"Gravitational Radiation from Post-Newtonian Sources
and Inspiralling Compact Binaries"
Luc Blanchet 
1 Introduction
1.1 Analytic approximations and wave generation formalism
1.2 The quadrupole moment formalism
1.3 Problem posed by compact binary systems
1.4 Post-Newtonian equations of motion
1.5 Post-Newtonian gravitational radiation
A Post-Newtonian Sources
2 Non-linear Iteration of the Vacuum Field Equations
2.1 Einstein’s field equations
2.2 Linearized vacuum equations
2.3 The multipolar post-Minkowskian solution
2.4 Generality of the MPM solution
2.5 Near-zone and far-zone structures
3 Asymptotic Gravitational Waveform
3.1 The radiative multipole moments
3.2 Gravitational-wave tails and tails-of-tails
3.3 Radiative versus source moments
4 Matching to a Post-Newtonian Source
4.1 The matching equation
4.2 General expression of the multipole expansion
4.3 Equivalence with the Will–Wiseman formalism
4.4 The source multipole moments
5 Interior Field of a Post-Newtonian Source
5.1 Post-Newtonian iteration in the near zone
5.2 Post-Newtonian metric and radiation reaction effects
5.3 The 3.5PN metric for general matter systems
5.4 Radiation reaction potentials to 4PN order
B Compact Binary Systems
6 Regularization of the Field of Point Particles
6.1 Hadamard self-field regularization
6.2 Hadamard regularization ambiguities
6.3 Dimensional regularization of the equations of motion
6.4 Dimensional regularization of the radiation field
7 Newtonian-like Equations of Motion
7.1 The 3PN acceleration and energy for particles
7.2 Lagrangian and Hamiltonian formulations
7.3 Equations of motion in the center-of-mass frame
7.4 Equations of motion and energy for quasi-circular orbits
7.5 The 2.5PN metric in the near zone
8 Conservative Dynamics of Compact Binaries
8.1 Concept of innermost circular orbit
8.2 Dynamical stability of circular orbits
8.3 The first law of binary point-particle mechanics
8.4 Post-Newtonian approximation versus gravitational self-force
9 Gravitational Waves from Compact Binaries
9.1 The binary’s multipole moments
9.2 Gravitational wave energy flux
9.3 Orbital phase evolution
9.4 Polarization waveforms for data analysis
9.5 Spherical harmonic modes for numerical relativity
10 Eccentric Compact Binaries
10.1 Doubly periodic structure of the motion of eccentric binaries
10.2 Quasi-Keplerian representation of the motion
10.3 Averaged energy and angular momentum fluxes
11 Spinning Compact Binaries
11.1 Lagrangian formalism for spinning point particles
11.2 Equations of motion and precession for spin-orbit effects
11.3 Spin-orbit effects in the gravitational wave flux and orbital phase

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 α β...μν... take space-time values 0, 1, 2, 3 and Latin indices ab...ij... 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 AB ... will refer to tetrad indices 0, 1, 2, 3 with ab ... the corresponding spatial values 1, 2, 3. Our signature is +2; hence the Minkowski metric reads ηαβ = diag(− 1,+1,+1, +1) = ηAB. As usual G and c are Newton’s constant and the speed of light.
3 Establishing the post-Newtonian 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 − e → 1) the Newtonian potential U can be numerically much larger than the estimate 2 2 2 𝒪(1∕c) ∼ v ∕c at the apastron of the orbit.
5 Whereas, the direct attack of the post-Minkowskian 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 1 2 3 (Gm ) = Ω a, where m = m1 + m2 is the total mass and Ω = 2π∕P is the orbital frequency. This law is also appropriately called the 1-2-3 law [319*].
8 This work entitled: “The last three minutes: Issues in gravitational-wave 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 scalar-tensor theories are known up to 2.5PN order [318].
10 The effective action should be equivalent, in the tree-level 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 gauge-fixing 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 leading-order contribution to the recoil as “Newtonian”, although it really corresponds to a 3.5PN subdominant radiation-reaction effect in the binary’s equations of motion.
13 Considering the coordinates xα as a set of four scalars, a simple calculation shows that
∂μhαμ = √−-g□gxα,
where □g ≡ gμν∇ μ∇ ν denotes the curved d’Alembertian operator. Hence the harmonic-coordinate condition tells that the coordinates xα themselves, considered as scalars, are harmonic, i.e., obey the vacuum (curved) d’Alembertian equation.
14 In d+ 1 space-time 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., hαμ = ημνhαν, h = ημνhμν, and ∂α = η αμ ∂μ. A parenthesis around a pair of indices denotes the usual symmetrization: T (αβ) = 12(T αβ + Tβα).
16 ℕ, ℤ, ℝ, and ℂ are the usual sets of non-negative integers, integers, real numbers, and complex numbers; Cp(Ω) is the set of p-times continuously differentiable functions on the open domain Ω (p ≤ +∞).
