General covariance and boundary symmetry algebras

TL;DR

The paper addresses how boundary symmetries in general relativity can produce nontrivial charges in the presence of radiation, where naive algebras develop field-dependent 2-cocycles. It shows that applying Wald-Zoupas covariance and stationarity resolves these issues for the current algebra and constrains central extensions, leaving only time-independent cocycles in charges. Through several case studies at future null infinity, including BMS, eBMS, gBMS, BMSW, and RBS, the authors construct covariant symplectic potentials and demonstrate when cocycles can be removed or persist. The results clarify the correct realization of asymptotic symmetry algebras in GR and point toward flux-based Hamiltonian generators on the radiative phase space, with potential implications for quantum gravity.

Abstract

In general relativity as well as gauge theories, non-trivial symmetries can appear at boundaries. In the presence of radiation some of the symmetries are not Hamiltonian vector fields, hence the definition of charges for the symmetries becomes delicate. It can lead to the problem of field-dependent 2-cocycles in the charge algebra, as opposed to the central extensions allowed in standard classical mechanics. We show that if the Wald-Zoupas prescription is implemented, its covariance requirement guarantees that the algebra of Noether currents is free of field-dependent 2-cocycles, and its stationarity requirement further removes central extensions. Therefore the charge algebra admits at most a time-independent field-dependent 2-cocycle, whose existence depends on the boundary conditions. We report on new results for asymptotic symmetries at future null infinity that can be derived with this approach.