Entanglement entropy of asymptotically flat non-extremal and extremal black holes with an island

TL;DR

This paper applies the island rule to compute entanglement entropy of both non-extremal and extremal Reissner-Nordström black holes, revealing different growth behaviors and addressing singularity issues with regular black holes.

Contribution

It extends the island rule to asymptotically flat black holes in thermal equilibrium, including extremal cases, and explores regular black holes to avoid singularities.

Findings

Non-extremal black holes show linear entropy growth, saturating after the Page time.

Extremal black holes exhibit logarithmic growth and constant entropy post-Page time.

Regular black holes can have finite entanglement entropy, avoiding singularity issues.

Abstract

The island rule for the entanglement entropy is applied to an eternal Reissner-Nordstr\"om black hole. The key ingredient is that the black hole is assumed to be in thermal equilibrium with a heat bath of an arbitrary temperature and so the generalized entropy is treated as being off-shell. Taking the on-shell condition to the off-shell generalized entropy, we find the generalized entropy and then obtain the entanglement entropy following the island rule. For the non-extremal black hole, the entanglement entropy grows linearly in time and can be saturated after the Page time as expected. The entanglement entropy also has a well-defined Schwarzschild limit. In the extremal black hole, the island prescription provides a logarithmically growing entanglement entropy in time and a constant entanglement entropy after the Page time. In the extremal black hole, the boundary of the island hits the curvature singularity where the semi-classical approximations appear invalid. To avoid encountering the curvature singularity, we apply this procedure to the Hayward black hole regular at the origin. Consequently, the presence of the island in extremal black holes can provide a finite entanglement entropy, which might imply non-trivial vacuum configurations of extremal black holes.