### B.4 Spin geodesic precession

The spin precession can be evaluated using the following equation:
Evaluating the surface integrals in and (appearing through the momentum-velocity relation),
we have up to
Note that there is no monopole-monopole coupling.
In our formalism, the above form is sufficient since we use , not the spin vector . To transform
the above equation into the usual form, we take a “crude” method; we shall treat one star, say, the star 1 as
if it felt only the gravitational field of the companion star. Motivated by the formalism on extended bodies
by Dixon [69], we introduce the intrinsic spin four-vector and the intrinsic spin tensor as

where is the totally anti-symmetric symbol with . is the proper volume element
satisfying . is a three-sphere surrounding star whose normal is and whose
radius is . is the spacelike four-vector satisfying . The four-momentum is
normalized as . The above definitions imply the spin supplementary condition
where is the intrinsic dipole moment. Now we construct a coordinate transformation from the near
zone to the Fermi normal coordinates (see e.g. Section 40.7 in [122]),
Then we express the intrinsic spin tensor in the Fermi normal coordinates in terms of the moments in the
near zone. Using , we have
Now defining the intrinsic center of mass by setting , we obtain
Notice that this relation provides the spin-orbit coupling force in the previous form (see Appendix B.1; thus
we get in the present treatment). Using Equation (228), we finally obtain the spin
geodesic equation in the usual form (we omit the quadrupole and term for simplicity),
where is the spatial part of the intrinsic spin four-vector . Equation (239) is the
geodesic precession equation, or called the de Sitter-Fokker precession (see e.g. Section 40.7 in [122]
and [36, 145]).