Abstract
The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general postNewtonian sources. The exterior field of the source is investigated by means of a combination of analytic postMinkowskian and multipolar approximations. The physical observables in the farzone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the postNewtonian source in the nearzone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many nonlinear multipole interactions, among them those associated with the tails (and tailsoftails of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third postNewtonian (3PN) order beyond the Newtonian acceleration. The gravitationalwave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail effects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary’s orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.
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Introduction
The theory of gravitational radiation from isolated sources, in the context of general relativity, is a fascinating science that can be explored by means of what was referred to in the French XVIIIth century as l’analyse sublime: the analytical (i.e. mathematical) method, and more specifically the resolution of partial differential equations. Indeed, the field equations of general relativity, when use is made of the harmoniccoordinate conditions, take the form of a quasilinear hyperbolic differential system of equations, involving the famous wave operator or d’Alembertian (denoted □), invented by d’Alembert in his Traité de dynamique of 1743.
Nowadays, the importance of the field lies in the exciting possibility of comparing the theory with contemporary astrophysical observations, made by a new generation of detectors — largescale optical interferometers LIGO, VIRGO, GEO and TAMA — that should routinely observe the gravitational waves produced by massive and rapidly evolving systems such as inspiralling compact binaries. To prepare these experiments, the required theoretical work consists of carrying out a sufficiently general solution of the Einstein field equations, valid for a large class of matter systems, and describing the physical processes of the emission and propagation of the waves from the source to the distant detector, as well as their backreaction onto the source.
Gravitationalwave generation formalisms
The basic problem we face is to relate the asymptotic gravitationalwave form h_{ij} generated by some isolated source, at the location of some detector in the wave zone of the source, to the stressenergy tensor T^{αβ} of the matter fields^{Footnote 1}. For general sources it is hopeless to solve the problem via a rigorous deduction within the exact theory of general relativity, and we have to resort to approximation methods, keeping in mind that, sadly, such methods are often not related in a very precise mathematical way to the first principles of the theory. Therefore, a general wavegeneration formalism must somehow manage the nonlinearity of the field equations by imposing some suitable approximation series in one or several small physical parameters. Of ourse the ultimate aim of approximation methods is to extract from the theory some firm predictions for the outcome of experiments such as VIRGO and LIGO. Some important approximations that we shall use in this article are the postNewtonian method (or nonlinear 1/cexpansion), the postMinkowskian method or nonlinear iteration (Gexpansion), the multipole decomposition in irreducible representations of the rotation group (or equivalently aexpansion in the source radius), and the farzone expansion (1/Rexpansion in the distance). In particular, the postNewtonian expansion has provided us in the past with our best insights into the problems of motion and radiation in general relativity. The most successful wavegeneration formalisms make a gourmet cocktail of all these approximation methods. For reviews on analytic approximations and applications to the motion and the gravitational wavegeneration see Refs. [143, 53, 54, 144, 150, 8, 13].
The postNewtonian approximation is valid under the assumptions of a weak gravitational field inside the source (we shall see later how to model neutron stars and black holes), and of slow internal motions. The main problem with this approximation is its domain of validity, which is limited to the near zone of the source — the region surrounding the source that is of small extent with respect to the wavelength of waves. A serious consequence is the a priori inability of the postNewtonian expansion to incorporate the boundary conditions at infinity, which determine the radiation reaction force in the source’s local equations of motion. The postMinkowskian expansion, by contrast, is uniformly valid, as soon as the source is weakly selfgravitating, over all spacetime. In a sense, the postMinkowskian method is more fundamental than the postNewtonian one; it can be regarded as an “upstream” approximation with respect to the postNewtonian expansion, because each coefficient of the postMinkowskian series can in turn be reexpanded in a postNewtonian fashion. Therefore, a way to take into account the boundary conditions at infinity in the postNewtonian series is first to perform the postMinkowskian expansion. Notice that the postMinkowskian method is also upstream (in the previous sense) with respect to the multipole expansion, when considered outside the source, and with respect to the farzone expansion, when considered far from the source.
The most “downstream” approximation that we shall use in this article is the postNewtonian one; therefore this is the approximation that dictates the allowed physical properties of our matter source. We assume mainly that the source is at once slowly moving and weakly stressed, and we abbreviate this by saying that the source is postNewtonian. For postNewtonian sources, the parameter defined from the components of the matter stressenergy tensor T^{αβ} and the source’s Newtonian potential U by
is much less than one. This parameter represents essentially a slow motion estimate ε ∼ v/c, where v denotes a typical internal velocity. By a slight abuse of notation, following Chandrasekhar et al. [40, 42, 41], we shall henceforth write ε ≡ 1/c, even though ε is dimensionless whereas c has the dimension of a velocity. The small postNewtonian remainders will be denoted \(\hat L\). Thus, 1/c ≪ 1 in the case of postNewtonian sources. We have U/c^{2}^{1/2} ≪ 1/c for sources with negligible selfgravity, and whose dynamics are therefore driven by nongravitational forces. However, we shall generally assume that the source is selfgravitating; in that case we see that it is necessarily weakly (but not negligibly) selfgravitating, i.e. \(\hat J\). Note that the adjective “slowmotion” is a bit clumsy because we shall in fact consider very relativistic sources such as inspiralling compact binaries, for which 1/c can be as large as 30% in the last rotations, and whose description necessitates the control of high postNewtonian approximations.
