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)
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  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
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:
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 :17, and
Then combining the near zone and the far zone contributions to the far zone field point, we have to the required order:
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
Evaluating the coefficients , we finally have
In this section, we have given a highly rough sketch about the method developed by Pati and Will  which is based on their previous work . Readers may consult [129, 130, 162, 163, 165] for more details.
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