Quantum evolution of the Hawking state for black holes

We give a general description of the evolving quantum state Schwarzschild black hole, in field theory approximation. Such time-dependent is based on introducing choice time slices. particular consider slices that smoothly cross horizon, and introduction "stationary" such simplifies analysis. This analysis goes beyond standard derivations Hawking radiation focus asymptotic excitations, gives an regular at with no explicit transplanckian dependence, can principle be generalized to incorporate interacting fields. It also expected useful connecting information-theoretic investigation hole evolution. The depends as well coordinates mode bases; these choices different "pictures" analogous Schr\"odinger. Evolution does have simpler appearance energy eigenbasis, but basis singular horizon; evolution modes has more complicated appearance, whose properties may inferred by comparing eigenbasis. In description, quanta are produced atmosphere, scales comparable horizon size. approach argued extend asymptotics, anti de Sitter space. latter context, this provides hamiltonian compared large-$N$ dynamics proposed dual CFT.