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

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.

48 pages + 4 appendices, 13 figures; minor comments added v2, JHEP version; v3 corrected formulae and plot for heat capacity and fixed typo in Appendix A (original conclusions unaltered)

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