List of Footnotes
1  A few errata have been published in this intricate field; all formulas take into account the latest changes.  
2  In this article Greek indices take spacetime values 0, 1, 2, 3 and Latin indices spatial values 1, 2, 3. Cartesian coordinates are assumed throughout and boldface notation is often used for ordinary Euclidean vectors. In Section 11 upper Latin letters will refer to tetrad indices 0, 1, 2, 3 with the corresponding spatial values 1, 2, 3. Our signature is +2; hence the Minkowski metric reads . As usual and are Newton’s constant and the speed of light.  
3  Establishing the postNewtonian expansion rigorously has been the subject of numerous mathematically oriented works, see e.g., [361*, 362*, 363*].  
4  Note that for very eccentric binaries (with say ) the Newtonian potential can be numerically much larger than the estimate at the apastron of the orbit.  
5  Whereas, the direct attack of the postMinkowskian expansion, valid at once inside and outside the source, faces some calculational difficulties [408, 136].  
6  The TT coordinate system can be extended to the near zone of the source as well; see for instance Ref. [282].  
7  Namely , where is the total mass and is the orbital frequency. This law is also appropriately called the 123 law [319*].  
8  This work entitled: “The last three minutes: Issues in gravitationalwave measurements of coalescing compact binaries” is sometimes coined the “3mn Caltech paper”.  
9  All the works reviewed in this section concern general relativity. However, let us mention here that the equations of motion of compact binaries in scalartensor theories are known up to 2.5PN order [318].  
10  The effective action should be equivalent, in the treelevel approximation, to the Fokker action [207], for which the field degrees of freedom (i.e., the metric), that are solutions of the field equations derived from the original matter + field action with gaugefixing term, have been inserted back into the action, thus defining the Fokker action for the sole matter fields.  
11  This reference has an eloquent title: “Feynman graph derivation of the Einstein quadrupole formula”.  
12  In absence of a better terminology, we refer to the leadingorder contribution to the recoil as “Newtonian”, although it really corresponds to a 3.5PN subdominant radiationreaction effect in the binary’s equations of motion.  
13  Considering the coordinates as a set of four scalars, a simple calculation shows that


14  In spacetime dimensions, only one coefficient in this expression is modified; see Eq. (175) below.  
15  See Eqs. (3.8) in Ref. [71*] for the cubic and quartic terms. We denote e.g., , , and . A parenthesis around a pair of indices denotes the usual symmetrization: .  
16  , , , 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 ().  
17  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 write explicitly the “carrier” letter or ), or . Always understood in expressions such as Eq. (34*) are summations over the indices ranging from 1 to 3. The derivative operator is a shorthand for . The function (for any spacetime indices ) 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. [403*] and Appendices A and B in Ref. [57*] 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 , for instance and ; an expansion into STF tensors is equivalent to the usual expansion in spherical harmonics , see Eqs. (75) below. Similarly, we denote where , and . The LeviCivita antisymmetric symbol is denoted (with ). Parenthesis refer to symmetrization, . Superscripts indicate successive time derivations.  
18  The constancy of the center of mass – rather than a linear variation with time – results from our assumption of stationarity before the date , see Eq. (29*). Hence, .  
19  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 (35*) – (37), there is a maximal multipolarity at any postMinkowskian order , which grows linearly with .  
20  We employ the Landau symbol for remainders with its standard meaning. Thus, when means that when . Furthermore, we generally assume some differentiability properties such as .  
21  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 .  
22  The STF tensorial coefficient can be computed as . Our notation is related to that used in Refs. [403*, 272*] by .  
23  The function is given in terms of the Legendre polynomial by


24  We pose until the end of this section.  
25  The equation (85) has been obtained using a not so well known mathematical relation between the Legendre functions and
polynomials:


