The Role of Elliptic Operators in the Initial-Value Problem for General Relativity

TL;DR

This paper explores how elliptic operators relate to the initial-value problem in general relativity, reformulating ADM equations as coupled systems and analyzing their integral equation representations.

Contribution

It introduces a new formulation of ADM equations using elliptic operators and integral equations, providing insights into their spectral properties and non-linearities.

Findings

Reformulation of ADM equations as coupled first-order systems

Analysis of integral equation formulations for these systems

Insights into spectral properties of associated elliptic operators

Abstract

The Arnowitt-Deser-Misner (ADM) equations are deeply intertwined with discrete spectral resolutions of an elliptic operator of Laplace type associated with the spacelike hypersurfaces which foliate the space-time manifold, and the non-linearities of the four-dimensional hyperbolic theory are mapped into the potential term occurring in this operator. The ADM equations are here re-expressed as a coupled first-order system for the induced metric and the trace-free part of the extrinsic-curvature tensor, and their formulation in terms of integral equations is studied.