"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

11 Spinning Compact Binaries

The post-Newtonian templates have been developed so far for compact binary systems which can be described with great precision by point masses without spins. Here by spin, we mean the intrinsic (classical) angular momentum S of the individual compact body. However, including the effects of spins is essential, as the astrophysical evidence indicates that stellar-mass black holes [2, 390, 311, 227, 323] and supermassive black holes [188, 101, 102] (see Ref. [364] for a review) can be generically close to maximally spinning. The presence of spins crucially affects the dynamics of the binary, in particular leading to orbital plane precession if they are not aligned with the orbital angular momentum (see for instance [138, 8]), and thereby to strong modulations in the observed signal frequency and phase.

In recent years an important effort has been undertaken to compute spin effects to high post-Newtonian order in the dynamics and gravitational radiation of compact binaries:

  1. Dynamics. The goal is to obtain the equations of motion and related conserved integrals of the motion, the equations of precession of the spins, and the post-Newtonian metric in the near zone. For this step we need a formulation of the dynamics of particles with spins (either Lagrangian or Hamiltonian);
  2. Radiation. The mass and current radiative multipole moments, including tails and all hereditary effects, are to be computed. One then deduces the gravitational waveform and the fluxes, from which we compute the secular evolution of the orbital phase. This step requires plugging the previous dynamics into the general wave generation formalism of Part A.

We adopt a particular post-Newtonian counting for spin effects that actually refers to maximally spinning black holes. In this convention the two spin variables Sa (a = 1,2) have the dimension of an angular momentum multiplied by a factor c, and we pose

2 Sa = Gm aχa, (366 )
where m a is the mass of the compact body, and χ a is the dimensionless spin parameter, which equals one for maximally spinning Kerr black holes. Thus the spins Sa of the compact bodies can be considered as “Newtonian” quantities [there are no c’s in Eq. (366*)], and all spin effects will carry (at least) an explicit 1∕c factor with respect to non-spin effects. With this convention any post-Newtonian estimate is expected to be appropriate (i.e., numerically correct) in the case of maximal rotation. One should keep in mind that spin effects will be formally a factor 1 ∕c smaller for non-maximally spinning objects such as neutron stars; thus in this case a given post-Newtonian computation will actually be a factor 1∕c more accurate.

As usual we shall make a distinction between spin-orbit (SO) effects, which are linear in the spins, and spin-spin (SS) ones, which are quadratic. In this article we shall especially review the SO effects as they play the most important role in gravitational wave detection and parameter estimation. As we shall see a good deal is known on spin effects (both SO and SS), but still it will be important in the future to further improve our knowledge of the waveform and gravitational-wave phasing, by computing still higher post-Newtonian SO and SS terms, and to include at least the dominant spin-spin-spin (SSS) effect [305*]. For the computations of SSS and even SSSS effects see Refs. [246, 245, 296, 305, 413].

The SO effects have been known at the leading level since the seminal works of Tulczyjew [411*, 412*], Barker & O’Connell [27*, 28*] and Kidder et al. [275*, 271*]. With our post-Newtonian counting such leading level corresponds to the 1.5PN order. The SO terms have been computed to the next-to-leading level which corresponds to 2.5PN order in Refs. [394*, 194*, 165*, 292*, 352*, 241*] for the equations of motion or dynamics, and in Refs. [53*, 54*] for the gravitational radiation field. Note that Refs. [394, 194, 165*, 241*] employ traditional post-Newtonian methods (both harmonic-coordinates and Hamiltonian), but that Refs. [292, 352] are based on the effective field theory (EFT) approach. The next-to-next-to-leading SO level corresponding to 3.5PN order has been obtained in Refs. [242, 244] using the Hamiltonian method for the equations of motion, in Ref. [297] using the EFT, and in Refs. [307*, 90*] using the harmonic-coordinates method. Here we shall focus on the harmonic-coordinates approach [307*, 90*, 89*, 306*] which is in fact best formulated using a Lagrangian, see Section 11.1. With this approach the next-to-next-to-leading SO level was derived not only for the equations of motion including precession, but also for the radiation field (energy flux and orbital phasing) [89*, 306*]. An analytic solution for the SO precession effects will be presented in Section 11.2. Note that concerning the radiation field the highest known SO level actually contains specific tail-induced contributions at 3PN [54*] and 4PN [306*] orders, see Section 11.3.

