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

Abstract

Abstract We study the semi-classical thermodynamics of two-dimensional de Sitter space (dS2) in Jackiw-Teitelboim (JT) gravity coupled to conformal matter. We extend the quasi-local formalism of Brown and York to dS2, 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 dS2 backgrounds, without invoking the replica trick. We discuss the implications of our findings for static patch de Sitter holography.

Keywords

• 2D Gravity
• Gauge-Gravity Correspondence
• Models of Quantum Gravity