Edge modes as dynamical frames: charges from post-selection in generally covariant theories

TL;DR

The paper develops a covariant phase space framework in which gravitational edge modes are reinterpreted as dynamical reference frames that define subregions in a gauge-invariant way. By post-selecting global dynamics, it constructs a conserved regional presymplectic structure and derives integrable, first-class diffeomorphism charges for the spacetime sector, while frame reorientations on relational spacetime yield physically meaningful boundary charges under appropriate boundary conditions. The approach provides explicit boundary actions and demonstrates their realization in Einstein–Hilbert gravity, clarifying connections to Brown–York charges and previous edge-mode proposals. It highlights relational spacetime symmetries, boundary algebras, and the role of boundary conditions in shaping charges, with potential implications for quantum gravity and holography.

Abstract

We develop a framework based on the covariant phase space formalism that identifies gravitational edge modes as dynamical reference frames. They enable the identification of the associated spacetime region and the imposition of boundary conditions in a gauge-invariant manner. While recent proposals considered the finite region in isolation and sought the maximal symmetry algebra compatible with that perspective, we regard it as a subregion embedded in a global spacetime and study the symmetries consistent with such an embedding. This clarifies that the frame, although appearing as "new" for the subregion, is built out of the field content of the complement. Given a global variational principle, this also permits us to invoke a systematic post-selection procedure, previously used in gauge theory [arXiv:2109.06184], to produce consistent dynamics for a subregion with timelike boundary. Requiring the subregion presymplectic structure to be conserved by the dynamics leads to an essentially unique prescription and unambiguous Hamiltonian charges. Unlike other proposals, this has the advantage that all spacetime diffeomorphisms acting on the subregion remain gauge and integrable, thus generating a first-class constraint algebra. By contrast, diffeomorphisms acting on the frame-dressed spacetime are physical, and those that are parallel to the boundary are integrable. Further restricting to ones preserving the boundary conditions yields an algebra of conserved charges. These record changes in the relation between the region and its complement as measured by frame reorientations. Finally, we explain how the boundary conditions and presymplectic structure can be encoded into boundary actions. While our formalism applies to any generally covariant theory, we illustrate it on general relativity, and conclude with a detailed comparison of our findings to earlier works. [abridged]

Figures (5)

• Figure 1: Given a set of reference events $\{ E(A), A=0, \ldots, d \}$, one can introduce a dynamical reference frame $U$ in terms of the geodesic distances $U^{A}(p):= d(p, E(A))$.
• Figure 2: If ${\mathcal{M}}$ has a time-like boundary $\Gamma_0$, a dynamical reference frame $U(p):= (y(p),z(p))$ can be defined in terms of field-dependent tangential ($y$) and radial ($z$) coordinates.
• Figure 3: Background subregions and hypersurfaces are first defined in the space of observabes $(\mathfrak{m}, U^\star g)$ (right). Their preimages by $U$ in the spacetime manifold $({\mathcal{M}}, g)$ are dynamical (left).
• Figure 4: A subregion $m$ on relational spacetime, is deliniated by boundaries labelled as shown.
• Figure 5: The boundary charges computed on $s_1$ and $s_2$ only differ by an integral over the portion of the time-like boundary $\gamma_{s_1 s_2}$ they delimit, as specified by equation \ref{['eq:balance_on-shell']}.