The SS effects are known at the leading level corresponding to 2PN order from Barker & O’Connell [27, 28] in the equations of motion (see [271*, 351*, 110] for subsequent derivations), and from Refs. [275, 271*] in the radiation field. Next-to-leading SS contributions are at 3PN order and have been obtained with Hamiltonian [387, 389*, 388, 247, 241], EFT [354, 356, 355, 293, 299] and harmonic-coordinates [88*] techniques (with [88] obtaining also the next-to-leading SS terms in the gravitational-wave flux). With SS effects in a compact binary system one must make a distinction between the spin squared terms, involving the coupling between the two same spins S1 or S2, and the interaction terms, involving the coupling between the two different spins S1 and S2. The spin-squared terms S2 1 and S2 2 arise due to the effects on the dynamics of the quadrupole moments of the compact bodies that are induced by their spins [347]. They have been computed through 2PN order in the fluxes and orbital phase in Refs. [217, 218*, 314]. The interaction terms S1 × S2 can be computed using a simple pole-dipole formalism like the one we shall review in Section 11.1. The interaction terms S1 × S2 between different spins have been derived to next-to-next-to-leading 4PN order for the equations of motion in Refs. [294, 298] (EFT) and [243] (Hamiltonian). In this article we shall generally neglect the SS effects and refer for these to the literature quoted above.

11.1 Lagrangian formalism for spinning point particles

Some necessary material for constructing a Lagrangian for a spinning point particle in curved spacetime is presented here. The formalism is issued from early works [239*, 19*] and has also been developed in the context of the EFT approach [351*]. Variants and alternatives (most importantly the associated Hamiltonian formalism) can be found in Refs. [389, 386, 25]. The formalism yields for the equations of motion of spinning particles and the equations of precession of the spins the classic results known in general relativity [411*, 412*, 310*, 331*, 135*, 409*, 179*].

Let us consider a single spinning point particle moving in a given curved background metric gαβ(x ). The particle follows the worldline α y (τ), with tangent four-velocity α α u = dy ∕dτ, where τ is a parameter along the representative worldline. In a first stage we do not require that the four-velocity be normalized; thus τ needs not be the proper time elapsed along the worldline. To describe the internal degrees of freedom associated with the particle’s spin, we introduce a moving orthonormal tetrad e α(τ) A along the trajectory, which defines a “body-fixed” frame.77 The rotation tensor ω αβ associated with the tetrad is defined by

De Aα αβ ------= − ω eAβ, (367 ) d τ
where β D∕d τ ≡ u ∇ β is the covariant derivative with respect to the parameter τ along the worldline; equivalently, we have
β ωαβ = eAαDe-A-. (368 ) d τ
Because of the normalization of the tetrad the rotation tensor is antisymmetric: ω αβ = − ωβα.

We look for an action principle for the spinning particle. Following Refs. [239*, 351] and the general spirit of effective field theories, we require the following symmetries to hold:

  1. The action is a covariant scalar, i.e., behaves as a scalar with respect to general space-time diffeomorphisms;
  2. It is a global Lorentz scalar, i.e., stays invariant under an arbitrary change of the tetrad vectors: eAα(τ) −→ ΛBAe B α(τ) where ΛBA is a constant Lorentz matrix;
  3. It is reparametrization-invariant, i.e., its form is independent of the parameter τ used to follow the particle’s worldline.

In addition to these symmetries we need to specify the dynamical degrees of freedom: These are chosen to be the particle’s position yα and the tetrad eA α. Furthermore we restrict ourselves to a Lagrangian depending only on the four-velocity uα, the rotation tensor ωαβ, and the metric gαβ. Thus, the postulated action is of the type

[ ] ∫ +∞ ( ) I yα,eA α = d τL u α,ωαβ,gαβ . (369 ) −∞
These assumptions confine the formalism to a “pole-dipole” model and to terms linear in the spins. An important point is that such a model is universal in the sense that it can be used for black holes as well as neutrons stars. Indeed, the internal structure of the spinning body appears only at the quadratic order in the spins, through the rotationally induced quadrupole moment.

As it is written in (369*), i.e., depending only on Lorentz scalars, L is automatically a Lorentz scalar. By performing an infinitesimal coordinate transformation, one easily sees that the requirement that the Lagrangian be a covariant scalar specifies its dependence on the metric to be such that (see e.g., Ref. [19])

∂L α β α βγ 2----- = p u + S γω . (370 ) ∂g αβ
We have defined the conjugate linear momentum pα and the antisymmetric spin tensor Sαβ by

Note that the right-hand side of Eq. (370*) is necessarily symmetric by exchange of the indices α and β. Finally, imposing the invariance of the action (369*) by reparametrization of the worldline, we find that the Lagrangian must be a homogeneous function of degree one in the velocity uα and rotation tensor ωαβ. Applying Euler’s theorem to the function L (uα,ωα β) immediately gives

α 1 αβ L = pαu + 2-Sαβω , (372 )
where the functions pα(u, ω) and S αβ(u,ω) must be reparametrization invariant. Note that, at this stage, their explicit expressions are not known. They will be specified only when a spin supplementary condition is imposed, see Eq. (379*) below.

