Before proceeding, let us recall that the “standard” post-Newtonian approximation, as it was used
until, say, the early 1980’s (see for instance Refs. [2, 142, 143, 172]), is plagued with some
apparently inherent difficulties, which crop up at some high post-Newtonian order. The first
problem is that in higher approximations some divergent Poisson-type integrals appear. Indeed the
post-Newtonian expansion replaces the resolution of a hyperbolic-like d’Alembertian equation
by a perturbatively equivalent hierarchy of elliptic-like Poisson equations. Rapidly it is found
during the post-Newtonian iteration that the right-hand side of the Poisson equations acquires a
non-compact support (it is distributed over all space), and that as a result the standard Poisson
integral diverges at the bound of the integral at spatial infinity, i.e. , with
.
The second problem is related with the a priori limitation of the approximation to the near zone,
which is the region surrounding the source of small extent with respect to the wavelength of the
emitted radiation: . The post-Newtonian expansion assumes from the start that all
retardations
are small, so it can rightly be viewed as a formal near-zone expansion, when
. In particular, the fact which makes the Poisson integrals to become typically divergent,
namely that the coefficients of the post-Newtonian series blow up at “spatial infinity”, when
, has nothing to do with the actual behaviour of the field at infinity. However, the serious
consequence is that it is not possible, a priori, to implement within the post-Newtonian iteration the
physical information that the matter system is isolated from the rest of the universe. Most
importantly, the no-incoming radiation condition, imposed at past null infinity, cannot be taken
into account, a priori, into the scheme. In a sense the post-Newtonian approximation is not
“self-supporting”, because it necessitates some information taken from outside its own domain of
validity.
Here we present, following Refs. [185, 41
], a solution of both problems, in the form of a general
expression for the near-zone gravitational field, developed to any post-Newtonian order, which has been
determined from implementing the matching equation (65
). This solution is free of the divergences of
Poisson-type integrals we mentionned above, and it incorporates the effects of gravitational radiation
reaction appropriate to an isolated system.
Theorem 7 The expression of the post-Newtonian field in the near zone of a post-Newtonian source, satisfying correct boundary conditions at infinity (no incoming radiation), reads
The first term represents a particular solution of the hierarchy of post-Newtonian equations, while the second one is a homogeneous multipolar solution of the wave equation, of the “anti-symmetric” type that is regular at the originMore precisely, the flat retarded d’Alembertian operator in Equation (93) is given by the standard
expression (21
) but with all retardations expanded (
), and with the finite part
procedure
involved for dealing with the bound at infinity of the Poisson-type integrals (so that all the integrals are
well-defined at any order of approximation),
Theorem 7 is furthermore to be completed by the information concerning the multipolar functions
parametrizing the anti-symmetric homogeneus solution, the second term of Equation (93
). Note
that this homogeneous solution represents the unique one for which the matching equation (65
) is satisfied.
The result is
Importantly, we find that the post-Newtonian expansion given by Theorem 7 is a functional not
only of the related expansion of the pseudo-tensor,
, but also, by Equation (95
), of its multipole
expansion
, which is valid in the exterior of the source, and in particular in the asymptotic regions
far from the source. This can be understood by the fact that the post-Newtonian solution (93
) depends on
the boundary conditions imposed at infinity, that describe a matter system isolated from the rest of the
universe.
Equation (93) is interesting for providing a practical recipe for performing the post-Newtonian iteration
ad infinitum. Moreover, it gives some insights on the structure of radiation reaction terms. Recall that the
anti-symmetric waves, regular in the source, are associated with radiation reaction effects. More precisely, it
has been shown [185] that the specific anti-symmetric wave given by the second term of Equation (93
) is
linked with some non-linear contribution due to gravitational wave tails in the radiation reaction force.
Such a contribution constitutes a generalization of the tail-transported radiation reaction term at the 4PN
order, i.e. 1.5PN order relative to the dominant radiation reaction order, as determined in Ref. [27
]. This
term is in fact required by energy conservation and the presence of tails in the wave zone (see,
e.g., Equation (97
) below). Hence, the second term of Equation (93
) is dominantly of order
4PN and can be neglected in computations of the radiation reaction up to 3.5PN order (as in
Ref. [164
]). The usual radiation reaction terms, up to 3.5PN order, which are linear in the
source multipole moments (for instance the usual radiation reaction term at 2.5PN order), are
contained in the first term of Equation (93
), and are given by the terms with odd powers of
in the post-Newtonian expansion (94
). It can be shown [41] that such terms take also
the form of some anti-symmetric multipolar wave, which turn out to be parametrized by the
same moments as in the exterior field, namely the moments which are the STF analogues of
Equations (68
).
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