Distributional sources in general relativity:

Abstract

Distributional sources in general relativity: Two point-like examples revisited

(Pantoja, Nelson; Rago, Hector)

Abstract

A regularization procedure, that allows one to relate singularities of curvature to those of the Einstein tensor without some of the shortcomings of previous approaches, is proposed. This regularization is obtained by requiring that (i) the density |detg|½Gab, associated to the Einstein tensor Gab of the regularized metric, be a distribution with support on a submanifold of codimension of at most one and (ii) the regularized metric be a continuous metric which coincides with the singular one everywhere except on this support. In this paper, the curvature and Einstein tensors of the geometries associated to point sources in the 2+1-dimensional gravity and the Schwarzschild spacetime are considered. In both examples the regularized metrics are shown to be continuous regular metrics, as defined by Geroch and Traschen, with well defined distributional curvature tensors at all the intermediate steps of the calculation. The limit in which the support of these curvature tensors tends to the singular region of the original spacetime is studied and the results are contrasted with the ones obtained in previous works.