Island in the Presence of Higher Derivative Terms

TL;DR

The paper extends the island formula to four-dimensional gravity with higher derivative terms and applies it to both asymptotically flat and AdS black holes coupled to a bath. The generalized entropy $S_{ m gen}$ includes Dong-type corrections to the gravitational area term and finite von Neumann entropy from matter fields, analyzed via a two-dimensional reduction; this framework is used to compute entanglement entropy for two-sided, one-sided, and AdS black holes in critical gravity. In all cases, islands emerge at late times and drive the entropy toward a Page-like saturation, even when the bulk theory may be non-unitary. The results suggest that the Page curve is a robust semiclassical feature of gravitational dynamics with islands, while clarifying the required boundary conditions and approximations in higher-derivative contexts.

Abstract

Using extended island formula we compute entanglement entropy of Hawking radiation for black hole solutions of certain gravitational models containing higher derivative terms. To be concrete we consider two different four dimensional models to compute entropy for both asymptotically flat and AdS black holes. One observes that the resultant entropy follows the Page curve, thanks to the contribution of the island, despite the fact that the corresponding gravitational models might be non-unitary.

Figures (3)

• Figure 1: Entanglement regions in the radiation part with the assumption that there is an island inside the black hole. The fictitious boundaries shown by violet lines at $r=r_0$ are the regions over which the gravity is negligible that are the radiation regions. At early times assuming that there is an island results in an imaginary solution for the location of island indicating that there is no island at early times and thus the whole contributions come from the matter von Neumann entropy.
• Figure 2: Entanglement regions in the radiation part with the assumption that there is an island inside the black hole. The fictitious boundary shown by violet lines at $r=r_0$ specifies the region over which the gravity is negligible, i.e. the radiation region.
• Figure 3: The AdS black hole+Bath system. The bath (colored region) plays the role of the environment around the evaporating black hole. This is necessary to include this environment since the boundary of AdS is reflecting and as a result of that the evaporation will be terminated at some point due to the expected equilibrium between the emission and the absorption processes.