Mass Inflation and the C2-inextendibility of Spherically Symmetric Charged Scalar Field Dynamical Black Holes

• Published in: Communications in mathematical physics Vol. 382; no. 2; pp. 1263 - 1341
• Main Author: Van de Moortel, Maxime
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2021; Springer Nature B.V
• Subjects: Angular momentum; Asymptotic properties; Black holes; Censorship; Classical and Quantum Gravitation; Complex Systems; Decay rate; Emission; Event horizon; Infinity; Mathematical and Computational Physics; Mathematical Physics; Naked singularities; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Scalars; Spacetime; Symmetry; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-020-03923-w

It has long been suggested that the Cauchy horizon of dynamical black holes is subject to a weak null singularity, under the mass inflation scenario. We study in spherical symmetry the Einstein–Maxwell–Klein–Gordon equations and while we do not directly show mass inflation , we obtain a “mass inflation/ridigity” dichotomy. More precisely, we prove assuming (sufficiently slow) decay of the charged scalar field on the event horizon, that the Cauchy horizon emanating from time-like infinity CH i + can be partitioned as CH i + = D ∪ S for two (possibly empty) disjoint connected sets D and S such that D (the dynamical set) is a future set on which the Hawking mass blows up (mass inflation scenario). S (the static set) is a past set isometric to a Reissner–Nordström Cauchy horizon i.e. the radiation is zero on S . As a consequence of this result, we prove that the entire Cauchy horizon CH i + is globally C 2 - inextendible ̲ , extending a previous local result established by the author. To this end, we establish a novel classification of Cauchy horizons into three types: dynamical ( S = ∅ ), static ( D = ∅ ) or mixed. As a side benefit, we prove that there exists a trapped neighborhood of the Cauchy horizon, thus the apparent horizon cannot cross the Cauchy horizon, which is a result of independent interest. Our main motivation is to prove the C 2 Strong Cosmic Censorship Conjecture for a realistic model of spherical collapse in which charged matter emulates the repulsive role of angular momentum. In our case, this model is the Einstein–Maxwell–Klein–Gordon system on space-times with one asymptotically flat end. As a consequence of the C 2 -inextendibility of the Cauchy horizon, we prove the following statements, in spherical symmetry: Two-ended asymptotically flat space-times are C 2 -future-inextendible i.e. C 2 Strong Cosmic Censorship is true for Einstein–Maxwell–Klein–Gordon, assuming the decay of the scalar field on the event horizon at the expected rate. In the one-ended case, under the same assumptions, the Cauchy horizon emanating from time-like infinity is C 2 -inextendible. This result suppresses the main obstruction to C 2 Strong Cosmic Censorship in spherical collapse. The remaining obstruction in the one-ended case is associated to “locally naked” singularities emanating from the center of symmetry, a phenomenon which is also related to the Weak Cosmic Censorship Conjecture.