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.

Modular Hamiltonian for de Sitter diamonds

“…Namely, it has been argued that the addition of any matter with its own entropy S M has the effect of decreasing the horizon area A dS , in such a way that the generalized entropy A dS /(4G N ) + S M decreases, see for example refs. [149][150][151]. In the context of the de Sitter holographic correspondence [152][153][154], the horizon entropy can be understood as the entanglement entropy between the right and left dual conformal field theories (CFTs) that appear in the correspondence, or between the past and future dual CFTs [155][156][157][158][159][160][161][162], and then the variation of the generalized entropy can be studied by considering the variation of the entanglement entropy of the CFTs when matter is added to de Sitter space.…”

An outsider’s perspective on information recovery in de Sitter space

“…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 [36,46].…”

“…The difference in sign can be attributed to the opposite sign in the first law of the horizons, see [46] 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.…”

Modave lectures on de Sitter space & holography

“…Modave Lecture Notes on de Sitter Space & Holography Damián A. Galante in [36,37], where in the latter it is shown that the first law ( 41) is recovered when shrinking the size of the York boundary towards the observer's worldline. See [38] for a recent discussion. Timelike boundaries might play an important role in defining holography in de Sitter space, so we will come back to the discussion of this problem in section 6.2.…”