Distributional geometry in general relativity

Abstract

In this article we look at the extent to which one can use classical linear distributional geometry in general relativity. We then go on to look at a non-linear theory of distributional geometry based on Colombeau algebras and show that this is compatible with the linear theory in situations where both may be used. For both the linear and non-linear theories of distributional geometry we will use a geometric coordinate free description. We conclude by looking at the example of the thin string limit of solutions of the field equations for an infinite length gravitating straight cosmic string described by a complex scalar field coupled to a U(1) gauge field. We show that within the Colombeau algebra this has a well-defined energy–momentum tensor and curvature which are associated to classical distributions.