Disembodied boundary data for Einstein's equations

TL;DR

This paper establishes a well-posed initial boundary value problem for Einstein's equations using geometric boundary data, enabling stable numerical simulations of black hole spacetimes with potential applicability to various formulations.

Contribution

It recasts Sommerfeld boundary conditions for Einstein's equations into a geometric form compatible with existing codes and discusses their potential application to different metric formulations.

Findings

Proved well-posedness for harmonic Einstein equations with geometric boundary data.

Boundary conditions can be directly applied to existing harmonic codes.

Discussed extension to other Einstein formulations like BSSN.

Abstract

A strongly well-posed initial boundary value problem based upon constraint-preserving boundary conditions of the Sommerfeld type has been established for the harmonic formulation of the vacuum Einstein's equations. These Sommerfeld conditions have been previously presented in a 4-dimensional geometric form. Here we recast the associated boundary data as 3-dimensional tensor fields intrinsic to the boundary. This provides a geometric presentation of the boundary data analogous to the 3-dimensional presentation of Cauchy data in terms of 3-metric and extrinsic curvature. In particular, diffeomorphisms of the boundary data lead to vacuum spacetimes with isometric geometries. The proof of well-posedness is valid for the harmonic formulation and its generalizations. The Sommerfeld conditions can be directly applied to existing harmonic codes which have been used in simulating binary black holes, thus ensuring boundary stability of the underlying analytic system. The geometric form of the boundary conditions also allows them to be formally applied to any metric formulation of Einstein's equations, although well-posedness of the boundary problem is no longer ensured. We discuss to what extent such a formal application might be implemented in a constraint preserving manner to 3+1 formulations, such as the Baumgarte-Shapiro-Shibata-Nakamura system which has been highly successful in binary black hole simulation.