Asymptotic Gravitational Waveform

3 Asymptotic Gravitational Waveform

3.1 The radiative multipole moments

The leading-order term $1∕R$ of the metric in radiative coordinates $(T,X )$ as given in Theorem 4, neglecting $𝒪 (1∕R2)$ , yields the operational definition of two sets of STF radiative multipole moments, mass-type $UL (U)$ and current-type $VL (U )$ . As we have seen, radiative coordinates are such that the retarded time $U ≡ T − R ∕c$ becomes asymptotically a null coordinate at future null infinity. The radiative moments are defined from the spatial components $ij$ of the metric in a transverse-traceless (TT) radiative coordinate system. By definition, we have [403 *]

We have formally re-summed the whole post-Minkowskian series in Eq. (56 *) from $n = 1$ up to $+∞$ . As before we denote for instance $N = N ⋅⋅⋅N L−2 i1 iℓ− 2$ and so on, where $N = (N ) i i$ and $N = X ∕R$ . The TT algebraic projection operator $𝒫ijab$ has already been defined at the occasion of the quadrupole-moment formalism in Eq. (2 *); and obviously the multipole decomposition (66) represents the generalization of the quadrupole formalism. Notice that the meaning of Eq. (66) is for the moment rather empty, because we do not yet know how to relate the radiative moments to the actual source parameters. Only at the Newtonian level do we know this relation, which is

where $Qij$ is the Newtonian quadrupole moment (3 *). Associated to the asymptotic waveform (66) we can compute by standard methods the total energy flux $ℱ = (dE ∕dU )GW$ and angular momentum flux $GW 𝒢i = (dJi∕dU )$ in gravitational waves [403 *]:

Next we introduce two unit polarization vectors $P$ and $Q$ , orthogonal and transverse to the direction of propagation $N$ (hence $NiNj + PiPj + QiQj = δij$ ). Our convention for the choice of $P$ and $Q$ will be clarified in Section 9.4. Then the two “plus” and “cross” polarization states of the asymptotic waveform are defined by

Although the multipole decomposition (66) is completely general, it will also be important, having in view the comparison between the post-Newtonian and numerical results (see for instance Refs. [107 *, 34 , 237 , 97 , 98 *]), to consider separately the various modes $(ℓ,m )$ of the asymptotic waveform as defined with respect to a basis of spin-weighted spherical harmonics of weight $− 2$ . Those harmonics are function of the spherical angles $(𝜃,ϕ)$ defining the direction of propagation $N$ , and given by

where $k1 = max (0,m − 2)$ and $k2 = min (ℓ + m, ℓ − 2 )$ . We thus decompose $h+$ and $h×$ onto the basis of such spin-weighted spherical harmonics, which means (see e.g., [107 , 272 *])

Using the orthonormality properties of these harmonics we can invert the latter decomposition and obtain the separate modes $hℓm$ from a surface integral,

where the overline refers to the complex conjugation. On the other hand, we can also relate $h ℓm$ to the radiative multipole moments $U L$ and $V L$ . The result is

where $ℓm U$ and $ℓm V$ denote the radiative mass and current moments in standard (non-STF) guise. These are related to the STF moments by

Here $ℓm αL$ denotes the STF tensor connecting together the usual basis of spherical harmonics $ℓm Y$ to the set of STF tensors $ˆNL = N ⟨i1 ...Niℓ⟩$ (where the brackets indicate the STF projection). Indeed both $ℓm Y$ and $ˆNL$ are basis of an irreducible representation of weight $ℓ$ of the rotation group; the two basis are related by²²

In Section 9.5 we shall present all the modes $(ℓ,m )$ of gravitational waves from inspiralling compact binaries up to 3PN order, and even 3.5PN order for the dominant mode $(2,2 )$ .

3.2 Gravitational-wave tails and tails-of-tails

We learned from Theorem 4 the general method which permits the computation of the radiative multipole moments $UL$ , $VL$ in terms of the source moments $IL, JL,...,ZL$ , or in terms of the intermediate canonical moments $M L$ , $S L$ discussed in Section 2.4. We shall now show that the relation between $UL$ , $VL$ and $ML$ , $SL$ (say) includes tail effects starting at the relative 1.5PN order.

