5.2 General expression of the multipole expansion
Theorem 5 Under the hypothesis of matching, Eq. (65), the multipole expansion of the solution of the
Einstein field equation outside a post-Newtonian source reads as
where the “multipole moments” are given by
Here,
denotes the post-Newtonian expansion of the stress-energy pseudo-tensor defined by Eq. (14).
Proof [6
, 11
]: 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
Eq. (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 Eq. (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 Eq. (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 Eq. (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 Eq. (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. [16], yields the second term in Eq. (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 Eq. (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 Eq. (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
Eq. (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 Eq. (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 Eq. (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 Eq. (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 bbb of this article) we must carefully follow the
rules for handling the
operator.
The two terms in the right-hand side of Eq. (67) depend separately on the length scale
that we
have introduced into the definition of the finite part, through the analytic-continuation factor
(see Eq. (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 [26
], but that this
is compensated by another
coming from the non-linear “tails of tails” at the 3PN
order.