Strong cosmic censorship in the presence of matter: the decisive effect of horizon oscillations on the black hole interior geometry

Motivated by the Strong Cosmic Censorship Conjecture in the presence of matter, we study the Einstein equations coupled with a charged/massive scalar field with spherically symmetric characteristic data relaxing to a Reissner-Nordström event horizon.Contrary to the vacuum case, the relaxation rate is conjectured to be slow (non-integrable), opening the possibility that the matter fields and the metric coefficients blow up in amplitude at the Cauchy horizon, not just in energy.We show that whether this blow-up in amplitude occurs or not depends on a novel oscillation condition on the event horizon which determines whether or not a resonance is excited dynamically:• If the oscillation condition is satisfied, then the resonance is not excited and we show boundedness and continuous extendibility of the matter fields and the metric across the Cauchy horizon.