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

TL;DR

The paper introduces surface-integral representations for PN source multipole moments in the outer near-zone and uses them to analyze a boosted Schwarzschild solution (BSS) at 3PN order. By comparing the BSS quadrupole with its binary-limit form, the authors fix the 3PN ambiguity parameter $\zeta$ to $-\frac{7}{33}$, arguing this value is mandated by Lorentz–Poincaré invariance. A second cross-check via the far-zone expansion and non-linear multipole interactions confirms the same $\zeta$, strengthening the consistency of the PN wave-generation formalism. The results tie the BSS limit to the physics of compact binaries, improving the theoretical basis for high-precision gravitational-wave templates and demonstrating the compatibility of Hadamard regularization with Poincaré invariance (and its equivalence to dimensional regularization in this context). Overall, the work provides robust, coordinate-aware tools for computing PN multipole moments and fixes a key 3PN ambiguity relevant to GW generation.

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: zeta=-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 zeta by a different method, based on the far-zone expansion of the BSS at fixed retarded time, and a calculation of the relevant non-linear multipole interactions in the external metric at the 3PN order.