The lowestorder wave generation formalism, in the Newtonian limit 1/c → 0, is the famous quadrupole formalism of Einstein [68] and Landau and Lifchitz [97]. This formalism can also be referred to as Newtonian because the evolution of the quadrupole moment of the source is computed using Newton’s laws of gravity. It expresses the gravitational field h ^{TT}_{ ij} in a transverse and traceless (TT) coordinate system, covering the far zone of the source^{Footnote 2}, as
where R = X is the distance to the source, N = X/R is the unit direction from the source to the observer, and \(\frac{1}{{\sqrt {2\pi } }}\) is the TT projection operator, with \(\hat {\mathcal{E}},\;\hat {\mathcal{C}}, \text{and} \hat {\mathcal{l}}_{z}\) being the projector onto the plane orthogonal to N. The source’s quadrupole moment takes the familiar Newtonian form
where ρ is the Newtonian mass density. The total gravitational power emitted by the source in all directions is given by the Einstein quadrupole formula
Our notation \({\mathcal{E}}, \text{and}\; \hat {\mathcal{l}}_{z}\) stands for the total gravitational “luminosity” of the source. The cardinal virtues of the EinsteinLandauLifchitz quadrupole formalism are its generality — the only restrictions are that the source be Newtonian and bounded — its simplicity, as it necessitates only the computation of the time derivatives of the Newtonian quadrupole moment (using the Newtonian laws of motion), and, most importantly, its agreement with the observation of the dynamics of the HulseTaylor binary pulsar PSR 1913+16 [140, 141, 139]. Indeed the prediction of the quadrupole formalism for the waves emitted by the binary pulsar system comes from applying Eq. (4) to a system of two point masses moving on an eccentric orbit (the classic reference is Peters and Mathews [117]; see also Refs. [71, 148]). Then, relying on the energy equation
where E is the Newtonian binary’s centerofmass energy, we deduce from Kepler’s third law the expression of the “observable”, that is, the change in the orbital period P of the pulsar, or P, as a function of P itself. From the binary pulsar test, we can say that the postNewtonian corrections to the quadrupole formalism, which we shall compute in this article, have already received, in the case of compact binaries, strong observational support (in addition to having, as we shall demonstrate, a sound theoretical basis).
The multipole expansion is one of the most useful tools of physics, but its use in general relativity is difficult because of the nonlinearity of the theory and the tensorial character of the gravitational interaction. In the stationary case, the multipole moments are determined by the expansion of the metric at spatial infinity, while, in the case of nonstationary fields, the moments, starting with the quadrupole, are defined at future null infinity. The multipole moments have been extensively studied in the linearized theory, which ignores the gravitational forces inside the source. Early studies have extended the formula (4) to include the currentquadrupole and massoctupole moments [111, 110], and obtained the corresponding formulas for linear momentum [111, 110, 1, 124] and angular momentum [116, 46]. The general structure of the infinite multipole series in the linearized theory was investigated by several works [126, 127, 119, 142], from which it emerged that the expansion is characterized by two and only two sets of moments: masstype and currenttype moments. Below we shall use a particular multipole decomposition of the linearized (vacuum) metric, parametrized by symmetric and tracefree (STF) mass and current moments, as given by Thorne [142]. The explicit expressions of the multipole moments (for instance in STF guise) as integrals over the source, valid in the linearized theory but irrespective of a slow motion hypothesis, are completely known [101, 39, 38, 57].
In the full nonlinear theory, the (radiative) multipole moments can be read off the coefficient of 1/R in the expansion of the metric when R → + ∞, with a null coordinate T — R/c = const.. The solutions of the field equations in the form of a farfield expansion (power series in 1/R) have been constructed, and their properties elucidated, by Bondi et al. [32] and Sachs [128]. The precise way under which such radiative spacetimes fall off asymptotically has been formulated geometrically by Penrose [114, 115] in the concept of an asymptotically simple spacetime (see also Ref. [76]). The resulting BondiSachsPenrose approach is very powerful, but it can answer a priori only a part of the problem, because it gives information on the field only in the limit where R → + ∞, which cannot be connected in a direct way to the actual behaviour of the source. In particular the multipole moments that one considers in this approach are those measured at infinity — we call them the radiative multipole moments. These moments are distinct, because of nonlinearities, from some more natural source multipole moments, which are defined operationally by means of explicit integrals extending over the matter and gravitational fields.
An alternative way of defining the multipole expansion within the complete nonlinear theory is that of Blanchet and Damour [14, 3], following pioneering work by Bonnor and collaborators [33, 34, 35, 81] and Thorne [142]. In this approach the basic multipole moments are the source moments, rather than the radiative ones. In a first stage, the moments are left unspecified, as being some arbitrary functions of time, supposed to describe an actual physical source. They are iterated by means of a postMinkowskian expansion of the vacuum field equations (valid in the source’s exterior). Technically, the postMinkowskian approximation scheme is greatly simplified by the assumption of a multipolar expansion, as one can consider separately the iteration of the different multipole pieces composing the exterior field (whereas, the direct attack of the postMinkowskian expansion, valid at once inside and outside the source, faces some calculational difficulties [147, 47]). In this “multipolarpostMinkowskian” formalism, which is physically valid over the entire weakfield region outside the source, and in particular in the wave zone (up to future null infinity), the radiative multipole moments are obtained in the form of some nonlinear functionals of the more basic source moments. A priori, the method is not limited to postNewtonian sources, however we shall see that, in the current situation, the closedform expressions of the source multipole moments can be established only in the case where the source is postNewtonian [6, 11]. The reason is that in this case the domain of validity of the postNewtonian iteration (viz. the near zone) overlaps the exterior weakfield region, so that there exists an intermediate zone in which the postNewtonian and multipolar expansions can be matched together. This is a standard application of the method of matched asymptotic expansions in general relativity [37, 36].
To be more precise, we shall show how a systematic multipolar and postMinkowskian iteration scheme for the vacuum Einstein field equations yields the most general physically admissible solution of these equations [14]. The solution is specified once we give two and only two sets of timevarying (source) multipole moments. Some general theorems about the nearzone and farzone expansions of that general solution will be stated. Notably, we find [3] that the asymptotic behaviour of the solution at future null infinity is in agreement with the findings of the BondiSachsPenrose [32, 128, 114, 115, 76] approach to gravitational radiation. However, checking that the asymptotic structure of the radiative field is correct is not sufficient by itself, because the ultimate aim is to relate the far field to the properties of the source, and we are now obliged to ask: What are the multipole moments corresponding to a given stressenergy tensor T^{αβ} describing the source? Only in the case of postNewtonian sources has it been possible to answer this question. The general expression of the moments was obtained at the level of the second postNewtonian (2PN) order in Ref. [6], and was subsequently proved to be in fact valid up to any postNewtonian order in Ref. [11]. The source moments are given by some integrals extending over the postNewtonian expansion of the total (pseudo) stressenergy tensor τ^{αβ}, which is made of a matter part described by T^{αβ} and a crucial nonlinear gravitational source term Λ^{αβ}. These moments carry in front a particular operation of taking the finite part (\(\left( {  \frac{{2399}}{{56}}q\;  \;\frac{{773}}{3}\pi } \right)\;{x^5}\) as we call it below), which makes them mathematically welldefined despite the fact that the gravitational part Λ^{αβ} has a spatially infinite support, which would have made the bound of the integral at spatial infinity singular (of course the finite part is not added a posteriori to restore the welldefiniteness of the integral, but is proved to be actually present in this formalism). The expressions of the moments had been obtained earlier at the 1PN level, albeit in different forms, in Ref. [16] for the masstype moments (strangely enough, the mass moments admit a compactsupport expression at 1PN order), and in Ref. [58] for the currenttype ones.
