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Lipschitz inextendibility of weak null singularities from curvature blow-up

Jan Sbierski

TL;DR

This paper proves that weak null singularities inside dynamical black holes are locally Lipschitz inextendible across the Cauchy horizon under an integrated curvature blow-up assumption, advancing the $C^{0,1}_{\mathrm{loc}}$-formulation of strong cosmic censorship near subextremal Kerr. The authors develop a novel approach that derives $C^{0,1}_{\mathrm{loc}}$-inextendibility from a bound/divergence type curvature condition by embedding a timelike geodesic problem into a null congruence, comparing $C^{0,1}_{\mathrm{loc}}$-extensions with the known $C^0$-extension from DafLuk17, and using Stokes’ theorem to obtain a finite boundary term, yielding a contradiction. The main theorem relies on a curvature blow-up bound along a sequence approaching the horizon and a structured set of Ricci-coefficient controls, establishing a sharp low-regularity barrier for extensions. A spherical toy-model demonstrates the necessity and sharpness of the curvature blow-up condition and highlights scenarios where the bound is violated yet Lipschitz extendibility is possible. Overall, the result provides a concrete mechanism linking curvature growth to geometric inextendibility, with implications for the global behavior of rotating black hole interiors and strong cosmic censorship.

Abstract

We prove the $C^{0,1}_{\mathrm{loc}}$-inextendibility of weak null singularities without any symmetry assumptions. The proof introduces a new strategy to infer $C^{0,1}_{\mathrm{loc}}$-inextendibility from the blow-up of curvature. The assumed blow-up is expected to be satisfied for weak null singularities in the interior of generic rotating black holes. Thus, we expect the result presented here to directly contribute to the resolution of the $C^{0,1}_{\mathrm{loc}}$-formulation of the strong cosmic censorship conjecture in a neighbourhood of subextremal Kerr.

Lipschitz inextendibility of weak null singularities from curvature blow-up

TL;DR

This paper proves that weak null singularities inside dynamical black holes are locally Lipschitz inextendible across the Cauchy horizon under an integrated curvature blow-up assumption, advancing the -formulation of strong cosmic censorship near subextremal Kerr. The authors develop a novel approach that derives -inextendibility from a bound/divergence type curvature condition by embedding a timelike geodesic problem into a null congruence, comparing -extensions with the known -extension from DafLuk17, and using Stokes’ theorem to obtain a finite boundary term, yielding a contradiction. The main theorem relies on a curvature blow-up bound along a sequence approaching the horizon and a structured set of Ricci-coefficient controls, establishing a sharp low-regularity barrier for extensions. A spherical toy-model demonstrates the necessity and sharpness of the curvature blow-up condition and highlights scenarios where the bound is violated yet Lipschitz extendibility is possible. Overall, the result provides a concrete mechanism linking curvature growth to geometric inextendibility, with implications for the global behavior of rotating black hole interiors and strong cosmic censorship.

Abstract

We prove the -inextendibility of weak null singularities without any symmetry assumptions. The proof introduces a new strategy to infer -inextendibility from the blow-up of curvature. The assumed blow-up is expected to be satisfied for weak null singularities in the interior of generic rotating black holes. Thus, we expect the result presented here to directly contribute to the resolution of the -formulation of the strong cosmic censorship conjecture in a neighbourhood of subextremal Kerr.
Paper Structure (12 sections, 4 theorems, 73 equations, 4 figures)

This paper contains 12 sections, 4 theorems, 73 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The spacetimes considered and gauge choice
  4. The class of spacetimes
  5. Localisation: relation to DafLuk17
  6. About the concept of a weak null singularity
  7. Frame and Ricci coefficients
  8. Causality
  9. The $C^{0,1}_{\mathrm{loc}}$-inextendibility of weak null singularities
  10. Auxiliary results
  11. The main theorem
  12. Examples and counterexamples

Key Result

Proposition 1.3

Let $\iota : M \hookrightarrow \tilde{M}$ be a $C^0$-extension of a time-oriented globally hyperbolic Lorentzian manifold $(M,g)$ with $g \in C^0$ and with Cauchy hypersurface $\Sigma$ -- and let $\tilde{p} \in \partial^+ \iota(M)$. For every $\delta >0$ there exists a chart $\tilde{\varphi} : \tild

Figures (4)

  • Figure 1: A Penrose-style diagram of a dynamical vacuum black hole settling down to a sub-extremal Kerr black hole in the exterior $I$. Very little is known about the singularity structure in the interior $II$ beyond the weak null singularity.
  • Figure 2: A Penrose-style diagram illustrating the interior of a perturbed sub-extremal Kerr black hole. The shaded region has been constructed in DafLuk17.
  • Figure 3: The curves and frames in $\tilde{M}$ used in Step 1.
  • Figure 4: The $C^1$-identification of $\tilde{W}_\leq$ with $\hat{W}_\leq$.

Theorems & Definitions (22)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Lemma 3.1
  • proof
  • Proposition 3.6
  • Remark 3.8
  • Remark 3.9
  • proof
  • Theorem 3.14
  • ...and 12 more