17 Our notation is the following: L = i1i2...iℓ denotes a multi-index, made of ℓ (spatial) indices. Similarly, we write for instance K = j1...jk (in practice, we generally do not need to write explicitly the “carrier” letter i or j), or aL− 1 = ai1...iℓ−1. Always understood in expressions such as Eq. (34*) are ℓ summations over the indices i1,...,iℓ ranging from 1 to 3. The derivative operator ∂L is a short-hand for ∂i1 ...∂iℓ. The function KL (for any space-time indices αβ) 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 and that δ 0 (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 ˆ KL ≡ KL, or sometimes with carets around the indices, KL ≡ K ⟨L ⟩. In particular, ˆnL = n⟨L⟩ is the STF projection of the product of unit vectors nL = ni1 ...niℓ, for instance 1 ˆnij = n⟨ij⟩ = nij − 3δij and 1 ˆnijk = n⟨ijk⟩ = nijk − 5(δijnk + δiknj +δjkni); an expansion into STF tensors ˆnL = nˆL (𝜃,ϕ) is equivalent to the usual expansion in spherical harmonics Ylm = Ylm(𝜃,ϕ), see Eqs. (75) below. Similarly, we denote xL = xi1 ...xiℓ = rlnL where r = |x|, and ˆxL = x⟨L⟩ = STF [xL]. The Levi-Civita antisymmetric symbol is denoted 𝜖ijk (with 𝜖123 = 1). Parenthesis refer to symmetrization, T(ij) = 12(Tij + Tji). Superscripts (q) indicate q successive time derivations.
18 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, see Eq. (29*). Hence, Pi = 0.
19 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 (35*) (37), there is a maximal multipolarity ℓmax(n) at any post-Minkowskian order n, which grows linearly with n.
20 We employ the Landau symbol o for remainders with its standard meaning. Thus, f(r) = o[g(r)] when r → 0 means that f(r)∕g(r) → 0 when r → 0. Furthermore, we generally assume some differentiability properties such as dnf (r)∕drn = o[g(r)∕rn].
21 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).
22 The STF tensorial coefficient αℓm L can be computed as αℓm = ∫ dΩNˆ Y-ℓm L L. Our notation is related to that used in Refs. [403*, 272*] by ℓm (2ℓ+1)!!-ℓm 𝒴L = 4πl! αL.
23 The function Qℓ is given in terms of the Legendre polynomial P ℓ by
1 ∫ 1 dzP (z) 1 ( x+ 1) ∑ℓ 1 Q ℓ(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 − ∞ to 1. The first equality is known as the Neumann formula for the Legendre function.
24 We pose c = 1 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:
∫ 1 1 ∘---------dzPℓ(z)---------- 2 −1 (xy− z)2 − (x2 − 1)(y2 − 1) = Q ℓ(x)Pℓ(y),
where 1 ≤ y < x is assumed. See Appendix A in Ref. [48*] for the proof. This relation constitutes a generalization of the Neumann formula (see the footnote 23).
26 The neglected remainders are indicated by o(1∕r) rather than 𝒪(1∕r2) because they contain powers of the logarithm of r; in fact they could be more accurately written as o(r𝜖−2) for some 𝜖 ≪ 1.
27 The canonical moment Mij differs from the source moment Iij 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 ijkl... carried by the moments themselves. Thus the dummy indices such as abc... are excluded from the STF projection.
29 Recall that our abbreviated notation ℱ 𝒫 includes the crucial regularization factor ^rB.
30 Recall that in actual applications we need mostly the mass-type moment IL and current-type one JL, 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 WL, ...,ZL 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 post-Minkowskian series over n = 1,...,∞ in Eq. (53*) has been formally summed up.
34 We mean the fully-fledge ℳ (τ); i.e., not the formal object ℳ (τ).
35 Though the latter integral is a priori divergent, its value can be determined by invoking complex analytic continuation in ℓ ∈ ℂ.
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 space-time curvature, see Eq. (377*).
37 Note, however, that the operation of order-reduction 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 ˆxijk in the vector potential denotes the STF combination ˆxijk = xijk − r2(xiδjk + xjδki + xkδij) 5 with xijk = xixjxk.
39 The function F(x) depends also on (coordinate) time t, through for instance its dependence on the velocities v1(t) and v2(t), but the time t is purely “spectator” in the regularization process, and thus will not be indicated. See the footnote 20 for the definition of the Landau symbol o for remainders.
40 The sum over k in Eq. (168*) is always finite since there is a maximal order a0 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 harmonic-coordinates 3PN equations of motion [69*, 71*] depend, in addition to λ, on two arbitrary constants r′1 and r′2 parametrizing some logarithmic terms. These constants are closely related to the constants s1 and s2 in the partie-finie integral (162*); see Ref. [71*] and Eq. (185*) below for the precise definition. However, r′1 and r′2 are not “physical” in the sense that they can be removed by a coordinate transformation.
43 One may wonder why the value of λ is a complicated rational fraction while ωstatic is so simple. This is because ωstatic was introduced precisely to measure the amount of ambiguities of certain integrals, while by contrast, λ was introduced [see Eq. (185*)] as an unknown constant entering the relation between the arbitrary scales r′1,r′2 on the one hand, and s1,s2 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 ξ happens to be amazingly related to the one for λ by a cyclic permutation of digits; compare 3ξ = − 9871∕3080 with λ = − 1987∕3080.
