Distributional Sources in General Relativity: Two Point-Like Examples Revisited

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| 1 2 G a b , associated to the Einstein tensor G a b 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.

Curvature singularity of the distributional Bañados, Teitelboim, and Zanelli black hole geometry

“…Now, it would be interesting to see if the distributional curvatures obtained above can also be obtained employing regularization methods. This is far from being obvious, since for conical singularities the results of the regularization approaches taken so far 6,[12][13][14]8 are in disagreement with the ones obtained following the procedure of Ref. 16 which is the one that we have employed.…”

“…5 On the other hand, besides the fact that the distributional analysis has not been considered hitherto, regularization procedures required to multiply distributions need not be invoked in the calculation of the distributional nonrotating BTZ black hole curvature tensor. Most of the distributional treatments of the (3ϩ1)-dimensional Schwarzschild [6][7][8] and Kerr black holes 7 rely on suitable regularization procedures; while general regularization procedures using Colombeau's generalized functions 9 to obtain the distributional curvature associated to the Schwarzschild 10 and Kerr geometries 11 and to a conical singularity [12][13][14] have been put forward ͑see also Refs. 6 and 15 for other approaches to conical singularities͒.…”

“…16 Recently, the metric associated to the (2ϩ1)-dimensional space-time around a point source 4 was demonstrated to be a third example. 8 In order to consider the coordinate dependence of the results, the distributional description of the BTZ black hole space-time is carried out in Schwarzschild coordinates and in Kerr-Schild coordinates. We find the distributional Ricci and Einstein tensor fields for the nonrotating BTZ black hole, which turn out to be equivalent in both coordinate systems.…”

“…22,6 -8,10 ͑The Schwarzschild metric is not a semiregular metric and cannot be handled with the methods used here to obtain its distributional curvature. 8 ͒ Now, let us consider the dependence of these results on the coordinate system. Note that whether or not a metric is semiregular depends in general on the differentiable structure imposed on the manifold.…”

The use of generalized functions and distributions in general relativity

“…Different authors ( [68,40,67]) using various regularization procedures have also assigned a distributional energy-momentum tensor to the Schwarzschild geometry. These approaches have been compared using the language of the special algebra in [34].…”