The wavegeneration formalism resulting from matching the exterior multipolar and postMinkowskian field [14, 3] to the postNewtonian source [6, 11] is able to take into account, in principle, any postNewtonian correction to both the source and radiative multipole moments (for any multipolarity of the moments). The relationships between the radiative and source moments include many nonlinear multipole interactions, because the source moments mix with each other as they “propagate” from the source to the detector. Such multipole interactions include the famous effects of wave tails, corresponding to the coupling between the nonstatic moments with the total mass M of the source. The nonlinear multipole interactions have been computed within the present wavegeneration formalism up to the 3PN order in Refs. [17, 12, 10]. Furthermore, the backreaction of the gravitationalwave emission onto the source, up to the 1.5PN order relative to the leading order of radiation reaction, has also been studied within this formalism [15, 5, 9]. Now, recall that the leading radiation reaction force, which is quadrupolar, occurs already at the 2.5PN order in the source’s equations of motion. Therefore the 1.5PN “relative” order in the radiation reaction corresponds in fact to the 4PN order in the equations of motion, beyond the Newtonian acceleration. It has been shown that the gravitational wave tails enter the radiation reaction at precisely the 1.5PN relative order, which means 4PN “absolute” order [15].
A different wavegeneration formalism has been devised by Will and Wiseman [152] (see also Refs. [151, 112]), after earlier attempts by Epstein and Wagoner [70] and Thorne [142]. This formalism has exactly the same scope as ours, i.e. it applies to any isolated postNewtonian sources, but it differs in the definition of the source multipole moments and in many technical details when properly implemented [152]. In both formalisms, the moments are generated by the postNewtonian expansion of the pseudotensor τ^{αβ}, but in the WillWiseman formalism they are defined by some compactsupport integrals terminating at some finite radius \(\left( {  \frac{{1041349}}{{18144}} + \frac{{171}}{{16}}{q^2}  \frac{{243}}{8}q  \frac{{785}}{6}\pi } \right){x^5}\) enclosing the source, e.g., the radius of the near zone). By contrast, in our case [6, 11], the moments are given by some integrals covering the whole space and regularized by means of the finite part \(y = \frac{C}{{{Q^2}}},\;{Q^2} = l_z^2 + {a^2}(1  {E^2})\). We shall prove the complete equivalence, at the most general level, between the two formalisms. What is interesting about both formalisms is that the source multipole moments, which involve a whole series of relativistic corrections, are coupled together, in the true nonlinear solution, in a very complicated way. These multipole couplings give rise to the many tail and related nonlinear effects, which form an integral part of the radiative moments at infinity and thereby of the observed signal.
Part A of this article is devoted to a presentation of the postNewtonian wave generation formalism. We try to state the main results in a form that is simple enough to be understood without the full details, but at the same time we outline some of the proofs when they present some interest on their own. To emphasize the importance of some key results, we present them in the form of mathematical theorems.
Problem posed by compact binary systems
Inspiralling compact binaries, containing neutron stars and/or black holes, are promising sources of gravitational waves detectable by the detectors LIGO, VIRGO, GEO and TAMA. The two compact objects steadily lose their orbital binding energy by emission of gravitational radiation; as a result, the orbital separation between them decreases, and the orbital frequency increases. Thus, the frequency of the gravitationalwave signal, which equals twice the orbital frequency for the dominant harmonics, “chirps” in time (i.e. the signal becomes higher and higher pitched) until the two objects collide and merge.
The orbit of most inspiralling compact binaries can be considered to be circular, apart from the gradual inspiral, because the gravitational radiation reaction forces tend to circularize the motion rapidly. For instance, the eccentricity of the orbit of the HulseTaylor binary pulsar is presently e_{0} = 0.617. At the time when the gravitational waves emitted by the binary system will become visible by the detectors, i.e. when the signal frequency reaches about 10 Hz (in a few hundred million years from now), the eccentricity will be e = 5.3 × 10^{6} — a value calculated from the Peters [116] law, which is itself based on the quadrupole formula (2).
