Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The formation of a weak null singularity in the interior of generic rotating black holes

Jonathan Luk, Jan Sbierski

Abstract

Given a characteristic initial value problem with smooth data representing a dynamical event horizon settling down to that of Kerr in the subextremal, strictly rotating range with suitable upper and lower bounds, we prove that a weak null singularity forms, across which the spacetime metric is continuously extendible but not Lipschitz extendible. The bulk of the proof is a stability argument showing that a dynamical Teukolsky field can be approximated by a linear Teukolsky field, whose linear instability was proved in previous works.

The formation of a weak null singularity in the interior of generic rotating black holes

Abstract

Given a characteristic initial value problem with smooth data representing a dynamical event horizon settling down to that of Kerr in the subextremal, strictly rotating range with suitable upper and lower bounds, we prove that a weak null singularity forms, across which the spacetime metric is continuously extendible but not Lipschitz extendible. The bulk of the proof is a stability argument showing that a dynamical Teukolsky field can be approximated by a linear Teukolsky field, whose linear instability was proved in previous works.

Paper Structure

This paper contains 104 sections, 95 theorems, 564 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Discussion of the proof
  3. Main strategy
  4. Principal null frame on dynamical spacetime
  5. Energy estimates for $\psi_0 - \psi_0^{\mathrm{lin}}$
  6. Propagation from $\Gamma$ to $\overline{V}$
  7. Related works
  8. Linear stability and instability results in black hole interiors
  9. Nonlinear spherically symmetric models
  10. Nonlinear results without symmetry assumptions
  11. Geometric inextendibility results
  12. The paper Gurriaran.nonlinear
  13. Organization of the paper
  14. The explicit Kerr background
  15. Kerr geometry in Boyer--Lindquist coordinates
  16. ...and 89 more sections

Key Result

Theorem 1.2

Fix $M,a$ such that $0 < |a| < M$. Consider the characteristic initial value problem for the vacuum Einstein equations with smoothSmoothness is assumed for convenience, but only finite regularity of order $I_0$ compatible with the precise assumptions in Section SecPreciseAssump is needed. initial da Then for $I_0$ big enough and $\epsilon>0$ small enough the following holds: $\blacktriangleleft$$

Figures (3)

  • Figure 1: Penrose-style diagram illustrating Theorem \ref{['thm:main']}.
  • Figure 2: A Penrose-style diagram illustrating the regions $\mathcal{M}$, $\overset{{\text{[{1}]}}}{\mathcal{M}}$, $\overset{{\text{[{2}]}}}{\mathcal{M}}$.
  • Figure 3: The region of integration

Theorems & Definitions (215)

  • Theorem 1.2
  • Lemma 2.10
  • Lemma 2.18
  • Definition 2.23: Kerr double null coordinates
  • Lemma 2.27
  • proof
  • Definition 2.28
  • Definition 2.30
  • Definition 2.32
  • Lemma 2.35
  • ...and 205 more