Tails are due to the back-scattering of multipolar waves off the Schwarzschild curvature generated by the total mass monopole $M$ of the source. They correspond to the non-linear interaction between $M$ and the multipole moments $M L$ and $S L$ , and are given by some non-local integrals, extending over the past history of the source. At the 1.5PN order we find [59 *, 44 *]

where $r0$ is the length scale introduced in Eq. (42 *), and the constants $κℓ$ and $πℓ$ are given by

Recall from the gauge vector $α ξ(1)$ found in Eq. (58 *) that the retarded time $U = T − R∕c$ in radiative coordinates is related to the retarded time $u = t − r∕c$ in harmonic coordinates by

Inserting $U$ as given by Eq. (78 *) into Eqs. (76) we obtain the radiative moments expressed in terms of “source-rooted” harmonic coordinates $(t,r)$ , e.g.,

The remainder $𝒪 (G2 )$ in Eq. (78 *) is negligible here. This expression no longer depends on the constant $r 0$ , i.e., we find that $r 0$ gets replaced by $r$ . If we now replace the harmonic coordinates $(t,r )$ to some new ones, such as, for instance, some “Schwarzschild-like” coordinates $′ ′ (t,r )$ such that $′ t = t$ and $′ 2 r = r + GM ∕c$ (and $′ 3 u = u − GM ∕c$ ), we get

where $κ′ℓ = κℓ + 1∕2$ . This shows that the constant $κℓ$ (and $πℓ$ as well) depends on the choice of source-rooted coordinates $(t,r)$ : For instance, we have $κ2 = 11∕12$ in harmonic coordinates from Eq. (77a), but $′ κ 2 = 17∕12$ in Schwarzschild coordinates [345 ].

The tail integrals in Eqs. (76) involve all the instants from $− ∞$ in the past up to the current retarded time $U$ . However, strictly speaking, they do not extend up to infinite past, since we have assumed in Eq. (29 *) that the metric is stationary before the date $− 𝒯$ . The range of integration of the tails is therefore limited a priori to the time interval $[− 𝒯 ,U ]$ . But now, once we have derived the tail integrals, thanks to the latter technical assumption of stationarity in the past, we can argue that the results are in fact valid in more general situations for which the field has never been stationary. We have in mind the case of two bodies moving initially on some unbound (hyperbolic-like) orbit, and which capture each other, because of the loss of energy by gravitational radiation, to form a gravitationally bound system around time $− 𝒯$ .

In this situation let us check, using a simple Newtonian model for the behaviour of the multipole moment $ML (U − τ)$ when $τ → + ∞$ , that the tail integrals, when assumed to extend over the whole time interval $[− ∞, U ]$ , remain perfectly well-defined (i.e., convergent) at the integration bound $τ = + ∞$ . Indeed it can be shown [180 ] that the motion of initially free particles interacting gravitationally is given by $i i i i x (U − τ) = V τ + W ln τ + X + o(1)$ , where $V i$ , $W i$ and $Xi$ denote constant vectors, and $o(1) → 0$ when $τ → + ∞$ . From that physical assumption we find that the multipole moments behave when $τ → + ∞$ like

where $AL$ , $BL$ and $CL$ are constant tensors. We used the fact that the moment $ML$ will agree at the Newtonian level with the standard expression for the $ℓ$ -th mass multipole moment $QL$ . The appropriate time derivatives of the moment appearing in Eq. (76a) are therefore dominantly like

which ensures that the tail integral is convergent. This fact can be regarded as an a posteriori justification of our a priori too restrictive assumption of stationarity in the past. Thus, this assumption does not seem to yield any physical restriction on the applicability of the final formulas. However, once again, we emphasize that the past-stationarity is appropriate for real astrophysical sources of gravitational waves which have been formed at a finite instant in the past.

To obtain the results (76), we must implement in details the post-Minkowskian algorithm presented in Section 2.3. Let us flash here some results obtained with such algorithm. Consider first the case of the interaction between the constant mass monopole moment $M$ (or ADM mass) and the time-varying quadrupole moment $Mij$ . This coupling will represent the dominant non-static multipole interaction in the waveform. For these moments we can write the linearized metric using Eq. (35 *) in which by definition of the “canonical” construction we insert the canonical moments $Mij$ in place of $Iij$ (notice that $M = I$ ). We must plug this linearized metric into the quadratic-order part $N αβ(h,h )$ of the gravitational source term (24) – (25 *) and explicitly given by Eq. (26). This yields many terms; to integrate these following the algorithm [cf. Eq. (45 *)], we need some explicit formulas for the retarded integral of an extended (non-compact-support) source having some definite multipolarity $ℓ$ . A thorough account of the technical formulas necessary for handling the quadratic and cubic interactions is given in the Appendices of Refs. [50 *] and [48 *]. For the present computation the most crucial formula, needed to control the tails, corresponds to a source term behaving like $2 1 ∕r$ :