The main point about modelling the inspiralling compact binary is that a model made of two structureless point particles, characterized solely by two mass parameters m_{1} and m_{2} (and possibly two spins), is sufficient. Indeed, most of the nongravitational effects usually plaguing the dynamics of binary star systems, such as the effects of a magnetic field, of an interstellar medium, and so on, are dominated by gravitational effects. However, the real justification for a model of point particles is that the effects due to the finite size of the compact bodies are small. Consider for instance the influence of the Newtonian quadrupole moments Q_{1} and Q_{2} induced by tidal interaction between two neutron stars. Let a_{1} and a_{2} be the radius of the stars, and L the distance between the two centers of mass. We have, for tidal moments,
where k_{1} and k_{2} are the star’s dimensionless (second) Love numbers [103], which depend on their internal structure, and are, typically, of the order unity. On the other hand, for compact objects, we can introduce their “compactness”, defined by the dimensionless ratios
which equal ∼ 0.2 for neutron stars (depending on their equation of state). The quadrupoles Q_{1} and Q_{2} will affect both sides of Eq. (5), i.e. the Newtonian binding energy E of the two bodies, and the emitted total gravitational flux \(\mathcal{O}(\epsilon{^2})\) as computed using the Newtonian quadrupole formula (4). It is known that for inspiralling compact binaries the neutron stars are not corotating because the tidal synchronization time is much larger than the time left till the coalescence. As shown by Kochanek [92] the best models for the fluid motion inside the two neutron stars are the socalled RocheRiemann ellipsoids, which have tidally locked figures (the quadrupole moments face each other at any instant during the inspiral), but for which the fluid motion has zero circulation in the inertial frame. In the Newtonian approximation we find that within such a model (in the case of two identical neutron stars) the orbital phase, deduced from Eq. (5), reads
where x — (Gmω/c^{3})^{2/3} is a standard dimensionless postNewtonian parameter ∼ 1/c^{2} (ω is the orbital frequency), and where k is the Love number and K is the compactness of the neutron star. The first term in the righthand side of (8) corresponds to the gravitationalwave damping of two point masses; the second term is the finitesize effect, which appears as a relative correction, proportional to (x/K)^{5}, to the latter radiation damping effect. Because the finitesize effect is purely Newtonian, its relative correction ∼ (x/K)^{5} should not depend on c; and indeed the factors 1/c^{2} cancel out in the ratio x/K. However, the compactness K of compact objects is by Eq. (7) of the order unity (or, say, one half), therefore the 1/c^{2} it contains should not be taken into account numerically in this case, and so the real order of magnitude of the relative contribution of the finitesize effect in Eq. (8) is given by x^{5} alone. This means that for compact objects the finitesize effect should be comparable, numerically, to a postNewtonian correction of order 5PN or 1/c^{10} (see Ref. [52] for the proof in the context of relativistic equations of motion). This is a much higher postNewtonian order than the one at which we shall investigate the gravitational effects on the phasing formula. Using k′ ≡ const. k ∼ 1 and K ∼ 0.2 for neutron stars (and the bandwidth of a VIRGO detector between 10 Hz and 1000 Hz), we find that the cumulative phase error due to the finitesize effect amounts to less that one orbital rotation over a total of ∼ 16, 000 produced by the gravitationalwave damping of point masses. The conclusion is that the finitesize effect can in general be neglected in comparison with purely gravitationalwave damping effects. But note that for noncompact or moderately compact objects (such as white dwarfs for instance) the Newtonian tidal interaction dominates over the radiation damping.
The inspiralling compact binaries are ideally suited for application of a highorder postNewtonian wave generation formalism. The main reason is that these systems are very relativistic, with orbital velocities as high as 0.3c in the last rotations (as compared to ∼ 10^{3}c for the binary pulsar), and it is not surprising that the quadrupolemoment formalism (2, 3, 4, 5) constitutes a poor description of the emitted gravitational waves, since many postNewtonian corrections play a substantial role. This expectation has been confirmed in recent years by several measurementanalyses [48, 49, 72, 50, 135, 121, 122, 96, 59], which have demonstrated that the postNewtonian precision needed to implement successively the optimal filtering technique in the LIGO/VIRGO detectors corresponds grossly, in the case of neutronstar binaries, to the 3PN approximation, or 1/c^{6} beyond the quadrupole moment approximation. Such a high precision is necessary because of the large number of orbital rotations that will be monitored in the detector’s frequency bandwidth (∼ 16, 000 in the case of neutron stars), giving the possibility of measuring very accurately the orbital phase of the binary. Thus, the 3PN order is required mostly to compute the time evolution of the orbital phase, which depends, via the energy equation (5), on the centerofmass binding energy E and the total gravitationalwave energy flux \(\mathcal{O}\left( {{{(v/c)}^n}} \right)\).
In summary, the theoretical problem posed by inspiralling compact binaries is twofold: On the one hand E, and on the other hand \({\psi _4} =  {C_{\alpha \beta \gamma \delta }}{n^\alpha }{\bar m^\beta }{n^\gamma }{\bar m^\delta }\), are to be deduced from general relativity with the 3PN precision or better. To obtain E we must control the 3PN equations of motion of the binary in the case of general, not necessarily circular, orbits. As for \(_s\mathcal{O}\) it necessitates the application of a 3PN wave generation formalism (actually, things are more complicated because the equations of motion are also needed during the computation of the flux). It is quite interesting that such a high order approximation as the 3PN one should be needed in preparation for LIGO and VIRGO data analysis. As we shall see, the signal from compact binaries contains at the 3PN order the signature of several nonlinear effects which are specific to general relativity. Therefore, we have here the possibility of probing, experimentally, some aspects of the nonlinear structure of Einstein’s theory [28, 29].
PostNewtonian equations of motion and radiation
By equations of motion we mean the explicit expression of the accelerations of the bodies in terms of the positions and velocities. In Newtonian gravity, writing the equations of motion for a system of N particles is trivial; in general relativity, even writing the equations in the case N = 2 is difficult. The first relativistic term, at the 1PN order, was derived by Lorentz and Droste [98]. Subsequently, Einstein, Infeld and Hoffmann [69] obtained the 1PN corrections by means of their famous “surfaceintegral” method, in which the equations of motion are deduced from the vacuum field equations, and which are therefore applicable to any compact objects (be they neutron stars, black holes, or, perhaps, naked singularities). The 1PNaccurate equations were also obtained, for the motion of the centers of mass of extended bodies, by Petrova [118] and Fock [73] (see also Ref. [109]).
The 2PN approximation was tackled by Otha et al. [105, 107, 106], who considered the postNewtonian iteration of the Hamiltonian of N pointparticles. We refer here to the Hamiltonian as the Fokkertype Hamiltonian, which is obtained from the matterplusfield ArnowittDeserMisner (ADM) Hamiltonian by eliminating the field degrees of freedom. The result for the 2PN and even 2.5PN equations of binary motion in harmonic coordinates was obtained by Damour and Deruelle [56, 55, 67, 51, 52], building on a nonlinear iteration of the metric of two particles initiated in Ref. [2]. The corresponding result for the ADMHamiltonian of two particles at the 2PN order was given in Ref. [63] (see also Refs. [130, 131]). Kopeikin [93] derived the 2.5PN equations of motion for two extended compact objects. The 2.5PNaccurate harmoniccoordinate equations as well as the complete gravitational field (namely the metric g_{αβ}) generated by two point masses were computed by Blanchet, Faye and Ponsot [25], following a method based on previous work on wave generation [6].
