Quasi-local energy and microcanonical entropy in two-dimensional nearly de Sitter gravity

TL;DR

This paper explores the semi-classical thermodynamics of two-dimensional de Sitter space in JT gravity, extending quasi-local formalism to define conserved charges and deriving the island rule without replica trick, with implications for de Sitter holography.

Contribution

It introduces a quasi-local formalism for $ ext{dS}_{2}$ in JT gravity, deriving the island rule from first principles and analyzing thermodynamics in the static patch.

Findings

The quasi-local first law relates entropy and energy in $ ext{dS}_{2}$.

On-shell Euclidean action equals minus the generalized entropy.

Extremization of entropy corresponds to minimizing the action.

Abstract

We study the semi-classical thermodynamics of two-dimensional de Sitter space ($\text{dS}_{2}$) in Jackiw-Teitelboim (JT) gravity coupled to conformal matter. We extend the quasi-local formalism of Brown and York to $\text{dS}_{2}$, where a timelike boundary is introduced in the static patch to uniquely define conserved charges, including quasi-local energy. The boundary divides the static patch into two systems, a cosmological system and a black hole system, the former being unstable under thermal fluctuations while the latter is stable. A semi-classical quasi-local first law is derived, where the Gibbons--Hawking entropy is replaced by the generalized entropy. In the microcanonical ensemble the generalized entropy is stationary. Further, we show the on-shell Euclidean microcanonical action of a causal diamond in semi-classical JT gravity equals minus the generalized entropy of the diamond, hence extremization of the entropy follows from minimizing the action. Thus, we provide a first principles derivation of the island rule for $U(1)$ symmetric $\text{dS}_{2}$ backgrounds, without invoking the replica trick. We discuss the implications of our findings for static patch de Sitter holography.