Equation of motion for relativistic compact binaries with the strong field point particle limit : Formulation, the first post-Newtonian and multipole terms

TL;DR

This work develops a strong-field point-particle framework to derive the equation of motion for relativistic compact binaries within the post-Newtonian hierarchy. By adopting surface-integral expressions over body zones in harmonic gauge, the authors obtain the Einstein-Infeld-Hoffman equation at $1$PN and identify a $2$PN subset that depends on the quadrupole moments and spins of the components, thereby incorporating tidal and spin effects without regularization. They demonstrate that boundary-size dependent terms cancel, yielding a well-defined dynamics governed by suitably defined masses, spins, and quadrupole moments that include strong self-gravity. The approach provides a consistent extension of PN EOM to objects with strong internal gravity and lays groundwork for completing higher-order terms ($2.5$PN and beyond) relevant for precise gravitational-wave templates.

Abstract

We derive the equation of motion for the relativistic compact binaries in the post-Newtonian approximation taking explicitly their strong internal gravity into account. For this purpose we adopt the method of the point particle limit where the equation of motion is expressed in terms of the surface integrals. We examine carefully the behavior of the surface integrals in the derivation. As a result, we obtain the Einstein-Infeld-Hoffman equation of motion at the first post-Newtonian (1PN) order, and a part of the 2PN order which depends on the quadrupole moments and the spins of component stars. Hence, it is found that the equation of motion in the post-Newtonian approximation is valid for the compact binaries by a suitable definition of the mass, spin and quadrupole moment.