Up to the 2PN level the equations of motion are conservative. Only at the 2.5PN order appears the first nonconservative effect, associated with the gravitational radiation reaction. The (harmoniccoordinate) equations of motion up to that level, as derived by Damour and Deruelle [56, 55, 67, 51, 52], have been used for the study of the radiation damping of the binary pulsar — its orbital P [52]. It is important to realize that the 2.5PN equations of motion have been proved to hold in the case of binary systems of strongly selfgravitating bodies [52]. This is via an “effacing” principle (in the terminology of Damour [52]) for the internal structure of the bodies. As a result, the equations depend only on the “Schwarzschild” masses, m_{1} and m_{2}, of the compact objects. Notably their compactness parameters K_{1} and K_{2}, defined by Eq. (7), do not enter the equations of motion, as has been explicitly verified up to the 2.5PN order by Kopeikin [93] and Grishchuk and Kopeikin [79], who made a “physical” computation, à la Fock, taking into account the internal structure of two selfgravitating extended bodies. The 2.5PN equations of motion have also been established by Itoh, Futamase and Asada [83, 84], who use a variant of the surfaceintegral approach of Einstein, Infeld and Hoffmann [69], that is valid for compact bodies, independently of the strength of the internal gravity.
The present state of the art is the 3PN approximation^{Footnote 3}. To this order the equations have been worked out independently by two groups, by means of different methods, and with equivalent results. On the one hand, Jaranowski and Schäfer [87, 88, 89], and Damour, Jaranowski and Schäfer [60, 62, 61], following the line of research of Refs. [105, 107, 106, 63], employ the ADMHamiltonian formalism of general relativity; on the other hand, Blanchet and Faye [21, 22, 20, 23], and de Andrade, Blanchet and Faye [66], founding their approach on the postNewtonian iteration initiated in Ref. [25], compute directly the equations of motion (instead of a Hamiltonian) in harmonic coordinates. The end results have been shown [62, 66] to be physically equivalent in the sense that there exists a unique “contact” transformation of the dynamical variables that changes the harmoniccoordinates Lagrangian obtained in Ref. [66] into a new Lagrangian, whose Legendre transform coincides exactly with the Hamiltonian given in Ref. [60]. The 3PN equations of motion, however, depend on one unspecified numerical coefficient, ω_{static}, in the ADMHamiltonian formalism and Λ in the harmoniccoordinates approach, which is due to some incompleteness of the Hadamard selffield regularization method. This coefficient has been fixed by means of a dimensional regularization in Ref. [61].
So far the status of the postNewtonian equations of motion is quite satisfying. There is mutual agreement between all the results obtained by means of different approaches and techniques, whenever it is possible to compare them: point particles described by Dirac deltafunctions, extended postNewtonian fluids, surfaceintegrals methods, mixed postMinkowskian and postNewtonian expansions, direct postNewtonian iteration and matching, harmonic coordinates versus ADMtype coordinates, and different processes or variants of the regularization of the self field of point particles. In Part B of this article, we shall present the most complete results for the 3PN equations of motion, and for the associated Lagrangian and Hamiltonian formulations (from which we deduce the centerofmass energy E).
The second subproblem, that of the computation of the energy flux \(\hat T\), has been carried out by application of the wavegeneration formalism described previously. Following earliest computations at the 1PN level [149, 30], at a time when the postNewtonian corrections in \({T_{nn}},{T_{\bar mn}},\;\text{and}\;{T_{\overline {mm} }}\) had a purely academic interest, the energy flux of inspiralling compact binaries was completed to the 2PN order by Blanchet, Damour and Iyer [18, 77], and, independently, by Will and Wiseman [152], using their own formalism (see Refs. [19, 27] for joint reports of these calculations). The preceding approximation, 1.5PN, which represents in fact the dominant contribution of tails in the wave zone, had been obtained in Refs. [153, 31] by application of the formula for tail integrals given in Ref. [17]. Higherorder tail effects at the 2.5PN and 3.5PN orders, as well as a crucial contribution of tails generated by the tails themselves (the socalled "tails of tails") at the 3PN order, were obtained by Blanchet [7, 10]. However, unlike the 1.5PN, 2.5PN and 3.5PN orders that are entirely composed of tail terms, the 3PN approximation also involves, besides the tails of tails, many nontail contributions coming from the relativistic corrections in the (source) multipole moments of the binary. These have been “almost” completed by Blanchet, Iyer and Joguet [26, 24], in the sense that the result still involves one unknown numerical coefficient, due to the use of the Hadamard regularization, which is a combination of the parameter λ in the equations of motion, and a new parameter θ coming from the computation of the 3PN quadrupole moment. In PartB of this article, we shall present the most uptodate results for the 3.5PN energy flux and orbital phase, deduced from the energy equation (5), supposed to be valid at this order.
The postNewtonian flux \({r_ \pm } = M \pm \sqrt {{M^2}  {a^2}} \), which comes from a “standard” postNewtonian calculation, is in complete agreement (up to the 3.5PN order) with the result given by the very different technique of linear blackhole perturbations, valid in the “testmass” limit where the mass of one of the bodies tends to zero (limit v → 0, where v = μ/m). Linear blackhole perturbations, triggered by the geodesic motion of a small mass around the black hole, have been applied to this problem by Poisson [120] at the 1.5PN order (following the pioneering work of Galt’sov et al. [75]), and by Tagoshi and Nakamura [135], using a numerical code, up to the 4PN order. This technique has culminated with the beautiful analytical methods of Sasaki, Tagoshi and Tanaka [129, 137, 138] (see also Ref. [102]), who solved the problem up to the extremely high 5.5PN order.
Part A: PostNewtonian Sources
Einstein’s Field Equations
The field equations of general relativity form a system of ten secondorder partial differential equations obeyed by the spacetime metric g_{αβ},
where the Einstein curvature tensor \(\bar R_{\ell m\omega }^{\text{in},\;\text{up}}\) is generated, through the gravitational coupling κ = 8πG/c^{4}, by the matter stressenergy tensor T^{αβ}. Among these ten equations, four govern, via the contracted Bianchi identity, the evolution of the matter system,
The spacetime geometry is constrained by the six remaining equations, which place six independent constraints on the ten components of the metric g_{αβ}, leaving four of them to be fixed by a choice of a coordinate system.
