### 5.3 Equivalence with the Will-Wiseman formalism

Recently, Will and Wiseman [152] (see also Refs. [151112]), extending previous work of Epstein and Wagoner [70] and Thorne [142], have obtained a different-looking multipole decomposition, with different definitions for the multipole moments of a post-Newtonian source. They find, instead of our multipole decomposition given by Eq. (67),
There is no operation in the first term, but instead the retarded integral is truncated, as indicated by the subscript , to extend only in the “far zone”: i.e.  in the notation of Eq. (21), where is a constant radius enclosing the source (). The near-zone part of the retarded integral is thereby removed, and there is no problem with the singularity of the multipole expansion at the origin. The multipole moments are then given, in contrast with our result (68), by an integral extending over the “near zone” only:
Since the integrand is compact-supported there is no problem with the bound at infinity and the integral is well-defined (no need of a ).

Let us show that the two different formalisms are equivalent. We compute the difference between our moment , defined by Eq. (68), and the Will-Wiseman moment , given by Eq. (77). For the comparison we split into far-zone and near-zone integrals corresponding to the radius . Since the finite part present in deals only with the bound at infinity, it can be removed from the near-zone integral, which is then seen to be exactly equal to . So the difference between the two moments is simply given by the far-zone integral:

Next, we transform this expression. Successively we write because we are outside the source, and from the matching equation (65). At this stage, we recall from our reasoning right after Eq. (74) that the finite part of an integral over the whole space of a quantity having the same structure as is identically zero by analytic continuation. The main trick of the proof is made possible by this fact, as it allows us to transform the far-zone integration in Eq. (78) into a near-zone one , at the price of changing the overall sign in front of the integral. So,
Finally, it is straightforward to check that the right-hand side of this equation, when summed up over all multipolarities , accounts exactly for the near-zone part that was removed from the retarded integral of (first term in Eq. (76)), so that the “complete” retarded integral as given by the first term in our own definition (67) is exactly reconstituted. In conclusion, the formalism of Ref. [152] is equivalent to the one of Refs. [611].