Generally Covariant Integral Formulation of Einstein's Field Equations

Abstract

As a basis for a generally covariant theory of Mach's principle, we express Einstein's field equations in integral form. The nonlinearity of these equations is reflected in the kernel of the integral representation, which is a functional of the metric tensor. The functional dependence is so constructed that, subject to supplementary conditions, the kernel may be regarded as remaining unchanged to the first order when a small change in the source produces a corresponding change in the potential. To obtain this kernel, a linear differential operator is derived by varying a particular form of Einstein's field equations. The elementary solution corresponding to this linear operator provides the kernel of an approximate integral representation which becomes exact in the limit of vanishing variations. This representation is in a certain sense unique. Our discussion is confined to a normal neighborhood of the field point.