26  The neglected remainders are indicated by rather than because they contain powers of the logarithm of ; in fact they could be more accurately written as for some .  
27  The canonical moment differs from the source moment by small 2.5PN and 3.5PN terms; see Eq. (97).  
28  In all formulas below the STF projection applies only to the “free” indices denoted carried by the moments themselves. Thus the dummy indices such as are excluded from the STF projection.  
29  Recall that our abbreviated notation includes the crucial regularization factor .  
30  Recall that in actual applications we need mostly the masstype moment and currenttype one , because the other moments simply parametrize a linearized gauge transformation.  
31  The work [65*] provided some alternative expressions for all the multipole moments (123) – (125), useful for some applications, in the form of surface integrals extending on the outer part of the source’s near zone.  
32  The moments have also a Newtonian limit, but which is not particularly illuminating.  
33  For this argument we assume the validity of the matching equation (103*) and that the postMinkowskian series over in Eq. (53*) has been formally summed up.  
34  We mean the fullyfledge ; i.e., not the formal object .  
35  Though the latter integral is a priori divergent, its value can be determined by invoking complex analytic continuation in .  
36  Of course the geodesic equations are appropriate for the motion of particles without spins; for spinning particles one has also to take into account the coupling of the spin to the spacetime curvature, see Eq. (377*).  
37  Note, however, that the operation of orderreduction is illicit at the level of the Lagrangian. In fact, it is known that the elimination of acceleration terms in a Lagrangian by substituting the equations of motion derived from that Lagrangian, results in a different Lagrangian whose equations of motion differ from those of the original Lagrangian by a gauge transformation [374].  
38  Recall the footnote 17 for our notation. In particular in the vector potential denotes the STF combination with .  
39  The function depends also on (coordinate) time , through for instance its dependence on the velocities and , but the time is purely “spectator” in the regularization process, and thus will not be indicated. See the footnote 20 for the definition of the Landau symbol for remainders.  
40  The sum over in Eq. (168*) is always finite since there is a maximal order of divergency in Eq. (159*).  
41  It was shown in Ref. [71*] that using one or the other of these derivatives results in some equations of motion that differ by a coordinate transformation, and the redefinition of some ambiguity parameter. This indicates that the distributional derivatives introduced in Ref. [70*] constitute some technical tools devoid of physical meaning besides precisely the appearance of Hadamard’s ambiguity parameters.  
42  Note also that the harmoniccoordinates 3PN equations of motion [69*, 71*] 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 (162*); see Ref. [71*] and Eq. (185*) below for the precise definition. However, and are not “physical” in the sense that they can be removed by a coordinate transformation.  
43  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 [see Eq. (185*)] 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 the ambiguous part of integrals.  
44  See however some comments on the latter work in Ref. [145], pp. 168 – 169.  
45  The result for happens to be amazingly related to the one for by a cyclic permutation of digits; compare with .  
46  In higher approximations there will be also IR divergences and one should really employ the dimensional version of Eq. (141*).  
47  We have . Notice that is closely linked to the volume of the sphere with dimensions (i.e.
embedded into Euclidean dimensional space):


48  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. [71*].  
49  Notice the dependence upon the irrational number . Technically, the terms arise from nonlinear interactions
involving some integrals such as


50  On the other hand, the ADMHamiltonian formalism provides a limited description of the gravitational radiation field, compared to what will be done using harmonic coordinates in Section 9.  
51  This parameter is an invariant in a large class of coordinate systems – those for which the metric becomes asymptotically Minkowskian far from the system: .  
52  Namely,


