Thermodynamics of a Spherically Symmetric Causal Diamond in Minkowski Spacetime

TL;DR

This paper calculates the thermodynamic properties of a spherical causal diamond in Minkowski space, revealing a connection between gravitational action, horizon area, and modular Hamiltonian fluctuations, with implications for spacetime geometry fluctuations.

Contribution

It provides a novel computation of the on-shell gravitational action for a causal diamond and links it to thermodynamic and quantum fluctuations of the modular Hamiltonian.

Findings

On-shell action proportional to horizon area $A_{\mathcal{B}}$

Mean and variance of modular Hamiltonian equal to $A_{\mathcal{B}}/4G_N$

Modular fluctuations induce measurable geometric phase shifts

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 $\langle K \rangle = \langle (\Delta K)^2 \rangle = \frac{A_{\mathcal{B}}}{4 G_N}$. 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.