A minimization problem for the lapse and the initial-boundary value problem for Einstein's field equations

TL;DR

This paper introduces a variational principle for the lapse function in Einstein's equations, proposing boundary conditions that aim to ensure well-posed initial-boundary value problems in numerical relativity.

Contribution

It develops a new gauge condition for the lapse and boundary conditions based on a variational principle, enhancing the mathematical well-posedness of Einstein's equations formulations.

Findings

The proposed gauge and boundary conditions lead to a well-posed problem in the weak field limit.

The method is applicable to various formulations of Einstein's equations.

The approach is expected to improve numerical stability in simulations.

Abstract

We discuss the initial-boundary value problem of General Relativity. Previous considerations for a toy model problem in electrodynamics motivate the introduction of a variational principle for the lapse with several attractive properties. In particular, it is argued that the resulting elliptic gauge condition for the lapse together with a suitable condition for the shift and constraint-preserving boundary conditions controlling the Weyl scalar Psi_0 are expected to yield a well posed initial-boundary value problem for metric formulations of Einstein's field equations which are commonly used in numerical relativity. To present a simple and explicit example we consider the 3+1 decomposition introduced by York of the field equations on a cubic domain with two periodic directions and prove in the weak field limit that our gauge condition for the lapse and our boundary conditions lead to a well posed problem. The method discussed here is quite general and should also yield well posed problems for different ways of writing the evolution equations, including first order symmetric hyperbolic or mixed first-order second-order formulations. Well posed initial-boundary value formulations for the linearization about arbitrary stationary configurations will be presented elsewhere.