We now derive the 3 PN momentum-velocity relation by calculating the integral at 3 PN order. From the definition of the integral, Equation (91),
of appears at the 4 PN or higher order field. Thus up to 3 PN order, affects the equations of motion only through the 3 PN momentum-velocity relation. For this reason, not but is necessary to derive the 3 PN equations of motion. The explicit expression for is given in .
Now with in hand, we obtain the momentum-velocity relation. It turns out that the part of the momentum velocity relation is a trivial identity12. Thus, defining the parts of and in the same way as for , we obtain13. One may set equal to zero up to 2.5 PN order. Alternatively, one may find it “natural” to see a three-momentum proportial to a three-velocity and take another choice,
Finally, it is important to realize that the nonzero dipole moment of order affects the 3 PN field and the 3 PN equations of motion in essentially the same manner as the Newtonian dipole moment affects the Newtonian field and equations of motion. From Equations (78, 79, 80) we see that appears only at as
Note that this change of the acceleration does not affect the existence of the conservation of the (Newtonian-sense) energy,
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