Geometric Boundary Data for the Gravitational Field

TL;DR

This paper introduces a local geometric interpretation of boundary data in general relativity, showing how boundary geometry determines unique solutions to Einstein's equations with boundary conditions.

Contribution

It presents a novel geometric boundary data framework for the initial-boundary value problem in general relativity, linking boundary geometry to solution uniqueness.

Findings

Boundary data determine solutions up to diffeomorphism.

Three boundary data pieces encode gravitational degrees of freedom.

Boundary geometry influences dynamical evolution.

Abstract

An outstanding issue in the treatment of boundaries in general relativity is the lack of a local geometric interpretation of the necessary boundary data. For the Cauchy problem, the initial data is supplied by the 3-metric and extrinsic curvature of the initial Cauchy hypersurface.. This Cauchy data determines a solution to Einstein's equations which is unique up to a diffeomorphism. Here, we show how three pieces of boundary data, which are associated locally with the geometry of the boundary, likewise determine a solution of the initial-boundary value problem which is unique up to a diffeomorphism. One piece of this data, constructed from the extrinsic curvature of the boundary, determines the dynamical evolution of the boundary. The other two pieces constitute a conformal class of rank-2, positive definite metrics, which represent the two gravitational degrees of freedom.