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4.3 Near-zone and far-zone structures

In our presentation of the post-Minkowskian algorithm (39View Equation, 40View Equation, 41View Equation, 42View Equation, 43View Equation) we have omitted a crucial recursive hypothesis, which is required in order to prove that at each post-Minkowskian order n, the inverse d’Alembertian operator can be applied in the way we did (and notably that the B-dependent retarded integral can be analytically continued down to a neighbourhood of B = 0). This hypothesis is that the “near-zone” expansion, i.e. when r --> 0, of each one of the post-Minkowskian coefficients hab n owns 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 r --> 0) is of the type: A N (- N,

sum h (x, t) = ^n rm(lnr)pF (t) + o(rN), (46) n L L,m,p,n
where m (- Z, with m0 < m < N (and m0 becoming more and more negative as n grows), p (- N with p < n - 1. The functions F L,m,p,n are multilinear functionals of the source multipole moments IL,...,ZL.

For the proof see Ref. [14Jump To The Next Citation Point]8. As we see, the near-zone expansion involves, besides the simple powers of r, some powers of the logarithm of r, with a maximal power of n- 1. As a corollary of that theorem, we find (by restoring all the powers of c in Eq. (46View Equation) and using the fact that each r goes into the combination r/c), that the general structure of the post-Newtonian expansion (c-- > +o o) is necessarily of the type

sum (ln-c)p hn(c) -~ cq , (47) p,q (- N
where p < n - 1 (and q > 2). The post-Newtonian expansion proceeds not only with the normal powers of 1/c but also with powers of the logarithm of c [14]. 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 r-- > +o o with u = t- r/c = const.: A N (- N,
( ) sum ^nL(lnr)p 1 hn(x,t) = ---rk---GL,k,p,n(u) + o rN- , (48)
where k,p (- N, with 1 < k < N, and where, likewise in the near-zone expansion (46View Equation), some powers of logarithms, such that p < n - 1, appear. The appearance of logarithms in the far-zone expansion of the harmonic-coordinates metric has been known since the work of Fock [74]. 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. [32Jump To The Next Citation Point], Sachs [128Jump To The Next Citation Point] and Penrose [114Jump To The Next Citation Point115Jump To The Next Citation Point], 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 [8299Jump To The Next Citation Point]. The following theorem, proved in Ref. [3Jump To The Next Citation Point], 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 [32128114Jump To The Next Citation Point115Jump To The Next Citation Point] approach to gravitational radiation.

Theorem 4 The most general multipolar-post-Minkowskian solution, stationary in the past (see Eq. (19View Equation)), admits some radiative coordinates (T,X), for which the expansion at future null infinity, R --> +o o with U =_ T - R/c = const., takes the form

( ) sum N^L- -1-- Hn(X, T ) = Rk KL,k,n(U ) + O RN . (49)
The functions KL,k,n are computable functionals of the source multipole moments. In radiative coordinates the retarded time U = T - R/c is a null coordinate in the asymptotic limit. The metric sum Habext = n>1GnHabn 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. (26View Equation, 27View Equation, 28View Equation), namely

ab ab b m H1 = h1 + @aq1 + @bqa1- jab@mq1, (50)
where the gauge vector a q1 is
( r ) qa1 = 2M j0a ln -- . (51) r0
This gauge transformation is non-harmonic:
am a 2M--0a @mH 1 = []q1 = r2 j . (52)
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 u = t - r/c coincides with a null coordinate at the linearized level9. 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 [99]. One can easily show that, at the dominant order when r --> + oo,
( 1) kmknHmn1 = O -2 , (53) r
where ka = (1,n) is the outgoing Minkowskian null vector. Given any n > 2, let us recursively assume that we have obtained all the previous radiative post-Minkowskian coefficients ab H m, i.e. A m < n - 1, and that all of them satisfy
( ) kmknHmnm = O -1 . (54) r2
From this induction hypothesis one can prove that the nth post-Minkowskian source term ab ab /\ n = /\n (H1, ...,Hn -1) is such that
kakb ( 1 ) /\abn = --2--sn (u,n) + O -3 . (55) r r
To the leading order this term takes the classic form of the stress-energy tensor for a swarm of massless particles, with sn 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 Eq. (55View Equation) that behave like 1/r2: See indeed the integration formula (103View Equation), which behaves like lnr/r at infinity. But now, thanks to the particular index structure of the term (55View Equation), we can correct for the effect by adjusting the gauge at the nth post-Minkowskian order. We pose, as a gauge vector,
[ka integral u ] qan = F P [] -r1et ---- dv sn(v,n) , (56) 2r2 - oo
where F P refers to the same finite part operation as in Eq. (39View Equation). 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 (55View Equation); see Ref. [3Jump To The Next Citation Point] for the details. Hence, to the nth post-Minkowskian order, we define the radiative metric as
ab ab ab a b b a ab m H n = U n + Vn + @ qn + @ qn - j @mqn. (57)
Here Uanb and Vnab denote the quantities that are the analogues of uanb and vanb, which were introduced into the harmonic-coordinates algorithm: See Eqs. (39View Equation, 40View Equation, 41View Equation, 42View Equation). In particular, these quantities are constructed in such a way that the sum ab ab Un + V n is divergence-free, so we see that the radiative metric does not obey the harmonic-gauge condition:
a integral u @ Ham = []qa = k--- dv s (v,n). (58) m n n 2r2 - oo n
The far-zone expansion of the latter metric is of the type (49View Equation), 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 [76], is proved by introducing the conformal factor _O_ = 1/r in radiative coordinates (see Ref. [3]). Finally, it can be checked that the metric so constructed, which is a functional of the source multipole moments IL, ..., ZL (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 (t,x) ---> (T, X), where (t,x) are the harmonic coordinates and (T,X) the radiative ones, together with a re-definition of the multipole moments.
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