### B.5 Remarks

We have derived the spin-orbit coupling force, the spin-spin coupling force, and the quadrupole-orbit
coupling force to the lowest order. The spin geodesic equation could also be derived in a cruder way. Our
results agree with the previous results modulo the definition of the center of mass. It should be noted,
however, that we have defined our multipole moments and as a volume integral of which
manifestly includes the effect of strong internal gravity of the star. Thus, our results support the
applicability of the spin-orbit, spin-spin, and quadrupole-orbit coupling forces to a relativistic
compact binary where the component stars have a strong internal gravitational field. On the effect
of these multipole moments in the orbital evolution and/or the gravitational waveform (see
e.g. [6, 7, 8, 9, 15, 44, 108, 109, 110, 112, 128, 132, 148]).