We now investigate the unconstrained variations of the action (369*) with respect to the dynamical variables α e A, α y and the metric. First, we vary it with respect to the tetrad α eA while keeping the position α y fixed. A worry is that we must have a way to distinguish intrinsic variations of the tetrad from variations which are induced by a change of the metric gαβ. This is conveniently solved by decomposing the variation δe β A according to

β ( αβ 1 αβ) δeA = eAα δ 𝜃 + 2δg , (373 )
in which we have introduced the antisymmetric tensor αβ A [α β] δ𝜃 ≡ e δeA, and where the corresponding symmetric part is simply given by the variation of the metric, i.e. eA(αδeAβ) ≡ 12δg αβ. Then we can consider the independent variations δ𝜃αβ and δg αβ. Varying with respect to δ𝜃αβ, but holding the metric fixed, gives the equation of spin precession which is found to be
DS-αβ-= ωαγSβγ − ω βγSαγ, (374 ) d τ
or, alternatively, using the fact that the right-hand side of Eq. (370*) is symmetric,
DS-αβ-= pαu β − p βuα. (375 ) dτ
We next vary with respect to the particle’s position yα while holding the tetrad eAα fixed. Operationally, this means that we have to parallel-transport the tetrad along the displacement vector, i.e., to impose
δyβ∇ βe α = 0. (376 ) A
A simple way to derive the result is to use locally inertial coordinates, such that the Christoffel symbols Γ αβγ = 0 along the particle’s worldline yα(τ); then, Eq. (376*) gives δeAα = δy β∂βeAα = − δyβΓ αβγeAγ = 0. The variation leads then to the well-known Mathisson–Papapetrou [310*, 331*, 135*] equation of motion
Dpα- 1-β μν d τ = − 2u RαβμνS , (377 )
which involves the famous coupling of the spin tensor to the Riemann curvature.78 With a little more work, the equation of motion (377*) can also be derived using an arbitrary coordinate system, making use of the parallel transport equation (376*). Finally, varying with respect to the metric while keeping αβ δ𝜃 = 0, gives the stress-energy tensor of the spinning particle. We must again take into account the scalarity of the action, as imposed by Eq. (370*). We obtain the standard pole-dipole result [411*, 412*, 310, 331, 135, 409, 179]:
∫ +∞ δ (x − y) ∫ +∞ δ (x − y) Tα β = d τp(αu β)-(4)√--------− ∇ γ dτS γ(αu β)-(4)√-------, (378 ) −∞ − g −∞ − g
where δ (x − y) (4) denotes the four-dimensional Dirac function. It can easily be checked that the covariant conservation law αβ ∇ βT = 0 holds as a consequence of the equation of motion (377*) and the equation of spin precession (375*).

Up to now we have considered unconstrained variations of the action (369*), describing the particle’s internal degrees of freedom by the six independent components of the tetrad e α A (namely a 4 × 4 matrix subject to the 10 constraints α β gαβeA eB = ηAB). To correctly account for the number of degrees of freedom associated with the spin, we must impose three supplementary spin conditions (SSC). Several choices are possible for a sensible SSC. Notice that in the case of extended bodies the choice of a SSC corresponds to the choice of a central worldline inside the body with respect to which the spin angular momentum is defined (see Ref. [271*] for a discussion). Here we adopt the Tulczyjew covariant SSC [411, 412]

Sαβp = 0. (379 ) β
As shown by Hanson & Regge [239] in the flat space-time case, it is possible to specify the Lagrangian in our original action (369*) in such a way that the constraints (379*) are directly the consequence of the equations derived from that Lagrangian. Here, for simplicity’s sake, we shall simply impose the constraints (379*) in the space of solutions of the Euler-Lagrange equations. From Eq. (379*) we can introduce the covariant spin vector S μ associated with the spin tensor by79
1 S αβ ≡ --𝜀αβμνpμS ν, (380 ) m
where we have defined the mass of the particle by m2 ≡ − gμνpμp ν. By contracting Eq. (375*) with pβ and using the equation of motion (377*), one obtains
2 1- γ β μν pα(pu) + m uα = 2 u R γμνS Sαβ, (381 )
where we denote μ (pu ) ≡ p μu. By further contracting Eq. (381*) with α u we obtain an explicit expression for (pu ), which can then be substituted back into (381*) to provide the relation linking the four-momentum pα to the four-velocity uα. It can be checked using (379*) and (381*) that the mass of the particle is constant along the particle’s trajectory: dm ∕dτ = 0. Furthermore the four-dimensional magnitude s of the spin defined by s2 ≡ gμνS S μ ν is also conserved: ds ∕dτ = 0.

