Thermodynamic ensembles with cosmological horizons

Abstract

The entropy of a de Sitter horizon was derived long ago by Gibbons and Hawking via a gravitational partition function. Since there is no boundary at which to define the temperature or energy of the ensemble, the statistical foundation of their approach has remained obscure. To place the statistical ensemble on a firm footing we introduce an artificial “York boundary”, with either canonical or microcanonical boundary conditions, as has been done previously for black hole ensembles. The partition function and the density of states are expressed as integrals over paths in the constrained, spherically reduced phase space of pure 3+1 dimensional gravity with a positive cosmological constant. Issues related to the domain and contour of integration are analyzed, and the adopted choices for those are justified as far as possible. The canonical ensemble includes a patch of spacetime without horizon, as well as configurations containing a black hole or a cosmological horizon. We study thermodynamic phases and (in)stability, and discuss an evolving reservoir model that can stabilize the cosmological horizon in the canonical ensemble. Finally, we explain how the Gibbons-Hawking partition function on the 4-sphere can be derived as a limit of well-defined thermodynamic ensembles and, from this viewpoint, why it computes the dimension of the Hilbert space of states within a cosmological horizon.

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

“…We derive the thermodynamic variables using the canonical Euclidean path integral for these two systems. A similar analysis was performed, for instance, for Schwarzschild black holes in [25], for two-dimensional black holes [75,76], and for SdS black holes in [74].…”

Black holes in dS$_3$

“…Euclidean action, quasi-local thermodynamics, and stability of quantum de Sitter: A first principles method for analyzing the thermodynamics of black holes is to directly compute the canonical partition function using a saddle-point approximation of the Euclidean gravitational path integral, à la Gibbons and Hawking [77]. However, the standard treatment by Gibbons-Hawking suffers from ambiguities for de Sitter spacetime, since Euclidean de Sitter has no asymptotic boundary where a temperature may be specified to define the canonical ensemble (see [78] for an in depth discussion on this point). Alternatively, one may adapt the quasi-local formalism of York [79] to de Sitter backgrounds and analyze quasi-local thermodynamics.…”

The minus sign in the first law of de Sitter horizons

“…The argument in [11] for this maximally mixed state interpretation of the Gibbons-Hawking entropy of "de Sitter space" was based on considerations of diffeomorphism invariance. In [19] this reasoning is supported by viewing this state as the limit of a canonical ensemble defined by conditions at a York boundary that shrinks to zero inside the horizon. This statistical interpretation of the minus sign is fundamental, but it does not directly clarify the thermodynamic interpretation of the first law.…”

“…Finally, a statistical explanation of the minus sign was given in [14] (see also [15,16] and [8]). It was argued there that, since the Gibbons-Hawking entropy of "de Sitter space" corresponds to the entropy of the maximally mixed state for a region surrounded by a cosmological horizon, where nothing is fixed but the topology [11], [17], [14] (see also [18][19][20]), any further specification of the state, such as the presence of a black hole or a certain amount of matter Killing energy, amounts to a "constraint" on the state. This results in a smaller entropy than that of the maximally mixed state.…”

“…To specify the thermodynamic ensemble we introduce a system boundary, inside the static patch, at which the temperature of the ensemble can be specified. This is a de Sitter version of the approach York and collaborators have taken to defining quasi-local thermodynamic ensembles containing black holes [21][22][23][24][25][26], and it has also been applied to the case of cosmological horizons [19,[27][28][29][30]. After the ensemble is defined in this way, we proceed to the limit in which the boundary shrinks to zero size, recovering the empty static patch as the background to be varied.…”