The Hawking–Penrose Singularity Theorem for $$C^1$$-Lorentzian Metrics

Abstract

We extend both the Hawking–Penrose theorem and its generalisation due to Galloway and Senovilla to Lorentzian metrics of regularity $$C^1$$ 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 $$C^1$$ 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 $$C^{1,1}$$ C 1 , 1 -settings. In addition, we develop refinements of the comparison techniques used in the proof of the $$C^{1,1}$$ C 1 , 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.

Synthetic versus distributional lower Ricci curvature bounds

“…The key to this analysis are certain versions of Friedrichs' Lemma, which provide improved convergence properties of commutators between differentiation and convolution operators. These turned out to be essential for generalising classical singularity theorems in Lorentzian geometry to metrics of lower regularity ( [24,25,13,12,20]). The versions we shall rely on here are the following (see [12,Lem.…”

“…Since the λ i are independent of x, by definition of ⋆ M (see ( 16)) we may interchange λ i and ⋆ M here. Thus, setting V := i λ i X i , (20) means that…”

“…Especially in physics, be it in classical electrodynamics, general relativity, or quantum field theory, this analytic approach is widely used to model singular sources or fields or to describe matched spacetimes (cf., e.g., [4,37] and references therein). In recent years, distributional Ricci bounds (in the shape of strong energy conditions) have also featured prominently in the generalisation of the classical singularity theorems of Penrose and Hawking to spacetime metrics of regularity below C 2 ([25, 24,13,12,20]). On the synthetic side, a generalisation of Hawking's singularity theorem to Lorentzian synthetic spaces was established by Cavalletti and Mondino in [6].…”

The singularity theorems of General Relativity and their low regularity extensions

“…However, in the context of the Hawking-Penrose theorem more general results are needed, which will ultimately force us to introduce a new condition, namely a nonbranching assumption for maximising causal geodesics, to be detailed in Section 4.8. This will finally allow us to discuss the recent C 1 -version of the Hawking-Penrose theorem [KOSS22] in Sections 4.9 and 4.10.…”

“…We may now finally proceed to establish the existence of appropriate approximating sequences of maximising geodesics, which is the missing link in our argument for (4.12). We only state the result in the timelike case, for the corresponding null-version in which we have to avoid the use of global hyperbolicity see [KOSS22,Prop. 34(ii)].…”