### 4.4 General form of the equations of motion

From the definition of the four-momentum,
and the conservation law (67), we have the evolution equations for the four-momentum:
Here we used the fact that the size and the shape of the body zone are defined to be fixed (in the near zone
coordinates), while the center of the body zone moves at the velocity of the star’s representative
point.
Substituting the momentum-velocity relation (86) into the spatial components of Equation (110), we
obtain the general form of equations of motion for star ,

All the right hand side terms in Equation (111) except the dipole moment are expressed as surface
integrals. We can specify the value of freely to determine the representative point of star
. Up to 2.5 PN order we take and simply call the center of mass of star
. Note that in order to obtain the spin-orbit coupling force in the same form as in previous
works [46, 108, 152], we have to make another choice for (see [94] and Appendix B.1). At 3 PN
order, yet another choice of the value of the dipole moment shall be examined (see Sections 7
and 8.2).
In Equation (111), rather than the mass of star appears. Hence we have to derive a relation
between the mass and . We shall derive that relation by solving the temporal component of the
evolution equations (110) functionally.

Then, since all the equations are expressed with surface integrals except to be specified, we can
derive the equations of motion for a strongly self-gravitating star using the post-Newtonian
approximation.