Entropy of causal diamond ensembles

TL;DR

This paper develops a statistical mechanics framework for gravitational causal diamonds by defining a canonical ensemble with a boundary, analyzing saddle points, and connecting to entropy calculations, with implications for different gravity theories.

Contribution

It introduces a canonical ensemble for causal diamonds with a boundary, providing a new approach to understanding their entropy in various gravity theories.

Findings

High-temperature regime yields an approximate saddle with a horizon.

Boundary terms in the Euclidean action relate to causal diamond entropy.

Exact saddles exist in theories with positive cosmological constant.

Abstract

We define a canonical ensemble for a gravitational causal diamond by introducing an artificial York boundary inside the diamond with a fixed induced metric and temperature, and evaluate the partition function using a saddle point approximation. For Einstein gravity with zero cosmological constant there is no exact saddle with a horizon, however the portion of the Euclidean diamond enclosed by the boundary arises as an approximate saddle in the high-temperature regime, in which the saddle horizon approaches the boundary. This high-temperature partition function provides a statistical interpretation of the recent calculation of Banks, Draper and Farkas, in which the entropy of causal diamonds is recovered from a boundary term in the on-shell Euclidean action. In contrast, with a positive cosmological constant, as well as in Jackiw-Teitelboim gravity with or without a cosmological constant, an exact saddle exists with a finite boundary temperature, but in these cases the causal diamond is determined by the saddle rather than being selected a priori.