Henceforth we shall restrict our attention to spin-orbit (SO) interactions, which are linear in the spins. We shall also adopt for the parameter τ along the particle’s worldline the proper time ------------ d τ ≡ ∘ − gμνdyμdy ν, so that gμνu μuν = − 1. Neglecting quadratic spin-spin (SS) and higher-order interactions, the linear momentum is simply proportional to the normalized four-velocity: 2 pα = mu α + 𝒪 (S ). Hence, from Eq. (375*) we deduce that 2 DS αβ∕dτ = 𝒪 (S ). The equation for the spin covariant vector S α then reduces at linear order to

DS-α-= 𝒪(S2 ). (382 ) dτ
Thus the spin covector is parallel transported along the particle’s trajectory at linear order in spin. We can also impose that the spin should be purely spatial for the comoving observer:
α S αu = 0. (383 )
From now on, we shall often omit writing the 𝒪 (S2 ) remainders.

In applications (e.g., the construction of gravitational wave templates for the compact binary inspiral) it is very useful to introduce new spin variables that are designed to have a conserved three-dimensional Euclidean norm (numerically equal to s). Using conserved-norm spin vector variables is indeed the most natural choice when considering the dynamics of compact binaries reduced to the frame of the center of mass or to circular orbits [90*]. Indeed the evolution equations of such spin variables reduces, by construction, to ordinary precession equations, and these variables are secularly constant (see Ref. [423*]).

A standard, general procedure to define a (Euclidean) conserved-norm spin spatial vector consists of projecting the spin covector S α onto an orthonormal tetrad e α A, which leads to the four scalar components (A = 0,1,2,3)

α SA = eA Sα. (384 )
If we choose for the time-like tetrad vector the four-velocity itself, e α = u α 0,80 the time component tetrad projection S0 vanishes because of the orthogonality condition (383*). We have seen that α 2 SαS = s is conserved along the trajectory; because of (383*) we can rewrite this as γ αβSαS β = s2, in which we have introduced the projector γαβ = gαβ + uαu β onto the spatial hypersurface orthogonal to uα. From the orthonormality of the tetrad and our choice e0α = u α, we have γ αβ = δabe α e β a b in which a,b = 1,2,3 refer to the spatial values of the tetrad indices, i.e., A = (0,a) and B = (0, b). Therefore the conservation law αβ 2 γ SαSβ = s becomes
δabS S = s2, (385 ) a b
which is indeed the relation defining a Euclidean conserved-norm spin variable Sa.81 However, note that the choice of the spin variable Sa is still somewhat arbitrary, since a rotation of the tetrad vectors can freely be performed. We refer to [165, 90*] for the definition of some “canonical” choice for the tetrad in order to fix this residual freedom. Such choice presents the advantage of providing a unique determination of the conserved-norm spin variable in a given gauge. This canonical choice will be the one adopted in all explicit results presented in Section 11.3.

The evolution equation (382*) for the original spin variable S α now translates into an ordinary precession equation for the tetrad components Sa, namely

dS --a-= Ω abSb, (386 ) dt
where the precession tensor Ωab is related to the tetrad components ωAB of the rotation tensor defined in (368*) by Ωab = zωab where we pose z ≡ dτ∕dt, remembering the redshift variable (276*). The antisymmetric character of the matrix Ωab guaranties that Sa satisfies the Euclidean precession equation
dS- dt = Ω × S, (387 )
where we denote S = (Sa), and Ω = (Ωa) with 1 bc Ωa = − 2𝜖abcΩ. As a consequence of (387*) the spin has a conserved Euclidean norm: S2 = s2. From now on we shall no longer make any distinction between the spatial tetrad indices ab ... and the ordinary spatial indices ij ... which are raised and lowered with the Kronecker metric. Explicit results for the equations of motion and gravitational wave templates will be given in Sections 11.2 and 11.3 using the canonical choice for the conserved-norm spin variable S.

