Non-extremal Island in de Sitter Gravity

TL;DR

This paper explores how to compute entanglement entropy in de Sitter gravity by introducing a holographic model, revealing that non-extremal islands are essential for physically consistent results.

Contribution

It proposes a doubly holographic model embedding a dS$_2$ brane in AdS$_3$ to address limitations of the island formula in de Sitter spacetime.

Findings

Non-extremal islands are necessary for correct entanglement entropy calculations.

The entanglement wedge includes the entire de Sitter gravitational space during the island phase.

Existence of minimal surfaces is related to the brane's extrinsic curvature.

Abstract

This paper investigates the challenges and resolutions in computing the entanglement entropy for the quantum field theory coupled to de Sitter (dS) gravity along a timelike boundary. The conventional island formula, originally designed to calculate the fine-grained entropy for a non-gravitational system coupled to anti-de Sitter (AdS) gravity, encounters difficulties in de Sitter gravitational spacetime, failing to provide a physically plausible extremal island. To overcome these problems, we introduce a doubly holographic model by embedding a dS$_2$ braneworld in an AdS$_3$ bulk spacetime. This approach facilitates the computation of entanglement entropy through holographic correlation functions, effectively circumventing the constraints of the island formula. We demonstrate that the correct recipe for calculating entanglement entropy with dS gravity involves the non-extremal island, whose boundary is instead defined at the edge of the dS gravitational region. Our findings indicate that, during the island phase, the entanglement wedge of the non-gravitational bath includes the entire dS gravitational space. Using the second variation formula, we further show that the existence of a locally minimal surface anchored on the gravitational brane is intrinsically linked to the extrinsic curvature of the brane.