# The Hawking–Penrose Singularity Theorem for C1,1-Lorentzian Metrics

@article{Graf2017TheHS, title={The Hawking–Penrose Singularity Theorem for C1,1-Lorentzian Metrics}, author={Melanie Graf and James D E Grant and Michael Kunzinger and Roland Steinbauer}, journal={Communications in Mathematical Physics}, year={2017}, volume={360}, pages={1009-1042} }

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…

## 22 Citations

Singularity theorems in Schwarzschild spacetimes

- Physics
- 2020

We present a review of the two prominent singularity theorems due to Penrose and Hawking, as well as their physical interpretation. Their usage is discussed in detail for the Schwarzschild spacetime…

Inextendibility of spacetimes and Lorentzian length spaces

- Mathematics, PhysicsAnnals of global analysis and geometry
- 2019

This work introduces appropriate notions of geodesics and timelike geodesic completeness and proves a general inextendibility result, and relates low-regularity inextendedibility to (synthetic) curvature blow-up.

Lorentzian length spaces

- Mathematics, MedicineAnnals of global analysis and geometry
- 2018

A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT(k)-spaces, based on triangle comparison, for Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.

A Lorentzian analog for Hausdorff dimension and measure

- Physics, Mathematics
- 2021

We define a one-parameter family of canonical volume measures on Lorentzian (pre-)length spaces. In the Lorentzian setting, this allows us to define a geometric dimension — akin to the Hausdorff…

Causality theory for closed cone structures with applications

- Mathematics, PhysicsReviews in Mathematical Physics
- 2019

We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular…

On geodesics in low regularity

- Physics, Mathematics
- 2018

We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new…

The future is not always open

- Mathematics, PhysicsLetters in mathematical physics
- 2020

The phenomena described here are relevant for recent synthetic approaches to low-regularity Lorentzian geometry where, in the absence of a differentiable structure, causality has to be based on locally Lipschitz curves.

Lorentz meets Lipschitz

- Mathematics, Physics
- 2020

We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a $\mathcal{C}^{1,1}$-parametrization and that they solve the geodesic equation in the sense of Filippov in this…

Lorentzian causality theory

- PhysicsLiving Reviews in Relativity
- 2019

I review Lorentzian causality theory paying particular attention to the optimality and generality of the presented results. I include complete proofs of some foundational results that are otherwise…

The Discovery and Significance of the RT-equations: Optimal Regularity and Uhlenbeck Compactness for General Relativity and Non-Riemannian Geometry

- 2021

The RT-equations are a novel system of elliptic partial differential equations foundational for geometric analysis in General Relativity and Mathematical Physics: Solutions of the RT-equations…

## References

SHOWING 1-10 OF 91 REFERENCES

Timelike Completeness as an Obstruction to C0-Extensions

- Physics, Mathematics
- 2017

The study of low regularity (in-)extendibility of Lorentzian manifolds is motivated by the question whether a given solution to the Einstein equations can be extended (or is maximal) as a weak…

The $C^0$-inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian Geometry

- Physics, Mathematics
- 2015

The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that…

FAST TRACK COMMUNICATION: Singularity theorems based on trapped submanifolds of arbitrary co-dimension

- Physics
- 2010

Standard singularity theorems are proven in Lorentzian manifolds of arbitrary dimension n if they contain closed trapped submanifolds of arbitrary co-dimension. By using the mean curvature vector to…

On Lorentzian causality with continuous metrics

- Physics, Mathematics
- 2012

We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as…

Definition and stability of Lorentzian manifolds with distributional curvature

- Physics, Mathematics
- 2007

Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain…

Lorentzian length spaces

- Mathematics, MedicineAnnals of global analysis and geometry
- 2018

A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT(k)-spaces, based on triangle comparison, for Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.

Volume comparison for hypersurfaces in Lorentzian manifolds and singularity theorems

- Mathematics, Physics
- 2013

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along…

Volume comparison for C 1 , 1-metrics

- 2016

The aim of this paper is to generalize certain volume comparison theorems (Bishop-Gromov and a recent result of Treude and Grant, Ann Global Anal Geom, 43:233– 251, 2013) for smooth Riemannian or…

Hawking's singularity theorem for $C^{1,1}$-metrics

- Physics, Mathematics
- 2014

We provide a detailed proof of Hawking's singularity theorem in the regularity class $C^{1,1}$, i.e., for spacetime metrics possessing locally Lipschitz continuous first derivatives. The proof uses…

Global Hyperbolicity for Spacetimes with Continuous Metrics

- Mathematics
- 2014

We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while…