The use of Generalised Functions and Distributions in General Relativity

TL;DR

This paper reviews the application of classical and generalized distribution theories in describing solutions to Einstein's equations in general relativity, highlighting the need for nonlinear generalized functions for certain cases.

Contribution

It introduces a mathematical framework of nonlinear generalized functions based on Colombeau algebras for use in general relativity.

Findings

Classical distribution theory is insufficient for some Einstein solutions.

Nonlinear generalized functions can describe weak singularities.

Certain solutions with weak singularities are distributional solutions.

Abstract

In this paper we review the extent to which one can use classical distribution theory in describing solutions of Einstein's equations. We show that there are a number of physically interesting cases which cannot be treated using distribution theory but require a more general concept. We describe a mathematical theory of nonlinear generalised functions based on Colombeau algebras and show how this may be applied in general relativity. We end by discussing the concept of singularity in general relativity and show that certain solutions with weak singularities may be regarded as distributional solutions of Einstein's equations.