### 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 [175176], 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 [131157]. 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 [53193175176] 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.