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The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions

Yvonne Choquet-Bruhat, Piotr T. Chruściel, José M. Martín-García

TL;DR

This work develops a complete framework for the characteristic Cauchy problem for the Einstein equations in arbitrary dimensions by employing a Minkowski-target wave-map gauge. It derives an explicit, hierarchical set of wave-map gauge constraints on a null cone and proves that satisfying these constraints is both necessary and sufficient for a reduced-wave Einstein solution to fulfill the full Einstein equations in vacuum. The authors establish local geometric uniqueness: any vacuum spacetime determined by null-cone data is locally isometric to a solution in wave-map gauge, ensuring the well-posedness of the characteristic initial-value problem near the cone vertex. The results extend foundational null-surface analyses to higher dimensions and set the stage for further global, asymptotic, and coupled-field extensions.

Abstract

We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary dimensions. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.

The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions

TL;DR

This work develops a complete framework for the characteristic Cauchy problem for the Einstein equations in arbitrary dimensions by employing a Minkowski-target wave-map gauge. It derives an explicit, hierarchical set of wave-map gauge constraints on a null cone and proves that satisfying these constraints is both necessary and sufficient for a reduced-wave Einstein solution to fulfill the full Einstein equations in vacuum. The authors establish local geometric uniqueness: any vacuum spacetime determined by null-cone data is locally isometric to a solution in wave-map gauge, ensuring the well-posedness of the characteristic initial-value problem near the cone vertex. The results extend foundational null-surface analyses to higher dimensions and set the stage for further global, asymptotic, and coupled-field extensions.

Abstract

We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary dimensions. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.

Paper Structure

This paper contains 57 sections, 16 theorems, 404 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions
  3. Ricci tensor and harmonicity functions
  4. Wave-map gauge
  5. Characteristic Cauchy problem
  6. The Cagnac-Dossa theorem
  7. Einstein equations in the wave-map gauge
  8. Null hypersurfaces, adapted coordinates
  9. Adapted coordinates
  10. Characteristic cones
  11. General properties
  12. Case $\hat{g}={ \if@compatibility \mathchar"0111 {} \mathchar"0111 }$, the Minkowski metric
  13. Limits at the vertex
  14. A lemma
  15. The affine-parameterization condition
  16. ...and 42 more sections

Key Result

Theorem 3.1

Consider the problem (22XI.1)--(22XI.2). Suppose that: 1. There is an open set $U\times W\subset\mathbf{R}^{n+1}\times\mathbf{R}^{N}$, $Y_{O}^{T}\subset U$ where the functions $A^{{ \if@compatibility \mathchar"0115 {} \mathchar"0115 }{ \if@compatibility \mathchar"0116 {} \mathchar"0116 }}$ are

Figures (1)

  • Figure 4.1: The cross-section $\Sigma_s$ of the light-cone $C_{O}$; $C_{O}^{s}$ is the shaded blue region. Two generators ${ \if@compatibility \mathchar"010D {} \mathchar"010D }_{\ell_{1}}$ and ${ \if@compatibility \mathchar"010D {} \mathchar"010D }_{\ell_{2}}$ are also shown.

Theorems & Definitions (24)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 1
  • Lemma 4.1
  • Theorem 5.1
  • Theorem 5.2
  • Lemma 7.1
  • Remark 2
  • Theorem 7.2
  • ...and 14 more