Initial boundary value problems for Einstein's field equations and geometric uniqueness

TL;DR

This paper discusses the unresolved issue of geometric uniqueness in initial boundary value problems for Einstein's equations, exploring approaches and conditions that could lead to a fully covariant formulation.

Contribution

It analyzes the causes of non-uniqueness in Einstein's initial boundary value problems and proposes potential conditions for achieving a covariant formulation.

Findings

Identifies key difficulties in ensuring geometric uniqueness.

Highlights the importance of boundary geometry conditions.

Suggests a class of problems with potential for covariant solutions.

Abstract

While there exist now formulations of initial boundary value problems for Einstein's field equations which are well posed and preserve constraints and gauge conditions, the question of geometric uniqueness remains unresolved. For two different approaches we discuss how this difficulty arises under general assumptions. So far it is not known whether it can be overcome without imposing conditions on the geometry of the boundary. We point out a natural and important class of initial boundary value problems which may offer possibilities to arrive at a fully covariant formulation.