The averaged null energy condition on holographic evaporating black holes

We examine the averaged null energy condition~(ANEC) for strongly coupled fields, along the event horizon of an evaporating black hole by using the AdS/CFT duality. First, we consider a holographic model of a $3$-dimensional evaporating black hole with a perturbed 4-dimensional black droplet geometry as the bulk dual, and investigate how negative energy flux going into the boundary black hole horizon appears. We show that the ingoing negative energy flux always appears at the boundary black hole horizon when the horizon area decreases. Second, we test the ANEC in a holographic model of a $4$-dimensional asymptotically flat dynamical black hole, which describes the formation and subsequent evaporation of a spherically symmetric black hole. By applying the "bulk-no-shortcut principle", we show that the ANEC is always satisfied when the local null energy is averaged with a wieght function along the incomplete null geodesic on the event horizon from beginning of the formation to the final instant of the black hole evaporation. Our results indicate that the total ingoing negative energy flux is compensated by a large amount of positive energy flux in the early stage of the black hole formation.