In this section, we give a simple and divergence free derivation of the equations of motion for a small compact object with mass around any sort of massive body, following the paper by Fukumoto and co-authors [78]. The object moves along the geodesic determined by the smooth part of the geometry around the object up to order . We note that the smooth part includes the gravitational waves emitted by the orbital motion of the object. Thus it generalizes the equivalence principle for such a compact object to the emission of gravitational waves. Also our derivation of the equations of motion provides a useful method to explicitly calculate the radiation reaction in the fast motion of a compact body.

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