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 (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.

Black holes in dS$_3$

“…Each of the first laws (5.41), (5.42), and (5.43) are precisely what happens for classical higher dimensional SdS, except here the classical entropies S h,c have been replaced by their generalized counterparts S gen,h,c . The semi-classical first laws above also hold in the twodimensional context, where one considers de Sitter JT gravity [13]. Consistent with the thermodynamics of the quantum BTZ solution [6], our observations here provide further evidence that, when semi-classical backreaction is accounted for, quantum black holes obey a first law of thermodynamics where the classical entropy is replaced by the generalized entropy.…”

“…Alternatively, one may adapt the quasi-local formalism of York [79] to de Sitter backgrounds and analyze quasi-local thermodynamics. This was recently accomplished in two-dimensional de Sitter JT gravity [13], where backreaction was accounted for exactly and the cosmological system was found to have a negative heat capacity and is thus thermodynamically unstable. It would be interesting to carry out a similar analysis for the quantum SdS solution, and see whether quantum de Sitter is thermodynamically stable.…”

“…a three-dimensional analog of the semi-classical first law in dS 2 explored in [13], 6 and a semi-classical generalization of the usual first law of higher dimensional SdS black holes [14]. Notably, the first laws (1.10) and (1.11) hold to all orders in backreaction and higher curvature corrections.…”

“…Carefully taking the limit rN ≈ rh,c , the temperature Th,c of both horizons approaches the Nariai temperature T N = 1/(2πr N ) (see App. B in [13]). The expression for T in terms of z is cumbersome, however, in the small ν limit we have…”

“…Recently it was proposed that the dual microscopic theory lives on the (stretched) cosmological horizon [46][47][48]. 13 Evidence for this comes from studying, in particular, the entropy deficit generated by nucleating a four-dimensional classical black hole in de Sitter space, where the nucleation rate is controlled by the deficit. As we will review, the form of the entropy deficit suggests dS gravity has a matrix theory interpretation [47,49,50], and the de Sitter static patch behaves as a holographic quantum mechanical system whose degrees of freedom are localized at the horizon.…”