Islands and Page curves of Reissner-Nordstr\"om black holes

TL;DR

This paper calculates the Page curve for Reissner-Nordstr"om black holes using quantum extremal surfaces, showing that including islands resolves the information paradox by reproducing the black hole entropy and ensuring finite radiation entropy.

Contribution

It applies the quantum extremal surface method to Reissner-Nordstr"om black holes, demonstrating the role of islands in resolving the information paradox in this context.

Findings

The island extends outside the horizon of the black hole.

Including islands reproduces the Bekenstein-Hawking entropy.

Entanglement entropy remains finite at late times.

Abstract

We apply the recently proposed quantum extremal surface construction to calculate the Page curve of the eternal Reissner-Nordstr\"om 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-Nordstr\"om 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-Nordstr\"om 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.