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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)
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 [129, 162, 165], 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
:
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 
17,
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
and
for 
.
Evaluating the coefficients 
, we finally have
, 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 [129, 130, 162, 163, 165] for more
details.
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