Islands in Kaluza-Klein black holes

Abstract

The newly proposed island formula for entanglement entropy of Hawking radiation is applied to spherically symmetric 4-dimensional eternal Kaluza-Klein (KK) black hole. The "charge" $Q$ of KK black holes quantifies its deviation from Schwarzschild black holes. The impact of $Q$ on the island is studied at late times. The late-time island, whose boundary is located outside but within a Planckian distance of the horizon, is slightly extended by $Q$. While the no-island entropy grows linearly, the late-time entanglement entropy is given by island configuration with twice the Bekenstein-Hawking entropy. Thus we reproduce the Page curve for the eternal KK black holes. Compared with Schwarzschild results, the Page time is delayed by a factor $(1+Q/r_h)$ and the scrambling time is prolonged by a factor $(1+Q/r_h)^{1/2}$. Moreover, the higher-dimensional generalization is presented. Skeptically, there are Planck length scales involved, in which a semi-classical description may break down.