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. 8. 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 Eq. (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 : , . 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. , Sachs  and Penrose [114, 115], 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 [82, 99]. The following theorem, proved in Ref. , 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 [32, 128, 114, 115] approach to gravitational radiation.
Theorem 4 The most general multipolar-post-Minkowskian solution, stationary in the past (see Eq. (19)), admits some radiative coordinates , for which the expansion at future null infinity, with , takes the formThe 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 [114, 115], 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 Eqs. (26, 27, 28), namely9. 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 . One can easily show that, at the dominant order when , cancel out the logarithms coming from the retarded integral of the source term (55); see Ref.  for the details. Hence, to the th post-Minkowskian order, we define the radiative metric as , is proved by introducing the conformal factor in radiative coordinates (see Ref. ). 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.
© Max Planck Society and the author(s)