The initial boundary value problem for the Einstein equations with totally geodesic timelike boundary

TL;DR

This paper proves the well-posedness of the Einstein equations with a totally geodesic timelike boundary, establishing geometric uniqueness and developing a hyperbolic formulation using the ADM system and specialized Sobolev spaces.

Contribution

It provides the first well-posedness result for Einstein equations with a totally geodesic boundary and introduces a hyperbolic formulation ensuring geometric uniqueness.

Findings

First well-posedness result for this boundary condition

Development of a hyperbolic system for Einstein equations

Use of anisotropic Sobolev spaces to handle boundary derivatives

Abstract

We prove the well-posedness of the initial boundary value problem for the Einstein equations with sole boundary condition the requirement that the timelike boundary is totally geodesic. This provides the first well-posedness result for this specific geometric boundary condition and the first setting for which geometric uniqueness in the original sense of Friedrich holds for the initial boundary value problem. Our proof relies on the ADM system for the Einstein vacuum equations, formulated with respect to a parallelly propagated orthonormal frame along timelike geodesics. As an independent result, we first establish the well-posedness in this gauge of the Cauchy problem for the Einstein equations, including the propagation of constraints. More precisely, we show that by appropriately modifying the evolution equations, using the constraint equations, we can derive a first order symmetric hyperbolic system for the connection coefficients of the orthonormal frame. The propagation of the constraints then relies on the derivation of a hyperbolic system involving the connection, suitably modified Riemann and Ricci curvature tensors and the torsion of the connection. In particular, the connection is shown to agree with the Levi-Civita connection at the same time as the validity of the constraints. In the case of the initial boundary value problem with totally geodesic boundary, we then verify that the vanishing of the second fundamental form of the boundary leads to homogeneous boundary conditions for our modified ADM system, as well as for the hyperbolic system used in the propagation of the constraints. An additional analytical difficulty arises from a loss of control on the normal derivatives to the boundary of the solution. To resolve this issue, we work with an anisotropic scale of Sobolev spaces and exploit the specific structure of the equations.