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

Grigorios Fournodavlos, Jacques Smulevici

TL;DR

The paper addresses the well-posedness of the initial boundary value problem for the Einstein vacuum equations with a timelike boundary that is totally geodesic (χ=0). It develops a modified ADM formulation in a parallelly propagated orthonormal frame along timelike geodesics, deriving a first-order symmetric hyperbolic system for the frame/connection and a propagation system for the constraints and curvature (with torsion) that recovers the Levi-Civita connection when satisfied. The main contributions are a local existence and uniqueness theory for the IBVP under totally geodesic boundary data, including propagation of constraints and geometric uniqueness in Friedrich’s sense for Λ=0, using anisotropic Sobolev spaces to handle boundary normal derivatives. The results provide a rigorous framework for geometric uniqueness and may support applications such as gravitational waves in a cavity, with potential extensions to more general Neumann-type boundary conditions. Overall, the work combines a careful gauge choice, constraint-based modifications, and hyperbolic-energy methods to obtain a robust, gauge-independent well-posedness theory for a physically and geometrically natural boundary condition in general relativity.

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.

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

TL;DR

The paper addresses the well-posedness of the initial boundary value problem for the Einstein vacuum equations with a timelike boundary that is totally geodesic (χ=0). It develops a modified ADM formulation in a parallelly propagated orthonormal frame along timelike geodesics, deriving a first-order symmetric hyperbolic system for the frame/connection and a propagation system for the constraints and curvature (with torsion) that recovers the Levi-Civita connection when satisfied. The main contributions are a local existence and uniqueness theory for the IBVP under totally geodesic boundary data, including propagation of constraints and geometric uniqueness in Friedrich’s sense for Λ=0, using anisotropic Sobolev spaces to handle boundary normal derivatives. The results provide a rigorous framework for geometric uniqueness and may support applications such as gravitational waves in a cavity, with potential extensions to more general Neumann-type boundary conditions. Overall, the work combines a careful gauge choice, constraint-based modifications, and hyperbolic-energy methods to obtain a robust, gauge-independent well-posedness theory for a physically and geometrically natural boundary condition in general relativity.

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.

Paper Structure

This paper contains 21 sections, 22 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The initial boundary value problem in General Relativity
  3. Geometric uniqueness
  4. The IBVP with totally geodesic boundary
  5. The hyperbolicity of the ADM system in a geodesic gauge
  6. Outline
  7. Notation
  8. Acknowledgements
  9. The ADM system in a geodesic gauge
  10. The modified ADM evolution equations and their hyperbolicity
  11. The differentiated system
  12. Local well-posedness of the reduced equations for the Cauchy problem
  13. Initial data
  14. Application to totally geodesic boundaries
  15. Choosing an orthogonal initial slice to the boundary
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $(\Sigma, h, {k})$ be a smooth initial data set for the Einstein vacuum equations such that $\Sigma$ is a $3$-manifold with boundary ${}{\partial\Sigma=\mathcal{S}}$. Assume that the corner conditions of Lemma lem:compcond.angle hold on ${}{\mathcal{S}}$, with respect to a given angle $\omega$,

Figures (2)

  • Figure 1: The classical solution in the domain of dependence $\mathcal{D}(\Sigma)$.
  • Figure 2: A local domain of dependence region near the boundary.

Theorems & Definitions (60)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 50 more