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An equation of motion for relativistic compact binaries is derived through the third post-Newtonian (3PN) 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 3PN 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. 5PN radiation reaction effect), is Lorentz invariant, and is unambiguous: there exist no undetermined parameters reported in the previous works. We shall show that our 3PN equation of motion agrees physically with the Blanchet-Faye 3PN equation of motion if =-1987/3080, where is the parameter which is undetermined within their framework. This value of is consistent with the result of Damour, Jaranowski, and Sch\"afer, who first completed a 3PN iteration of the ADM Hamiltonian in the ADMTT gauge using dimensional regularization.
Y. Itoh (Thu,) studied this question.