Geometric Boundary Data for the Gravitational Field

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.

Clarification given and typos corrected. Published version

References in corpus (5)

• Boundary conditions for coupled quasilinear wave equations with application to isolated systems
• Initial boundary value problems for Einstein's field equations and geometric uniqueness
• Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates
• Boundary Conditions for the Gravitational Field
• Geometrization of metric boundary data for Einstein's equations

Cited by in corpus (3)

• Key Elements of Robustness in Binary Black Hole Evolutions using Spectral Methods
• The initial boundary value problem and quasi-local Hamiltonians in General Relativity
• Spacetime positive mass theorems for initial data sets with noncompact boundary