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 $C^{1,1}$-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.

Minor amendments in v4: Removed non-equivalent condition from Def. 2.2 and adapted Lemma 3.5 and the proof of Lemma 3.6

References in corpus (6)

• Singularity Theorems and Their Consequences
• The 1965 Penrose singularity theorem
• Limit curve theorems in Lorentzian geometry
• Global hyperbolicity for spacetimes with continuous metrics
• The Penrose singularity theorem in regularity $C^{1,1}$
• Volume comparison for $C^{1,1}$ metrics

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