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4.4 General form of the equations of motion

From the definition of the four-momentum,
μ ∫ τμ PA (τ) = ε2 d3αA ΛN , (109 ) BA
and the conservation law (67View Equation), we have the evolution equations for the four-momentum:
μ ∮ ∮ dPA- = − ε−4 dSk Λk μ+ ε− 4vk dSk Λ τμ. (110 ) dτ ∂BA N A ∂BA N
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 (86View Equation) into the spatial components of Equation (110View Equation), we obtain the general form of equations of motion for star A,

dvi ∮ ∮ PAτ--A-= − ε− 4 dSk ΛkNi+ ε−4vkA dSk ΛτNi d τ ∂B(A∮ ∂∮BA ) − 4 i kτ k ττ +ε v A dSk ΛN − vA dSk ΛN ∂BA ∂BA dQiA- 2d2DiA- − dτ − ε dτ2 . (111 )
All the right hand side terms in Equation (111View Equation) except the dipole moment are expressed as surface integrals. We can specify the value of i D A freely to determine the representative point i zA(τ) of star A. Up to 2.5 PN order we take DiA = 0 and simply call ziA the center of mass of star A. Note that in order to obtain the spin-orbit coupling force in the same form as in previous works [46Jump To The Next Citation Point108Jump To The Next Citation Point152Jump To The Next Citation Point], we have to make another choice for zi A (see [94Jump To The Next Citation Point] and Appendix B.1). At 3 PN order, yet another choice of the value of the dipole moment i D A shall be examined (see Sections 7 and 8.2).

In Equation (111View Equation), PAτ rather than the mass of star A appears. Hence we have to derive a relation between the mass and PτA. We shall derive that relation by solving the temporal component of the evolution equations (110View Equation) functionally.

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

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