The Poincaré Structure and the Centre-of-Mass of Asymptotically Flat Spacetimes
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The asymptotic symmetries and the conserved quantities of asymptotically flat spacetimes are investigated by extending the canonical analysis of vacuum general relativity of Beig and Ó Murchadha. It is shown that the algebra of asymptotic Killing symmetries, defined with respect to a given foliation of the spacetime, depends on the fall-off. rate of the metric. It is only the Lorentz Lie algebra for slow fall-off, but it is the Poincaré algebra for 1/r or faster fall-off. value of the Beig–Ó Murchadha Hamiltonian with lapse and shift corresponding to asymptotic Killing vectors. While this energy-momentum and spatial angular momentum reproduce the familiar ADM energy-momentum and Regge–Teitelboim angular momentum, respectively, the centre-of-mass deviates from that of Beig and Ó Murchadha. The new centre-of-mass is conserved, and, together with the spatial angular momentum, form an anti-symmetric Lorentz tensor which transforms just in the correct way under asymptotic Poincaré transformations of the asymptotically Cartesian coordinate system.
KeywordsAngular Momentum Minkowski Spacetime Asymptotic Symmetry Spacelike Hypersurface Poisson Algebra
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