On the Boundary Conditions in General Relativity

Abstract

The attempt is made to derive a suitable set of boundary conditions in general relativity from the standpoint that Einstein's field equations to be constructed from the combined metric tensor, which is defined as a linear combination of two metric tensors specifying two adjacent regions by making use of the step-function, must be free from any kind of delta-singularities. The analysis is performed in a system of coordinates relative to which the boundary surface of the two regions is at rest. Then it is shown that the metric tensor and its first derivative must satisfy the boundary conditions which are identical with the counterparts in O'Brien-Synge's approach. However, in order that Einstein's equations thus obtainecf may be consistent with the law of conservation, the energy-momentum tensor must satisfy a new boundary condition as well as the remaining ones in O'Brien-Synge's approach.