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 $\mathit{\text{while}}\mathit{\text{we}}\mathit{\text{do}}\mathit{\text{not}}\mathit{\text{directly}}\mathit{\text{show}}\mathit{\text{mass}}\mathit{\text{inflation}}$, 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 is $$\mathcal{C}\mathcal{H}$_{i^+}= $\mathcal{D}$ \cup $\mathcal{S}$$fortwo(possiblyempty)disjointconnectedsets$$\mathcal{D}$$and$$\mathcal{S}$$ such that: _$\mathcal{D}$ (the dynamical set) is a past set on which the Hawking mass blows up (mass inflation scenario). _$\mathcal{S}$ (the static set) is a future set isometric to a Reissner–Nordstr"{o}m Cauchy horizon i.e.\ the radiation is zero on $\mathcal{S}$. As a consequence, we establish a novel classification of Cauchy horizons into three types: dynamical ($\mathcal{S}=\mathrm{\varnothing }$), static ($\mathcal{D}=\mathrm{\varnothing }$) or mixed, and prove that ${\mathcal{C}\mathcal{H}}_{i^{+}}$ is globally $C^2$-inextendible. Our main motivation is 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 the Einstein-Maxwell-Klein-Gordon system on one-ended space-times. As a result, we prove in spherical symmetry that:

• 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, 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.