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 C1,1-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for C1,1-metrics, and of C0-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.