Dressed Subsystems in Classical Gravity

This paper considers the problem of consistently defining subsystems in gravitational theories. It is argued that a subsystem is a spacetime subregion in which the observables form a closed Poisson algebra. In a generally covariant theory, the location of the subregion must be determined in relation to other degrees of freedom. It is proposed that these degrees of freedom should live within the region, so that an observer can determine its edge by only measuring fields inside of it. This turns out to be equivalent to the property that observables in the subregion generate field-dependent gauge transformations on the causal complement. Furthermore, it is demonstrated that this is \textit{exactly} what is necessary for the observables to form a Poisson algebra and thus to constitute a consistent subsystem. Observables in spacelike separated "dressed subsystems" are shown to commute. Several examples are given in the context of General Relativity. Along the way, new perspectives on the covariant phase space formalism are introduced that clarify well-known issues, such as the factorization of subregions in gauge theories and the unambiguous definition of Noether charges associated with one-sided boosts. Finally, prospects for extending these results to a perturbative quantum setting are discussed.