We have adopted the generalized Fermi normal coordinates (GFC)  as the star’s reference coordinates to address this problem. A question specific to our formalism is whether the differences between the multipole moments defined in Equations (81, 82, 83) and the multipole moments in GFC give purely monopole terms. If so, we of course have to take into account such terms in the field to compute the equations of motion for two intrinsically spherical star binaries. This problem is addressed in the Appendix C of  and the differences are mainly attributed to the shape of the body zone. The body zone which is spherical in the near zone coordinates (NZC) is not spherical in the GFC mainly because of a kinematic effect (Lorentz contraction). To derive the 3 PN equations of motion, it turns out that it is sufficient to compute the difference in the STF quadrupole moment up to 1 PN order. The result is
As is obvious from Equation (106), this difference appears even if the companion star does not exist. We note that we could derive the 3 PN metric for an isolated star moving at a constant velocity using our method explained in this section by simply letting the mass of the companion star be zero. Actually, the above is a necessary term which makes the so-obtained 3 PN metric equal to the Schwarzschild metric boosted at the constant velocity in harmonic coordinates.
The reason why the influence of a body’s Lorentz contraction appears starting from 3 PN order is as follows. The body’s Lorentz contraction appears as an apparent deformation of the body, namely, apparent quadrupole moment as the leading order in the frame where the body is moving. The body’s (real) quadrupole affects the field from 2 PN order through (see Equation (78))
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