Islands in Linear Dilaton Black Holes

TL;DR

This paper constructs novel linear dilaton black hole solutions, analyzes their entanglement entropy evolution with islands, and explores RG flow effects, demonstrating how islands resolve entropy divergence issues and modify the Page curve.

Contribution

It introduces new four-dimensional linear dilaton black holes, studies entanglement entropy with islands, and generalizes findings along the RG flow, including charged black holes.

Findings

Entanglement entropy grows before Page time and saturates after islands appear.

Islands significantly modify entropy, making it finite and consistent with Bekenstein-Hawking bounds.

RG flow causes the entanglement entropy to run, decreasing towards IR at late times.

Abstract

We derive a novel four-dimensional black hole with planar horizon that asymptotes to the linear dilaton background. The usual growth of its entanglement entropy before Page's time is established. After that, emergent islands modify to a large extent the entropy, which becomes finite and is saturated by its Bekenstein-Hawking value in accordance with the finiteness of the von Neumann entropy of eternal black holes. We demonstrate that viewed from the string frame, our solution is the two-dimensional Witten black hole with two additional free bosons. We generalize our findings by considering a general class of linear dilaton black hole solutions at a generic point along the $\sigma$-model renormalization group (RG) equations. For those, we observe that the entanglement entropy is "running" i.e. it is changing along the RG flow with respect to the two-dimensional worldsheet length scale. At any fixed moment before Page's time the aforementioned entropy increases towards the infrared (IR) domain, whereas the presence of islands leads the running entropy to decrease towards the IR at later times. Finally, we present a four-dimensional charged black hole that asymptotes to the linear dilaton background as well. We compute the associated entanglement entropy for the extremal case and we find that an island is needed in order for it to follow the Page curve.