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

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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 compact binaries by a suitable definition of the mass, spin, and quadrupole moment.