Evidence that the Rate of Evolution of a Black Hole Interior Has a Holographic Dual

A ``large'' AdS black hole can attain equilibrium with its own Hawking radiation, and in that condition it is thought to be dual to a strongly coupled field theory, also at equilibrium. But the interior of the black hole is by no means static: the geometry of spatial sections lying inside the event horizon evolves at some rate. This prompts the obvious question: can this rate possibly have a holographic dual? We present circumstantial evidence that such a dual does exist. We do this by making concrete proposals for two objects: first, a rough measure (already suggested in the literature) of the rate at which the unitary evolution of the state vector describing equilibrated strongly coupled matter evolves; and, second, a rough measure of the rate at which the interior (just under the horizon) evolves (in the case of the AdS$_5$-Kerr black hole). We then study how these two very different objects change as two physical parameters describing the exterior of the bulk black hole (the specific angular momentum and the temperature) are varied. We find that they change in remarkably similar ways, as holographic duals should.