11.2 Equations of motion and precession for spin-orbit effects

The previous formalism can be generalized to self-gravitating systems consisting of two (or more generally N) spinning point particles. The metric generated by the system of particles, interacting only through gravitation, is solution of the Einstein field equations (18*) with stress-energy tensor given by the sum of the individual stress-energy tensors (378*) for each particles. The equations of motion of the particles are given by the Mathisson–Papapetrou equations (377*) with “self-gravitating” metric evaluated at the location of the particles thanks to a regularization procedure (see Section 6). The precession equations of each of the spins are given by

dSa- = Ω × S , (388 ) dt a a
where a = 1,2 labels the particles. The spin variables Sa are the conserved-norm spins defined in Section 11.1. In the following it is convenient to introduce two combinations of the individuals spins defined by (with Xa ≡ ma ∕m and ν ≡ X X 1 2)82

We shall investigate the case where the binary’s orbit is quasi-circular, i.e., whose radius is constant apart from small perturbations induced by the spins (as usual we neglect the gravitational radiation damping effects). We denote by x = y1 − y2 and v = dx ∕dt the relative position and velocity.83 We introduce an orthonormal moving triad {n, λ,ℓ } defined by the unit separation vector n = x ∕r (with r = |x|) and the unit normal ℓ to the instantaneous orbital plane given by ℓ = n × v ∕|n × v|; the orthonormal triad is then completed by λ = ℓ × n. Those vectors are represented on Figure 4*, which shows the geometry of the system. The orbital frequency Ω is defined for general orbits, not necessarily circular, by v = r˙n + rΩ λ where ˙r = n ⋅ v represents the derivative of r with respect to the coordinate time t. The general expression for the relative acceleration a ≡ dv∕dt decomposed in the moving basis {n,λ, ℓ} is

a = (¨r − rΩ2 )n + (rΩ˙+ 2˙rΩ )λ + rϖ Ωℓ. (390 )
Here we have introduced the orbital plane precession ϖ of the orbit defined by ϖ ≡ − λ ⋅ dℓ∕dt. Next we impose the restriction to quasi-circular precessing orbits which is defined by the conditions ˙ 5 r˙= Ω = 𝒪 (1∕c ) and 10 ¨r = 𝒪 (1∕c ) so that 2 2 2 10 v = r Ω + 𝒪(1∕c ); see Eqs. (227). Then λ represents the direction of the velocity, and the precession frequency ϖ is proportional to the variation of ℓ in the direction of the velocity. In this way we find that the equations of the relative motion in the frame of the center-of-mass are
( 1 ) a = − r Ω2n + r ϖΩ ℓ + 𝒪 -5 . (391 ) c
Since we neglect the radiation reaction damping there is no component of the acceleration along λ. This equation represents the generalization of Eq. (226*) for spinning quasi-circular binaries with no radiation reaction. The orbital frequency Ω will contain spin effects in addition to the non-spin terms given by (228), while the precessional frequency ϖ will entirely be due to spins.

Here we report the latest results for the spin-orbit (SO) contributions into these quantities at the next-to-next-to-leading level corresponding to 3.5PN order [307*, 90*]. We project out the spins on the moving orthonormal basis, defining S = Snn + S λλ + Sℓℓ and similarly for Σ. We have


which has to be added to the non-spin terms (228) up to 3.5PN order. We recall that the ordering post-Newtonian parameter is γ = Gm2- rc. On the other hand the next-to-next-to-leading SO effects into the precessional frequency read


where this time the ordering post-Newtonian parameter is x ≡ (GmcΩ3-)2∕3. The SO terms at the same level in the conserved energy associated with the equations of motion will be given in Eq. (415) below. In order to complete the evolution equations for quasi-circular orbits we need also the precession vectors Ωa of the two spins as defined by Eq. (388*). These are given by


We obtain Ω2 from Ω1 simply by exchanging the masses, Δ → − Δ. At the linear SO level the precession vectors Ω a are independent of the spins.84

We now investigate an analytical solution for the dynamics of compact spinning binaries on quasi-circular orbits, including the effects of spin precession [54*, 306*]. This solution will be valid whenever the radiation reaction effects can be neglected, and is restricted to the linear SO level.

View Image
Figure 4: Geometric definitions for the precessional motion of spinning compact binaries [54*, 306*]. We show (i) the source frame defined by the fixed orthonormal basis {x, y,z }; (ii) the instantaneous orbital plane which is described by the orthonormal basis {xℓ,yℓ,ℓ}; (iii) the moving triad {n, λ, ℓ} and the associated three Euler angles α, ι and Φ; (v) the direction of the total angular momentum J which coincides with the z–direction. Dashed lines show projections into the xy plane.

