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 (dS 2 ) in Jackiw-Teitelboim (JT) gravity coupled to conformal matter. We extend the quasilocal formalism of Brown and York to 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 dS 2 backgrounds, without invoking the replica trick. We discuss the implications of our findings for static patch de Sitter holography.

On the Euclidean Action of de Sitter Black Holes and Constrained Instantons

“…In the limit r b,c → r N we find κN = √ d − 1/L (see Appendix B in [45]). Thus the Bousso-Hawking normalization gives the expected (nonvanishing) surface gravity for the Nariai black hole.…”

“…We often express SdS quantities in terms of the dimensionless ratio µ ≡ M/M N , for which the range (2.3) becomes 0 ≤ µ ≤ 1. In the Nariai limit the SdS geometry reduces to dS 2 × S d−2 , where the curvature radius of two-dimensional de Sitter space is L = L/ √ d − 1, and the curvature radius of the sphere is equal to r N , which are equal in d = 4 [9,10,27,45]. In the near-Nariai limit for a spherical reduction of four-dimensional SdS the Einstein action reduces to the action of de Sitter JT gravity, which was studied in [45][46][47].…”

“…Here Θ is the conjugate quantity to the cosmological constant in an extended version of the first law: T b δS b +T c δS c + Θ 8πG δΛ = 0. It can also be defined in a geometric way as the "Killing volume" [45,49]…”

“…In order to properly interpret this as a thermodynamic entropy, and exp(−I E ) as the saddle point approximation to a thermal partition function, one should introduce a timelike boundary at some fixed radius r = R [50]. At this boundary either the temperature or the (quasi-local) energy should be fixed depending on whether the thermodynamic ensemble is the canonical or microcanonical ensemble, see [36,42,45,55]. In the present paper, however, we are interested in using the Euclidean action to compute the pair creation rate of black holes, and not in defining different thermodynamic ensembles for SdS.…”

“…by letting the matter stress-energy tensor T µν be nonzero in the perturbed solution, then the first law for SdS takes the form[12,45] …”