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Black holes without spacelike singularities

Mihalis Dafermos

TL;DR

The paper analyzes the interior of near-Reissner–Nordström black holes within the Einstein–Maxwell–real scalar field framework, showing that sufficiently small two-ended perturbations produce a globally bifurcate null boundary CH^+ across which the spacetime extends as a $C^0$ metric and the scalar field extends continuously. With a dynamical decay lower bound on the scalar on horizons, Hawking mass inflates along CH^+, rendering a $C^2$ extension impossible and the scalar field non-$H^1_{loc}$ there, thereby providing a precise weak null-singularity description and reinforcing a $C^0$-based failure of strong cosmic censorship in this regime. The results emphasize that no spacelike singularities arise on an open set in the solution moduli space around RN, and they connect the gravitational instability picture to recent work on impulsive gravitational waves, while also exploring cosmological analogues that challenge censorship in spacetimes with a positive cosmological constant.

Abstract

It is shown that for small, spherically symmetric perturbations of asymptotically flat two-ended Reissner-Nordström data for the Einstein-Maxwell-real scalar field system, the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continuously. Under additional assumptions, it is shown that the Hawking mass blows up identically along this bifurcate null hypersurface, and thus the metric cannot be extended twice differentiably, in fact, cannot be extended in a weaker sense characterized at the level of the Christoffel symbols. The proof combines estimates obtained in previous work with an elementary Cauchy stability argument. There are no restrictions on the size of the support of the scalar field, and the result applies to both the future and past boundary of spacetime. In particular, it follows that for an open set in the moduli space of solutions around Reissner-Nordström, there is no spacelike component of either the future or the past singularity.

Black holes without spacelike singularities

TL;DR

The paper analyzes the interior of near-Reissner–Nordström black holes within the Einstein–Maxwell–real scalar field framework, showing that sufficiently small two-ended perturbations produce a globally bifurcate null boundary CH^+ across which the spacetime extends as a metric and the scalar field extends continuously. With a dynamical decay lower bound on the scalar on horizons, Hawking mass inflates along CH^+, rendering a extension impossible and the scalar field non- there, thereby providing a precise weak null-singularity description and reinforcing a -based failure of strong cosmic censorship in this regime. The results emphasize that no spacelike singularities arise on an open set in the solution moduli space around RN, and they connect the gravitational instability picture to recent work on impulsive gravitational waves, while also exploring cosmological analogues that challenge censorship in spacetimes with a positive cosmological constant.

Abstract

It is shown that for small, spherically symmetric perturbations of asymptotically flat two-ended Reissner-Nordström data for the Einstein-Maxwell-real scalar field system, the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continuously. Under additional assumptions, it is shown that the Hawking mass blows up identically along this bifurcate null hypersurface, and thus the metric cannot be extended twice differentiably, in fact, cannot be extended in a weaker sense characterized at the level of the Christoffel symbols. The proof combines estimates obtained in previous work with an elementary Cauchy stability argument. There are no restrictions on the size of the support of the scalar field, and the result applies to both the future and past boundary of spacetime. In particular, it follows that for an open set in the moduli space of solutions around Reissner-Nordström, there is no spacelike component of either the future or the past singularity.

Paper Structure

This paper contains 26 sections, 19 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Strong cosmic censorship
  3. The mass inflation scenario and weak null singularities
  4. Null dust models
  5. The Einstein--Maxwell--real scalar field model
  6. The numerical and heuristic studies
  7. The proof of the mass-inflation/weak null singularities scenario
  8. Global, bifurcate weak null singularities
  9. The Einstein--Maxwell--(real)-scalar field model
  10. The spherically symmetric reduction
  11. The general framework
  12. The trapped region, event horizons, and outermost apparent horizons
  13. The Hawking mass
  14. Persistence of the null boundary: $\mathcal{CH}^+_{A,B}\ne \emptyset$
  15. $\mathcal{CH}^+_{A,B}$ is (globally) singular
  16. ...and 11 more sections

Key Result

Theorem 1

For arbitrary asymptotically flat spherically symmetric data for the Einstein--Maxwell--real scalar field system for which the scalar field decays suitably at spatial infinity $i^0$, then if the charge is non-vanishing and the event horizon $\mathcal{H}^+$ is asymptotically subextremal, it follows t where $\mathcal{CH}^+$ is a non-empty piece of null boundary. Moreover, the spacetime can be contin

Theorems & Definitions (32)

  • Conjecture : Strong cosmic censorship
  • Theorem 1: Cauchy horizon stability md:cbh
  • Theorem 2: Mass inflation md:cbh
  • Theorem 3
  • Conjecture 1: A. Ori
  • Theorem : Mch
  • Proposition 3.1
  • Proposition 4.1
  • Theorem 4: md:cbh
  • Theorem 5: M.D.--Rodnianski dr1
  • ...and 22 more