Symmetry charges on reduced phase space and BMS algebra

This paper studies the reduced phase space formulation (relational formalism) of gravity coupling to the Brown-Kucha\v r dust for asymptotic flat spacetimes. A set of boundary conditions for the asymptotic flatness are formulated for Dirac observables on the reduced phase space. The physical Hamiltonian generates the time translation of the dust clock. We compute the boundary term of the physical Hamiltonian, which is identical to the ADM mass. We construct a set of the symmetry charges on the reduced phase space, which are conserved by the physical Hamiltonian evolution. The symmetry charges generate transformations preserving the asymptotically flat boundary condition. Under the reduced-phase-space Poisson bracket, the symmetry charges form an infinite dimensional Lie algebra $\mathscr{A}_G$ after adding a central charge. A suitable quotient of $\mathscr{A}_G$ closely relates to the BMS algebra at spatial infinity by Henneaux and Troessaert.