The minus sign in the first law of de Sitter horizons

Abstract

Due to a well-known, but curious, minus sign in the Gibbons-Hawking first law for the static patch of de Sitter space, the entropy of the cosmological horizon is reduced by the addition of Killing energy. This minus sign raises the puzzling question how the thermodynamics of the static patch should be understood. We argue the confusion arises because of a mistaken interpretation of the matter Killing energy as the total internal energy, and resolve the puzzle by introducing a system boundary at which a proper thermodynamic ensemble can be specified. When this boundary shrinks to zero size the total internal energy of the ensemble (the Brown-York energy) vanishes, as does its variation. Part of this vanishing variation is thermalized, captured by the horizon entropy variation, and part is the matter contribution, which may or may not be thermalized. If the matter is in global equilibrium at the de Sitter temperature, the first law becomes the statement that the generalized entropy is stationary.

Interpolating geometries and the stretched dS2 horizon

“…finding that with Dirichlet boundary conditions, the black hole horizon of the near-Nariai geometry has positive specific heat while the cosmological horizon has negative specific heat [27]. This also resonates with the results of [20,39]. In consequence, even though the cosmological saddle has lower free energy, it is thermally unstable in the setup under consideration.…”

“…In this paper, we explore mechanisms for stabilising a portion of the dS 2 static patch within a near-AdS 2 world with ordinary boundary behaviour for the dilaton. We study Dirichlet boundary conditions on the geometry at a finite proper distance from the horizon, following the method of York [14,15] applied to the de Sitter case [16][17][18][19][20]. 1 The temperature is fixed to be the Tolman temperature [26], which is the proper length of the Euclidean boundary S 1 .…”

“…One approach is to set up a type of timelike Dirichlet boundary near the cosmological horizon, as explored in[16][17][18][19][20]. However, at least in four and higher spacetime dimensions, caution must be exercised since this leads to various instabilities and issues with well-posedness of the Dirichlet problem[36,37].…”

The price of curiosity: information recovery in de Sitter space

“…where the upper sign corresponds to Rindler space and the lower sign to de Sitter space. The difference in sign can be attributed to the opposite sign in the first law of the horizons, see [45] for a recent discussion. Thus, the non-equilibrium state (2.32) contains a positive flux of Killing energy with respect to a de Sitter observer, but a negative one from the perspective of a Rindler observer.…”

“…This asymptotic region gives us an outside perspective of the cosmological horizon, evading some of the thorny issues of defining a subsystem in de Sitter space where the island formula can be applied to. The use of auxiliary subsystems has recently proven to be useful to study de Sitter thermodynamics [36,45].…”