Time in gravitational subregions and in closed universes

Abstract

What are gauge-invariant local observables in a subregion in quantum gravity? How does one even define such a subregion non-perturbatively? We study these questions in JT gravity. One can define a subregion by specifying the value of the dilaton at the boundary of the region. We study conformal matter correlators in such a subregion. There is a gravitational constraint associated with York time evolution within the causal diamond of the subregion. This constraint can be leveraged to construct gauge-invariant observables in quantum gravity, using a crossed product construction. The extrinsic curvature of Cauchy slices acts as the physical clock. This is a simple example of how gauge-invariant observables can be obtained by dressing to features of a spacetime (or other fields), without the need for introducing an external observer. The entropy associated with this algebra of observables is not an area, or any boundary term. We show that gravitational constraints only give boundary formulas for entropy when gauging isometric diffeomorphisms. York time flow is merely a conformal isometry, not an actual isometry, and thus leads to bulk contributions to entropy. We repeat our construction for Milne-type closed Big-Bang universes, which may be of independent interest.

Figures (9)

• Figure 1: One of the setups we study are AdS closed universes with a Milne-type big-bang and big-crunch. There is a gravitational constraint associated with conformal time flow, closely related with York time flow. In quantum gravity, where York time of the geometry becomes a dynamical (quantized) variable, we can use York time as physical clock with respect to which matter (the remainder of the universe) evolves. We dress local matter operators to this clock instead of the clock of some dynamical observer. A physical statement is: place some local matter operator when the York time equals $K$. See section \ref{['sect:2obcl']}.
• Figure 2: The second setup we consider is the causal diamond associated with a spatial subregion in global AdS. The subregion is defined by specifying the value $\Phi_\text{bdy}$ of the dilaton at the boundaries of the subregion. These are the black dots. The blue dot shows where the dilaton attains a minimum $\Phi_h$. The boost isometry may be used to place the two endpoints of the subregion at the same global time coordinate. This diamond can be sliced using curves with a fixed extrinsic curvature $K$ (or conformal time $s$). The associated generator is a constraint, allowing us to dress local matter operators to this curvature clock. The phase space consists of the geodesic volume of the subregion, the global time of the endpoints, conformal time, and its generator. See section \ref{['sect:3diamons']}.
• Figure 3: We investigate the classical solutions of AdS JT gravity, which are Milne-type big-bang universes with finite proper time $-\pi/2<T<\pi/2$.
• Figure 4: Bra-ket wormhole contour for the Milne spacetime preparing the CFT in a thermal state with an inverse temperature $\beta=2\pi$. One can consider alternative bra-ket wormhole contours $\mathcal{C}_{n\pi}$Chen:2020tes, resulting in inverse temperature $\beta=n\pi$ for conformal matter. We briefly speculate on the consequences of this freedom in the discussion section \ref{['sect:concl']}. Should one be summing over $n$ in the no-boundary state? Maybe not JEVwipHeld:2026hujBlommaert:2025bgdAguilar-Gutierrez:2023ril.
• Figure 5: The CFT thermal state prepared by the bra-ket wormhole can be realized as a thermofield double on two copies of a closed universe. We call the original universe "closed universe" and the auxiliary copy the "reservoir". We only turn on gravity for the first copy. The red arrow indicates the flow of modular time.