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

Abstract

In this first 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 signature (up to an overall 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.

Significant reworking of this series of papers on the IBVP in GR. This work is now first in the series. Results the same as in v1, but some proofs corrected via the stronger use of energy estimates. Additional improvements in exposition. 44 pages