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]^{10}.
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

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 levelhttp://www.livingreviews.org/lrr-2006-4 |
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