Time in gravitational subregions and in closed universes

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.