Applications to General Relativity

Abstract

The aim of this chapter is a detailed study of applicability and applications of distributional geometry as developed in the previous chapters—with its special focus, of course, on the generalized pseudo-Riemannian geometry of Section 3.2.5—to the theory of general relativity. We shall begin with a general introduction into our theme in Section 5.1, where we also introduce our notational conventions concerning general relativity. Section 5.2 is divided into two parts. The first one is concerned with a brief overview of applications of linear distributional geometry to relativity with a strong focus on a “no-go”-theorem by Geroch and Traschen (cf. [Ger87]). In the second part of Section 5.2 we use the nonlinear distributional geometry of Chapter 3 to define the generalized curvature tensor as well its contractions appearing in the field equations of general relativity. Moreover, we present a guideline for applying this setting to the study of singular spacetime metrics and discuss consistency with respect to the classical theory. Finally in Section 5.3 we give a complete distributional description of impulsive gravitational pp-waves based upon a variety of concepts introduced so far.