Abstract
We explore the Hawking evaporation of two-dimensional anti-de Sitter (AdS), dilatonic black hole coupled with conformal matter, and derive the Page curve for the entanglement entropy of radiation. We first work in a semiclassical approximation with backreaction. We show that the end-point of the evaporation process is AdS with a vanishing dilaton, i.e., a regular, singularity-free, zero-entropy state. We explicitly compute the entanglement entropies of the black hole and the radiation as functions of the horizon radius, using the conformal field theory (CFT) dual to AdS gravity. We use a simplified toy model, in which evaporation is described by the forming and growing of a negative mass configuration in the positive-mass black hole interior. This is similar to the “islands” proposal, recently put forward to explain the Page curve for evaporating black holes. The resulting Page curve for AdS black holes is in agreement with unitary evolution. The entanglement entropy of the radiation initially grows, closely following a thermal behavior, reaches a maximum at half-way of the evaporation process, and then goes down to zero, following the Bekenstein–Hawking entropy of the black hole. Consistency of our simplified model requires a non-trivial identification of the central charge of the CFT describing AdS gravity with the number of species of fields describing Hawking radiation.
Highlights
Since the discovery of Hawking radiation [1,2], the information paradox for an evaporating black hole has been one of the most intriguing puzzles of fundamental theoretical physics
We investigate the semiclassical dynamics of JT black holes, coupled to conformal matter in the form of N massless scalar fields, including the backreaction of the geometry, and derive the Page curve for their entanglement entropy (EE) in a closed form
Equation (51) is instead obtained using a single conformal field theory (CFT) defined in the maximally extended space-time by tracing the degrees of freedom (DOF) in half of it; Holographic entanglement entropy formulas [33,46,47] give the EE of maximally extended AdS space-time in terms of the area of co-dimension two minimal surfaces, which can be identified with the event horizon
Summary
Since the discovery of Hawking radiation [1,2], the information paradox for an evaporating black hole has been one of the most intriguing puzzles of fundamental theoretical physics. The AdS/CFT correspondence implies that any gravitational bulk process in D-dimensions, such as black hole evaporation, can be holographically described in terms of a ( D − 1)-dimensional conformal field theory on the boundary, for which the evolution of quantum states is unitary. Unlike the semiclassical entropy formula used in the aforementioned papers, the EE formula used in this paper allows us to capture contributions of purely quantum mechanical correlations between the interior and the exterior of the black hole, in line with the ER=EPR spirit as well These correlations arise in a natural and simple way in our model, without resorting to higher-dimensional duals.
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