Boundary conditions for hyperbolic formulations of the Einstein equations

TL;DR

This paper discusses boundary conditions for hyperbolic formulations of Einstein's equations, emphasizing the importance of boundary projections and demonstrating their application in the Einstein-Christoffel formulation under spherical symmetry.

Contribution

It establishes that boundary conditions derived from the projection of Einstein equations are necessary for hyperbolic formulations, with explicit validation in spherical symmetry.

Findings

Projection-based boundary conditions are necessary for Einstein equations.

Application to Einstein-Christoffel formulation confirms validity.

Boundary conditions improve well-posedness of the initial-boundary value problem.

Abstract

In regards to the initial-boundary value problem of the Einstein equations, we argue that the projection of the Einstein equations along the normal to the boundary yields necessary and appropriate boundary conditions for a wide class of equivalent formulations. We explicitly show that this is so for the Einstein-Christoffel formulation of the Einstein equations in the case of spherical symmetry.