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The interior of charged black holes and the problem of uniqueness in general relativity

Mihalis Dafermos

TL;DR

The work analyzes the interior of charged black holes in the spherically symmetric Einstein–Maxwell–scalar field system with Price-law tails along an outgoing null surface, proving a mass-inflation scenario where the Hawking mass diverges on a Cauchy horizon CH^+, while the metric admits a continuous C^0 extension across CH^+.A red-shift/no-shift/blue-shift region decomposition is developed to control geometric and scalar-field quantities in the interior, enabling a bootstrap argument that yields both stability (C^0 extendibility) and blow-up (mass inflation) phenomena depending on decay rates of the scalar hair.Under Price-law bounds that hold for generic collapse data (non-extremal, nontrapped), the Hawking mass blows up along CH^+ identically, implying CH^+ cannot be extended as a C^1 metric; this yields a violation of strong cosmic censorship in Christodoulou’s C^0 formulation for the Einstein–Maxwell–scalar system in spherical symmetry.The analysis is reinforced by recent results showing Price-law decay with exponent approaching 3 (via Rodnianski) for collapse data with compactly supported scalar fields, which situates mass inflation as a robust interior feature and underscores the delicate balance between red-shift stabilization and blue-shift-driven blow-up in black-hole interiors.

Abstract

We consider a spherically symmetric characteristic initial value problem for the Einstein-Maxwell-scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal domain of development has a future boundary, over which the spacetime is extendible as a continuous metric, but along which the Hawking mass blows up identically; thus, the spacetime is inextendible as a differentiable metric. In view of recent results of the author in collaboration with I. Rodnianski (gr-qc/0309115), which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the continuous extendibility result applies to the collapse of complete asymptotically flat spacelike data where the scalar field is compactly supported on the initial hypersurface. This shows that under Christodoulou's C^0 formulation, the strong cosmic censorship conjecture is false for this system.

The interior of charged black holes and the problem of uniqueness in general relativity

TL;DR

The work analyzes the interior of charged black holes in the spherically symmetric Einstein–Maxwell–scalar field system with Price-law tails along an outgoing null surface, proving a mass-inflation scenario where the Hawking mass diverges on a Cauchy horizon CH^+, while the metric admits a continuous C^0 extension across CH^+.A red-shift/no-shift/blue-shift region decomposition is developed to control geometric and scalar-field quantities in the interior, enabling a bootstrap argument that yields both stability (C^0 extendibility) and blow-up (mass inflation) phenomena depending on decay rates of the scalar hair.Under Price-law bounds that hold for generic collapse data (non-extremal, nontrapped), the Hawking mass blows up along CH^+ identically, implying CH^+ cannot be extended as a C^1 metric; this yields a violation of strong cosmic censorship in Christodoulou’s C^0 formulation for the Einstein–Maxwell–scalar system in spherical symmetry.The analysis is reinforced by recent results showing Price-law decay with exponent approaching 3 (via Rodnianski) for collapse data with compactly supported scalar fields, which situates mass inflation as a robust interior feature and underscores the delicate balance between red-shift stabilization and blue-shift-driven blow-up in black-hole interiors.

Abstract

We consider a spherically symmetric characteristic initial value problem for the Einstein-Maxwell-scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal domain of development has a future boundary, over which the spacetime is extendible as a continuous metric, but along which the Hawking mass blows up identically; thus, the spacetime is inextendible as a differentiable metric. In view of recent results of the author in collaboration with I. Rodnianski (gr-qc/0309115), which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the continuous extendibility result applies to the collapse of complete asymptotically flat spacelike data where the scalar field is compactly supported on the initial hypersurface. This shows that under Christodoulou's C^0 formulation, the strong cosmic censorship conjecture is false for this system.

Paper Structure

This paper contains 14 sections, 19 theorems, 361 equations.

Table of Contents

  1. Introduction
  2. The Einstein-Maxwell-scalar field equations in spherical symmetry
  3. The initial value problem
  4. The maximal future development
  5. Monotonicity
  6. Remarks on the analysis of black holes
  7. The red-shift region
  8. A "no-shift" region
  9. The stable blue-shift region
  10. Beyond the stable blue shift region
  11. $C^0$ extension of the metric
  12. Mass inflation
  13. $BV$ estimates for $\phi$
  14. Two causal constructions

Key Result

Theorem 1.1

Fix constants $0<e<\varpi_+$, $C>0$, $p>\frac{1}{2}$. Define $r_+=\varpi_++\sqrt{\varpi_+^2-e^2}$, and fix a constant $r_0<r_+$. Let $r$ and $\phi$ be functions defined on the union $\mathcal{C}_{out}\cup_{\{p\}}\mathcal{C}_{in}$ of two connected $C^\infty$$1$-dimensional manifolds, each with bounda and, defining assume Parametrizing $\mathcal{C}_{in}$ by $[0,u_0)$ for $u_0<r_0$, assume that $r$

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • Proposition 4.3
  • ...and 23 more