Generator algebra of the asymptotic Poincaré group in the general theory of relativity

The Poisson brackets of the generators of the Hamiltonian formalism for general relativity are obtained with allowance for surface terms of arbitrary form. For Minkowski space there exists the asymptotic Poincaré group, which is the semidirect product of the Poincaré group and an infinite subgroup for which the algebra of generators with surface terms closes. A criterion invariant with respect to the choice of the coordinate system on the hypersurfaces is obtained for realization of the Poincaré group in asymptotically flat space-time. The “background” fiat metric on the hypersurfaces and Poincaré group that preserve it are determined nonuniquely; however, the numerical values of the generators do not depend on the freedom of this choice on solutions of the constraint equations. For an asymptotically Galilean metric, the widely used boundary conditions are determined more accurately. A prescription is given for application of the Arnowitt–Deser–Misner decomposition in the case of a slowly decreasing contribution from coordinate and time transformations.