53  From the thermodynamic relation (235*) we necessarily have the relations 

54  In all of Section 8 we pose .  
55  Note that this is an iterative process because the masses in Eq. (247*) are themselves to be replaced by the irreducible masses.  
56  In Ref. [51*] it was assumed that the corotation condition was given by the leadingorder result . The 1PN correction in Eq. (247*) modifies the 3PN terms in Eq. (250*) with respect to the result of Ref. [51*].  
57  One should not confuse the circularorbit radius with the constant entering the logarithm at the 3PN order in Eq. (228) and which is defined by Eq. (221*).  
58  This tendency is in agreement with numerical and analytical selfforce calculations [24, 287].  
59  The first law (280*) has also been generalized for binary systems of point masses moving along generic stable bound (eccentric) orbits in Ref. [286].  
60  In the case of extended material bodies, would represent the baryonic mass of the bodies.  
61  Since there are logarithms in this expansion we use the Landau symbol for remainders; see the footnote 20.  
62  In addition, the wave generation formalism will provide the waveform itself, see Sections 9.4 and 9.5.  
63  The STF projection applies only on “living” indices but not on the summed up indices and .  
64  The same argument shows that the nonlinear multipole interactions in Eq. (89) as well as those in Eqs. (97) and (98) do not contribute to the flux for circular orbits.  
65  Or, rather, as shown in the Appendix of Ref. [87*].  
66  See Section 10 for the generalization of the flux of energy to eccentric binary orbits.  
67  Notice the “strange” postNewtonian order of this time variable: .  
68  This procedure for computing analytically the orbital phase corresponds to what is called in the jargon the “Taylor T2 approximant”. We refer to Ref. [98] for discussions on the usefulness of defining several types of approximants for computing (in general numerically) the orbital phase.  
69  Notice the obvious fact that the polarization waveforms remain invariant when we rotate by the separation direction between the particles and simultaneously exchange the labels of the two particles, i.e., when we apply the transformation . Moreover, due to the parity invariance, the ’s are unchanged after the replacement , while the ’s being the projection of on a tensorial product of two vectors of inverse parity types, is changed into its opposite.  
70  The dependence on and will no longer be indicated but is always understood as implicit in what follows.  
71  Comparing with Eqs. (338) we have also


72  Note that this postNewtonian parameter is precisely specified by Eq. (344a), while we only intended to define in Eq. (1*) as representing a postNewtonian estimate.  
73  More precisely, , , , are composed of 2PN and 3PN terms, but , , , start only at 3PN order.  
74  On the other hand, for the computation of the gravitational waveform of eccentric binary orbits up to the 2PN order in the Fourier domain, see Refs. [401, 402].  
75  Recall that the fluxes are defined in a general way, for any matter system, in terms of the radiative multipole moments by the expressions (68).  
76  The second of these formulas can alternatively be written with the standard Legendre polynomial
as


77  The tetrad is orthonormal in the sense that , where denotes a Minkowski metric. The indices are the internal Lorentz indices, while as usual are the spacetime covariant indices. The inverse dual tetrad , defined by , satisfies . We have also the completeness relation .  
78  Our conventions for the Riemann tensor follow those of MTW [319].  
79  The fourdimensional LeviCivita tensor is defined by and ; here denotes the completely antisymmetric LeviCivita symbol such that . For convenience in this section we pose .  
80  Because of this choice, it is better to consider that the tetrad is not the same as the one we originally employed to construct the action (369*).  
81  Beware that here we employ the usual slight ambiguity in the notation when using the same carrier letter to denote the tetrad components (384*) and the original spin covector. Thus, should not be confused with the spatial components (with ) of the covariant vector .  
82  Notation adopted in Ref. [271]; the inverse formulas read


83  Note that the individual particle’s positions in the frame of the centerofmass (defined by the cancellation of the centerofmass integral of motion: ) are related to the relative position and velocity and by some expressions similar to Eqs. (224) but containing spin effects starting at order 1.5PN.  
84  Beware of our inevitably slightly confusing notation: is the binary’s orbital frequency and refers to the spinorbit terms therein; is the precession frequency of the th spin while is the precession frequency of the orbital plane; and defined earlier in Eqs. (244*) and (284) is the rotation frequency of the th black hole. Such different notions nicely mix up in the first law of spinning binary black holes in Section 8.3; see Eq. (282*) and the corotation condition (285*).  
85  Recall from Eq. (366*) that in our convention the spins have the dimension of an angular momentum times .  
86  In this section we can neglect the gauge multipole moments .  
87  Notice that the spinorbit contributions due to the absorption by the blackhole horizons have to be added to this postNewtonian result [349, 392, 5, 125]. 