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 ![d3Σ [αuA β] = 0](article1096x.gif)
. 
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]).
), we finally obtain the spin
geodesic equation in the usual form (we omit the quadrupole and 
![ij dM-A- −2 [i j] −2 [ij] dτ = − 2ε vAP A − 2ε R A . (227 )](article1083x.gif){kind=small}
![( ) ( ) -d- M ij1 + 2v[1iDj1] = ε2m2- 2n[1i2M j1]kvk1 − 4v[i1M j1]knk12 − 4n [i12M j1]kvk2 + 4v[2iM j1]knk12 dτ r12 26m2- k [i j]k + ε r3 n12n12I1 12 ( ) + ε22m2- nk ni Zk[jl]l− nk nj Zk [il]l+ nk ni Zj [kl]l− nk ni Zj [kl]l r312 12 12 1 12 12 1 12 12 1 12 12 1 + 𝒪 (ε3). (228 )](article1087x.gif)
![2ε−6 ∫ ℳ μAν = √---- d3Σ ρσ[AμΛνN]ρ, (229 ) − g ℬA 1- ρσ α 𝒮Aμ = 2εαρσμℳ A u A, (230 )](article1092x.gif)
![[( 1 2m ) 1 ] ℳ ˆıˆτ1 = 1 + -ε2v21 − ε2---2 δij − -ε2vi1vj1 Dj1 − ε2M i1kvk1 + 𝒪 (ε3), (236 ) 2 r12 2 ˆıˆj ij 2 k [i j]k 3 ℳ 1 = M 1 + ε v1v1M 1 + 𝒪 (ε ). (237 )](article1110x.gif)
![[( ) ] d ⃗𝒮1 2m2 3 3 ---- = ε -2- 2⃗v2 − -⃗v1 × ⃗n12 × 𝒮⃗1 + 𝒪 (ε ), (239 ) dτ r12 2](article1115x.gif){kind=small}