### 5.3 Equivalence with the Will-Wiseman formalism

Recently, Will and Wiseman [152] (see also Refs. [151, 112]), 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. [6, 11].