varying ways they affect our senses.
Of course, Eddington’s attitude, which is reminiscent of that of Clifford, is contrary to Riemann’s ideas, since, according to Riemann, an extension could have no geometry, hence no curvature and no metrical field, in the absence of matter, which would act as an active moulding agent. Furthermore, Eddington’s views lead us into certain difficulties when we attempt to apply the principle of action.
The reason for this is plain. If we regard a certain curvature of the metrical field of space-time as being synonymous with matter, there will be no reason to consider this curvature twice: once as representative of the action of matter, and once again as representative of the action of the gravitational field. By so doing, we should be merely duplicating our results. Hence, according to Eddington’s views, there would be but one action, that of the metrical field; and we could no longer add together the two separate actions in order to obtain a total action. Calculation then shows that in regions where matter exists the principle of stationary action would break down, so that a rigid application of the principle would connote that the world must be empty of matter. Eddington feels no objection to this limited validity of the principle of action, and defends his attitude with technical arguments which we cannot reproduce here.