A model problem for the initial-boundary value formulation of Einstein's field equations

TL;DR

This paper investigates a simplified model problem based on Maxwell's equations to develop and analyze boundary conditions that are compatible with constraints and well-posedness, aiming to inform formulations of Einstein's equations.

Contribution

It introduces a family of Sommerfeld-type boundary conditions for a Maxwell model that preserve constraints and are well posed, providing insights for Einstein's equations.

Findings

Boundary conditions are shown to be constraint-preserving.

The initial-boundary value problem is proven to be well posed.

Results suggest potential generalization to Einstein's equations.

Abstract

In many numerical implementations of the Cauchy formulation of Einstein's field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In particular, they should be compatible with the constraints, yield a well posed initial-boundary value formulation and incorporate some physically desirable properties like, for instance, minimizing reflections of gravitational radiation. Motivated by the problem in General Relativity, we analyze a model problem, consisting of a formulation of Maxwell's equations on a spatially compact region of spacetime with timelike boundaries. The form in which the equations are written is such that their structure is very similar to the Einstein-Christoffel symmetric hyperbolic formulations of Einstein's field equations. For this model problem, we specify a family of Sommerfeld-type constraint-preserving boundary conditions and show that the resulting initial-boundary value formulations are well posed. We expect that these results can be generalized to the Einstein-Christoffel formulations of General Relativity, at least in the case of linearizations about a stationary background.