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

We develop a framework based on the covariant phase space formalism that identifies gravitational edge modes as dynamical reference frames. Previously considered in gauge theory [arXiv:2109.06184], this construction relies on a systematic post-selection procedure. Given a global variational principle, it produces consistent dynamics for a spacetime subregion bounded by a codimension-one timelike submanifold. By serving as dynamical reference frames for the gauge group, edge modes enable the imposition of gauge-invariant boundary conditions. In generally covariant theories, where the gauge group is a group of diffeomorphisms, such reference frames play an even more fundamental role, as they allow one to identify the subregion in a gauge-invariant manner. This leads to the introduction of a frame-dressed spacetime, that we call relational spacetime, and which serves as the primary arena in which to perform post-selection. Just like in gauge theory, requiring the subregion presymplectic structure to be conserved by the dynamics leads to an essentially unique prescription and unambiguous Hamiltonian charges. Gauge diffeomorphisms are integrable, irrespective of how they behave in the vicinity of the time-like boundary, and generate a first-class constraint algebra. By contrast, diffeomorphisms acting on relational spacetime are in general physical, and those that are parallel to the timelike boundary are integrable. Upon further restriction to relational diffeomorphisms that preserve the boundary conditions, we obtain a subalgebra of conserved charges. Finally, we explain how the boundary conditions and the conserved presymplectic structure can both be encoded into suitable boundary actions. While our formalism applies to any generally covariant theory, we illustrate it on vacuum general relativity, and conclude with a detailed comparison of our findings to earlier works.Comment: 52+10 pages, 4 figures, 2 table