On the roots of the Poincaré structure of asymptotically flat spacetimes

The analysis of canonical vacuum general relativity by Beig and Ó Murchadha (1987 Ann. Phys., NY 174 463–498) is extended in numerous ways. The weakest possible power-type fall-off conditions for the energy–momentum tensor of the matter fields, 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 law T^ab_;b = 0 and the evolution equations for the geometry. 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 Poincaré 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–Ó Murchadha Hamiltonians (and, in particular, the constraint algebra) and the asymptotic Killing vectors is clarified. Their Hamiltonian H[K^a] is shown to be constant in time modulo constraints for those asymptotic Killing vectors K^a that are defined with respect to the foliation by the constant time slices.

The energy–momentum and angular momentum are defined by the boundary term Q[K^a] in H[K^a] even in the presence of matter. Although the energy–momentum is well defined even for slightly faster than r^−1/2 fall-off, we show that the angular momentum and centre-of-mass are finite only if the metric falls off as 1/r or faster. Q[K^a] is constant in time for those K^a that are asymptotic Killing vectors with respect to the foliation by the constant time slices. If the foliation corresponds to proper time evolution (i.e., its lapse tends to 1 at infinity), then Q[K^a] reproduces the ADM energy, the spatial momentum and spatial angular momentum, but the centre-of-mass deviates from that of Beig and Ó Murchadha by the spatial momentum times the coordinate time. The spatial angular momentum and the new centre-of-mass form an anti-symmetric Lorentz tensor, which transforms in the expected way under Poincaré transformations.