Mathematical Aspects of Einstein's Theory of General relativity

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

Recently, we have developed a new point of view towards general relativity, where the analogy with electro-magnetic fields completely breaks down. The primary variables are no longer conventional fields but are instead non-local structures; in particular, they are three dimensional surfaces in space-time with the special property that light can travel along these surfaces, the surfaces are referred to as null or characteristic surfaces. It has been possible (though it was not at all obvious how to do so) to give a formulation of GR where the basic equations are for the determination of these surfaces. In turn, when these surfaces are determined the geometry of the space-time turns out to be identical with the conventional view derived from Einstein's general relativity. In other words, general relativity has been reformulated as a theory of three dimensional surfaces. We plan to continue working on the elucidation of these ideas and in particular to develop an approximation scheme to examine the differential equations defining these surfaces. An additional project ? closely related but distinctly different from the theory of surfaces - is the application of GR to the astrophysically important subject of gravitational lensing. Almost always GR has been applied to lensing via a rather radical simplifying approximation - the so-called thin lens approximation using linear theory on a simple background. We have developed an exact formulation of lensing with an associated perturbation theory that is an improvement over the thin lens version. We strongly believe that there will be observationally relevant corrections to the thin lens predictions both in intergalactic lensing and in galactic microlensing. Preliminary papers on this have already been accepted for publication. In the context of the exact approach to lensing we also plan to study image distortion by lenses in a nonperturbative fashion. Both our two projects, the reformulation of GR as a theory of 3-surfaces in four space and the theory of gravitational lensing rely heavily on the study of properties of null geodesics. This in turn has led us, in a peripheral project, to study and work on Penrose's Theory of Twistors and their application to GR.