Islands and Page curves of Reissner-Nordström black holes

TL;DR

This work applies the quantum extremal surface island prescription to four-dimensional Reissner-Nordström black holes in a fixed background, neglecting backreaction and greybody factors, to derive the Page curve for Hawking radiation. Without islands, the radiation entropy grows linearly with time, recreating the information paradox; with a single island, the generalized entropy becomes time-independent at late times and the entanglement entropy saturates to a finite value near the Bekenstein-Hawking entropy, yielding a unitary Page curve. The analysis identifies a late-time island just outside the outer horizon, computes the Page time and scrambling time, and demonstrates consistency with unitarity and fast scrambling while avoiding firewall-like conclusions under the given approximations. The results bolster the claim that island contributions can resolve the information paradox in realistic 4D black hole spacetimes, within the adopted s-wave and no-backreaction framework.

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.

Figures (3)

• Figure 1: Penrose diagram for the eternal Reissner–Nordström black hole without islands. Hawking radiation is assumed to lie on the region $R_+$ and $R_-$. $b_{\pm}$ are the boundary surfaces of $R_+$ and $R_-$, respectively.
• Figure 2: Penrose diagram for the eternal Reissner–Nordström black hole with the assumption of island. The islands extend to the outside of the horizons of the black holes. The boundaries of islands are located at $a_+$ and $a_-$. The points $b_\pm$ are the boundaries of the left and right radiation regions $R_-$ and $R_+$.
• Figure 3: The Page curve for the eternal Reissner–Nordström black hole with the assumption that the higher order terms in $c \, G_{\rm N}/r_{\rm h}^{2}$ are ignored. The orange dashed line shows the result without islands. The solid line represents the quantitative result with island.