[Paper Review] The interior of charged black holes and the problem of uniqueness in general relativity

Mihalis Dafermos|arXiv (Cornell University)|Jul 3, 2003

Cosmology and Gravitation Theories7 references10 citations

TL;DR

This paper rigorously establishes the mass inflation scenario in spherically symmetric charged black holes by proving that the maximal future development of asymptotically flat initial data for the Einstein-Maxwell-scalar field system admits a $C^0$-extendible future boundary where the Hawking mass blows up identically, rendering the spacetime inextendible as a $C^1$ metric. The result confirms the heuristic mass inflation conjecture of Israel and Poisson and shows that the strong cosmic censorship conjecture fails under Christodoulou's $C^0$ formulation for this system.

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.

Motivation & Objective

• To rigorously validate the mass inflation scenario proposed by Israel and Poisson in the context of spherically symmetric charged black holes.
• To establish the existence of a $C^0$-extendible future boundary with infinite Hawking mass, implying inextendibility as a $C^1$ metric.
• To demonstrate that the strong cosmic censorship conjecture fails under Christodoulou's $C^0$ formulation for the Einstein-Maxwell-scalar field system.
• To confirm Price's law decay of scalar field radiation on the event horizon as an upper bound, validating the physical relevance of the initial data.
• To resolve the long-standing question of uniqueness in general relativity by showing that smooth initial data can lead to non-unique, singular extensions.

Proposed method

• Formulates a spherically symmetric double characteristic initial value problem for the Einstein-Maxwell-scalar field equations with data satisfying the Price law decay on the initial outgoing null characteristic.
• Analyzes the maximal future development using a detailed geometric and analytic framework, focusing on the behavior of the Hawking mass and the metric regularity along the future boundary.
• Applies BV (bounded variation) estimates for the scalar field to control the growth of curvature invariants and establish the blow-up of the Hawking mass.
• Employs a recursive construction of spacetime rectangles and uses integral estimates involving the null normals $\nu$ and $\lambda$ to control the decay of $r^{-1}$ terms.
• Utilizes a 'zigzag' causal construction to propagate information across regions with varying causal structure, ensuring global control over the evolution.
• Relies on prior results by Dafermos and Rodnianski establishing Price’s law as an upper bound for scalar field decay in spherically symmetric collapse.

Experimental results

Research questions

• RQ1 Does the mass inflation scenario, as proposed by Israel and Poisson, occur in a fully nonlinear, dynamical setting of gravitational collapse?
• RQ2 Can the future boundary of the maximal development of asymptotically flat initial data be extended as a $C^0$ metric, and if so, what is the regularity of the metric beyond it?
• RQ3 Does the Hawking mass blow up identically along the Cauchy horizon, implying inextendibility as a $C^1$ metric?
• RQ4 Is the strong cosmic censorship conjecture valid under Christodoulou’s $C^0$ formulation for the Einstein-Maxwell-scalar field system?
• RQ5 Can the heuristic mass inflation scenario be rigorously derived from a well-posed initial value problem with physically motivated data?

Key findings

• The maximal future development of the initial data has a future boundary that is extendible as a $C^0$ metric, but not as a $C^1$ metric.
• The Hawking mass blows up identically along the Cauchy horizon, confirming the mass inflation scenario in a rigorous, nonlinear setting.
• The spacetime is inextendible as a $C^1$ metric, indicating a curvature singularity at the Cauchy horizon despite $C^0$ regularity of the metric.
• The $C^0$ extendibility result applies to the collapse of complete asymptotically flat spacelike initial data with compactly supported scalar fields, under the validity of Price’s law.
• The strong cosmic censorship conjecture is false under Christodoulou’s $C^0$ formulation for this system, as smooth initial data lead to non-unique extensions.
• The analysis confirms that the Cauchy horizon is a $C^1$-singular boundary, resolving the long-standing question of whether such singularities occur generically in charged black hole interiors.