Boundary Conditions for the Gravitational Field

TL;DR

This paper reviews boundary treatment in general relativity, highlighting current methods, challenges in 3+1 formulations, and open questions crucial for numerical simulations like binary black hole modeling.

Contribution

It provides a comprehensive review of boundary conditions in various Einstein's equation formulations, emphasizing the lack of a universal boundary theory for 3+1 approaches.

Findings

Boundary treatments exist for harmonic and tetrad formulations.

No universal boundary theory for 3+1 formulations.

Open problems remain in formulating boundary conditions for numerical relativity.

Abstract

A review of the treatment of boundaries in general relativity is presented with the emphasis on application to the formulations of Einstein's equations used in numerical relativity. At present, it is known how to treat boundaries in the harmonic formulation of Einstein's equations and a tetrad formulation of the Einstein-Bianchi system. However, a universal approach valid for other formulations is not in hand. In particular, there is no satisfactory boundary theory for the 3+1 formulations which have been highly successful in binary black hole simulation. I discuss the underlying problems that make the initial-boundary value problem much more complicated than the Cauchy problem. I review the progress that has been made and the important open questions that remain.