  The analysis of vacuum general relativity by R. Beig and N. O Murchadha (Ann.
Phys. vol 174, 463 (1987)) is extended in numerous ways. The weakest possible
power-type fall-off conditions for the energy-momentum tensor, the metric, the
extrinsic curvature, the lapse and the shift are determined, which, together
with the parity conditions, are preserved by the energy-momentum conservation
and the evolution equations. The algebra of the asymptotic Killing vectors,
defined with respect to a foliation of the spacetime, is shown to be the
Lorentz Lie algebra for slow fall-off of the metric, but it is the Poincare
algebra for 1/r or faster fall-off. It is shown that the applicability of the
symplectic formalism already requires the 1/r (or faster) fall-off of the
metric. The connection between the Poisson algebra of the Beig-O Murchadha
Hamiltonians and the asymptotic Killing vectors is clarified. The value H[K^a]
of their Hamiltonian is shown to be conserved in time if K^a is an asymptotic
Killing vector defined with respect to the constant time slices. The angular
momentum and centre-of-mass, defined by the value of H[K^a] for asymptotic
rotation-boost Killing vectors K^a, are shown to be finite only for 1/r or
faster fall-off of the metric. Our center-of-mass expression is the difference
of that of Beig and O Murchadha and the spatial momentum times the coordinate
time. The spatial angular momentum and this centre-of-mass form a Lorentz
tensor, which transforms in the correct way under Poincare transformations.
