Well-posed geometric boundary data in General Relativity, II: Dirichlet boundary data

TL;DR

This paper proves the local well-posedness of the initial-boundary value problem for vacuum Einstein equations with Dirichlet boundary conditions, under a convexity assumption on the Brown-York stress tensor.

Contribution

It establishes the well-posedness of Einstein's equations with boundary data, extending previous results to the Lorentzian setting with specific boundary stress tensor conditions.

Findings

Well-posedness proven under convexity assumption.

Boundary stress tensor must have same sign as boundary metric.

Results extend Riemannian boundary data analogs.

Abstract

In this second work in a series, we prove the local-in-time well-posedness of the IBVP for the vacuum Einstein equations with Dirichlet boundary data on a finite timelike boundary, provided the Brown-York stress tensor of the boundary is a Lorentz metric of the same sign as the induced Lorentz metric on the boundary. This is a convexity-type assumption which is an exact analog of a similar result in the Riemannian setting. This assumption on the (extrinsic) Brown-York tensor cannot be dropped in general.