# Gravitational Thermodynamics of Causal Diamonds in (A)dS

## TL;DR

This paper extends gravitational thermodynamics to all causal diamonds in maximally symmetric spacetimes, deriving a Smarr formula and first law involving geometric and matter variations, and discusses quantum corrections and entanglement equilibrium.

## Contribution

It generalizes thermodynamic relations for causal diamonds beyond static patches, incorporating conformal Killing vectors and quantum effects.

## Key findings

- Established a Smarr formula for causal diamonds.
- Derived a first law relating area, volume, cosmological constant, and matter.
- Recovered entanglement equilibrium as a quantum correction result.

## Abstract

The static patch of de Sitter spacetime and the Rindler wedge of Minkowski spacetime are causal diamonds admitting a true Killing field, and they behave as thermodynamic equilibrium states under gravitational perturbations. We explore the extension of this gravitational thermodynamics to all causal diamonds in maximally symmetric spacetimes. Although such diamonds generally admit only a conformal Killing vector, that seems in all respects to be sufficient. We establish a Smarr formula for such diamonds and a "first law" for variations to nearby solutions. The latter relates the variations of the bounding area, spatial volume of the maximal slice, cosmological constant, and matter Hamiltonian. The total Hamiltonian is the generator of evolution along the conformal Killing vector that preserves the diamond. To interpret the first law as a thermodynamic relation, it appears necessary to attribute a negative temperature to the diamond, as has been previously suggested for the special case of the static patch of de Sitter spacetime. With quantum corrections included, for small diamonds we recover the "entanglement equilibrium" result that the generalized entropy is stationary at the maximally symmetric vacuum at fixed volume, and we reformulate this as the stationarity of free conformal energy with the volume not fixed.