### 4.7 First post-Newtonian equations of motion

Next, at 1 PN order, we need and . The term in the retardation expansion series of , Equation (76), gives no contribution at the 1 PN order by the constancy of the mass , i.e. .

Now we obtain from the Newtonian momentum-velocity relation:

We evaluate the surface integrals in the evolution equation for at 1 PN order. The result for star is

where we used the Newtonian equations of motion. From this equation we have the mass-energy relation at 1 PN order,

Then we have to calculate the 1 PN order . The result is . As depends on , we ignore it (see Section 4.5). As a result we obtain the momentum-velocity relation at 1 PN order, from Equation (86).

Now as for , we first calculate the surface integrals , , and from Equations (91) and (92). We then find that they depend on , hence we ignore them and obtain

To derive and , we have to evaluate non-compact support integrals for and , and the term in Equation (76) for :

with
The evaluation of the twice retardation expansion term (the last term in Equation (121)) is straightforward. For the non-compact support integrals, it is sufficient to consider the following integral:
To evaluate the above Poisson integrals, we use Equation (101), thus we need to find super-potentials for the integrands. For this purpose, it is convenient to transform the tensorial integrands into scalar integrands with differentiation operators,
Then it is relatively easy to find the super-potentials for these scalars. The results are and where  [77]. Equation (101) for each integrand then becomes
The second last term of Equation (128) and the terms abbreviated as in the above two equations arise from the surface integrals in Equation (101). Since they depend on , we ignore it (see Section 4.8). Substituting Equations (128) and (129) back into Equation (125), we can compute the non-compact support integrals.

Using the above results and the Newtonian equations of motion for the twice retardation expansion term, we finally obtain

where .

Evaluating the surface integrals in Equation (111) as in the Newtonian case, we obtain the 1 PN equations of motion,

where we defined the relative velocity as and we used the Newtonian equations of motion as well as Equation (119).

Finally let us give a summary of our procedure (see Figure 4). With the  PN order equations of motion and in hand, we first derive the  PN evolution equation for . Then we solve it functionally and obtain the mass-energy relation at  PN order. Next we calculate at  PN order and derive the momentum-velocity relation at  PN order. Then we calculate and . With the  PN mass-energy relation, the  PN momentum-velocity relation, , and , we next derive the  PN deviation field . Finally we evaluate the surface integrals which appear in the right hand side of Equation (111) and obtain the  PN equations of motion. In the above calculations we use the  PN order equations of motion to reduce the order of the equations of motion whenever an acceleration appears in the right hand of the resulting equations of motion. For instance, when we meet in the right hand side of the equations motion and we have to evaluate this up to , then using the Newtonian equations of motion, we replace it by . Basically we shall derive the 3 PN equations of motion with the procedure as described above.