Stability and instability of the sub-extremal Reissner-Nordstr\"om black hole interior for the Einstein-Maxwell-Klein-Gordon equations in spherical symmetry

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We show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole -approaching a sub-extremal Reissner-Nordstr\"om background fast enough at infinity- 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-Nordstr\"om 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 spacetime 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.