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5.5 Post-Newtonian field in the near zone

Theorem 6 solves in principle the question of the generation of gravitational waves by extended post-Newtonian sources. However, note that this result has to be completed by the definition of an explicit algorithm for the post-Newtonian iteration, analogous to the post-Minkowskian algorithm we defined in Section 4, so that the source multipole moments, which contain the full post-Newtonian expansion of the pseudo-tensor ab t, can be completely specified. Such a systematic post-Newtonian iteration scheme, valid (formally) to any post-Newtonian order, has been implemented [185Jump To The Next Citation Point41Jump To The Next Citation Point] using matched asymptotic expansions. The solution of this problem yields, in particular, some general expression, valid up to any order, of the terms associated with the gravitational radiation reaction force inside the post-Newtonian source14.

Before proceeding, let us recall that the “standard” post-Newtonian approximation, as it was used until, say, the early 1980’s (see for instance Refs. [2142143172]), is plagued with some apparently inherent difficulties, which crop up at some high post-Newtonian order. The first problem is that in higher approximations some divergent Poisson-type integrals appear. Indeed the post-Newtonian expansion replaces the resolution of a hyperbolic-like d’Alembertian equation by a perturbatively equivalent hierarchy of elliptic-like Poisson equations. Rapidly it is found during the post-Newtonian iteration that the right-hand side of the Poisson equations acquires a non-compact support (it is distributed over all space), and that as a result the standard Poisson integral diverges at the bound of the integral at spatial infinity, i.e. r =_ |x|--> +o o, with t = const.

The second problem is related with the a priori limitation of the approximation to the near zone, which is the region surrounding the source of small extent with respect to the wavelength of the emitted radiation: r« c. The post-Newtonian expansion assumes from the start that all retardations r/c are small, so it can rightly be viewed as a formal near-zone expansion, when r --> 0. In particular, the fact which makes the Poisson integrals to become typically divergent, namely that the coefficients of the post-Newtonian series blow up at “spatial infinity”, when r --> +o o, has nothing to do with the actual behaviour of the field at infinity. However, the serious consequence is that it is not possible, a priori, to implement within the post-Newtonian iteration the physical information that the matter system is isolated from the rest of the universe. Most importantly, the no-incoming radiation condition, imposed at past null infinity, cannot be taken into account, a priori, into the scheme. In a sense the post-Newtonian approximation is not “self-supporting”, because it necessitates some information taken from outside its own domain of validity.

Here we present, following Refs. [185Jump To The Next Citation Point41Jump To The Next Citation Point], a solution of both problems, in the form of a general expression for the near-zone gravitational field, developed to any post-Newtonian order, which has been determined from implementing the matching equation (65View Equation). This solution is free of the divergences of Poisson-type integrals we mentionned above, and it incorporates the effects of gravitational radiation reaction appropriate to an isolated system.

Theorem 7 The expression of the post-Newtonian field in the near zone of a post-Newtonian source, satisfying correct boundary conditions at infinity (no incoming radiation), reads

[ { } ] --ab 16pG [-- ] + sum oo (- )l Rab (t- r/c) - Rab (t + r/c) h = ---4-- F P [] -r1et tab + -----@L --L--------------L---------- . (93) c l=0 l! 2 r
The first term represents a particular solution of the hierarchy of post-Newtonian equations, while the second one is a homogeneous multipolar solution of the wave equation, of the “anti-symmetric” type that is regular at the origin r = 0 located in the source.

More precisely, the flat retarded d’Alembertian operator in Equation (93View Equation) is given by the standard expression (21View Equation) but with all retardations expanded (r/c --> 0), and with the finite part F P procedure involved for dealing with the bound at infinity of the Poisson-type integrals (so that all the integrals are well-defined at any order of approximation),

[ ] + sum o o n( )n integral FP []-re1t tab = _ --1- (--)- -@-- F P d3x'|x - x'|n -1tab(x',t). (94) 4p n=0 n! c @t
The existence of the solution (94View Equation) shows that the problem of divergences of the post-Newtonian expansion is simply due to the fact that the standard Poisson integral does not constitute the correct solution of the Poisson equation in the context of post-Newtonian expansions. So the problem is purely of a technical nature, and is solved once we succeed in finding the appropriate solution to the Poisson equation.

Theorem 7 is furthermore to be completed by the information concerning the multipolar functions ab R L (u) parametrizing the anti-symmetric homogeneus solution, the second term of Equation (93View Equation). Note that this homogeneous solution represents the unique one for which the matching equation (65View Equation) is satisfied. The result is

1 integral integral + oo RabL (u) = - ---F P d3x'^x'L dz gl(z)M(t ab)(x',u- z|x'|/c), (95) 4p 1
where M(t ab) denotes the multipole expansion of the pseudo-tensor (in the sense of Equation (62View Equation)), and where we denote g (z) = - 2d (z) l l, with d (z) l being given by Equation (82View Equation)15.

Importantly, we find that the post-Newtonian expansion hab given by Theorem 7 is a functional not only of the related expansion of the pseudo-tensor, -ab t, but also, by Equation (95View Equation), of its multipole expansion ab M(t ), which is valid in the exterior of the source, and in particular in the asymptotic regions far from the source. This can be understood by the fact that the post-Newtonian solution (93View Equation) depends on the boundary conditions imposed at infinity, that describe a matter system isolated from the rest of the universe.

Equation (93View Equation) is interesting for providing a practical recipe for performing the post-Newtonian iteration ad infinitum. Moreover, it gives some insights on the structure of radiation reaction terms. Recall that the anti-symmetric waves, regular in the source, are associated with radiation reaction effects. More precisely, it has been shown [185] that the specific anti-symmetric wave given by the second term of Equation (93View Equation) is linked with some non-linear contribution due to gravitational wave tails in the radiation reaction force. Such a contribution constitutes a generalization of the tail-transported radiation reaction term at the 4PN order, i.e. 1.5PN order relative to the dominant radiation reaction order, as determined in Ref. [27Jump To The Next Citation Point]. This term is in fact required by energy conservation and the presence of tails in the wave zone (see, e.g., Equation (97View Equation) below). Hence, the second term of Equation (93View Equation) is dominantly of order 4PN and can be neglected in computations of the radiation reaction up to 3.5PN order (as in Ref. [164Jump To The Next Citation Point]). The usual radiation reaction terms, up to 3.5PN order, which are linear in the source multipole moments (for instance the usual radiation reaction term at 2.5PN order), are contained in the first term of Equation (93View Equation), and are given by the terms with odd powers of 1/c in the post-Newtonian expansion (94View Equation). It can be shown [41] that such terms take also the form of some anti-symmetric multipolar wave, which turn out to be parametrized by the same moments as in the exterior field, namely the moments which are the STF analogues of Equations (68View Equation).

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