## 7 Third Post-Newtonian Momentum-Velocity Relation

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),

we find that the calculation required is almost the same as that in the equation for . Namely, it turns out that we do not need to use the direct-integration method to compute the integral. Therefore it is straightforward to evaluate the surface integrals in the definition of . Here we split into and as
with
and we show only :
where is the quantity up to order inclusively. Here it should be understood that in the last expression is evaluated with the Newtonian acceleration.

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 [91].

Now with in hand, we obtain the momentum-velocity relation. It turns out that the part of the momentum velocity relation is a trivial identity. Thus, defining the parts of and in the same way as for , we obtain

As explained in the previous Sections 3.4 and 4.4, we define the representative point of star by choosing the value of . In other words, one can freely choose in principle. 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,
Henceforth, we shall define by this equation.

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

Then the corresponding acceleration becomes
The last term compensates the integral contribution appearing through the momentum-velocity relation (163).

Note that this change of the acceleration does not affect the existence of the conservation of the (Newtonian-sense) energy,