where $F$ is any smooth function representing a time derivative of the quadrupole moment, and $Q ℓ$ denotes the Legendre function of the second kind.²³ Note that there is no need to include a finite part operation $ℱ 𝒫$ in Eq. (83 *) as the integral is convergent. With the help of this and other formulas we obtain successively the objects defined in this algorithm by Eqs. (45 *) – (48) and finally obtain the quadratic metric (49 *) for that multipole interaction. The result is [60 *]²⁴

The metric is composed of two types of terms: “instantaneous” ones depending on the values of the quadrupole moment at the retarded time $u = t − r$ , and “hereditary” tail integrals, depending on all previous instants $t − rx < u$ .

Let us investigate now the cubic interaction between two mass monopoles $M$ with the mass quadrupole $Mij$ . Obviously, the source term corresponding to this interaction will involve [see Eq. (40b)] cubic products of three linear metrics, say $hM × hM × hMij$ , and quadratic products between one linear metric and one quadratic, say $h 2 × hM M ij$ and $hM × hMM ij$ . The latter case is the most tricky because the tails present in $hMMij$ , which are given explicitly by Eqs. (84), will produce in turn some tails of tails in the cubic metric $hM2Mij$ . The computation is rather involved [48 *] but can now be performed by an algebraic computer programme [74 *, 197 *]. Let us just mention the most difficult of the needed integration formulas for this calculation:²⁵

where $F(−1)$ is the time anti-derivative of $F$ . With this formula and others given in Ref. [48 *] we are able to obtain the closed algebraic form of the cubic metric for the multipole interaction $M × M × Mij$ , at the leading order when the distance to the source $r → ∞$ with $u = const$ . The result is²⁶

where all the moments $Mab$ are evaluated at the instant $u − τ = t − r − τ$ . Notice that the logarithms in Eqs. (86) contain either the ratio $τ∕r$ or $τ ∕r0$ . We shall discuss in Eqs. (93 *) – (94 *) below the interesting fate of the arbitrary constant $r0$ .

From Theorem 4, the presence of logarithms of $r$ in Eqs. (86) is an artifact of the harmonic coordinates $xα$ , and it is convenient to gauge them away by introducing radiative coordinates $X α$ at future null infinity. For controling the leading $1∕R$ term at infinity, it is sufficient to take into account the linearized logarithmic deviation of the light cones in harmonic coordinates: $α α α 2 X = x + G ξ(1) + 𝒪 (G )$ , where $ξα(1)$ is the gauge vector defined by Eq. (58 *) [see also Eq. (78 *)]. With this coordinate change one removes the logarithms of $r$ in Eqs. (86) and we obtain the radiative (or Bondi-type [93 ]) logarithmic-free expansion

where the moments are evaluated at time $U − τ = T − R − τ$ . It is trivial to compute the contribution of the radiative moments corresponding to that metric. We find the “tail of tail” term which will be reported in Eq. (91) below.

3.3 Radiative versus source moments

We first give the result for the radiative quadrupole moment $Uij$ expressed as a functional of the intermediate canonical moments $ML$ , $SL$ up to 3.5PN order included. The long calculation follows from implementing the explicit MPM algorithm of Section 2.3 and yields various types of terms:

The instantaneous (i.e., non-hereditary) piece $inst U ij$ up to 3.5PN order reads

The Newtonian term in this expression contains the Newtonian quadrupole moment $Qij$ and recovers the standard quadrupole formalism [see Eq. (67 *)];
The hereditary tail integral $Utail ij$ is made of the dominant tail term at 1.5PN order in agreement with Eq. (76a) above:

The length scale $r0$ is the one that enters our definition of the finite-part operation $ℱ 𝒫$ [see Eq. (42 *)] and it enters also the relation between the radiative and harmonic retarded times given by Eq. (78 *);
The hereditary tail-of-tail term appears dominantly at 3PN order [48 *] and is issued from the radiative metric computed in Eqs. (87):
Finally the memory-type hereditary piece $Umeijm$ contributes at orders 2.5PN and 3.5PN and is given by

The 2.5PN non-linear memory integral – the first term inside the coefficient of $G ∕c5$ – has been obtained using both post-Newtonian methods [42 , 427 *, 406 , 60 *, 50 ] and rigorous studies of the field at future null infinity [128 ]. The expression (92) is in agreement with the more recent computation of the non-linear memory up to any post-Newtonian order in Refs. [189 *, 192 ].

