Einstein’s Field Equations

Abstract

In the previous chapter we examined the kinematical framework of the general theory of relativity and the effect of gravitational fields on physical systems. The core of the theory, however, consists of Einstein’s field equations, which relate the metric field to matter. After a discussion of the physical meaning of the curvature tensor, we shall first give a simple physical motivation of the field equations and will show then that they are determined by only a few natural requirements¹.

“There is something else that I have learned from the theory of gravitation: No collection of empirical facts, however comprehensive, can ever lead to the setting up of such complicated equations. A theory can be tested by experience, but there is no way from experience to the formulation of a theory. Equations of such complexity as are the equations of the gravitational field can be found only through the discovery of a logically simple mathematical condition which determines the equations completely or [at least] almost completely. Once one has those sufficiently strong formal conditions, one requires only little knowledge of facts for the setting up of a theory; in the case of the equations of gravitation it is the four-dimensionality and the symmetric tensor as expression for the structure of space which, together with the invariance concerning the continuous transformation-group, determine the equations almost completely.” A. Einstein