### 4.2 Near zone contribution

We shall evaluate the near zone contribution as follows. First, we make a retarded expansion of Equation (75) and change the integral region to a spatial hypersurface,
Note that the above integral depends on the arbitrary length in general. The cancellation between the dependent terms in the far zone contribution and those in the near zone contribution was shown by [129] through all the post-Newtonian order. In the following, we shall omit the terms which have negative powers of (). In other words, we simply let whenever it gives a convergent result. On the other hand, we shall retain terms having positive powers of (), and logarithmic terms () to confirm that the final result, in the end of calculation, is independent of (and for logarithmic terms to keep the arguments of logarithm non-dimensional).

Second we split the integral into two parts: a contribution from the body zone , and from elsewhere, . Schematically we evaluate the following two types of integrals (we omit indices of the field),

where . We shall deal with these two contributions successively.

#### 4.2.1 Body zone contribution

As for the body zone contribution, we make a multipole expansion using the scaling of the integrand, i.e.  in the body zone. For example, the part in Equation (76), , gives

Here the operator denotes a symmetric and tracefree (STF) operation on the indices in the brackets. See [129150165] for some useful formulas of STF tensors. Also we define . To derive the 3 PN equations of motion, up to and as well as up to are required.

In the above equations we define the multipole moments as

where we introduced a collective multi-index and . Then , , and . We simply call the four-momentum of the star , the momentum, and the energy. Also we call the dipole moment of the star , and the quadrupole moment of the star .

Then we transform these moments into more convenient forms. By the conservation law (67), we have

where , an overdot denotes a time derivative with respect to , and . Using these equations and noticing that the body zone remains unchanged (in the near zone coordinates), i.e. , we have
where
and
The operators and attached to the indices denote anti-symmetrization and symmetrization. is the spin of the star and Equation (86) is the momentum-velocity relation. Thus our momentum-velocity relation is a direct analog of the Newtonian momentum-velocity relation (see Section 3.4). In general, we have
and
where is the number of indices in the multi-index .

Now, from the above equations, especially Equation (88), we find that the body zone contributions, , are of order . Note that if we can not or do not assume the (nearly) stationarity of the initial data for the stars, then, instead of Equation (88) we have

where we used the dynamical time (see Section 3.3). In this case the lowest order metric differs from the Newtonian form. From our (almost) stationarity assumption the remaining motion inside a star, apart from the spinning motion, is caused only by the tidal effect by the companion star and from Equation (88); it appears at 3 PN order [81].

To obtain the lowest order , we have to evaluate the surface integrals . Generally, in the moments and appear formally at the order and . Thus and appear as

where we omitted irrelevant terms and numerical coefficients. Thus one may expect that and appear at the order for any and we have to calculate an infinite number of moments. In fact, this is not the case and it was shown in [91] that only terms of contribute to the 3 PN equations of motion. The important thing here is that and are at most in .

Finally, since the order of is higher than that of , we conclude that .

#### 4.2.2 contribution

As far as the contribution is concerned, since the integrand is at least quadratic in the small deviation field , we make the post-Newtonian expansion in the integrand. Then, basically, with the help of (super-)potentials which satisfy , denoting the Laplacian, we have for each integral (see e.g. the term in Equation (77))

Equation (101) can be derived without using Dirac delta distributions (see Appendix B of [91]). For the terms in Equation (77), we use potentials many times to convert all the volume integrals into surface integrals and “” terms.

In fact finding the super-potentials is one of the most formidable tasks especially when we proceed to high post-Newtonian orders. Fortunately, up to 2.5 PN order, all the required super-potentials are available. At 3 PN order, there appear many integrands for which we could not find the required super-potentials. To obtain the 3 PN equations of motion, we devise an alternative method similar to the method employed by Blanchet and Faye [28]. The details of the method will be explained later.

Now the lowest order integrands can be evaluated with the body zone contribution , and since is , we find

where we expanded in an series,
Similarly, in the following we expand in an series. From these equations we find that the deviation field in , , is . (It should be noted that in the body zone is assumed to be of order unity and within our method we can not calculate there explicitly. To obtain in the body zone, we have to know the internal structure of the star.)