List of Footnotes

1 In this article Greek indices take the values 0,1,2,3 and Latin 1,2,3. Our signature is +2. G and c 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. [81Jump To The Next Citation Point] 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 point-particle binaries [136Jump To The Next Citation Point, 137Jump To The Next Citation Point, 138Jump To The Next Citation Point, 174Jump To The Next Citation Point, 148Jump To The Next Citation Point, 164Jump To The Next Citation Point] and extended fluid bodies [14Jump To The Next Citation Point, 18Jump To The Next Citation Point]; 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 [27Jump To The Next Citation Point].
5 See also Equation (140View Equation) for the expression in d+ 1 space-time dimensions.
6 N, Z, R, and C are the usual sets of non-negative integers, integers, real numbers, and complex numbers; Cp(_O_) is the set of p-times continuously differentiable functions on the open domain _O_ (p < + oo).
7 Our notation is the following: L = i1i2...il denotes a multi-index, made of l (spatial) indices. Similarly we write for instance P = j1...jp (in practice, we generally do not need to consider the carrier letter i or j), or aL- 1 = ai1...il- 1. Always understood in expressions such as Equation (25View Equation) are l summations over the l indices i1,...,il ranging from 1 to 3. The derivative operator @L is a short-hand for @i1 ...@il. The function KL is symmetric and trace-free (STF) with respect to the l indices composing L. This means that for any pair of indices ip,iq (- L, we have K...ip...iq...= K...iq...ip... and that dipiqK...ip...iq...= 0 (see Ref. [210Jump To The Next Citation Point] and Appendices A and B in Ref. [26Jump To The Next Citation Point] for reviews about the STF formalism). The STF projection is denoted with a hat, so KL =_ K^L, or sometimes with carets around the indices, KL =_ K<L>. In particular, ^nL = n<L> is the STF projection of the product of unit vectors nL = ni1 ...nil; an expansion into STF tensors ^nL = ^nL(h,f) is equivalent to the usual expansion in spherical harmonics Ylm = Ylm(h,f). Similarly, we denote xL = xi1 ...xil = rlnL and ^xL = x<L>. Superscripts like (p) indicate p successive time-derivations.
8 The constancy of the center of mass Xi - rather than a linear variation with time - results from our assumption of stationarity before the date - T. Hence, Pi = 0.
9 This assumption is justified because we are ultimately interested in the radiation field at some given finite post-Newtonian 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 (26View Equation, 27View Equation, 28View Equation), there is a maximal multipolarity lmax(n) at any post-Minkowskian order n, which grows linearly with n.
10 The o and O Landau symbols for remainders have their standard meaning.
11 In this proof the coordinates are considered as dummy variables denoted (t,r). At the end, when we obtain the radiative metric, we shall denote the associated radiative coordinates by (T,R).
12 Recall that in actual applications we need mostly the mass-type moment IL and current-type one JL, because the other moments parametrize a linearized gauge transformation.
13 This function approaches the Dirac delta-function (hence its name) in the limit of large multipoles: lim l-->+o o dl(z) = d(z). Indeed the source looks more and more like a point mass as we increase the multipolar order l.
14 An alternative approach to the problem of radiation reaction, besides the matching procedure, is to work only within a post-Minkowskian iteration scheme (which does not expand the retardations): see, e.g., Ref. [69].
15 Notice that the normalization integral + oo 1 dzgl(z) = 1 holds as a consequence of the corresponding normalization (83View Equation) for dl(z), together with the fact that integral + oo - oo dzgl(z) = 0 by analytic continuation in the variable l (- C.
16 At the 3PN order (taking into account the tails of tails), we find that r 0 does not completely cancel out after the replacement of U by the right-hand side of Equation (100View Equation). The reason is that the moment M L also depends on r 0 at the 3PN order. Considering also the latter dependence we can check that the 3PN radiative moment U L is actually free of the unphysical constant r 0.
17 The computation of the third term in Equation (106View Equation), which corresponds to the interaction between two quadrupoles, Mab × Mcd, can be found in Ref. [21Jump To The Next Citation Point].
18 The function Q l is given in terms of the Legendre polynomial P l by
1 integral 1 dzP (z) 1 (x +1 ) sum l 1 Ql(x) = - ----l--= -Pl(x)ln ----- - -Pl-j(x)Pj-1(x). 2 - 1 x - z 2 x -1 j=1 j
In the complex plane there is a branch cut from - oo to 1. The first equality is known as the Neumann formula for the Legendre function.
