4.3 Near-zone and far-zone structures
In our presentation of the post-Minkowskian algorithm (39, 40, 41, 42, 43) we have omitted a crucial
recursive hypothesis, which is required in order to prove that at each post-Minkowskian order
, the
inverse d’Alembertian operator can be applied in the way we did (and notably that the
-dependent
retarded integral can be analytically continued down to a neighbourhood of
). This hypothesis is
that the “near-zone” expansion, i.e. when
, of each one of the post-Minkowskian coefficients
has a certain structure. This hypothesis is established as a theorem once the mathematical induction
succeeds.
Theorem 3 The general structure of the expansion of the post-Minkowskian exterior metric in the
near-zone (when
) is of the type:
,
where
, with
(and
becoming more and more negative as
grows),
with
. The functions
are multilinear functionals of the source multipole moments
.
For the proof see Ref. [26
].
As we see, the near-zone expansion involves, besides the simple powers of
, some powers of the logarithm
of
, with a maximal power of
. As a corollary of that theorem, we find (by restoring all the
powers of
in Equation (46) and using the fact that each
goes into the combination
), that
the general structure of the post-Newtonian expansion (
) is necessarily of the type
where
(and
). The post-Newtonian expansion proceeds not only with the normal powers
of
but also with powers of the logarithm of
[26].
Paralleling the structure of the near-zone expansion, we have a similar result concerning the
structure of the far-zone expansion at Minkowskian future null infinity, i.e. when
with
:
,
where
, with
, and where, likewise in the near-zone expansion (46), some powers of
logarithms, such that
, appear. The appearance of logarithms in the far-zone expansion of the
harmonic-coordinates metric has been known since the work of Fock [113]. One knows also
that this is a coordinate effect, because the study of the “asymptotic” structure of space-time
at future null infinity by Bondi et al. [53
], Sachs [193
], and Penrose [175
, 176
], has revealed
the existence of other coordinate systems that avoid the appearance of any logarithms: the
so-called radiative coordinates, in which the far-zone expansion of the metric proceeds with simple
powers of the inverse radial distance. Hence, the logarithms are simply an artifact of the use of
harmonic coordinates [131, 157
]. The following theorem, proved in Ref. [12
], shows that our
general construction of the metric in the exterior of the source, when developed at future null
infinity, is consistent with the Bondi-Sachs-Penrose [53, 193, 175
, 176
] approach to gravitational
radiation.
Theorem 4 The most general multipolar-post-Minkowskian solution, stationary in the past (see
Equation (19)), admits some radiative coordinates
, for which the expansion at future null infinity,
with
, takes the form
The functions
are computable functionals of the source multipole moments. In radiative
coordinates the retarded time
is a null coordinate in the asymptotic limit. The metric
is asymptotically simple in the sense of Penrose [175, 176], perturbatively to any
post-Minkowskian order.
Proof : We introduce a linearized “radiative” metric by performing a gauge transformation of the
harmonic-coordinates metric defined by Equations (26, 27, 28), namely
where the gauge vector
is
This gauge transformation is non-harmonic:
Its effect is to “correct” for the well-known logarithmic deviation of the harmonic coordinates’
retarded time with respect to the true space-time characteristic or light cones. After the change
of gauge, the coordinate
coincides with a null coordinate at the linearized
level.
This is the requirement to be satisfied by a linearized metric so that it can constitute the linearized
approximation to a full (post-Minkowskian) radiative field [157]. One can easily show that, at the dominant
order when
,
where
is the outgoing Minkowskian null vector. Given any
, let us recursively assume
that we have obtained all the previous radiative post-Minkowskian coefficients
, i.e.
,
and that all of them satisfy
From this induction hypothesis one can prove that the
th post-Minkowskian source term
is such that
To the leading order this term takes the classic form of the stress-energy tensor for a swarm of massless
particles, with
being related to the power in the waves. One can show that all the problems with the
appearance of logarithms come from the retarded integral of the terms in Equation (55) that behave
like
: See indeed the integration formula (109), which behaves like
at infinity.
But now, thanks to the particular index structure of the term (55), we can correct for the
effect by adjusting the gauge at the
th post-Minkowskian order. We pose, as a gauge vector,
where
refers to the same finite part operation as in Equation (39). This vector is such that the
logarithms that will appear in the corresponding gauge terms cancel out the logarithms coming from the
retarded integral of the source term (55); see Ref. [12
] for the details. Hence, to the
th
post-Minkowskian order, we define the radiative metric as
Here
and
denote the quantities that are the analogues of
and
, which were
introduced into the harmonic-coordinates algorithm: See Equations (39, 40, 41, 42). In particular, these
quantities are constructed in such a way that the sum
is divergence-free, so we see that the
radiative metric does not obey the harmonic-gauge condition:
The far-zone expansion of the latter metric is of the type (49), i.e. is free of any logarithms, and the
retarded time in these coordinates tends asymptotically toward a null coordinate at infinity. The property of
asymptotic simplicity, in the mathematical form given by Geroch and Horowitz [121], is proved by
introducing the conformal factor
in radiative coordinates (see Ref. [12]). Finally, it can be
checked that the metric so constructed, which is a functional of the source multipole moments
(from the definition of the algorithm), is as general as the general harmonic-coordinate metric of
Theorem 2, since it merely differs from it by a coordinate transformation
, where
are the harmonic coordinates and
the radiative ones, together with a re-definition of the multipole
moments.