Entanglement Islands, Page curves and Phase Transitions of Kerr-AdS Black Holes

TL;DR

The paper addresses the information paradox for Kerr–AdS black holes by applying the island formula to a near-horizon 2D reduction and coupling to a non-gravitational bath to model evaporation. The main approach yields a Page-curve behaviour: without islands the radiation entropy grows linearly and violates unitarity, while including a quantum extremal island renders the late-time entropy constant, giving a Page curve consistent with unitary evolution. The study also analyzes the impact of thermodynamic phase transitions on the Page curve in two ensembles; in the canonical ensemble a first-order transition leaves a sharp discontinuity in the Page curve, whereas in a fixed $\zeta$ ensemble no such transition occurs and Page curves remain continuous. These results strengthen the island proposal as a resolution to the information paradox in rotating AdS black holes and reveal a link between black hole thermodynamics and entanglement entropy evolution.

Abstract

We study the Page curve and information paradox for Kerr AdS black hole in light of entanglement entropy by employing the recently proposed island paradigm. By incorporating the island rule, we show that the entanglement entropy of Kerr AdS black hole grows linearly at early times and declines to a constant value at late times in agreement with the well established Page curve. The novelty of this study resides in the investigation of influence of phase transitions on the page curve in two different ensembles. We find that a first order phase transition results in a sharp discontinuity in the Page curve. We study the evaporation process in different scenarios and find that in all the situations, the Page curve doesn't violate the unitary principle of quantum mechanics.

Figures (9)

• Figure 1: The Penrose diagram of a non extremal Kerr $\text{AdS}_4$ black hole. Here, $r_{\pm}$ denotes the outer and inner horizon respectively, $\mathcal{J}_{\pm}$ is future and past null infinity respectively, and $\mathcal{I}_{o}$ is spacelike infinity
• Figure 2: (a) The Penrose diagram of an eternal Kerr $\text{AdS}_4$ black hole without island.The black and red regions denote the black hole and the beth region respectively. The radiation regions are denoted by $R_{\pm}$ on the right and left wedges and $b_{\pm}$ are the boundaries of $R_{\pm}$ (b) The Penrose diagram of an eternal Kerr $\text{AdS}_4$ black hole with an island. The boundary of the island is denoted as $a_{\pm}$ The radiation regions are denoted by $R_{\pm}$ on the right and left wedges and $b_{\pm}$ are the boundaries of $R_{\pm}$.
• Figure 3: The Page curve of a non extremal Kerr $\text{AdS}_4$ black hole represented by the solid line. The red line and blue line represents the entanglement entropy without and with island respectively.
• Figure 4: (a) The plot of free energy versus temperature showing the swallow-tail behaviour for below critical values of the spin parameter. (b) The variation of the Hawking temperature with the horizon radius. Below the critical temperature, three black hole branches are observed.
• Figure 5: The Page curves of eternal Kerr AdS black holes. (a) Page curves for spin parameter $a=0.11$ (blue), $a=0.05$ (red) and $a=0.01$ (green) for large black hole. (b) Page curves for $a=0.08$ with blue line for intermediate black hole and orange line for small black hole. (c) Page curve for small black hole $r_+=0.12$, $a=0.05$ (blue) and $a=0.09$ (red)