Mass Inflation and the C-2-inextendibility of Spherically Symmetric Charged Scalar Field Dynamical Black Holes

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 CHi+ can be partitioned as CHi+ = D boolean OR S for two (possibly empty) disjoint connected sets D and S such that center dot D (the dynamical set) is a future set on which the Hawking mass blows up (mass inflation scenario). center dot S (the static set) is a past set isometric to a Reissner-Nordstrom Cauchy horizon i.e. the radiation is zero on S. As a consequence of this result, we prove that the entire Cauchy horizonCH(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 = phi), static (D = phi) 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: 1. 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. 2. 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.