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The coexistence of null and spacelike singularities inside spherically symmetric black holes

Maxime Van de Moortel

TL;DR

This work analyzes the interior dynamics of dynamically formed, charged spherically symmetric black holes in the Einstein–Maxwell–Klein–Gordon system. It proves that, under a breakdown of the Cauchy horizon, the spacetime boundary must contain a spacelike singularity with Kasner-type geometry, whose exponents are positive and asymptotically degenerate to $(1,0,0)$ along the null–spacelike transition, providing the first quantitative null–spacelike transition inside a black hole. The authors develop a robust framework of a priori and quantitative estimates, both under and beyond the Cauchy horizon, and introduce a variegated Kasner-formation mechanism via $v$-dependent exponents, enabling detailed control of the metric and scalar field. The results bridge interior black hole dynamics with Strong Cosmic Censorship in the spherically symmetric setting and set the stage for global constructions of one- and two-ended black holes with coexisting null and spacelike singularities in companion work. The methods, notably AVTD-type analysis, Kasner casting, and carefully designed initial data, offer a blueprint for understanding deterministic breakdowns in gravitational collapse beyond spherical symmetry.$

Abstract

In our previous work [Van de Moortel, The breakdown of weak null singularities, Duke Mathematical Journal 172 (15), 2957-3012, 2023], we showed that dynamical black holes formed in charged spherical collapse generically feature both a null weakly singular Cauchy horizon and a stronger (presumably spacelike) singularity, confirming a longstanding conjecture in the physics literature. However, this previous result, based on a contradiction argument, did not provide quantitative estimates on the stronger singularity. In this study, we adopt a new approach by analyzing local initial data inside the black hole that are consistent with a breakdown of the Cauchy horizon. We prove that the remaining portion is spacelike and obtain sharp spacetime estimates near the null-spacelike transition. Notably, we show that the Kasner exponents of the spacelike portion are positive, in contrast to the well-known Oppenheimer-Snyder model of gravitational collapse. Moreover, these exponents degenerate to (1,0,0) towards the null-spacelike transition. Our result provides the first quantitative instances of a null-spacelike singularity transition inside a black hole. In our companion paper, we moreover apply our analysis to carry out the construction of a large class of asymptotically flat one or two-ended black holes featuring coexisting null and spacelike singularities.

The coexistence of null and spacelike singularities inside spherically symmetric black holes

TL;DR

This work analyzes the interior dynamics of dynamically formed, charged spherically symmetric black holes in the Einstein–Maxwell–Klein–Gordon system. It proves that, under a breakdown of the Cauchy horizon, the spacetime boundary must contain a spacelike singularity with Kasner-type geometry, whose exponents are positive and asymptotically degenerate to along the null–spacelike transition, providing the first quantitative null–spacelike transition inside a black hole. The authors develop a robust framework of a priori and quantitative estimates, both under and beyond the Cauchy horizon, and introduce a variegated Kasner-formation mechanism via -dependent exponents, enabling detailed control of the metric and scalar field. The results bridge interior black hole dynamics with Strong Cosmic Censorship in the spherically symmetric setting and set the stage for global constructions of one- and two-ended black holes with coexisting null and spacelike singularities in companion work. The methods, notably AVTD-type analysis, Kasner casting, and carefully designed initial data, offer a blueprint for understanding deterministic breakdowns in gravitational collapse beyond spherical symmetry.$

Abstract

In our previous work [Van de Moortel, The breakdown of weak null singularities, Duke Mathematical Journal 172 (15), 2957-3012, 2023], we showed that dynamical black holes formed in charged spherical collapse generically feature both a null weakly singular Cauchy horizon and a stronger (presumably spacelike) singularity, confirming a longstanding conjecture in the physics literature. However, this previous result, based on a contradiction argument, did not provide quantitative estimates on the stronger singularity. In this study, we adopt a new approach by analyzing local initial data inside the black hole that are consistent with a breakdown of the Cauchy horizon. We prove that the remaining portion is spacelike and obtain sharp spacetime estimates near the null-spacelike transition. Notably, we show that the Kasner exponents of the spacelike portion are positive, in contrast to the well-known Oppenheimer-Snyder model of gravitational collapse. Moreover, these exponents degenerate to (1,0,0) towards the null-spacelike transition. Our result provides the first quantitative instances of a null-spacelike singularity transition inside a black hole. In our companion paper, we moreover apply our analysis to carry out the construction of a large class of asymptotically flat one or two-ended black holes featuring coexisting null and spacelike singularities.

Paper Structure

This paper contains 60 sections, 36 theorems, 388 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Connections to the Strong Cosmic Censorship Conjecture
  3. The mathematical statement of Strong Cosmic Censorship
  4. Connection to black holes
  5. Perturbations of the Kerr black hole for the Einstein equations in vacuum
  6. Spherically-symmetric version of Strong Cosmic Censorship
  7. Inextendibily across the singularities obtained in our result
  8. Local aspects of the black hole interior terminal boundary near $i^+$
  9. Gravitational collapse and local aspects of the black hole interior terminal boundary near $\Gamma$
  10. Mathematical setting
  11. Locally-naked singularities
  12. Weak Cosmic Censorship
  13. Trapped surface conjecture
  14. Global aspects of the black hole interior in gravitational collapse
  15. Breakdown of the weak null singularity
  16. ...and 45 more sections

Key Result

Theorem 1

The interior of a spherically symmetric dynamical black hole solution of 1.1--5.1 admits a weakly singular Cauchy horizon $\mathcal{CH}_{i^+}$ which must break down and give rise to an additional singularity.

Figures (6)

  • Figure 1: Penrose diagram of a solution obtained in Theorem \ref{['thm.I']} with bifurcate initial data $\underline{C}_{in} \cup C_{out}$.
  • Figure 2: Penrose diagram of the gravitational collapse (=one-ended) spacetimes obtained in Theorem \ref{['thm.II']}. $\mathcal{CH}_{i^+}$ is a weak null singularity and $\mathcal{S}$ a strong spacelike singularity.
  • Figure 3: Local structure of the black hole terminal boundary near $i^+$ for a spherically-symmetric solution of \ref{['1.1']}--\ref{['5.1']}.
  • Figure 4: General Penrose diagram of a one-ended spherically-symmetric black hole solution after Kommemi.
  • Figure 5: The impossible Penrose diagram if $\mathcal{CH}_{i^+}$ is weakly singular, as a consequence of the result of breakdown.
  • ...and 1 more figures

Theorems & Definitions (75)

  • Theorem : breakdown
  • Theorem I
  • Remark 1.1
  • Remark 1.2
  • Theorem II
  • Remark 1.3
  • Conjecture 1.2: Strong Cosmic Censorship
  • Theorem 1.3: Moi
  • Conjecture 1.4: Spherical trapped surface conjecture, KommemiJonathanICM
  • Theorem III: Two-ended black holes with coexisting singularities, our companion paper bif2
  • ...and 65 more