A clock is just a way to tell the time: gravitational algebras in cosmological spacetimes

TL;DR

This work shows that nontrivial diffeomorphism-invariant algebras arise in semiclassical quantum gravity on cosmological backgrounds without external clocks, by treating the inflaton or a black hole’s timeshift as physical clocks. Using Bunch–Davies weights and modular theory, the authors construct Type II factors (Type II$_\infty$ for inflaton and SdS, with a related Type III structure for SdS’s full algebra) whose traces reproduce generalized entropy, $S_{\rm gen}=A_{\rm hor}/4G+S_{\rm QFT}$. The analysis highlights the role of out-of-equilibrium dynamics in defining gauge-invariant observables and extends the framework to generic wedges bounded by extremal surfaces, where a crossed-product structure can emerge without symmetries. Collectively, the results deepen the connection between horizon entropy, modular flows, and gauge-invariant observables in quantum gravity, linking no-boundary proposals to a universal Type II entropy for gravitational algebras.

Abstract

We study the algebra of observables in semiclassical quantum gravity for cosmological backgrounds, focusing on two key examples: slow-roll inflation and evaporating Schwarzschild-de Sitter black holes. In both cases, we demonstrate the existence of a nontrivial algebra of diffeomorphism-invariant observables \emph{without} the introduction of an external clock system or the presence of any asymptotic gravitational charges. Instead, the rolling inflaton field and the evaporating black hole act as physical clocks that allow a definition of gauge-invariant observables at $G = 0$. The resulting algebras are both Type II$_\infty$ factors, but neither is manifestly a crossed product algebra. We establish a connection between the Type II entropy of these algebras and generalized entropies for appropriate states. Our work extends previous results on Type II gravitational algebras and highlights the crucial role of out-of-equilibrium dynamics for defining gauge-invariant observables in semiclassical canonically quantised gravity. We also briefly discuss the construction of gauge-invariant algebras for compact wedges bounded by extremal surfaces in generic spacetimes (i.e. in the absence of any Killing symmetry). In contrast to the inflaton and black hole cases, this algebra does end up being a simple crossed product. No clock or asymptotic charges are required because of the absence of any symmetry in the classical background.

Figures (9)

• Figure 1: A Penrose diagram for global de Sitter space. The compact $\tau=0$ Cauchy slice is shown in red. We consider the QFT algebra $\mathcal{A}$ of operators localised in the static patch centered around $\chi = 0$. The commutant algebra $\mathcal{A}'$ describes operators in the opposite static patch.
• Figure 2: The Bunch-Davies wavefunctional $\Psi_{BD}(\phi(\Omega_3))$ is defined by a path integral on a Euclidean hemisphere with Dirichlet boundary conditions $\phi(\Omega_3)$.
• Figure 3: The formal density matrix of the Bunch-Davies weight on the algebra $\mathcal{A}$ is computed by a Euclidean path integral over the full sphere, with fixed boundary conditions on either side of a cut at the static patch centered on $\chi = 0$. This can be reinterpreted as a sequence of operators $\exp(-\varepsilon \boldsymbol{h})$ generating Euclidean rotations (i.e. the analytic continuation of Lorentzian boosts). The full density matrix describes (up to normalisation) a Euclidean rotation by $2\pi$.
• Figure 4: The metric near a Cauchy slice $\Sigma$ (the dark-blue region) of a Schwarzschild (or other asymptotically flat) black hole (left) can be smoothly connected to time-independent metrics on an asymptotically flat space (the light-blue region on the right). This allows us to construct an unambiguous Hilbert space $\mathcal{H}_{\rm QFT}$ for the black hole.
• Figure 5: Penrose diagrams for Schwarzschild-de Sitter black hole, with the red slice being a spacelike geodesics that winds around the closed universe. The left and right edges are periodically identified as shown. The dashed lines indicate the direction of the boosts. On the left, the black hole bifurcation surface is placed in the middle, while on the right the cosmological bifurcation surface is placed there.