## 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,