Time-like hypersurfaces of prescribed mean extrinsic curvature

TL;DR

This paper derives hyperbolic equations related to time-like hypersurfaces with prescribed mean extrinsic curvature, simplifying boundary conditions in Einstein's vacuum equations and enabling better control of spacetime evolution.

Contribution

It introduces a gauge-based approach to analyze hyperbolic equations for time-like hypersurfaces with prescribed mean extrinsic curvature, independent of Einstein's equations.

Findings

Hyperbolic equations are independent of Einstein's equations.

Boundary conditions with constant mean extrinsic curvature simplify analysis.

Totally geodesic boundaries further simplify the problem.

Abstract

The results on the initial boundary value problem for Einstein's vacuum field equation obtained in \cite{friedrich:nagy} rely on an unusual gauge. One of the defining gauge source functions represents the mean extrinsic curvature of the time-like leaves of a foliation that includes the boundary and covers a neighbourhoood of it. The others steer the development of a frame field and coordinates on the leaves. In general their combined action is needed to control in the context of the reduced field equations the evolution of the leaves. In this article are derived the hyperbolic equations implicit in that gauge. It is shown that the latter are independent of the Einstein equations and well defined on arbitrary space-times. The analysis simplifies if boundary conditions with constant mean extrinsic curvature are stipulated. It simplifies further if the boundary is required to be totally geodesic.