Gravitational radiation and the validity of the far-zone quadrupole formula in the Newtonian limit of general relativity

Details Gravitational radiation and the validity of the far-zone quadrupole formula in the Newtonian limit of general relativity Gravitational radiation and the validity of the far-zone quadrupole formula in the Newtonian limit of general relativity

Nov 1985

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1985phrvd..32.2557f&link_type=abstract

Physical Review D (Particles and Fields), Volume 32, Issue 10, 15 November 1985, pp.2557-2565

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We examine the gravitational radiation emitted by a sequence of spacetimes whose near-zone Newtonian limit we have previously studied. The spacetimes are defined by initial data which scale in a Newtonian fashion: the density as ɛ2, velocity as ɛ, pressure as ɛ4, where ɛ is the sequence parameter. We asymptotically approximate the metric at an event which, as ɛ-->0, remains a fixed number of gravitational wavelengths distant from the system and a fixed number of wave periods to the future of the initial hypersurface. We show that the radiation behaves like that of linearized theory in a Minkowski spacetime, since the mass of the metric vanishes as ɛ-->0. We call this Minkowskian far-zone limiting manifold FM; it is a boundary of the sequence of spacetimes, in which the radiation carries an energy flux given asymptotically by the usual far-zone quadrupole formula (the Landau-Lifshitz formula), as measured both by the Isaacson average stress-energy tensor in FM or by the Bondi flux on I+FM. This proves that the quadrupole formula is an asymptotic approximation to general relativity. We study the relation between I+ɛ, the sequence of null infinities of the individual manifolds, and I+FM; and we examine the gauge-invariance of FM under certain gauge transformations. We also discuss the relation of this calculation with similar ones in the frame-work of matched asymptotic expansions and others based on the characteristic initial-value problem.

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