Remarks on the distributional Schwarzschild geometry

TL;DR

This paper rigorously analyzes the Schwarzschild geometry with distributional methods, extending the solution to include singularities and comparing different approaches in the literature.

Contribution

It introduces a mathematically rigorous framework using Colombeau's generalized functions to handle nonlinearities in the distributional Schwarzschild geometry.

Findings

Distributional Schwarzschild solution extended to include singularity

Energy-momentum tensor represented as delta distribution at r=0

Unified discussion of various approaches in the literature

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 delta-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.