The singularity theorems of General Relativity and their low regularity extensions

Abstract

On the occasion of Sir Roger Penrose's 2020 Nobel Prize in Physics, we review the singularity theorems of General Relativity, as well as their recent extension to Lorentzian metrics of low regularity. The latter is motivated by the quest to explore the nature of the singularities predicted by the classical theorems. Aiming at the more mathematically minded reader, we give a pedagogical introduction to the classical theorems with an emphasis on the analytical side of the arguments. We especially concentrate on focusing results for causal geodesics under appropriate geometric and initial conditions, in the smooth and in the low regularity case. The latter comprise the main technical advance that leads to the proofs of $C^1$-singularity theorems via a regularisation approach that allows to deal with the distributional curvature. We close with an overview on related lines of research and a future outlook.

Final Version. Invited Review article for Jahresbericht der Deutschen Mathematiker-Vereinigung, 50 pages

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• Singularity Theorems and Their Consequences
• The 1965 Penrose singularity theorem
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• Existence and absence of Killing horizons in static solutions with symmetries
• Hawking-type singularity theorems for worldvolume energy inequalities
• How Much NEC Breaking Can the Universe Endure?
• Tipler Naked Singularities in $N$ Dimensions
• Fake Schwarzschild and Kerr black holes