In most of this paper we adopt the conditions of harmonic, or de Donder, coordinates. We define, as a basic variable, the gravitationalfield amplitude
where g^{αβ} denotes the contravariant metric (satisfying g^{αμ} g_{μβ} = δ ^{α}_{ β} ), where g is the determinant of the covariant metric, g = det(g_{αβ}), and where η^{αβ} represents an auxiliary Minkowskian metric. The harmoniccoordinate condition, which accounts exactly for the four equations (10) corresponding to the conservation of the matter tensor, reads
The equations (11, 12) introduce into the definition of our coordinate system a preferred Minkowskian structure, with Minkowski metric η_{αβ}. Of course, this is not contrary to the spirit of general relativity, where there is only one physical metric g_{αβ} without any flat prior geometry, because the coordinates are not governed by geometry (so to speak), but rather are chosen by researchers when studying physical phenomena and doing experiments. Actually, the coordinate condition (12) is especially useful when we view the gravitational waves as perturbations of spacetime propagating on the fixed Minkowskian manifold with the background metric η_{αβ}. This view is perfectly legitimate and represents a fructuous and rigorous way to think of the problem when using approximation methods. Indeed, the metric η_{αβ}, originally introduced in the coordinate condition (12), does exist at any finite order of approximation (neglecting higherorder terms), and plays in a sense the role of some “prior” flat geometry.
The Einstein field equations in harmonic coordinates can be written in the form of inhomogeneous flat d’Alembertian equations,
where □ ≡ □_{η} = η^{μν}∂_{μ}∂_{ν}. The source term, τ^{αβ}, can rightly be interpreted as the stressenergy pseudotensor (actually, τ^{αβ} is a Lorentz tensor) of the matter fields, described by T^{αβ}, and the gravitational field, given by the gravitational source term Λ^{αβ}, i.e.
The exact expression of Λ^{αβ}, including all nonlinearities, reads
As is clear from this expression, Λ^{αβ} is made of terms at least quadratic in the gravitationalfield strength h and its first and second spacetime derivatives. In the following, for the highest postNewtonian order that we consider (3PN), we need the quadratic, cubic and quartic pieces of Λ^{αβ}. With obvious notation, we can write them as
These various terms can be straightforwardly computed from Eq. (15); see Eqs. (3.8) in Ref. [22] for explicit expressions.
As said above, the condition (12) is equivalent to the matter equations of motion, in the sense of the conservation of the total pseudotensor τ^{αβ},
In this article, we look for the solutions of the field equations (13, 14, 15, 17) under the following four hypotheses:

1.
The matter stressenergy tensor T^{αβ} is of spatially compact support, i.e. can be enclosed into some timelike world tube, say r ≤ a, where r = x is the harmoniccoordinate radial distance. Outside the domain of the source, when r > a, the gravitational source term, according to Eq. (17), is divergencefree,
$${\partial _\mu }{\Lambda ^{\alpha \mu }} = 0\;\;\;(when\;r > a).$$((18)) 
2.
The matter distribution inside the source is smooth^{Footnote 4}: \(\mathcal{E} = \mu \hat {\mathcal{E}},{l_z} = \mu {\hat l_z}\). We have in mind a smooth hydrodynamical “fluid” system, without any singularities nor shocks (a priori), that is described by some Eulerian equations including high relativistic corrections. In particular, we exclude from the start any black holes (however we shall return to this question when we find a model for describing compact objects).

3.
The source is postNewtonian in the sense of the existence of the small parameter defined by Eq. (1). For such a source we assume the legitimacy of the method of matched asymptotic expansions for identifying the inner postNewtonian field and the outer multipolar decomposition in the source’s exterior near zone.

4.
The gravitational field has been independent of time (stationary) in some remote past, i.e. before some finite instant \(C = {\mu ^2}\hat C\) in the past, in the sense that
$$\frac{\partial }{{\partial t}}[{h^{\alpha \beta }}(x,t)] = 0\;\;\;when\;t \leqslant  \mathcal{T}.$$((19))
The latter condition is a means to impose, by brute force, the famous noincoming radiation condition, ensuring that the matter source is isolated from the rest of the Universe and does not receive any radiation from infinity. Ideally, the noincoming radiation condition should be imposed at past null infinity. We shall later argue (see Section 6) that our condition of stationarity in the past (19), although much weaker than the real noincoming radiation condition, does not entail any physical restriction on the general validity of the formulas we derive.
Subject to the condition (19), the Einstein differential field equations (13) can be written equivalently into the form of the integrodifferential equations
containing the usual retarded inverse d’Alembertian operator, given by
extending over the whole threedimensional space \(\dot t = dt/d\tau \).
Linearized Vacuum Equations
In what follows we solve the field equations (12, 13), in the vacuum region outside the compactsupport source, in the form of a formal nonlinearity or postMinkowskian expansion, considering the field variable h^{αβ} as a nonlinear metric perturbation of Minkowski spacetime. At the linearized level (or firstpostMinkowskian approximation), we write:
where the subscript “ext” reminds us that the solution is valid only in the exterior of the source, and where we have introduced Newton’s constant G as a bookkeeping parameter, enabling one to label very conveniently the successive postMinkowskian approximations. Since h^{αβ} is a dimensionless variable, with our convention the linear coefficient h ^{αβ}_{1} in Eq. (22) has the dimension of the inverse of G — a mass squared in a system of units where ħ = c = 1. In vacuum, the harmoniccoordinate metric coefficient h_{1}^{αβ} satisfies
We want to solve those equations by means of an infinite multipolar series valid outside a timelike world tube containing the source. Indeed the multipole expansion is the correct method for describing the physics of the source as seen from its exterior (r > a). On the other hand, the postMinkowskian series is physically valid in the weakfield region, which surely includes the exterior of any source, starting at a sufficiently large distance. For postNewtonian sources the exterior weakfield region, where both multipole and postMinkowskian expansions are valid, simply coincides with the exterior r > a. It is therefore quite natural, and even, one would say inescapable when considering general sources, to combine the postMinkowskian approximation with the multipole decomposition. This is the original idea of the “doubleexpansion” series of Bonnor [33], which combines the Gexpansion (or mexpansion in his notation) with the aexpansion (equivalent to the multipole expansion, since the lth order multipole moment scales like a^{l} with the source radius).
