Surface-integral expressions for the multipole moments of post-Newtonian sources and the boosted Schwarzschild solution

Abstract

New expressions for the multipole moments of an isolated post-Newtonian source, in the form of surface integrals in the outer near-zone, are derived. As an application we compute the ‘source’ quadrupole moment of a Schwarzschild solution boosted to uniform velocity, at the third post-Newtonian (3PN) order. We show that the consideration of this boosted Schwarzschild solution (BSS) is enough to uniquely determine one of the ambiguity parameters in the recent computation of the gravitational wave generation by compact binaries at 3PN order: ζ = −7/33. We argue that this value is the only one for which the Poincaré invariance of the 3PN wave generation formalism is realized. As a check, we confirm the value of ζ by a different method, based on the far-zone expansion of the BSS at fixed retarded time, and a calculation of the relevant nonlinear multipole interactions in the external metric at the 3PN order.