Abstract
We apply the recently proposed quantum extremal surface construction to calculate the Page curve of the eternal Reissner-Nordström black holes in four dimensions ignoring the backreaction and the greybody factor. Without the island, the entropy of Hawking radiation grows linearly with time, which results in the information paradox for the eternal black holes. By extremizing the generalized entropy that allows the contributions from the island, we find that the island extends to the outside the horizon of the Reissner-Nordström black hole. When taking the effect of the islands into account, it is shown that the entanglement entropy of Hawking radiation at late times for a given region far from the black hole horizon reproduces the Bekenstein-Hawking entropy of the Reissner-Nordström black hole with an additional term representing the effect of the matter fields. The result is consistent with the finiteness of the entanglement entropy for the radiation from an eternal black hole. This facilitates to address the black hole information paradox issue in the current case under the above-mentioned approximations.
Highlights
JHEP04(2021)103 at the end of black hole evaporation at the Planck scale
When taking the effect of the islands into account, it is shown that the entanglement entropy of Hawking radiation at late times for a given region far from the black hole horizon reproduces the Bekenstein-Hawking entropy of the ReissnerNordström black hole with an additional term representing the effect of the matter fields
The presence of replica wormholes as the saddle points in the Euclidean path integral leads to the island formula for the eternal black holes and for the evaporating black holes [7, 64,65,66]
Summary
We carry out the calculation of the entanglement entropy in the four dimensional Reissner-Nordström geometry without/with involving the islands. For Reissner-Nordström black holes, the Hawking radiation and Schwinger effect both act as the emission channels of charged pairs. For a two-dimensional CFT, it is just the length of the minimal curve in the bulk This formula is applied when no island is formed. The above equations, eq (2.1), (2.2) and (2.3) allow us to compute the entanglement entropy for disjoint union of intervals This general RT formula will be useful when one or more islands appear. With the approach to calculate the entropy, we apply the quantum extremal surface with the islands neglecting the backreaction of the radiation on the black hole metric. Include the configurations of the islands in the generalized entropy and take the minimal values of all such saddle points. The information will not be conserved and the information paradox will remain to be an conflicting issue of gravity and quantum mechanics for Reissner-Nordström black holes
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