Mass inflation and the \(C^2\)-inextendibility of spherically symmetric charged scalar field dynamical black holes

Abstract

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 we prove, assuming (sufficiently slow) decay of the charged scalar field on the event horizon that - either mass inflation occurs on the entire Cauchy horizon emanating from time-like infinity \(\mathcal{CH}_{i^+}\), - or \(\mathcal{CH}_{i^+}\) is isometric to a Reissner-Nordstr\"{o}m Cauchy horizon i.e. the radiation is zero on the Cauchy horizon. In both cases, we prove that \(\mathcal{CH}_{i^+}\) is globally \(C^2\)-inextendible. To this end, we establish a novel classification of Cauchy horizons into three types: dynamical, static 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, 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.