Island in Charged Black Holes

TL;DR

This work extends the island paradigm to higher-dimensional, charged black holes in a doubly-holographic framework, where a Planck brane on which the black hole is realized exchanges information with flat baths. It provides an analytic Page curve in the weak-tension limit and a numerical backreacted analysis in four dimensions using a DGP term to augment brane degrees of freedom, demonstrating conditions under which the Page time is nonzero. The key finding is that increasing brane tension or incorporating a brane-localized curvature term can guarantee a positive initial entropy difference and recover a Page curve, with near-extremal cases exhibiting linear growth in Hawking temperature. Together, these results broaden the applicability of holographic island physics to charged, higher-dimensional black holes and suggest concrete levers to control information flow in doubly-holographic setups.

Abstract

We study the information paradox for the eternal black hole with charges on a doubly-holographic model in general dimensions, where the charged black hole on a Planck brane is coupled to the baths on the conformal boundaries. In the case of weak tension, the brane can be treated as a probe such that its backreaction to the bulk is negligible. We analytically calculate the entanglement entropy of the radiation and obtain the Page curve with the presence of an island on the brane. For the near-extremal black holes, the growth rate is linear in the temperature. Taking both Dvali-Gabadadze-Porrati term and nonzero tension into account, we obtain the numerical solution with backreaction in four-dimensional spacetime and find the quantum extremal surface at $t=0$. To guarantee that a Page curve can be obtained in general cases, we propose two strategies to impose enough degrees of freedom on the brane such that the black hole information paradox can be properly described by the doubly-holographic setup.

Figures (8)

• Figure 1: (a): The $d$-dimensional charged black hole is in equilibrium with two flat baths (colored in gray). The conformal boundary is located at $\sigma=0$, while the black hole and baths are distributed in the region with $\sigma>0$ and $\sigma<0$, respectively. Two candidates of the QES at time $t$ are plotted in different colors. (b): A sketch of the $(d+1)$-dimensional dual of the $d$-dimensional holographic system. Here the $d$-dimensional black hole is described equivalently by the Planck brane $\bm{pl}$ in the $(d+1)$-dimensional ambient spacetime. The QES is measured by an ordinary HRT surface (which is plotted as a curve either in blue or rose gold) in the $(d+1)$-dimensional spacetime.
• Figure 2: A simple setup of Randall-Sundrum braneTakayanagi:2011zk. Here the Planck brane is anchored on the conformal boundary at $(z,w)=(0,0)$ and penetrates into the bulk with an angle $\theta$.
• Figure 3: (a): For $\{d,T_h/\mu,\theta,w_b/\mu\}=\{3, 0.45757,2\pi/5, 2\}$, two candidates $\gamma_{tr}$ and $\gamma_{pl}$ are colored in rose gold and blue, respectively. Half of the island is colored in red. (b): The trivial surface $\gamma_{tr}$ is colored in rose gold at time $t$, where $z_{max}$ is the turning point with $z'|_{z_{max}}=0$.
• Figure 4: (a): For $d=3$ and $\theta = \pi/2$, the relations between the density difference and the end point of the HRT surface are plotted at different temperatures. (b): The location of QES $z_{_{QES}}$ as a function of $w_b \mu$, with $\{d,\theta\}=\{3,\pi/2\}$.
• Figure 5: For $\{d,w_b,L,\theta\}=\{3,1,1,9\pi/20\}$, the Page curves are plotted for different Hawking temperatures. The dashed curve in blue connects the Page time $t=t_P$ of each Page curve. In the plot, The Newton constant is fixed to be $\frac{L^2}{4G_N^{(4)}}=1$.