Equation of motion for relativistic compact binaries with the strong field point particle limit: Third post-Newtonian order

TL;DR

This paper derives a third post-Newtonian order equation of motion for relativistic compact binaries using a strong field point particle limit, ensuring Lorentz invariance and unambiguity, and compares it with previous results.

Contribution

It introduces a new 3 PN equation of motion for compact binaries that is Lorentz invariant, unambiguous, and agrees with prior work when a specific parameter is set.

Findings

Derived a 3 PN equation of motion using strong field point particle limit.

Ensured the equation admits a conserved energy and is Lorentz invariant.

Found agreement with previous 3 PN results for a specific parameter value.

Abstract

An equation of motion for relativistic compact binaries is derived through the third post-Newtonian (3 PN) approximation of general relativity. The strong field point particle limit and multipole expansion of the stars are used to solve iteratively the harmonically relaxed Einstein equations. We take into account the Lorentz contraction on the multipole moments defined in our previous works. We then derive a 3 PN acceleration of the binary orbital motion of the two spherical compact stars based on a surface integral approach which is a direct consequence of local energy momentum conservation. Our resulting equation of motion admits a conserved energy (neglecting the 2.5 PN radiation reaction effect), is Lorentz invariant and is unambiguous: there exist no undetermined parameter reported in the previous works. We shall show that our 3 PN equation of motion agrees physically with the Blanchet and Faye 3 PN equation of motion if $\lambda = - 1987/3080$, where $\lambda$ is the parameter which is undetermined within their framework. This value of $\lambda$ is consistent with the result of Damour, Jaranowski, and Sch\"afer who first completed a 3 PN iteration of the ADM Hamiltonian in the ADMTT gauge using the dimensional regularization.