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

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, we show 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 Ψ_0 yields a priori estimates on the metric, connection and curvature fields. These estimates are expected to be useful for obtaining awell-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, an a priori specified shift vector 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.

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