Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance

TL;DR

This work develops a fully non-perturbative, gauge-invariant framework for gravity based on dynamical reference frames and relational observables. Central to the construction is the universal dressing space and relational atlases, which enable frame-dependent yet gauge-invariant descriptions of bulk physics and locality. The authors prove bulk microcausality for relational observables, establish a non-perturbative notion of relational locality, and show equivalence between covariant, canonical, and single-integral formulations. They illustrate the framework with geodesic dressings, parametrised field theory, Brown–Kuchař dust, and minimal-surface dressings, and discuss implications for quantum gravity and holography. Overall, the paper presents a physically meaningful update to general covariance via dynamical frame covariance and relational locality, reconciling bulk local dynamics with gauge invariance and opening routes to quantum generalisations and AQFT-style treatments.

Abstract

We describe a completely general and fully non-perturbative framework for constructing dynamical reference frames in generally covariant theories, and for understanding the gauge-invariant observables that they yield. Our approach makes use of a 'universal dressing space', which contains as a subset every possible dynamical frame. We describe examples of such frames, including matter frames, a popular construction based on boundary-anchored geodesics and one using minimal surfaces -- but our formalism does not depend on the existence of a boundary. The class of observables we construct generalises and unifies the dressed and relational approaches to constructing gravitational observables, including single-integral and canonical power-series constructions. All these (possibly gravitationally charged) relational observables describe physics in a precise sense relative to the dynamical frame and respect a notion of 'relational' locality based on the relationships between fields. By using 'relational atlases', i.e. collections of dynamical frames glued together by field-dependent maps (which are relational observables too), we can construct relationally local observables throughout spacetime. This further establishes a framework for dynamical frame covariance that permits us to change between arbitrary relational frame perspectives. Relational locality obeys many desirable properties: we prove that it satisfies microcausality in the bulk (in tension with previous work done mainly in a perturbative setting which we comment on), and show that it permits a relational version of local bulk dynamics. Relational locality is therefore arguably more physically meaningful than the ordinary notion of locality. Thus, our formalism -- which we argue to be an updated, gauge-invariant version of general covariance -- refutes the commonly claimed non-existence of local gravitational bulk physics.

Figures (11)

• Figure 2.1: Shooting a geodesic in from the boundary $\partial\mathcal{M}$ to construct a dressing in $\mathcal{M}$.
• Figure 2.2: Different choices of sets of values taken by the parameters $\tau$, $z$ and $W$ lead to different sets of geodesic dressings. Each such set may be viewed as a different frame. For example, here we choose a maximum length $\tau_1$, and a vector field $W_1$ in a boundary subregion $\mathcal{U}_1$. Then we vary over all $z\in\mathcal{U}_1$ and $\tau\in[0,\tau_1)$, and set $W=W_1(z)$ at $z$. The region of spacetime $\mathcal{N}_1[g]$ accessible by the frame is called its image.
• Figure 2.3: A vector $v$ on $\mathscr{O}$ is an infinitesimal change in the parameters of a dressing, $(\tau,z,W)\to(\tau',z',W')$. Pushing $v$ forward to spacetime with the frame yields a spacetime vector $V[g]$ that gives the corresponding infinitesimal change in the end of the geodesic.
• Figure 2.4: Given two frames of geodesics specified by vector fields $W_1,W_2$ in boundary subregions $\mathcal{U}_1,\mathcal{U}_2$, the map $R_{1\to 2}[g]$ implementing the change of frames involves going along one geodesic tangent to $W_1-\hat{n}$ from $\mathcal{U}_1$ into the bulk, and then along a second geodesic to $\mathcal{U}_2$ where it is tangent to $W_2-\hat{n}$. This gives a map $\tau_1,z_1\mapsto \tau_2,z_2$.
• Figure 3.1: A frame $\mathscr{R}$ is a subset of the universal dressing space $\mathscr{D}$. A parametrisation of the frame is a bijection $R$ from a space of local orientations $\mathscr{O}$ to $\mathscr{R}$. For each kinematical field configuration $\phi$, the image of the frame is the subset $\mathcal{N}[\phi]=\pi[\phi](\mathscr{R})$ of spacetime $\mathcal{M}$ obtained by evaluating all the dressings in the frame. If the frame is injective, then we can invert $R[\phi]$ on its image to obtain $(R[\phi])^{-1}$, the dynamical frame field on the region of spacetime $\mathcal{N}[\phi]$.