In the following, we will extensively employ the total angular momentum of the system, that we denote by J, and which is conserved when radiation-reaction effects are neglected,

dJ --- = 0. (395 ) dt
It is customary to decompose the conserved total angular momentum J as the sum of the orbital angular momentum L and of the two spins,85
S J = L + --. (396 ) c
This split between L and S = S1 + S2 is specified by our choice of spin variables, here the conserved-norm spins defined in Section 11.1. Note that although L is called the “orbital” angular momentum, it actually includes both non-spin and spin contributions. We refer to Eq. (4.7) in [90*] for the expression of L at the next-to-next-to-leading SO level for quasi-circular orbits.

Our solution will consist of some explicit expressions for the moving triad {n, λ,ℓ } at the SO level in the conservative dynamics for quasi-circular orbits. With the previous definitions of the orbital frequency Ω and the precessional frequency ϖ we have the following system of equations for the time evolution of the triad vectors,


Equivalently, introducing the orbital rotation vector Ω ≡ Ω ℓ and spin precession vector ϖ ≡ ϖn, these equations can be elegantly written as


Next we introduce a fixed (inertial) orthonormal basis {x, y,z} as follows: z is defined to be the normalized value J ∕J of the total angular momentum J; y is orthogonal to the plane spanned by z and the direction N = X ∕R of the detector as seen from the source (notation of Section 3.1) and is defined by y = z × N ∕|z × N |; and x completes the triad – see Figure 4*. Then, we introduce the standard spherical coordinates (α,ι) of the vector ℓ measured in the inertial basis {x, y,z}. Since ι is the angle between the total and orbital angular momenta, we have

|J × ℓ| sin ι = ---J---, (399 )
where J ≡ |J|. The angles (α, ι) are referred to as the precession angles.

We now derive from the time evolution of our triad vectors those of the precession angles (α,ι), and of an appropriate phase Φ that specifies the position of n with respect to some reference direction in the orbital plane denoted x ℓ. Following Ref. [13], we pose


such that {xℓ,yℓ,ℓ} forms an orthonormal basis. The motion takes place in the instantaneous orbital plane spanned by n and λ, and the phase angle Φ is such that (see Figure 4*):


from which we deduce

e−iΦ = xℓ ⋅ (n + iλ ) = ∘Jλ-−-iJn-. (402 ) J2n + J2λ
Combining Eqs. (402*) with (399*) we also get
− iΦ J-λ −-iJn sinιe = J . (403 )
By identifying the right-hand sides of (397) with the time-derivatives of the relations (401) we obtain the following system of equations for the variations of α, ι and Φ,

On the other hand, using the decompositon of the total angular momentum (396*) together with the fact that the components of L projected along n and λ are of the order 𝒪(S ), see e.g., Eq. (4.7) in Ref. [90*], we deduce that sin ι is itself a small quantity of order 𝒪 (S). Since we also have ϖ = 𝒪 (S), we conclude by direct integration of the sum of Eqs. (404a) and (404c) that

Φ + α = ϕ + 𝒪 (S2), (405 )
in which we have defined the “carrier” phase as
∫ ϕ ≡ Ωdt = Ω(t − t0) + ϕ0, (406 )
with ϕ 0 the value of the carrier phase at some arbitrary initial time t 0. An important point we have used when integrating (406*) is that the orbital frequency Ω is constant at linear order in the spins. Indeed, from Eq. (392) we see that only the components of the conserved-norm spin vectors along ℓ can contribute to Ω at linear order. As we show in Eq. (409c) below, these components are in fact constant at linear order in spins. Thus we can treat Ω as a constant for our purpose.

The combination Φ + α being known by Eq. (405*), we can further express the precession angles ι and α at linear order in spins in terms of the components Jn and Jλ; from Eqs. (399*) and (403*):


where we denote by LNS the norm of the non-spin (NS) part of the orbital angular momentum L.

It remains to obtain the explicit time variation of the components of the two individual spins a Sn, Saλ and Saℓ (with a = 1,2). Using Eqs. (407) together with the decomposition (396*) and the explicit expression of L in Ref. [90*], we shall then be able to obtain the explicit time variation of the precession angles (α, ι) and phase Φ. Combining (388*) and (397) we obtain


where Ωa is the norm of the precession vector of the a-th spin as given by (394), and the precession frequency ϖ is explicitly given by (393). At linear order in spins these equations translate into


We see that, as stated before, the spin components along ℓ are constant, and so is the orbital frequency Ω given by (392). At the linear SO level, the equations (409) can be decoupled and integrated as


Here Sa⊥ and Sa∥ denote two quantities for each spins, that are constant up to terms 𝒪 (S2 ). The phase of the projection perpendicular to the direction ℓ of each of the spins is given by

0 ψa = (Ω − Ωa )(t − t0) + ψ a, (411 )
where ψ0 a is the constant initial phase at the reference time t0.

