The Hawking-Penrose Singularity Theorem for C1-Lorentzian Metrics

We extend both the Hawking-Penrose theorem and its generalisation due to Galloway and Senovilla to Lorentzian metrics of regularity C-1. For metrics of such low regularity, two main obstacles have to be addressed. On the one hand, the Ricci tensor now is distributional, and on the other hand, unique solvability of the geodesic equation is lost. To deal with the first issue in a consistent way, we develop a theory of tensor distributions of finite order, which also provides a framework for the recent proofs of the theorems of Hawking and of Penrose for C-1-metrics (Graf in Commun Math Phys 378(2):1417-1450, 2020). For the second issue, we study geodesic branching and add a further alternative to causal geodesic incompleteness to the theorem, namely a condition of maximal causal non-branching. The genericity condition is re-cast in a distributional form that applies to the current reduced regularity while still being fully compatible with the smooth and C-1,C-1-settings. In addition, we develop refinements of the comparison techniques used in the proof of the C-1,C-1-version of the theorem (Graf in Commun Math Phys 360:1009-1042, 2018). The necessary results from low regularity causality theory are collected in an appendix.

Automated summary

This study extends the Hawking-Penrose theorem and its generalisation to Lorentzian metrics of regularity C-1. The authors address the issues of distributional Ricci tensor and lost unique solvability of geodesic equation. They develop a theory of tensor distributions of finite order to deal with the first issue and study geodesic branching and causality non-branching for the second issue. The study also provides refinements of comparison techniques used in the proof.