[Paper Review] The breakdown of weak null singularities inside black holes

Maxime Van de Moortel|arXiv (Cornell University)|Dec 23, 2019

Cosmology and Gravitation Theories4 citations

TL;DR

This paper proves that weak null singularities—predicted by mass inflation in charged black hole interiors—cannot close off spacetime; instead, they inevitably break down and give way to a stronger, $ r=0 $ singularity. Using a contradiction argument in spherically symmetric Einstein–Maxwell–Klein–Gordon spacetimes, it shows that the Cauchy horizon cannot be the final boundary, thus confirming the $ r=0 $ singularity conjecture for one-ended black holes.

ABSTRACT

It is widely expected that generic black holes have a non-empty but weakly singular Cauchy horizon, due to mass inflation. Indeed this has been proven by the author in the spherical collapse of a charged scalar field, under decay assumptions of the field in the black exterior which are conjectured to be generic. A natural question then arises: can this weakly singular Cauchy horizon close off the space-time, or does the weak null singularity necessarily "break down", giving way to a different type of singularity? The main result of this paper is to prove that the Cauchy horizon cannot ever "close off" the space-time. As a consequence, the weak null singularity breaks down and transitions to a different singularity for which the area-radius $r$ extends to $0$.

Motivation & Objective

• To resolve whether weak null singularities in black hole interiors can close off spacetime, preventing further evolution.
• To establish the necessity of a stronger $ r=0 $ singularity when weak null singularities are present.
• To prove the $ r=0 $ singularity conjecture in the context of spherically symmetric charged scalar field collapse.
• To show that the Penrose diagram with a closed-off Cauchy horizon (Figure 1) is impossible under weak singularity conditions.

Proposed method

• Uses a contradiction argument assuming the Penrose diagram of Figure 1, where the Cauchy horizon $ \mathcal{CH}_{i^{+}} $ closes off spacetime and is weakly singular.
• Applies the null energy condition and continuity of the area-radius function $ r $ to derive a contradiction at the endpoint of $ \mathcal{CH}_{i^{+}} $.
• Relies on prior classification of possible Penrose diagrams and stability estimates from earlier works on mass inflation and Cauchy horizon formation.
• Employs the Einstein–Maxwell–Klein–Gordon equations in spherical symmetry to model charged scalar field collapse.
• Uses red-shift estimates and decay assumptions on the scalar field to control behavior near the horizon and center.
• Applies the concept of terminal indecomposable pasts and boundary components to identify the first singularity.

Experimental results

Research questions

• RQ1 Can a weak null singularity at the Cauchy horizon close off spacetime, preventing the formation of a $ r=0 $ singularity?
• RQ2 Is it possible for a black hole interior to be bounded solely by a weakly singular Cauchy horizon without a stronger $ r=0 $ singularity?
• RQ3 Does the presence of a weak null singularity necessarily lead to a breakdown and transition to a stronger singularity?
• RQ4 What is the structure of the spacetime boundary when mass inflation occurs in a one-ended black hole?

Key findings

• Weak null singularities in the interior of charged black holes cannot close off spacetime; the Penrose diagram with a closed Cauchy horizon (Figure 1) is ruled out.
• The Cauchy horizon must break down, leading to the formation of a $ r=0 $ singularity, which is a terminal indecomposable past with compact intersection with the initial hypersurface.
• The endpoint of the Cauchy horizon cannot be the only boundary point; a non-empty set $ \mathcal{S}^{1}_{\Gamma} \cup \mathcal{CH}_{\Gamma} \cup \mathcal{S}^{2}_{\Gamma} \cup \mathcal{S} $ must exist, implying a stronger singularity.
• A contradiction arises if one assumes $ \mathcal{S}_{i^{+}} = \emptyset $ and $ \mathcal{CH}_{i^{+}} $ closes off spacetime, due to continuity of $ r $ at the endpoint.
• The result holds under the null energy condition and decay assumptions on the scalar field, which are conjectured to be generic.
• The $ r=0 $ singularity conjecture is proven: one-ended black holes with no locally naked singularity must feature a $ r=0 $ singularity in addition to a weakly singular Cauchy horizon.