1  In this article Greek indices take the values and Latin . Our signature is +2. and are Newton’s constant and the speed of light.  
2  The TT coordinate system can be extended to the near zone of the source as well; see for instance Ref. [151].  
3  See Ref. [81] for the proof of such an “effacement” principle in the context of relativistic equations of motion.  
4  Let us mention that the 3.5PN terms in the equations of motion are also known, both for pointparticle binaries [136, 137, 138, 174, 148, 164] and extended fluid bodies [14, 18]; they correspond to 1PN “relative” corrections in the radiation reaction force. Known also is the contribution of wave tails in the equations of motion, which arises at the 4PN order and represents a 1.5PN modification of the gravitational radiation damping [27].  
5  See also Equation (140) for the expression in spacetime dimensions.  
6  , , , and are the usual sets of nonnegative integers, integers, real numbers, and complex numbers; is the set of times continuously differentiable functions on the open domain ().  
7  Our notation is the following: denotes a multiindex, made of (spatial) indices. Similarly we write for instance (in practice, we generally do not need to consider the carrier letter or ), or . Always understood in expressions such as Equation (25) are summations over the indices ranging from 1 to 3. The derivative operator is a shorthand for . The function is symmetric and tracefree (STF) with respect to the indices composing . This means that for any pair of indices , we have and that (see Ref. [210] and Appendices A and B in Ref. [26] for reviews about the STF formalism). The STF projection is denoted with a hat, so , or sometimes with carets around the indices, . In particular, is the STF projection of the product of unit vectors ; an expansion into STF tensors is equivalent to the usual expansion in spherical harmonics . Similarly, we denote and . Superscripts like indicate successive timederivations.  
8  The constancy of the center of mass  rather than a linear variation with time  results from our assumption of stationarity before the date . Hence, .  
9  This assumption is justified because we are ultimately interested in the radiation field at some given finite postNewtonian precision like 3PN, and because only a finite number of multipole moments can contribute at any finite order of approximation. With a finite number of multipoles in the linearized metric (26, 27, 28), there is a maximal multipolarity at any postMinkowskian order , which grows linearly with .  
10  The and Landau symbols for remainders have their standard meaning.  
11  In this proof the coordinates are considered as dummy variables denoted . At the end, when we obtain the radiative metric, we shall denote the associated radiative coordinates by .  
12  Recall that in actual applications we need mostly the masstype moment and currenttype one , because the other moments parametrize a linearized gauge transformation.  
13  This function approaches the Dirac deltafunction (hence its name) in the limit of large multipoles: . Indeed the source looks more and more like a point mass as we increase the multipolar order .  
14  An alternative approach to the problem of radiation reaction, besides the matching procedure, is to work only within a postMinkowskian iteration scheme (which does not expand the retardations): see, e.g., Ref. [69].  
15  Notice that the normalization holds as a consequence of the corresponding normalization (83) for , together with the fact that by analytic continuation in the variable .  
16  At the 3PN order (taking into account the tails of tails), we find that does not completely cancel out after the replacement of by the righthand side of Equation (100). The reason is that the moment also depends on at the 3PN order. Considering also the latter dependence we can check that the 3PN radiative moment is actually free of the unphysical constant .  
17  The computation of the third term in Equation (106), which corresponds to the interaction between two quadrupoles, , can be found in Ref. [21].  
18  The function is given in terms of the Legendre polynomial by


19  Equation (112) has been obtained using a not so well known mathematical relation between the Legendre functions and
polynomials:


20  Actually, such a metric is valid up to 3.5PN order.  
21  It has been possible to “integrate directly” all the quartic contributions in the 3PN metric. See the terms composed of and in the first of Equations (115).  
22  The function depends also on time , through for instance its dependence on the velocities and , but the (coordinate) time is purely “spectator” in the regularization process, and thus will not be indicated.  
23  It was shown in Ref. [38] that using one or the other of these derivatives results in some equations of motion that differ by a mere coordinate transformation. This result indicates that the distributional derivatives introduced in Ref. [36] constitute merely some technical tools which are devoid of physical meaning.  
24  Note also that the harmoniccoordinates 3PN equations of motion as they have been obtained in Refs. [37, 38] depend, in addition to , on two arbitrary constants and parametrizing some logarithmic terms. These constants are closely related to the constants and in the partiefinie integral (124); see Ref. [38] for the precise definition. However, and are not “physical” in the sense that they can be removed by a coordinate transformation.  
25  One may wonder why the value of is a complicated rational fraction while is so simple. This is because was introduced precisely to measure the amount of ambiguities of certain integrals, while, by contrast, was introduced as an unknown constant entering the relation between the arbitrary scales on the one hand, and on the other hand, which has a priori nothing to do with ambiguities of integrals.  
26  See some comments on this work in Ref. [84], pp. 168  169.  
27  The result for happens to be amazingly related to the one for by a cyclic permutation of digits; compare with .  
28  The work [34] provided also some new expressions for the multipole moments of an isolated postNewtonian source, alternative to those given by Theorem 6, in the form of surface integrals extending on the outer part of the source’s near zone.  
29  We have . Notice that is closely linked to the volume of the sphere with dimensions
(i.e. embedded into Euclidean dimensional space):


30  When working at the level of the equations of motion (not considering the metric outside the worldlines), the effect of shifts can be seen as being induced by a coordinate transformation of the bulk metric as in Ref. [38].  
31  Notice also the dependence upon . Technically, the terms arise from nonlinear interactions involving some
integrals such as


32  Note that in the result published in Ref. [95] the following terms are missing:


33  Actually, in the present computation we do not need the radiationreaction 2.5PN term in these relations; we give it only for completeness.  
34  In this section we pose , and the two individual black hole masses are denoted and .  
35  We are following the discussion in Ref. [24]. Note that the arguments of this section are rather biased toward the author’s own work [23, 24].  
36  Actually, the postNewtonian series could be only asymptotic (hence divergent), but nevertheless it should give excellent results provided that the series is truncated near some optimal order of approximation. In this discussion we assume that the 3PN order is not too far from that optimum.  
37  When computing the gravitationalwave flux in Ref. [45] we preferred to call the Hadamardregularization constants and , in order to distinguish them from the constants and that were used in our previous computation of the equations of motion in Ref. [38]. Indeed these regularization constants need not neccessarily be the same when employed in different contexts.  
38  For circular orbits there is no difference at this order between , and , .  
39  All formulas incorporate the changes in some equations following the published Errata (2005) to the works [16, 19, 45, 40, 4].  
40  Generalizing the flux formula (231) to point masses moving on quasi elliptic orbits dates back to the work of Peters and Mathews [178] at Newtonian order. The result was obtained in [217, 49] at 1PN order, and then further extended by Gopakumar and Iyer [122] up to 2PN order using an explicit quasiKeplerian representation of the motion [99, 197]. No complete result at 3PN order is yet available.  
41  Notice the “strange” postNewtonian order of this time variable: .  
42  We neglect the nonlinear memory (DC) term present in the Newtonian plus polarization . See Wiseman and Will [222] and Arun et al. [4] for the computation of this term. 
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