  We study the inner horizons of rotating and charged black holes in anti-de
Sitter space. These black holes have a classical analytic extension through the
inner horizon to additional asymptotic regions. If this extension survives in
the quantum theory, it requires particular analytic properties in a dual CFT,
which give a prescription for calculating correlation functions for operators
placed on any asymptotic boundary of the maximally extended spacetime. We show
that for charged black holes in three or greater dimensions, and rotating black
holes in four or greater dimensions, these analytic properties are inconsistent
in the dual CFT, implying the absence of an analytic extension for quantum
fields past the inner horizon. Thus, we find that strong cosmic censorship
holds for all AdS black holes except rotating BTZ. To further study the latter
case, we insert classical perturbations near the boundary at late times,
producing shockwaves traveling along the inner horizon. We holographically
compute CFT correlators in this background that probe a high energy scattering
process near the inner horizon and argue that the shockwave does not
destabilize the inner horizon violently enough to prevent signaling between
different asymptotic regions of the Penrose diagram. This provides evidence
that the rotating BTZ black hole does violate the strong cosmic censorship
conjecture.
