The minus sign in the first law of de Sitter horizons

TL;DR

The minus sign in the Gibbons–Hawking first law for the de Sitter static patch poses a thermodynamic puzzle. By introducing a York boundary to define a well-posed quasi-local ensemble and distinguishing Brown–York energy from matter Killing energy, the authors show that the horizon entropy variation accounts for the thermalized portion while the matter contribution may be non-thermalized; in the boundaryless limit the standard GH first law is recovered. The framework clarifies the role of generalized entropy and demonstrates, with examples in 3D SdS, how the minus sign emerges from energy accounting rather than an exotic negative temperature. Additionally, the York-boundary perspective links competing proposals for static-patch holography, suggesting a unifying quasi-local thermodynamic description of de Sitter horizons with potential holographic interpretations.

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.

Figures (5)

• Figure 1: A York boundary (dashed curve) at radius $r=R$ in the dS static patch. The boundary splits the static patch into two systems: the "pole patch", i.e., the white region between the pole and the boundary, and the "horizon patch", i.e., the shaded region from the boundary to the horizon (the terminology comes from Coleman:2021nor).
• Figure 2: Matter infalling across the past horizon $\mathcal{H}$ of a dS static patch.
• Figure 3: The Penrose diagram of Schwarzschild-de Sitter spacetime. The York boundary lies at $r=R$ between the black hole and cosmological horizon, and hence defines two distinct thermal systems: the white region between the black hole horizon and the boundary, and the shaded region between the boundary and the cosmological horizon.
• Figure 4: Heat capacity of black hole system (left) and cosmological system (right) as a function of the boundary radius $R$ for $d=4$. The graphs are meaningful only between the horizon radii (thick circles), where both heat capacities vanish. The minimum on the right lies at the Nariai radius $L/\sqrt{3}.$ Here $r_{b}=0.1$ and $L=1$.
• Figure 5: The phase diagram for cosmological horizon patches in $d=4$ spacetime dimensions, with boundary size $R$ and inverse temperature $\beta$ (figure adapted from Banihashemi:2022jys). The temperature on the upper (semicircular) boundary of the metastable $r_c=L$ region is the GH Tolman temperature at $R$. In the metastable region the configuration with $r_c=L$ is a local (endpoint) minimum of the action, while the absolute (endpoint) minimum has either $r_c = R$ or $r_c= \widehat{R}$, where $\widehat{R}$ is the radius of the other horizon when one horizon has radius $R$. The dotted path describes the $R\to 0$ sequence of metastable de Sitter patches considered in the main body of this paper. Not shown are the black hole and thermal de Sitter phases, which have higher action along the dashed path Banihashemi:2022jys.