Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics

Melanie Graf, James D. E. Grant, Michael Kunzinger, Roland Steinbauer

Abstract

We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for $C^{1,1}$-metrics, and of $C^0$-trapped submanifolds. By regularisation, we show that, under these weak conditions, causal geodesics necessarily become non-maximising. This requires a detailed analysis of the matrix Riccati equation for the approximating metrics, which may be of independent interest.

The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics

Abstract

We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of -regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for -metrics, and of -trapped submanifolds. By regularisation, we show that, under these weak conditions, causal geodesics necessarily become non-maximising. This requires a detailed analysis of the matrix Riccati equation for the approximating metrics, which may be of independent interest.

Paper Structure

This paper contains 14 sections, 44 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Regularisation results
  4. Conjugate points for smooth metrics
  5. Maximising geodesics
  6. Initial conditions
  7. The hypersurface case
  8. Submanifolds of codimension $1<m<n$
  9. Trapped points
  10. Proof of the main result
  11. Causality results in $C^{1,1}$
  12. Limit curves and the structure of $\partial J^+(S)$
  13. Cauchy development and Cauchy horizon
  14. Strong causality

Key Result

Theorem 1.1

HP Let $(M, g)$ be a spacetime with $g$ a $C^2$-metric with the following properties: Then $(M, g)$ is causally geodesically incomplete.

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Hawking--Penrose for $C^{1,1}$-metrics
  • Theorem 2.6
  • Proposition 3.1
  • ...and 77 more