On the roots of the Poincare structure of asymptotically flat spacetimes

TL;DR

This paper extends the analysis of asymptotically flat spacetimes in general relativity, clarifying the conditions for the Poincare structure, and refining the definitions of conserved quantities like energy, momentum, and angular momentum.

Contribution

It establishes minimal fall-off conditions for the metric and related fields that preserve the Poincare algebra and clarifies the connection between Hamiltonian conserved quantities and asymptotic symmetries.

Findings

The Poincare algebra applies for 1/r or faster fall-off of the metric.

Conserved quantities like angular momentum are finite under these fall-off conditions.

The spatial angular momentum and center-of-mass form a Lorentz tensor.

Abstract

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.