46 In higher approximations there will be also IR divergences and one should really employ the d-dimensional version of Eq. (141*).
47 We have limd→3 &tidle;k = 1. Notice that &tidle;k is closely linked to the volume Ωd −1 of the sphere with d− 1 dimensions (i.e. embedded into Euclidean d-dimensional space):
&tidle;kΩd− 1 =-4π--. d − 2
48 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. [71*].
49 Notice the dependence upon the irrational number π2. Technically, the π2 terms arise from non-linear interactions involving some integrals such as
1 ∫ d3x π2 π- r2r2= r-. 1 2 12
50 On the other hand, the ADM-Hamiltonian 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: gαβ → diag(− 1,1,1,1).
52 Namely,
Schw 2 [ 1− 2x ] E = μc √1-−-3x − 1 .
53 From the thermodynamic relation (235*) we necessarily have the relations
j4(ν) = − 5e4(ν)+ 64, 7 35
2 4988 656 j5(ν) = − 3e5(ν)− 945 − 135ν.
54 In all of Section 8 we pose G = 1 = c.
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 leading-order result ωa = Ω. 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 circular-orbit radius r0 with the constant r′0 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 self-force 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, μa would represent the baryonic mass of the bodies.
61 Since there are logarithms in this expansion we use the Landau o-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 ijkl⋅⋅⋅ but not on the summed up indices a and b.
64 The same argument shows that the non-linear 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, 𝒪(c−5lnc) 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” post-Newtonian order of this time variable: +8 Θ = 𝒪(c ).
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 π the separation direction between the particles and simultaneously exchange the labels of the two particles, i.e., when we apply the transformation (ψ,Δ ) → (ψ + π,− Δ). Moreover, due to the parity invariance, the H+’s are unchanged after the replacement i → π − i, while the H×’s being the projection of hTT ij on a tensorial product of two vectors of inverse parity types, is changed into its opposite.
70 The dependence on E and J will no longer be indicated but is always understood as implicit in what follows.
71 Comparing with Eqs. (338) we have also
W (ℓ) ( 1 ) v = ℓ+--K--+ 𝒪 c4 .
72 Note that this post-Newtonian parameter 𝜀 is precisely specified by Eq. (344a), while we only intended to define 𝜖 in Eq. (1*) as representing a post-Newtonian estimate.
73 More precisely, ft, fϕ, gt, gϕ are composed of 2PN and 3PN terms, but it, iϕ, ht, hϕ 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 Pl as
∫ 2π ( ) du-------1----l+1-= ----1--l+1Pl ∘--1----. 0 2π (1− etcosu) (1− e2t) 2 1− e2t
77 The tetrad is orthonormal in the sense that α β gαβeA eB = ηAB, where ηAB = diag(− 1,1,1,1) denotes a Minkowski metric. The indices AB ⋅⋅⋅ = 0,1,2,3 are the internal Lorentz indices, while as usual αβ ...μν⋅⋅⋅ = 0,1,2,3 are the space-time covariant indices. The inverse dual tetrad A e α, defined by β A β eA e α = δα, satisfies A B ηABe αe β = gαβ. We have also the completeness relation β B B eA e β = δA.
78 Our conventions for the Riemann tensor Rαβμν follow those of MTW [319].
79 The four-dimensional Levi-Civita tensor is defined by √--- 𝜀αβμν ≡ − g𝜖αβμν and √--- 𝜀αβμν ≡ − 𝜖αβμν∕ − g; here 𝜖αβμν = 𝜖αβμν denotes the completely anti-symmetric Levi-Civita symbol such that 𝜖0123 = 𝜖0123 = 1. For convenience in this section we pose c = 1.
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 S to denote the tetrad components (384*) and the original spin covector. Thus, Sa should not be confused with the spatial components Si (with i = 1,2,3) of the covariant vector Sα.
82 Notation adopted in Ref. [271]; the inverse formulas read
S1 = X1S − νΣ,
S2 = X2S + νΣ.
83 Note that the individual particle’s positions ya in the frame of the center-of-mass (defined by the cancellation of the center-of-mass integral of motion: G = 0) are related to the relative position and velocity x and v by some expressions similar to Eqs. (224) but containing spin effects starting at order 1.5PN.
84 Beware of our inevitably slightly confusing notation: Ω is the binary’s orbital frequency and ΩSO refers to the spin-orbit terms therein; Ωa is the precession frequency of the a-th spin while ϖ is the precession frequency of the orbital plane; and ωa defined earlier in Eqs. (244*) and (284) is the rotation frequency of the a-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 c.
86 In this section we can neglect the gauge multipole moments WL, ⋅⋅⋅,ZL.
87 Notice that the spin-orbit contributions due to the absorption by the black-hole horizons have to be added to this post-Newtonian result [349, 392, 5, 125].