The multipolarpostMinkowskian method will be implemented systematically, using STFharmonics to describe the multipole expansion [142], and looking for a definite algorithm for the approximation scheme [14]. The solution of the system of equations (23, 24) takes the form of a series of retarded multipolar waves^{Footnote 5}
where r = x, and where the functions \({c_0} = (\ell  1)\ell (\ell + 1)(\ell + 2)  12iM\omega \) are smooth functions of the retarded time \(R_{\ell m\omega }^{\text{in}}\), which become constant in the past, when \(X_{\ell m\omega }^{\text{in}}\). It is evident, since a monopolar wave satisfies □(K_{L}(u)/r) = 0 and the d’Alembertian commutes with the multiderivative ∂_{L}, that Eq. (25) represents the most general solution of the wave equation (23) (see Section 2 in Ref. [14] for a proof based on the EulerPoissonDarboux equation). The gauge condition (24), however, is not fulfilled in general, and to satisfy it we must algebraically decompose the set of functions K ^{00}_{ L} , K ^{0 i}_{ L} , K ^{ij}_{ L} into ten tensors which are STF with respect to all their indices, including the spatial indices i, ij. Imposing the condition (24) reduces the number of independent tensors to six, and we find that the solution takes an especially simple “canonical” form, parametrized by only two moments, plus some arbitrary linearized gauge transformation [142, 14].
Theorem 1 The most general solution of the linearized field equations (23, 24), outside some timelike world tube enclosing the source (r > a), and stationary in the past (see Eq. (19)), reads
The first term depends on two STFtensorial multipole moments, I_{L} (u) and J_{L} (u), which are arbitrary functions of time except for the laws of conservation of the monopole: I = const., and dipoles: I_{i} = const., J_{i} = const.. It is given by
The other terms represent a linearized gauge transformation, with gauge vector ϕ ^{α}_{1} of the type (25), and parametrized for four other multipole moments, say W_{L}(u), X_{L}(u), Y_{L}(u) and Z_{L}(u).
The conservation of the lowestorder moments gives the constancy of the total mass of the source, M ≡ I = const., centerofmass position^{Footnote 6}, X_{i} ≡ I_{i}/I = const., total linear momentum P_{i} ≡ I ^{(1)}_{ i} = 0, and total angular momentum, S_{i} ≡ J_{i} = const.. It is always possible to achieve X_{i} = 0 by translating the origin of our coordinates to the center of mass. The total mass M is the ArnowittDeserMisner (ADM) mass of the Hamiltonian formulation of general relativity. Note that the quantities M, X_{i}, P_{i} and S_{i} include the contributions due to the waves emitted by the source. They describe the “initial” state of the source, before the emission of gravitational radiation.
The multipole functions I_{L}(u) and J_{L}(u), which thoroughly encode the physical properties of the source at the linearized level (because the other moments W_{L}, ..., Z_{L} parametrize a gauge transformation), will be referred to as the masstype and currenttype source multipole moments. Beware, however, that at this stage the moments are not specified in terms of the stressenergy tensor T^{αβ} of the source: the above theorem follows merely from the algebraic and differential properties of the vacuum equations outside the source.
For completeness, let us give the components of the gaugevector ϕ ^{α}_{1} entering Eq. (26):
Because the theory is covariant with respect to nonlinear diffeomorphisms and not merely with respect to linear gauge transformations, the moments W_{L}, ..., Z_{L} do play a physical role starting at the nonlinear level, in the following sense. If one takes these moments equal to zero and continues the calculations one ends up with a metric depending on I_{L} and J_{L} only, but that metric will not describe the same physical source as the one constructed from the six moments I_{L}, ..., Z_{L}. In other words, the two nonlinear metrics associated with the sets of multipole moments {I_{L}, J_{L}, 0, ..., 0} and {I_{L}, J_{L}, W_{L}, ..., Z_{L}} are not isometric. We point out in Section 4.2 below that the full set of moments {I_{L}, J_{L}, W_{L}, ..., Z_{L}} is in fact physically equivalent to some reduced set {M_{L}, S_{L}, 0, ..., 0}, but with some moments M_{L}, S_{L} that differ from I_{L}, J_{L} by nonlinear corrections (see Eq. (90)). All the multipole moments I_{L}, J_{L}, W_{L}, X_{L}, Y_{L}, Z_{L} will be computed in Section 5.
Nonlinear Iteration of the Field Equations
By Theorem 1 we know the most general solution of the linearized equations in the exterior of the source. We then tackle the problem of the postMinkowskian iteration of that solution. We consider the full postMinkowskian series
where the first term is composed of the result given by Eqs. (26, 27, 28). In this article, we shall always understand the infinite sums such as the one in Eq. (29) in the sense of formal power series, i.e. as an ordered collection of coefficients, e.g., \(R_{\ell m\omega }^{\text{in}}\;\text{at}\;r \sim {r_0}\). We do not attempt to control the mathematical nature of the series and refer to the mathematicalphysics literature for discussion (in the present context, see notably Refs. [45, 64]).
The postMinkowskian solution
We insert the ansatz (29) into the vacuum Einstein field equations (12, 13), i.e. with τ^{αβ} = c^{4}/(16πG)Λ^{αβ}, and we equate term by term the factors of the successive powers of our bookkeeping parameter G. We get an infinite set of equations for each of the \(R_{\ell m\omega }^{\text{in}}\),
The righthand side of the wave equation (30) is obtained from inserting the previous iterations, up to the order n — 1, into the gravitational source term. In more details, the series of equations (30) reads
The quadratic, cubic and quartic pieces of Λ^{αβ} are defined by Eq. (16).