Be careful to note that the latter post-Newtonian orders correspond to “relative” orders when counted in the local radiation-reaction force, present in the equations of motion: For instance, the 1.5PN tail integral in Eq. (90) is due to a 4PN radiative effect in the equations of motion [58 *]; similarly, the 3PN tail-of-tail integral is expected to be associated with some radiation-reaction terms occurring at the 5.5PN order.

Note that $Uij$ , when expressed in terms of the intermediate moments $ML$ and $SL$ , shows a dependence on the (arbitrary) length scale $r0$ ; cf. the tail and tail-of-tail contributions (90) – (91). Most of this dependence comes from our definition of a radiative coordinate system as given by (78 *). Exactly as we have done for the 1.5PN tail term in Eq. (79 *), we can remove most of the $r0$ ’s by inserting $U = u − 2GM-ln(r∕r ) c3 0$ back into (89) – (92), and expanding the result when $c → ∞$ , keeping the necessary terms consistently. In doing so one finds that there remains a $r0$ -dependent term at the 3PN order, namely

However, the latter dependence on $r0$ is fictitious and should in fine disappear. The reason is that when we compute explicitly the mass quadrupole moment $Mij$ for a given matter source, we will find an extra contribution depending on $r0$ occurring at the 3PN order which will cancel out the one in Eq. (93 *). Indeed we shall compute the source quadrupole moment $I ij$ of compact binaries at the 3PN order, and we do observe on the result (300 *) – (301) below the requested terms depending on $r0$ , namely²⁷

where $Qij = μˆxij$ denotes the Newtonian quadrupole, $r12$ is the separation between the particles, and $m$ is the total mass differing from the ADM mass $M$ by small post-Newtonian corrections. Combining Eqs. (93 *) and (94 *) we see that the $r0$ -dependent terms cancel as expected. The appearance of a logarithm and its associated constant $r0$ at the 3PN order was pointed out in Ref. [7 *]; it was rederived within the present formalism in Refs. [58 *, 48 ]. Recently a result equivalent to Eq. (93 *) was obtained by means of the EFT approach using considerations related to the renormalization group equation [222 ].

The previous formulas for the 3.5PN radiative quadrupole moment permit to compute the dominant mode $(2,2)$ of the waveform up to order 3.5PN [197 *]; however, to control the full waveform one has also to take into account the contributions of higher-order radiative moments. Here we list the most accurate results we have for all the moments that permit the derivation of the waveform up to order 3PN [74 *]:²⁸

For all the other multipole moments in the 3PN waveform, it is sufficient to assume the agreement between the radiative and canonical moments, namely

In a second stage of the general formalism, we must express the canonical moments ${ML, SL }$ in terms of the six types of source moments ${I ,J ,W ,X ,Y ,Z } L L L L L L$ . For the control of the $(2,2)$ mode in the waveform up to 3.5PN order, we need to relate the canonical quadrupole moment $Mij$ to the corresponding source quadrupole moment $Iij$ up to that accuracy. We obtain [197 *]

Here, for instance, $W$ denotes the monopole moment associated with the moment $WL$ , and $Yi$ is the dipole moment corresponding to $YL$ . Notice that the difference between the canonical and source moments starts at the relatively high 2.5PN order. For the control of the full waveform up to 3PN order we need also the moments $Mijk$ and $Sij$ , which admit similarly some correction terms starting at the 2.5PN order:

The remainders in Eqs. (98) are consistent with the 3PN approximation for the full waveform. Besides the mass quadrupole moment (97), and mass octopole and current quadrupole moments (98), we can state that, with the required 3PN precision, all the other moments $ML$ , $SL$ agree with their source counterparts $I L$ , $J L$ :

With those formulas we have related the radiative moments ${UL, VL }$ parametrizing the asymptotic waveform (66) to the six types of source multipole moments ${IL,JL, WL, XL, YL, ZL }$ . What is missing is the explicit dependence of the source moments as functions of the actual parameters of some matter source. We come to grips with this important question in the next section.

	Abstract
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
	Acknowledgments
	References
	Footnotes
	Updates
	Figures
	Tables