  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.
