Mathematical Aspects of Einstein's Theory of General Relativity

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

This project will involve work on several related, but different topics, all of which are associated with mathematical aspects of Einstein's General Theory of Relativity. The first and probably the most significant of the topics is the study of what is referred to as families of null surfaces, families of three-dimensional surfaces embedded in space-time on which high frequency electromagnetic waves travel. It turns out that General Relativity can be reformulated as equations for these surfaces. This forms a rich research area which borders on both physical and mathematical issues and their intimate relationships. The second topic deals with a practical aspect of properties of these null surfaces. The entire theory of gravitational lensing - now a basic astrophysical tool - can be formulated in terms of null surfaces and their properties. Using these null surfaces one can go beyond the (usual) thin lens predictions and find corrections to the thin lens predictions that could be tested. A third topic is a return to the long neglected study of the physical meaning and applications of certain mathematical expressions that appear in algebraically special Einstein-Maxwell fields. The first topic will clarify some beautiful and deep relations between ideas associated with the propagation of high frequency electromagnetic waves in space-time and very abstract mathematical theorems concerning ordinary and partial differential equation theory. It also should shed light, via a very unconventional point of view, on the structure of the Einstein equations. The second topic should or could be useful for the explanation of certain anomalies in the observations of magnifications in gravitational lensing. There is evidence that the last topic will clarify the potentially deep relationship between intrinsic magnetic moments and spin angular momentum that appears to arise naturally in general relativity.