The Einstein field equations

As we have already indicated more than once, the basic idea of Einstein's theory of gravitation consists of geometrizing the gravitational force, that is, mapping all properties of the gravitational force and its influence upon physical processes onto the properties of a Riemannian space. While up until the present we have concerned ourselves only with the mathematical structure of such a space and the influence of a given Riemannian space upon physical laws, we want now to turn to the essential physical question. Gravitational fields are produced by masses – so how are the properties of the Riemannian space calculated from the distribution of matter? Here, in the context of General Relativity, ‘matter’ means everything that can produce a gravitational field (i.e. that contributes to the energy-momentum tensor), for example, not only atomic nuclei and electrons, but also the electromagnetic field.

Of course one cannot derive logically the required new fundamental physical law from the laws already known; however, one can set up several very plausible requirements. We shall do this in the following and discover, surprisingly, that once one accepts the Riemannian space, the Einstein field equations follow almost directly.

The following requirements appear reasonable.

1. (a) The field equations should be tensor equations (independence of coordinate systems of the laws of nature).

2. (b) Like all other field equations of physics they should be partial differential equations of at most second order for the functions to be determined (the components of the metric tensor grrmn), which are linear in the highest derivatives.

3. […]