List Of Footnotes

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. [95 ].
3		Let us mention that the 3.5PN terms in the equations of motion are also known, both for point-particle 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		$N$ , $Z$ , $R$ 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 $_O_$ ( $p < + oo$ ).
5		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 Eq. (25 ) 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. [142 ] and Appendices A and B in Ref. [14 ] 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.
6		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, $Pi = 0$ .
7		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 (26 , 27 , 28 ), there is a maximal multipolarity $lmax(n)$ at any post-Minkowskian order $n$ , which grows linearly with $n$ .
8		The $o$ and $O$ Landau symbols for remainders have their standard meaning.
9		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 $(T,R)$ .
10		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.
11		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 $l$ .
12		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. [43 ].
13		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 Eq. (94 ). 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 $r 0$ .
14		The computation of the third term in Eq. (100 ), which corresponds to the interaction between two quadrupoles, $Mab × Mcd$ , can be found in Ref. [12 ].
15		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.
16		Eq. (106 ) 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. [10 ] for the proof. This relation constitutes a generalization of the Neumann formula (see footnote after Eq. (103 )).
17		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 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 $r'1$ and $r2'$ are closely related to the constants $s1$ and $s2$ in the partie-finie integral (119 ). See Ref. [22 ] for the precise definition.
20		Notice also the dependence upon $p2$ . Technically, the $p2$ terms arise from non-linear interactions involving some integrals such as $integral 3 2 1- d2-x2-= p--. p r1r2 r12$
21		Note that in the result published in Ref. [60 ] the following terms are missing: $G2 ( 55 193 )(N12P2)2P2 c6r2-- - 12m1 - -48 m2 ---m-m---1 + 1 <--> 2. 12 1 2$ This misprint has been corrected in an Erratum [60 ].
22		Actually, in the present computation we do not need the radiation-reaction 2.5PN terms in these relations; we give them only for completeness.
23		When computing the gravitational-wave flux in Ref. [26 ] 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. [22 ]. Indeed these regularization constants need not neccessarily need to be the same when employed in different contexts.
24		Notice the strange post-Newtonian order of this time variable: $Q = O(c+8)$ .