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

Abstract

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 ${\ensuremath{\epsilon}}^{2}$, velocity as \ensuremath{\epsilon}, pressure as ${\ensuremath{\epsilon}}^{4}$, where \ensuremath{\epsilon} is the sequence parameter. We asymptotically approximate the metric at an event which, as \ensuremath{\epsilon}\ensuremath{\rightarrow}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 \ensuremath{\epsilon}\ensuremath{\rightarrow}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 ${\mathit{I}}_{\mathrm{FM}}^{+}$. This proves that the quadrupole formula is an asymptotic approximation to general relativity. We study the relation between ${\mathit{I}}_{\mathrm{\ensuremath{\epsilon}}}^{+}$, the sequence of null infinities of the individual manifolds, and ${\mathit{I}}_{\mathrm{FM}}^{+}$; 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.