### 5.2 General expression of the multipole expansion

Theorem 5 Under the hypothesis of matching, Equation (65), the multipole expansion of the solution of the Einstein field equation outside a post-Newtonian source reads
where the “multipole moments” are given by
Here, denotes the post-Newtonian expansion of the stress-energy pseudo-tensor defined by Equation (14).
Proof  [1520]: First notice where the physical restriction of considering a post-Newtonian source enters this theorem: the multipole moments (68) depend on the post-Newtonian expansion , rather than on itself. Consider , namely the difference between , which is a solution of the field equations everywhere inside and outside the source, and the first term in Equation (67), namely the finite part of the retarded integral of the multipole expansion :
From now on we shall generally abbreviate the symbols concerning the finite-part operation at by a mere . According to Equation (20), is given by the retarded integral of the pseudo-tensor . So,
In the second term the finite part plays a crucial role because the multipole expansion is singular at . By contrast, the first term in Equation (70), as it stands, is well-defined because we are considering only some smooth field distributions: . There is no need to include a finite part in the first term, but a contrario there is no harm to add one in front of it, because for convergent integrals the finite part simply gives back the value of the integral. The advantage of adding “artificially” the in the first term is that we can re-write Equation (70) into the much more interesting form
in which we have also used the fact that because has a compact support. The interesting point about Equation (71) is that appears now to be the (finite part of a) retarded integral of a source with spatially compact support. This follows from the fact that the pseudo-tensor agrees numerically with its own multipole expansion when (same equation as (63)). Therefore, can be obtained from the known formula for the multipole expansion of the retarded solution of a wave equation with compact-support source. This formula, given in Appendix B of Ref. [28], yields the second term in Equation (67),
but in which the moments do not yet match the result (68); instead,
The reason is that we have not yet applied the assumption of a post-Newtonian source. Such sources are entirely covered by their own near zone (i.e. ), and, in addition, the integral (73) has a compact support limited to the domain of the source. In consequence, we can replace the integrand in Equation (73) by its post-Newtonian expansion, valid over all the near zone, i.e.
Strangely enough, we do not get the expected result because of the presence of the second term in Equation (74). Actually, this term is a bit curious, because the object it contains is only known in the form of the formal series whose structure is given by the first equality in Equation (66) (indeed and have the same type of structure). Happily (because we would not know what to do with this term in applications), we are now going to prove that the second term in Equation (74) is in fact identically zero. The proof is based on the properties of the analytic continuation as applied to the formal structure (66) of . Each term of this series yields a contribution to Equation (74) that takes the form, after performing the angular integration, of the integral , and multiplied by some function of time. We want to prove that the radial integral is zero by analytic continuation (). First we can get rid of the logarithms by considering some repeated differentiations with respect to ; thus we need only to consider the simpler integral . We split the integral into a “near-zone” integral and a “far-zone” one , where is some constant radius. When is a large enough positive number, the value of the near-zone integral is , while when is a large negative number, the far-zone integral reads the opposite, . Both obtained values represent the unique analytic continuations of the near-zone and far-zone integrals for any except . The complete integral is equal to the sum of these analytic continuations, and is therefore identically zero (, including the value ). At last we have completed the proof of Theorem 5:

The latter proof makes it clear how crucial the analytic-continuation finite part is, which we recall is the same as in our iteration of the exterior post-Minkowskian field (see Equation (39)). Without a finite part, the multipole moment (75) would be strongly divergent, because the pseudo-tensor has a non-compact support owing to the contribution of the gravitational field, and the multipolar factor behaves like when . In applications (Part B of this article) we must carefully follow the rules for handling the operator.

The two terms in the right-hand side of Equation (67) depend separately on the length scale that we have introduced into the definition of the finite part, through the analytic-continuation factor (see Equation (36)). However, the sum of these two terms, i.e. the exterior multipolar field itself, is independent of . To see this, the simplest way is to differentiate formally with respect to . The independence of the field upon is quite useful in applications, since in general many intermediate calculations do depend on , and only in the final stage does the cancellation of the ’s occur. For instance, we shall see that the source quadrupole moment depends on starting from the 3PN level [45], but that this is compensated by another coming from the non-linear “tails of tails” at the 3PN order.