Finally we can give in an explicit way, to linear SO order, the triad {n (t),λ (t),ℓ(t)} in terms of some reference triad {n0, λ0,ℓ0} at the reference time t0 in Eqs. (411*) and (406*). The best way to express the result is to introduce the complex null vector m ≡ √12(n + iλ ) and its complex conjuguate m--; the normalization is chosen so that m ⋅ m-= 1. We obtain


The precession effects in the dynamical solution for the evolution of the basis vectors {n, λ, ℓ} are given by the second terms in these equations. They depend only in the combination iα sinιe and its complex conjugate sinιe−iα, which follows from Eqs. (407) and the known spin and non-spin contributions to the total angular momentum J. One can check that precession effects in the above dynamical solution (412) for the moving triad start at order 𝒪 (1∕c3).

11.3 Spin-orbit effects in the gravitational wave flux and orbital phase

Like in Section 9 our main task is to control up to high post-Newtonian order the mass and current radiative multipole moments UL and VL which parametrize the asymptotic waveform and gravitational fluxes far away from the source, cf. Eqs. (66) – (68). The radiative multipole moments are in turn related to the source multipole moments IL and JL through complicated relationships involving tails and related effects; see e.g., Eqs. (76).86

The source moments have been expressed in Eqs. (123) in terms of some source densities Σ, Σi and Σij defined from the components of the post-Newtonian expansion of the pseudo-tensor, denoted -αβ τ. To lowest order the (PN expansion of the) pseudo-tensor reduces to the matter tensor T αβ which has compact support, and the source densities Σ, Σi, Σij reduce to the compact support quantities σ, σ i σ ij given by Eqs. (145). Now, computing spin effects, the matter tensor αβ T has been found to be given by (378*) in the framework of the pole-dipole approximation suitable for SO couplings (and sufficient also for SS interactions between different spins). Here, to give a flavor of the computation, we present the lowest order spin contributions (necessarily SO) to the general mass and current source multipole moments (∀ ℓ ∈ ℕ):


Paralleling the similar expressions (304) for the Newtonian approximation to the source moments in the non-spin case, we posed ℓ−1 ℓ ℓ−1 σℓ(ν) ≡ X 2 + (− ) X 1 with Xa = ma ∕m [see also Eqs. (305)]. In Eqs. (413) we employ the notation (389) for the two spins and the ordinary cross product × of Euclidean vectors. Thus, the dominant level of spins is at the 1.5PN order in the mass-type moments IL, but only at the 0.5PN order in the current-type moments JL. It is then evident that the spin part of the current-type moments will always dominate over that of the mass-type moments. We refer to [53*, 89*] for higher order post-Newtonian expressions of the source moments. If we insert the expressions (413) into tail integrals like (76), we find that some spin contributions originate from tails starting at the 3PN order [54*].

Finally, skipping details, we are left with the following highest-order known result for the SO contributions to the gravitational wave energy flux, which is currently 4PN order [53, 89*, 306]:87


We recall that S ℓ ≡ ℓ ⋅ S and Σℓ ≡ ℓ ⋅ Σ, with S and Σ denoting the combinations (389), and the individual spins are the specific conserved-norm spins that have been introduced in Section 11.1. The result (414) superposes to the non-spin contributions given by Eq. (314). Satisfyingly it is in complete agreement in the test-mass limit where ν → 0 with the result of black-hole perturbation theory on a Kerr background obtained in Ref. [396].

Finally we can compute the spin effects in the time evolution of the binary’s orbital frequency Ω. We rely as in Section 9 on the equation (295*) balancing the total emitted energy flux ℱ with the variation of the binary’s center-of-mass energy E. The non-spin contributions in E have been provided for quasi-circular binaries in Eq. (232); the SO contributions to next-to-next-to-leading order are given by [307, 90]


Using E and ℱ expressed as functions of the orbital frequency Ω (through x) and of the spin variables (through Sℓ and Σ ℓ), we transform the balance equation into

