Singularity theorems for $C^1$-Lorentzian metrics

Abstract

Continuing recent efforts in extending the classical singularity theorems of General Relativity to low regularity metrics, we give a complete proof of both the Hawking and the Penrose singularity theorem for $C^1$-Lorentzian metrics - a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. The proofs make use of careful estimates of the curvature of approximating smooth metrics and certain stability properties of long existence times for causal geodesics. On the way we also prove that for globally hyperbolic spacetimes with a $C^1$-metric causal geodesic completeness is $C^1$-fine stable. This improves a similar older stability result of Beem and Ehrlich where they also used the $C^1$-fine topology to measure closeness but still required smoothness of all metrics. Lastly, we include a brief appendix where we use some of the same techniques in the Riemannian case to give a proof of the classical Myers Theorem for $C^1$-metrics.

40 pages; v3: reviewer corrections, added some remarks, corrected typos; To appear in CMP

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• A Lorentzian analog for Hausdorff dimension and measure
• The Hawking-Penrose singularity theorem for $C^1$-Lorentzian metrics
• Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds
• A Note on the Gannon-Lee Theorem
• Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes
• On the asymptotic assumptions for Milne-like spacetimes
• A conformal infinity approach to asymptotically ${\text{AdS}}_2\times S^{n-1}$ spacetimes
• Synthetic versus distributional lower Ricci curvature bounds
• Hawking-type singularity theorems for worldvolume energy inequalities
• Spacetime singularities and curvature blow-ups
• Volume Singularities in General Relativity
• How Much NEC Breaking Can the Universe Endure?
• Optimal transport on null hypersurfaces and the null energy condition
• Radial Gravitational Collapse Causes Timelike Incompleteness