Hadamard regularization of the third post-Newtonian gravitational wave generation of two point masses

TL;DR

This paper derives the third post-Newtonian gravitational wave moments for binary systems using Hadamard regularization, identifying key ambiguity parameters and setting the stage for further dimensional regularization calculations.

Contribution

It provides explicit 3PN mass quadrupole and dipole moments for general orbits using Hadamard regularization, including the resolution of ambiguity parameters.

Findings

Identified three ambiguity parameters in the 3PN quadrupole moment.

Established a relation between ambiguity parameters xi and kappa.

Set the foundation for complete 3PN gravitational wave calculations with dimensional regularization.

Abstract

Continuing previous work on the 3PN-accurate gravitational wave generation from point particle binaries, we obtain the binary's 3PN mass-type quadrupole and dipole moments for general (not necessarily circular) orbits in harmonic coordinates. The final expressions are given in terms of their ``core'' parts, resulting from the application of the pure Hadamard-Schwartz (pHS) self-field regularization scheme, and augmented by an ``ambiguous'' part. In the case of the 3PN quadrupole we find three ambiguity parameters, xi, kappa and zeta, but only one for the 3PN dipole, in the form of the particular combination xi+kappa. Requiring that the dipole moment agree with the center-of-mass position deduced from the 3PN equations of motion in harmonic coordinates yields the relation xi+kappa=-9871/9240. Our results will form the basis of the complete calculation of the 3PN radiation field of compact binaries by means of dimensional regularization.