Thermodynamics of a Spherically Symmetric Causal Diamond in Minkowski Spacetime

TL;DR

This paper treats a finite spherically symmetric causal diamond in (d+2)-dim Minkowski spacetime within a semiclassical gravitational framework. By computing the on-shell action and identifying the Euclidean path integral with a thermal partition function, the authors use the replica trick to show that the modular Hamiltonian K has mean and fluctuations both proportional to the horizon area, AB/(4 G_N). They further relate fluctuations in K to fluctuations in the replica index n through a Legendre transform, and show these modular fluctuations induce concrete geometric perturbations, such as a shift of the stretched horizon and a Shapiro-like phase shift for photons crossing the diamond. The results bolster a thermodynamic and information-theoretic view of causal horizons, with explicit, gauge-invariant observables and clear connections to entanglement entropy and density-of-states arguments, while suggesting paths to deeper microscopic understandings via quantum gravity formalisms.

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

Figures (2)

• Figure 1: A causal diamond with radius $L$, and area given by $A \equiv \Omega_dL^d$, where $\Omega_d$ is the area of a unit $d$-sphere. The complement region to the causal diamond is shaded red and is traced out when obtaining the reduced density matrix $\hat{\rho}$ associated to the causal diamond.
• Figure 2: A spacetime diagram of a causal diamond of size $L$ in $d+2$ dimensions with $d$ angular directions suppressed. Under modular fluctuations, the causal horizons of the diamond can become deformed to a stretched horizon ${\mathcal{H}}_s$, with its midpoint labeled to be at $u=u_{\text{mid}}$. The separation between ${\mathcal{H}}_s$ and the bifurcate horizon ${\mathcal{B}}$ is given by $s_{\mathcal{B}} = 2 L \sqrt{\frac{|\Delta K|}{K d}}$, where $\Delta K$ is the fluctuation of the modular Hamiltonian.