Remarks on the distributional Schwarzschild geometry

Abstract

This work is devoted to a mathematical analysis of the distributional Schwarzschild geometry. The Schwarzschild solution is extended to include the singularity; the energy momentum tensor becomes a δ-distribution supported at r = 0. Using generalized distributional geometry in the sense of Colombeau's (special) construction the nonlinearities are treated in a mathematically rigorous way. Moreover, generalized function techniques are used as a tool to give a unified discussion of various approaches taken in the literature so far; in particular we comment on geometrical issues.

Generalized pseudo-Riemannian geometry

“…Finally, we would like to emphasize that the framework developed in the previous sections for the first time allows a comprehensive and consistent interpretation of the calculations given, e.g., in [6], [14], [22], [23].…”

“…This program has been carried out for a conical metric (representing a cosmic string) by Clarke, Vickers and Wilson (see [6], [42], [43] for a treatment in the full setting of Colombeau's construction) rigorously assigning to it a distributional curvature and (via the field equations) the heuristically expected energy-momentum tensor. Moreover, the nonlinear generalized function setting was used in [2], [3] to calculate the energy momentum tensor of the extended Kerr geometry as well as in [14] to unify several distributional approaches to the Schwarzschild geometry. Finally, a complete distributional description of impulsive pp-wave spacetimes was achieved in [39], [22], [23].…”

On Time-Dependent Black Holes and Cosmological Models From a Kaluza–klein Mechanism

“…A rigorous correct treatment of point mass distributions has been provided based on Colombeau's [22], [23], [24], [25], [26], [27], [28] theory of nonlinear distributions, generalized functions and nonlinear calculus. This permits the proper multiplication of distributions since the old Schwarz theory of linear distributions is invalid in nonlinear theories like General Relativity.…”

“…In the same fashion that the Kantowski-Sachs metric can be obtained from the Schwarzschild metric after the exchange of variables t ↔ r, it is warranted to explore the cosmological implications of seeing the universe as a dynamical black hole by starting with the metric obtained from the Vaidya metric after the exchange of variables t ↔ r ds 2 = − (1 − 2m(ṽ) t ) dṽ 2 + 2 dt dṽ + t 2 dΩ 2 (7.23) and 24) where the t-relatives of the advanced/retarded temporal coordinates and the temporal tortoise coordinate are now given by : v = r + t * ,ũ = r − t * , t * = t + 2GM ln | t 2GM − 1|. (7.25) For recent work on the role of Entropy and the Universe as a black hole see [43].…”

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

“…22,7,11,10 In the following we shall prove that the nonrotating BTZ solution in Kerr-Schild coordinates is a semiregular metric.…”