( ) Ω˙SO = − ---ℱ---- . (416 ) dE ∕d Ω SO

However, in writing the latter equation it is important to justify that the spin quantities Sℓ and Σ ℓ are secularly constant, i.e., do not evolve on a gravitational radiation reaction time scale so we can neglect their variations when taking the time derivative of Eq. (415). Fortunately, this is the case of the conserved-norm spin variables, as proved in Ref. [423] up to relative 1PN order, i.e., considering radiation reaction effects up to 3.5PN order. Furthermore this can also be shown from the following structural general argument valid at linear order in spins [54, 89]. In the center-of-mass frame, the only vectors at our disposal, except for the spins, are n and v. Recalling that the spin vectors are pseudovectors regarding parity transformation, we see that the only way SO contributions can enter scalars such as the energy E or the flux ℱ is through the mixed products (n,v,Sa), i.e., through the components a Sℓ. Now, the same reasoning applies to the precession vectors Ωa in Eqs. (388*): They must be pseudovectors, and, at linear order in spin, they must only depend on n and v; so that they must be proportional to ℓ, as can be explicitly seen for instance in Eq. (394). Now, the time derivative of the components along ℓ of the spins are given by dSa∕dt = S ⋅ (d ℓ∕dt + ℓ × Ω ) ℓ a a. The second term vanishes because Ωa ∝ ℓ, and since dℓ∕dt = 𝒪 (S), we obtain that a S ℓ is constant at linear order in the spins. We have already met an instance of this important fact in Eq. (409c). This argument is valid at any post-Newtonian order and for general orbits, but is limited to spin-orbit terms; furthermore it does not specify any time scale for the variation, so it applies to short time scales such as the orbital and precessional periods, as well as to the long gravitational radiation reaction time scale (see also Ref. [218] and references therein for related discussions).

Table 4: Spin-orbit contributions to the number of gravitational-wave cycles 𝒩cycle [defined by Eq. (319*)] for binaries detectable by ground-based detectors LIGO-VIRGO. The entry frequency is fseismic = 10 Hz and the terminal frequency is ---c3-- fISCO = 63∕2πGm. For each compact object the magnitude χa and the orientation κa of the spin are defined by Sa = Gm2a χaSˆa and κa = Sˆa ⋅ ℓ; remind Eq. (366*). The spin-spin (SS) terms are neglected.
PN order   1.4M ⊙ + 1.4M ⊙ 10M ⊙ + 1.4M ⊙ 10M ⊙ + 10M ⊙
1.5PN (leading SO) 65.6κ1χ1 + 65.6κ2χ2 114.0κ1χ1 + 11.7κ2χ2 16.0κ1χ1 + 16.0κ2 χ2
2.5PN (1PN SO) 9.3κ1χ1 + 9.3κ2χ2 33.8κ1χ1 + 2.9κ2χ2 5.7κ1χ1 + 5.7κ2χ2
3PN (leading SO-tail) − 3.2 κ1χ1 − 3.2κ2χ2 − 13.2κ1χ1 − 1.3κ2χ2 − 2.6κ1χ1 − 2.6κ2χ2
3.5PN (2PN SO) 1.9κ1χ1 + 1.9κ2χ2 11.1κ1χ1 + 0.8κ2χ2 1.7κ1χ1 + 1.7κ2χ2
4PN (1PN SO-tail) − 1.5 κ1χ1 − 1.5κ2χ2 − 8.0κ1χ1 − 0.7κ2χ2 − 1.5κ1χ1 − 1.5κ2χ2

We are then allowed to apply Eq. (416*) with conserved-norm spin variables at the SO level. We thus obtain the secular evolution of Ω and from that we deduce by a further integration (following the Taylor approximant T2) the secular evolution of the carrier phase ∫ ϕ ≡ Ωdt:


This expression, when added to the expression for the non-spin terms given by Eq. (318), and considering also the SS terms, constitutes the main theoretical input needed for the construction of templates for spinning compact binaries. However, recall that in the case of precessional binaries, for which the spins are not aligned or anti-aligned with the orbital angular momentum, we must subtract to the carrier phase ϕ the precessional correction α arising from the precession of the orbital plane. Indeed the physical phase variable Φ which is defined in Figure 4*, has been proved to be given by Φ = ϕ − α at linear order in spins, cf. Eq. (405*). The precessional correction α can be computed at linear order in spins from the results of Section 11.2.

As we have done in Table 3 for the non-spin terms, we evaluate in Table 4 the SO contributions to the number of gravitational-wave cycles accumulated in the bandwidth of LIGO-VIRGO detectors, Eq. (319*). The results show that the SO terms up to 4PN order can be numerically important, for spins close to maximal and for suitable orientations. They can even be larger than the corresponding non-spin contributions at 3.5PN order and perhaps at 4PN order (but recall that the non-spin terms at 4PN order are not known); compare with Table 3. We thus conclude that the SO terms are relevant to be included in the gravitational wave templates for an accurate extraction of the binary parameters. Although numerically smaller, the SS terms are also relevant; for these we refer to the literature quoted at the beginning of Section 11.

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