Mathematical Relativity Seminar

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

The classical singularity theorems of General Relativity show that a Lorentzian manifold with a $C^2$-metric that satisfies physically reasonable conditions cannot be geodesically complete. One of the questions left unanswered by the classical singularity theorems is whether one could extend such a spacetime with a lower regularity Lorentzian metric such that the extension still satisfies these physically reasonable conditions and does no longer contain any incomplete causal geodesics. In other words, the question is if a lower differentiability of the metric is sufficient for the theorems to hold. The natural differentiability class to consider here is $C^{1,1}$. This regularity corresponds, via the field equations, to a finite jump in the matter variables, a situation that is not a priori regarded as singular from the viewpoint of physics and it is the minimal condition that ensures unique solvability of the geodesic equations. Recent progress in low-regularity Lorentzian geometry has allowed one to tackle this question and show that, in fact, the classical singularity theorems of Penrose, Hawking, and Hawking-Penrose remain valid for $C^{1,1}$-metrics. In this talk I will focus on the Hawking-Penrose theorem, being the most recent and in a sense most general of the aforementioned results, and some of the methods from low regularity causality and comparison geometry that were employed in its proof. This is joint work with J. D. E. Grant, M. Kunzinger and R. Steinbauer.

Current organizers: José Natário, João Lopes Costa