5.1 The matching equation
We shall provide the answer in the case of a post-Newtonian source for which the post-Newtonian
parameter
defined by Equation (1) is small. The fundamental fact that permits the connection of the
exterior field to the inner field of the source is the existence of a “matching” region, in which both the
multipole and the post-Newtonian expansions are valid. This region is nothing but the exterior near zone,
such that
(exterior) and
(near zone). It always exists around post-Newtonian
sources.
Let us denote by
the multipole expansion of
(for simplicity, we suppress the space-time
indices). By
we really mean the multipolar-post-Minkowskian exterior metric that we have
constructed in Sections 3 and 4:
Of course,
agrees with its own multipole expansion in the exterior of the source,
By contrast, inside the source,
and
disagree with each other because
is a fully-fledged
solution of the field equations with matter source, while
is a vacuum solution becoming singular at
. Now let us denote by
the post-Newtonian expansion of
. We have already anticipated the
general structure of this expansion as given in Equation (47). In the matching region, where
both the multipolar and post-Newtonian expansions are valid, we write the numerical equality
This “numerical” equality is viewed here in a sense of formal expansions, as we do not control the
convergence of the series. In fact, we should be aware that such an equality, though quite natural and even
physically obvious, is probably not really justified within the approximation scheme (mathematically
speaking), and we take it as part of our fundamental assumptions.
We now transform Equation (64) into a matching equation, by replacing in the left-hand side
by its near-zone re-expansion
, and in the right-hand side
by its multipole expansion
.
The structure of the near-zone expansion (
) of the exterior multipolar field has been found in
Equation (46). We denote the corresponding infinite series
with the same overbar as for the
post-Newtonian expansion because it is really an expansion when
, equivalent to an
expansion when
. Concerning the multipole expansion of the post-Newtonian metric,
, we simply postulate its existence. Therefore, the matching equation is the statement that
by which we really mean an infinite set of functional identities, valid
, between the
coefficients of the series in both sides of the equation. Note that such a meaning is somewhat different from
that of a numerical equality like Equation (64), which is valid only when
belongs to some limited
spatial domain. The matching equation (65) tells us that the formal near-zone expansion of the multipole
decomposition is identical, term by term, to the multipole expansion of the post-Newtonian solution.
However, the former expansion is nothing but the formal far-zone expansion, when
, of each of the
post-Newtonian coefficients. Most importantly, it is possible to write down, within the present formalism,
the general structure of these identical expansions as a consequence of Theorem 3, Equation (46):
where the functions
. The latter expansion can be interpreted either as the
singular re-expansion of the multipole decomposition when
(first equality in Equation (66)), or the
singular re-expansion of the post-Newtonian series when
(second equality). We recognize the
beauty of singular perturbation theory, where two asymptotic expansions, taken formally outside their
respective domains of validity, are matched together. Of course, the method works because there exists,
physically, an overlapping region in which the two approximation series are expected to be numerically close
to the exact solution.