Distributional Sources in General Relativity: two point-like examples revisited

TL;DR

This paper introduces a new regularization method for relating curvature singularities to Einstein tensor distributions in general relativity, applied to point sources in 2+1 gravity and Schwarzschild spacetime, improving previous approaches.

Contribution

A novel regularization procedure that ensures the Einstein tensor's associated density is a distribution, enabling continuous metrics with well-defined distributional curvature tensors for point source geometries.

Findings

Regularized metrics are continuous with well-defined distributional curvature tensors.

The method successfully relates curvature singularities to Einstein tensor distributions.

Results differ from previous approaches, providing a more consistent regularization.

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 $|det {\bf g}|^{{1/2}}G^a_{b}$, associated to the Einstein tensor $G^a_{b}$ of the regularized metric, rather than the Einstein tensor itself, be a distribution and (ii) the regularized metric be a continuous metric with a discontinuous extrinsic curvature across a non-null hypersurface of codimension one. 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 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.