General form of the equations of motion

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 $A$ ,

dvi &#x222E; &#x222E; PA&#x03C4;--A-= &#x2212; &#x03B5;&#x2212; 4 dSk &#x039B;kNi+ &#x03B5;&#x2212;4vkA dSk &#x039B;&#x03C4;Ni d &#x03C4; &#x2202;B&#x0028;A&#x222E; &#x2202;&#x222E;BA &#x0029; &#x2212; 4 i k&#x03C4; k &#x03C4;&#x03C4; +&#x03B5; v A dSk &#x039B;N &#x2212; vA dSk &#x039B;N &#x2202;BA &#x2202;BA dQiA- 2d2DiA- &#x2212; d&#x03C4; &#x2212; &#x03B5; d&#x03C4;2 . &#x0028;111 &#x0029;

All the right hand side terms in Equation (111

) 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 [46

, 108

, 152

], we have to make another choice for $zi A$ (see [94

] 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 (111 ), $PAτ$ 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 (110 ) 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.