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.
Highlights
In this paper we explored aspects of the gravitational thermodynamics of causal diamonds in maximally symmetric spacetimes and their first order variations
Our starting point was the notion that the maximally symmetric diamonds behave as thermodynamic equilibrium states
This is initially motivated by the examples offered by the static patch of de Sitter spacetime and the Rindler wedge of Minkowski spacetime, which are special cases of causal diamonds admitting a true Killing field
Summary
Horizon thermodynamics was first discovered in the context of black holes [1,2,3], but the principles are far more universal than that. In the present paper we generalize the first law of causal diamonds to (Anti-)de Sitter spacetime – i.e. it applies to any maximally symmetric space – and we include variations of the cosmological constant and matter stress-energy, both using a fluid description as done for black holes by Iyer [30]. At fixed volume V and cosmological constant Λ this implies that the generalized entropy is stationary in a maximally symmetric vacuum This coincides with the entanglement equilibrium condition of [21], which was shown in that paper to be equivalent to the semiclassical Einstein equation. The appendices are devoted to establishing several properties of conformal Killing fields in (A)dS and in flat space, and of bifurcate conformal Killing horizons in general
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