Stability and Instability of the Sub-extremal Reissner–Nordström Black Hole Interior for the Einstein–Maxwell–Klein–Gordon Equations in Spherical Symmetry

We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole—approaching a sub-extremal Reissner–Nordström background fast enough—in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting:1.Stability We prove that spherically symmetric characteristic initial data to the Einstein–Maxwell–Klein–Gordon equations approaching a Reissner–Nordström background with a sufficiently decaying polynomial decay rate on the event horizon gives rise to a space–time possessing a Cauchy horizon in a neighbourhood of time-like infinity. Moreover, if the decay is even stronger, we prove that the space–time metric admits a continuous extension to the Cauchy horizon. This generalizes the celebrated stability result of Dafermos for Einstein–Maxwell-real-scalar-field in spherical symmetry.2.Instability We prove that for the class of space–times considered in the stability part, whose scalar field in addition obeys a polynomial averaged-L² (consistent) lower bound on the event horizon, the scalar field obeys an integrated lower bound transversally to the Cauchy horizon. As a consequence we prove that the non-degenerate energy is infinite on any null surface crossing the Cauchy horizon and the curvature of a geodesic vector field blows up at the Cauchy horizon near time-like infinity. This generalizes an instability result due to Luk and Oh for Einstein–Maxwell-real-scalar-field in spherical symmetry.This instability of the black hole interior can also be viewed as a step towards the resolution of the C² strong cosmic censorship conjecture for one-ended asymptotically flat initial data.