19 Equation (112View Equation) has been obtained using a not so well known mathematical relation between the Legendre functions and polynomials:
1 integral 1 dzP (z) - V~ ------2---l2------2-----= Ql(x)Pl(y) 2 -1 (xy - z) - (x - 1)(y - 1)
(where 1 < y < x is assumed). See Appendix A in Ref. [19Jump To The Next Citation Point] for the proof. This relation constitutes a generalization of the Neumann formula (see footnote after Equation (109View Equation)).
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 4 V and VX^ in the first of Equations (115View Equation).
22 The function F(x) depends also on time t, through for instance its dependence on the velocities v1(t) and v2(t), but the (coordinate) t time is purely “spectator” in the regularization process, and thus will not be indicated.
23 It was shown in Ref. [38Jump To The Next Citation Point] 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. [36Jump To The Next Citation Point] constitute merely some technical tools which are devoid of physical meaning.
24 Note also that the harmonic-coordinates 3PN equations of motion as they have been obtained in Refs. [37Jump To The Next Citation Point, 38Jump To The Next Citation Point] depend, in addition to c, on two arbitrary constants ' r1 and ' r2 parametrizing some logarithmic terms. These constants are closely related to the constants s1 and s2 in the partie-finie integral (124View Equation); see Ref. [38Jump To The Next Citation Point] for the precise definition. However, ' r1 and ' r2 are not “physical” in the sense that they can be removed by a coordinate transformation.
25 One may wonder why the value of c is a complicated rational fraction while wstatic is so simple. This is because wstatic was introduced precisely to measure the amount of ambiguities of certain integrals, while, by contrast, c was introduced as an unknown constant entering the relation between the arbitrary scales ' ' r1,r2 on the one hand, and s1,s2 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 q happens to be amazingly related to the one for c by a cyclic permutation of digits; compare 3q = -9871/3080 with c = -1987/3080.
28 The work [34] provided also some new expressions for the multipole moments of an isolated post-Newtonian 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 limd-->3 ~k = 1. Notice that ~k is closely linked to the volume _O_d-1 of the sphere with d - 1 dimensions (i.e. embedded into Euclidean d-dimensional space):
~k_O_d-1 = -4p--. d- 2
30 When working at the level of the equations of motion (not considering the metric outside the world-lines), the effect of shifts can be seen as being induced by a coordinate transformation of the bulk metric as in Ref. [38Jump To The Next Citation Point].
31 Notice also the dependence upon p2. Technically, the p2 terms arise from non-linear interactions involving some integrals such as
integral 1- -d3x = p2-. p r21r22 r12
32 Note that in the result published in Ref. [95Jump To The Next Citation Point] the following terms are missing:
G2 ( 55 193 )(N12P2)2P 2 c6r2- - 12 m1 - 48-m2 --m--m---1+ 1 <--> 2. 12 1 2
This misprint has been corrected in an Erratum [95Jump To The Next Citation Point].
33 Actually, in the present computation we do not need the radiation-reaction 2.5PN term in these relations; we give it only for completeness.
34 In this section we pose G = 1 = c, and the two individual black hole masses are denoted M1 and M2.
35 We are following the discussion in Ref. [24Jump To The Next Citation Point]. Note that the arguments of this section are rather biased toward the author’s own work [23, 24].
36 Actually, the post-Newtonian 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 gravitational-wave flux in Ref. [45Jump To The Next Citation Point] we preferred to call the Hadamard-regularization constants u1 and u2, in order to distinguish them from the constants s1 and s2 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 IL, JL and ML, SL.
39 All formulas incorporate the changes in some equations following the published Errata (2005) to the works [16, 19, 45, 40, 4Jump To The Next Citation Point].
40 Generalizing the flux formula (231View Equation) 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 quasi-Keplerian representation of the motion [99, 197]. No complete result at 3PN order is yet available.
41 Notice the “strange” post-Newtonian order of this time variable: Q = O(c+8).
42 We neglect the non-linear memory (DC) term present in the Newtonian plus polarization (0) H +. See Wiseman and Will [222] and Arun et al. [4Jump To The Next Citation Point] for the computation of this term.