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 paper derives the relativistic equations of motion for compact binary systems within the post-Newtonian framework, explicitly incorporating strong internal gravity effects through surface integral methods, including 1PN and some 2PN terms.

Contribution

It presents a novel derivation of the equations of motion for relativistic compact binaries using the strong field point particle limit and surface integrals, extending previous work to include quadrupole and spin effects.

Findings

Derived the 1PN Einstein-Infeld-Hoffman equation of motion.

Included partial 2PN terms depending on quadrupole moments and spins.

Validated the equations for compact binaries with appropriate definitions of mass, spin, and quadrupole.

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.