Gravitational Thermodynamics of Causal Diamonds in (A)dS

TL;DR

The paper extends gravitational thermodynamics to causal diamonds in maximally symmetric spacetimes by leveraging a conformal Killing vector to derive a Smarr formula and a first law that include cosmological-constant and matter variations. It reveals a consistent framework with a negative temperature interpretation and shows how quantum corrections lead to a generalized entropy that remains stationary in an entanglement-equilibrium sense. By connecting the conformal Killing energy to entanglement and York-time concepts, the work unifies local causal-diamond thermodynamics with semiclassical gravity and offers limiting-case insight for dS, Minkowski, Rindler, AdS-Rindler, and Wheeler-DeWitt patches. These results illuminate the thermodynamic character of spacetime regions and suggest holographic and Euclidean-path-integral avenues for future exploration.

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.

Figures (6)

• Figure 1: A causal diamond in a maximally symmetric spacetime for a ball-shaped spacelike region $\Sigma$. The past and future vertices of the diamond are denoted by $p$ and $p'$, respectively, and $\mathcal{H}$ is the null boundary. The dashed arrows are the flow lines of the conformal Killing vector $\zeta$, whose flow sends the boundary of the diamond into itself, and vanishes at $\partial \Sigma$, $p$ and $p'$.
• Figure 2: The $(x,s)$ coordinate chart of a maximally symmetric causal diamond. The coordinate $s \in (-\infty, \infty)$ is the conformal Killing time, defined as the function that vanishes on the maximal slice $\Sigma$ and satisfies $\zeta \cdot ds =1$. The coordinate $x \in [0, \infty)$ is spherically symmetric and satisfies $\zeta\cdot dx = 0$ and $|dx|=|ds|$. Constant $s$ and $x$ lines are plotted at equal coordinate intervals of $0.5$. See Appendix \ref{['appyork']} for a demonstration that $ds$ and $dx$ are everywhere orthogonal, and for the line element in these coordinates.
• Figure 3: A causal diamond in the de Sitter space static patch. The conformal Killing vector $\zeta$ turns into the timelike Killing vector $\xi$ if the boundary of the diamond coincides with the cosmological horizon $\mathcal{H}_C$.
• Figure 4: (a) A causal diamond associated to a ball at the center of Minkowski space. If one the normalizes the conformal Killing vector that preserves the diamond such that $\zeta^2 = -1$ at the center of the ball, then it becomes identical to the timelike Killing vector $\xi=\partial_t$ in the infinite-size limit. (b) A causal diamond whose edge touches the bifurcation surface of two Rindler horizons in Minkowski space. In the infinite-size limit the diamond coincides with the right Rindler wedge, and the conformal Killing vector $\zeta$ becomes the boost Killing vector $\xi$.
• Figure 5: Sequence of causal diamond edges anchored at the origin and with growing radius, shown on a time slice of compactified 2+1 dimensional Minkowski spacetime. In the infinite radius limit, the entire Rindler horizon subtends an infinitesimal angle of the diamond's edge, and the remainder of the edge lies at spatial infinity.