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

Zhongshan An, Michael T. Anderson

TL;DR

This work addresses local-in-time well-posedness of the vacuum Einstein initial boundary value problem with Dirichlet boundary data on a finite timelike boundary, under a convexity-type Assumption $(*)$ that the Brown-York tensor $\Pi = H g_{\mathcal{C}} - A$ induces a Lorentz metric with the same signature as the boundary metric. The authors implement a Nash–Moser implicit function approach in Fréchet spaces, working with a gauged formulation that yields a tame Fredholm linearization and overcoming loss-of-derivatives that impede standard hyperbolic methods. They identify ${\mathcal E}^*$, the open Fréchet manifold of smooth vacuum solutions satisfying $(*)$, and prove that the data-to-solution map $\Phi^H$ is a smooth tame open embedding (hence locally bijective) onto the target of compatible initial and Dirichlet boundary data, establishing local existence and uniqueness for data in appropriate function spaces. The results extend to arbitrary dimension and nonzero $\Lambda$, and clarify how Dirichlet boundary data can be well-posed in a geometric setting under the convexity condition, contrasting with Part I where conformal/mean curvature data do not yield well-posedness without additional corner data. The analysis relies on a detailed setup of Sobolev-type spaces, corner compatibility, and a wave-gauge formulation to enable Nash–Moser techniques for the nonlinear Einstein equations.

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.

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

TL;DR

This work addresses local-in-time well-posedness of the vacuum Einstein initial boundary value problem with Dirichlet boundary data on a finite timelike boundary, under a convexity-type Assumption that the Brown-York tensor induces a Lorentz metric with the same signature as the boundary metric. The authors implement a Nash–Moser implicit function approach in Fréchet spaces, working with a gauged formulation that yields a tame Fredholm linearization and overcoming loss-of-derivatives that impede standard hyperbolic methods. They identify , the open Fréchet manifold of smooth vacuum solutions satisfying , and prove that the data-to-solution map is a smooth tame open embedding (hence locally bijective) onto the target of compatible initial and Dirichlet boundary data, establishing local existence and uniqueness for data in appropriate function spaces. The results extend to arbitrary dimension and nonzero , and clarify how Dirichlet boundary data can be well-posed in a geometric setting under the convexity condition, contrasting with Part I where conformal/mean curvature data do not yield well-posedness without additional corner data. The analysis relies on a detailed setup of Sobolev-type spaces, corner compatibility, and a wave-gauge formulation to enable Nash–Moser techniques for the nonlinear Einstein equations.

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.
Paper Structure (4 sections, 2 theorems, 47 equations)

This paper contains 4 sections, 2 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Initial Material
  3. The basic mappings, spaces and norms.
  4. Gauge fixing

Key Result

Theorem 1.1

The space ${\mathcal{E}}^*$ is a smooth Frechet manifold and the map is a smooth open embedding into the Frechet manifold $\mathcal{I}_0 \times_c Met({\mathcal{C}})$. Thus ${\rm Im} \Phi$ is an open set in $\mathcal{I}_0 \times_c Met({\mathcal{C}})$ and $\Phi$ is a smooth diffeomorphism onto its image.

Theorems & Definitions (6)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4