The minus sign in the first law of de Sitter horizons

TL;DR

This paper clarifies the negative sign in the de Sitter horizon's first law by redefining the thermodynamic ensemble with a boundary, resolving the entropy-energy interpretation puzzle.

Contribution

It introduces a boundary-based framework to correctly interpret the thermodynamics of de Sitter horizons, resolving the minus sign puzzle in the first law.

Findings

The total internal energy (Brown-York energy) vanishes as the boundary shrinks to zero.

The horizon entropy variation accounts for thermalized matter contributions.

The first law implies stationarity of generalized entropy in equilibrium.

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.