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Generic linearized curvature singularity at the perturbed Kerr Cauchy horizon

Sebastian Gurriaran

TL;DR

The paper analyzes linearized gravitational perturbations of subextremal Kerr black holes by solving the spin-$2$ Teukolsky equation for $\widehat{\psi}_{-2}$ in the interior and near the Cauchy horizon CH$_{+}$. It develops a novel radial-time decomposition of the Teukolsky operator and an accompanying ODE method, combining with Price’s law-type horizon data to extract exact $u^{-8}$-scale asymptotics for the $\ell=2$ mode and sharp bounds for higher angular modes. The main contribution is a precise, coordinate-invariant description of how perturbations lead to a curvature singularity at the perturbed CH$_{+}$, consistent with a coordinate-invariant Kretschmann blow-up when paired with the spin-$+2$ interior behavior. This provides a rigorous linearized manifestation of Strong Cosmic Censorship in Kerr spacetimes and sharp quantitative insight into the oscillatory blow-up profile of the curvature near CH$_{+}$.

Abstract

We prove the precise asymptotics of the spin $-2$ Teukolsky field in the interior and along the Cauchy horizon of a subextremal Kerr black hole. Together with the oscillatory blow-up asymptotics of the spin $+2$ Teukolsky field proven in our previous work arXiv:2409.02670, our result suggests that generic perturbations of a Kerr black hole build up to form a coordinate-independent curvature singularity at the Cauchy horizon. This supports the Strong Cosmic Censorship conjecture in Kerr spacetimes. Unlike in the spin $+2$ case, the spin $-2$ Teukolsky field is regular on the Cauchy horizon and the first term in its asymptotic development vanishes. As a result, the derivation of a precise lower bound for the spin $-2$ field is more delicate than in the spin $+2$ case, and relies on a novel ODE method based on a decomposition of the Teukolsky operator between radial and time derivatives.

Generic linearized curvature singularity at the perturbed Kerr Cauchy horizon

TL;DR

The paper analyzes linearized gravitational perturbations of subextremal Kerr black holes by solving the spin- Teukolsky equation for in the interior and near the Cauchy horizon CH. It develops a novel radial-time decomposition of the Teukolsky operator and an accompanying ODE method, combining with Price’s law-type horizon data to extract exact -scale asymptotics for the mode and sharp bounds for higher angular modes. The main contribution is a precise, coordinate-invariant description of how perturbations lead to a curvature singularity at the perturbed CH, consistent with a coordinate-invariant Kretschmann blow-up when paired with the spin- interior behavior. This provides a rigorous linearized manifestation of Strong Cosmic Censorship in Kerr spacetimes and sharp quantitative insight into the oscillatory blow-up profile of the curvature near CH.

Abstract

We prove the precise asymptotics of the spin Teukolsky field in the interior and along the Cauchy horizon of a subextremal Kerr black hole. Together with the oscillatory blow-up asymptotics of the spin Teukolsky field proven in our previous work arXiv:2409.02670, our result suggests that generic perturbations of a Kerr black hole build up to form a coordinate-independent curvature singularity at the Cauchy horizon. This supports the Strong Cosmic Censorship conjecture in Kerr spacetimes. Unlike in the spin case, the spin Teukolsky field is regular on the Cauchy horizon and the first term in its asymptotic development vanishes. As a result, the derivation of a precise lower bound for the spin field is more delicate than in the spin case, and relies on a novel ODE method based on a decomposition of the Teukolsky operator between radial and time derivatives.

Paper Structure

This paper contains 61 sections, 43 theorems, 446 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Kerr black holes and Strong Cosmic Censorship
  3. The Kerr black hole interior
  4. Strong Cosmic Censorship conjecture
  5. Teukolsky equations
  6. Coordinate-independent curvature singularity
  7. Rough version of the main theorem and consequences
  8. Perturbations of black hole interiors
  9. Price's law-type results
  10. Previous results on the interior of perturbed black holes
  11. Structure of the proof
  12. Overview of the paper
  13. Acknowledgments
  14. Preliminaries
  15. Notations
  16. ...and 46 more sections

Key Result

Theorem 1.1

We consider an initially smooth and compactly supported linearized gravitational perturbation of a Kerr black hole such that $0<|a|<M$. We denote $\widehat{\psi}_{-2}$ the spin $-2$ Teukolsky scalar obtained in a principal null frame regular at $\mathcal{CH_+}$. Then $\widehat{\psi}_{-2}$ is regular where the constants $C_m$ are defined in eq:cm, depend only on the initial data and black hole para

Figures (4)

  • Figure 1: The Penrose diagram of Kerr spacetime, with the exterior of the black hole in white, the interior of the black hole in the darker shaded grey region, and the region with infinitely many extensions of Kerr in the lightest shaded grey region.
  • Figure 2: The range of the coordinates $u,{\underline{u}},r^*$ in the Kerr black hole interior.
  • Figure 3: Examples of constant $u,{\underline{u}},w,\underline{w}$ hypersurfaces and their causal nature.
  • Figure 4: Subregions $\mathbf{I},\mathbf{II}$ and $\mathbf{III}$ of the Kerr black hole interior.

Theorems & Definitions (108)

  • Theorem 1.1: Main theorem, rough version
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • ...and 98 more