## A Far Zone Contribution

A formal retardation expansion gives a divergent integral when we evaluate the gravitational field. Schematically the field at the field point is given by (we omit the indices of the field for simplicity)

This integral is divergent if we set the upper bound of the integral to infinity. In our formalism we take the upper bound of the integral as and keep the dependence in the field. In the last step of the derivation, we let go to infinity if and only if this procedure gives finite result at least up to the relevant post-Newtonian order. At a high order of the retardation expansion approximation, however, some terms must have a , and/or a dependence. In this case, we can not let go infinite.

To solve this problem, it is important to realize that the field at consists of two contributions: the retarded integral inside the near zone (the near zone contribution) and the retarded integral outside the near zone (the far zone contribution).

Since is introduced artificially, we expect that the far zone contribution to the near zone field must have some dependent terms which cancel out completely the dependent terms in the near zone. This expectation is the case and was proved to any post-Newtonian order by Pati and Will [129] within their formalism. They have shown that the total field (which is the sum of the near zone contribution plus the far zone contribution) is finite and independent of .

In this section we calculate the far zone contribution to the 3 PN equations of motion to make this article self-contained. We entirely follow [129162165], and the result is the same as theirs: The far zone field does not have any influence on the equations of motion up to 3 PN order inclusively.

For the near zone field, we write the field as

where is the field in the near zone, is the near zone integral contribution to the near zone field, and is the far zone integral contribution to the near zone field. Our task here is to evaluate to the 3 PN order. In turn, this means that we should derive to lower order because the integrand in the retarded integral for consists of .

Only in this section, we do not use our bookkeeping parameter while it should be understood that is of order , and of order .

Now we evaluate the near zone contribution to the far zone field using a multipole expansion:

where is a collective multi-index, is the distance from the near zone center to the field point, and . The near zone multipole moments are defined as

Next, the far zone contribution to the far zone field point can be evaluated as follows. The key idea would be an introduction of a new time and may be seen from the following transformation of the retarded integral where the radial integral is transformed into a temporal integral from the past infinity to :

where
We then make STF decomposition of the integrand in the retarded integral as , where . The integrand in Equation (193) becomes a summation of terms, each of which consists of (some algebraic combinations of) near zone multipole moments (defined by Equation (192)) multiplied by terms which explicitly depend on , , and . Roughly speaking, the idea is that we do integral by parts many times, each time increasing the number of the time derivatives of the multipole moments, and assume that the system is sufficiently stationary in the past so that the contributions from the past infinity disappear. We then have for the far zone contribution to the far zone field point:
with , and
for and
for . Other quantities in the above equations are given by
Here and are the Legendre function of the first and the second kind. represents an angular defect due to the fact that the far zone integral does not cover the whole spacetime due to the near zone. The function and the retarded time multi-derivatives of the STF coefficient comes from the recursive integrals by parts.

Then combining the near zone and the far zone contributions to the far zone field point, we have to the required order:

where , , , , , , , , and .

Next we evaluate the far zone contribution to the near zone field point. A transformation of the retarded integral (193) is again used and similar arguments below Equation (193) lead to the following formulae that we use

with
for and
for .

Evaluating the coefficients , we finally have

where we reintroduce our bookkeeping parameter , and is written with the near zone quadrupole moment as . We note that the fields of and of are the 3 PN fields in our formalism. Finally assuming that in the distant past the binary was sufficiently stationary, we find that the far zone contribution becomes a function of time only at the 3 PN order. It turns out that only the spatial derivative of those 3 PN fields contributes to the 3 PN equations of motion. Thus the far zone contribution does not affect the equations of motion up to 3 PN order inclusively.

In this section, we have given a highly rough sketch about the method developed by Pati and Will [129] which is based on their previous work [165]. Readers may consult [129130162163165] for more details.