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. [95].  
3  Let us mention that the 3.5PN terms in the equations of motion are also known, both for pointparticle binaries [85, 86, 113] and extended fluid bodies [5, 9]; 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 [15].  
4  , , 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 ().  
5  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 Eq. (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. [142] and Appendices A and B in Ref. [14] 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.  
6  The constancy of the center of mass  rather than a linear variation with time  results from our assumption of stationarity before the date . Hence, .  
7  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 .  
8  The and Landau symbols for remainders have their standard meaning.  
9  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 .  
10  Recall that in actual applications we need mostly the masstype moment and currenttype one , because the other moments parametrize a linearized gauge transformation.  
11  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 .  
12  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. [43].  
13  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 Eq. (94). 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 .  
14  The computation of the third term in Eq. (100), which corresponds to the interaction between two quadrupoles, , can be found in Ref. [12].  
15  The function is given in terms of the Legendre polynomial by


16  Eq. (106) has been obtained using a not so well known mathematical relation between the Legendre functions and
polynomials:


17  It has been possible to “integrate directly” all the quartic contributions in the 3PN metric. See the terms composed of and in Eq. (109).  
18  It was shown in Ref. [22] 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. [20] constitute merely some technical tools which are devoid of physical meaning.  
19  The constants and are closely related to the constants and in the partiefinie integral (119). See Ref. [22] for the precise definition.  
20  Notice also the dependence upon . Technically, the terms arise from nonlinear interactions involving some
integrals such as


21  Note that in the result published in Ref. [60] the following terms are missing:


22  Actually, in the present computation we do not need the radiationreaction 2.5PN terms in these relations; we give them only for completeness.  
23  When computing the gravitationalwave flux in Ref. [26] 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. [22]. Indeed these regularization constants need not neccessarily need to be the same when employed in different contexts.  
24  Notice the strange postNewtonian order of this time variable: . 
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