Thermodynamics of a Spherically Symmetric Causal Diamond in Minkowski Spacetime

Abstract

We compute a gravitational on-shell action of a finite, spherically symmetric causal diamond in $(d+2)$-dimensional Minkowski spacetime, finding it is proportional to the area of the bifurcate horizon $A_{\mathcal{B}}$. We then identify the on-shell action with the saddle point of the Euclidean gravitational path integral, which is naturally interpreted as a partition function. This partition function is thermal with respect to a modular Hamiltonian $K$. Consequently, we determine, from the on-shell action using standard thermodynamic identities, both the mean and variance of the modular Hamiltonian, finding $⟨K⟩=⟨(\Delta K{)}^2⟩=\frac{A_{\mathcal{B}}}{4G_N}$. Finally, we show that modular fluctuations give rise to fluctuations in the geometry, and compute the associated phase shift of massless particles traversing the diamond under such fluctuations.

Comments: 52 pages, 2 figures; v2: minor clarifications, version to appear in JHEP