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On the Local Existence for the Characteristic Initial Value Problem in General Relativity

Jonathan Luk

Abstract

Given a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere $S$ with smooth characteristic initial data, a theorem of Rendall shows that the vacuum Einstein equations can be solved in a small neighborhood of $S$ in the future of $S$. We show that in fact the vacuum Einstein equations can be solved in a neighborhood in the future of the cones, as long as the constraint equations are initially satisfied on the null cones. The proof is based on energy type estimates and relies heavily on the null structure of the Einstein equations in the double null foliation.

On the Local Existence for the Characteristic Initial Value Problem in General Relativity

Abstract

Given a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere with smooth characteristic initial data, a theorem of Rendall shows that the vacuum Einstein equations can be solved in a small neighborhood of in the future of . We show that in fact the vacuum Einstein equations can be solved in a neighborhood in the future of the cones, as long as the constraint equations are initially satisfied on the null cones. The proof is based on energy type estimates and relies heavily on the null structure of the Einstein equations in the double null foliation.

Paper Structure

This paper contains 21 sections, 37 theorems, 356 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic Setup
  3. Canonical Coordinate System
  4. The Equations
  5. Initial Data
  6. Integration and Norms
  7. Statement of the Theorem
  8. Statement of the Theorem
  9. Rendall's Theorem
  10. Basic Estimates
  11. Estimates for Metric Components
  12. Estimates for Transport Equations
  13. Sobolev Embedding
  14. Commutation Formulae
  15. The General Elliptic Estimates for the Hodge System
  16. ...and 6 more sections

Key Result

Theorem 1

Given regular characteristic initial data that satisfy the constraint equations, there exists a regular solution to the Einstein equations (unique in the double null foliation) in a neighborhood to the future of the null cones. Moreover, the size of the neighborhood can be made to depend only on the

Figures (4)

  • Figure 1: Basic Setup
  • Figure 2: Region of Existence in Rendall's Theorem
  • Figure 3: Improved Region of Existence
  • Figure 4: Basic Setup

Theorems & Definitions (72)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Theorem 2
  • Remark 1
  • Theorem 3: Rendall Rendall
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 62 more