The Hawking-Penrose Singularity Theorem for C 1,1-Lorentzian Metrics

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.