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Extremal black hole formation as a critical phenomenon

Christoph Kehle, Ryan Unger

Abstract

In this paper, we prove that extremal black holes arise on the threshold of gravitational collapse. More precisely, we construct smooth one-parameter families of smooth, spherically symmetric solutions to the Einstein-Maxwell-Vlasov system which interpolate between dispersion and collapse and for which the critical solution is an extremal black hole. Physically, these solutions can be understood as beams of gravitationally self-interacting collisionless charged particles fired into Minkowski space from past infinity. Depending on the precise value of the parameter, we show that the Vlasov matter either disperses due to the combined effects of angular momentum and electromagnetic repulsion, or undergoes gravitational collapse. At the critical value of the parameter, an extremal Reissner-Nordström black hole is formed. No naked singularities occur as the extremal threshold is crossed. We call this critical phenomenon extremal critical collapse and the present work constitutes the first rigorous result on the black hole formation threshold in general relativity.

Extremal black hole formation as a critical phenomenon

Abstract

In this paper, we prove that extremal black holes arise on the threshold of gravitational collapse. More precisely, we construct smooth one-parameter families of smooth, spherically symmetric solutions to the Einstein-Maxwell-Vlasov system which interpolate between dispersion and collapse and for which the critical solution is an extremal black hole. Physically, these solutions can be understood as beams of gravitationally self-interacting collisionless charged particles fired into Minkowski space from past infinity. Depending on the precise value of the parameter, we show that the Vlasov matter either disperses due to the combined effects of angular momentum and electromagnetic repulsion, or undergoes gravitational collapse. At the critical value of the parameter, an extremal Reissner-Nordström black hole is formed. No naked singularities occur as the extremal threshold is crossed. We call this critical phenomenon extremal critical collapse and the present work constitutes the first rigorous result on the black hole formation threshold in general relativity.
Paper Structure (73 sections, 43 theorems, 467 equations, 16 figures)

This paper contains 73 sections, 43 theorems, 467 equations, 16 figures.

Table of Contents

  1. Introduction
  2. The Einstein--Maxwell--Vlasov system
  3. The problem of critical collapse
  4. Extremal critical collapse
  5. Stability of extremal critical collapse
  6. Extremal critical collapse of a charged scalar field and in vacuum
  7. The third law of black hole thermodynamics and event horizon jumping at extremality
  8. Outline of the paper
  9. \ref{['sec:spherical-symmetry']}.
  10. \ref{['sec:EMV-system']}.
  11. \ref{['sec:dust-intro']}.
  12. \ref{['sec:Vlasov-beams']}.
  13. \ref{['sec:third-law-and-h-jumping']}.
  14. Spherically symmetric charged spacetimes
  15. Double null gauge
  16. ...and 58 more sections

Key Result

Theorem 1

There exist extremal black holes on the threshold between collapsing and dispersing smooth configurations of charged matter. More precisely, for any mass $M>0$, fundamental charge $\mathfrak e\ne 0$, and particle mass $0\le \mathfrak m\le \mathfrak m_0$, where $0<\mathfrak m_0\ll 1$ depends only on In addition, for every $\lambda\in [0,1]$, $\mathcal{D}_\lambda$ is past causally geodesically comp

Figures (16)

  • Figure 1: Penrose diagrams of the one-parameter family $\{\mathcal{D}_\lambda\}$ from \ref{['thm:main']} in the case of massive particles. The dark gray region depicts the physical space support of the Vlasov matter beam. The region of spacetime to the left of the beam is exactly Minkowski space and the region to the right of the beam is exactly Reissner--Nordström with the parameter ratio as depicted. In every case, the beam "bounces" before it hits the center $\{r=0\}$ due to the repulsive effects of angular momentum and the electromagnetic field. When $\lambda<\lambda_*$, the beam bounces before a black hole is formed. We note already that the beams actually have more structure than is depicted here in these "zoomed out" pictures. See already \ref{['fig:zoomed-1']}.
  • Figure 2: Penrose diagrams of the one-parameter family $\{\mathcal{D}_\lambda\}$ from \ref{['thm:main']} in the massless case. In the ingoing (resp., outgoing) phases, the massless beams are entirely contained in slabs of finite advanced (resp., retarded) time. Therefore, for sufficiently early advanced times and sufficiently late retarded times, the solutions are vacuum and isometric to Minkowski space.
  • Figure 3: A cartoon depiction of the conjectured structure of a neighborhood of moduli space near an interpolating family $\{\Psi_\lambda\}$ from \ref{['thm:main']}. We have suppressed infinitely many dimensions and emphasize the codimension-one property of the critical "submanifold" $\mathfrak B_\mathrm{crit}$ which consists of asymptotically extremal black holes in accordance with \ref{['conj:massless-stab', 'conj:massive-stab']}. The interpolating family $\{\Psi_\lambda'\}$ is a small perturbation of $\{\Psi_\lambda\}$ which also crosses $\mathfrak B_\mathrm{crit}$ and exhibits extremal critical collapse. Locally, $\mathfrak B$ is foliated by "hypersurfaces" $\mathfrak B(\mathfrak r)$ consisting of black hole spacetimes with asymptotic parameter ratio $\mathfrak r$ close to $1$.
  • Figure 4: Penrose diagram of a counterexample to the third law of black hole thermodynamics in the Einstein--Maxwell--Vlasov model from \ref{['thm:third-law-Vlasov']}. The broken curve $\mathcal{A}'$ is the outermost apparent horizon of the spacetime. This view is zoomed in on the Vlasov beam that charges up the subextremal black hole to extremality. We refer to \ref{['fig:third-law-Vlasov-proof']} in \ref{['sec:Vlasov-third-law']} for diagrams of the entire spacetime.
  • Figure 5: Penrose diagram of a normalized development $\mathcal{U}$ of a time symmetric seed $\mathcal{S}$. The spacelike hypersurface $\{\tau=0\}$ is totally geodesic, i.e., time symmetric, and the outgoing cone $\{u=-r_2\}$ has $f=0$. To the left of the support of $f$, the spacetime is vacuum: both the Hawking mass $m$ and charge $Q$ vanish identically. For the significance of the cone $\{v=r_0\}$, see already \ref{['rk:Mink-corner']}.
  • ...and 11 more figures

Theorems & Definitions (153)

  • Theorem 1
  • Corollary 1
  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8: Stationary solutions and the extremal limit
  • ...and 143 more