Let us now proceed by induction. Some n being given, we assume that we succeeded in constructing, from the linearized coefficient h_{1}, the sequence of postMinkowskian coefficients h_{2}, h_{3}, ..., h_{n1}, and from this we want to infer the next coefficient h_{n}. The righthand side of Eq. (30), Λ ^{αβ}_{ n} , is known by induction hypothesis. Thus the problem is that of solving a wave equation whose source is given. The point is that this wave equation, instead of being valid everywhere in \(X_{\ell m\omega }^{\text{in}} (z)\), is correct only outside the matter (r > a), and it makes no sense to solve it by means of the usual retarded integral. Technically speaking, the righthand side of Eq. (30) is composed of the product of many multipole expansions, which are singular at the origin of the spatial coordinates r = 0, and which make the retarded integral divergent at that point. This does not mean that there are no solutions to the wave equation, but simply that the retarded integral does not constitute the appropriate solution in that context.
What we need is a solution which takes the same structure as the source term Λ_{n}^{αβ}, i.e. is expanded into multipole contributions, with a singularity at r = 0, and satisfies the d’Alembertian equation as soon as r > 0. Such a particular solution can be obtained, following the suggestion in Ref. [14], by means of a mathematical trick in which one first “regularizes” the source term And by multiplying it by the factor r^{B}, where \(X_{\ell m\omega }^{\text{in}}\). Let us assume, for definiteness, that Λ ^{αβ}_{ n} is composed of multipolar pieces with maximal multipolarity l_{max}. This means that we start the iteration from the linearized metric (26, 27, 28) in which the multipolar sums are actually finite^{Footnote 7}. The divergences when r → 0 of the source term are typically powerlike, say 1/r_{k} (there are also powers of the logarithm of r), and with the previous assumption there will exist a maximal order of divergency, say k_{max}. Thus, when the real part of B is large enough, i.e. \(X_{\ell}\), the “regularized” source term r^{B}Λ ^{αβ}_{ n} is regular enough when r → 0 so that one can perfectly apply the retarded integral operator. This defines the Bdependent retarded integral
where the symbol □ ^{1}_{ret} stands for the retarded integral (21). It is convenient to introduce inside the regularizing factor some arbitrary constant length scale r_{0} in order to make it dimensionless. Everywhere in this article we pose
The fate of the constant r_{0} in a detailed calculation will be interesting to follow, as we shall see, because it provides some check that the calculation is going well. Now the point for our purpose is that the function I^{αβ}(B) on the complex plane, which was originally defined only when \({\xi _\ell }\), admits a unique analytic continuation to all values of \({X _\ell }\) except at some integer values. Furthermore, the analytic continuation of I^{αβ} (B) can be expanded, when B → 0 (namely the limit of interest to us) into a Laurent expansion involving in general some multiple poles. The key idea, as we shall prove, is that the finite part, or the coefficient of the zeroth power of B in that expansion, represents the particular solution we are looking for. We write the Laurent expansion of I^{αβ}(B), when B → 0, in the form
where \({\xi _\ell }\), and the various coefficients ι ^{αβ}_{ p} are functions of the field point (x, t). When p_{0} ≤ 1 there are poles; p_{0}, which depends on n, refers to the maximal order of the poles. By applying the box operator onto both sides of (37), and equating the different powers of B, we arrive at
As we see, the case p = 0 shows that the finitepart coefficient in Eq. (37), namely ι ^{αβ}_{0} , is a particular solution of the requested equation: □ι ^{αβ}_{0} = Ant. Furthermore, we can prove that this term, by its very construction, owns the same structure made of a multipolar expansion singular at r=0.
Let us forget about the intermediate name ι ^{αβ}_{0} , and denote, from now on, the latter solution by u ^{αβ}_{ n} ≡ ι ^{αβ}_{0} , or, in more explicit term
where the finitepart symbol \(\mathcal{O}(\epsilon)\) means the previously detailed operations of considering the analytic continuation, taking the Laurent expansion, and picking up the finitepart coefficient when B → 0. The story is not complete, however, because u ^{αβ}_{ n} does not fulfill the constraint of harmonic coordinates (31); its divergence, say w ^{α}_{ n} — ∂_{μ}u ^{αμ}_{ n} , is different from zero in general. From the fact that the source term is divergencefree in vacuum, ∂_{μ}Λ ^{αμ}_{ n} = 0 (see Eq. (18)), we find instead
The factor B comes from the differentiation of the regularization factor \(\mathcal{O}(\epsilon^2)\). So, w ^{α}_{ n} is zero only in the special case where the Laurent expansion of the retarded integral in Eq. (40) does not develop any simple pole when B → 0. Fortunately, when it does, the structure of the pole is quite easy to control. We find that it necessarily consists of a solution of the sourcefree d’Alembertian equation, and, what is more (from its stationarity in the past), the solution is a retarded one. Hence, taking into account the index structure of w ^{α}_{ n} , there must exist four STFtensorial functions of the retarded time u = t — r/c, say N_{L}(u), P_{L} (u), Q_{L}(u) and R_{L}(u), such that
From that expression we are able to find a new object, say v_{n}, which takes the same structure as w ^{α}_{ n} (a retarded solution of the sourcefree wave equation) and, furthermore, whose divergence is exactly the opposite of the divergence of u ^{αβ}_{ n} , i.e. ∂_{μ}v ^{αμ}_{ n} = w ^{α}_{ n} . Such a v ^{αβ}_{ n} is not unique, but we shall see that it is simply necessary to make a choice for v ^{αβ}_{ n} (the simplest one) in order to obtain the general solution. The formulas that we adopt are
Notice the presence of antiderivatives, denoted, e.g., by \(\mathcal{O}(\epsilon)\); there is no problem with the limit v → — ∞ since all the corresponding functions are zero when \(\mathcal{O}(\epsilon^{2\ell+2})\). The choice made in Eqs. (42) is dictated by the fact that the 00 component involves only some monopolar and dipolar terms, and that the spatial trace ii is monopolar: v ^{ii}_{ n} = 3r^{1}P. Finally, if we pose
we see that we solve at once the d’Alembertian equation (30) and the coordinate condition (31). That is, we have succeeded in finding a solution of the field equations at the nth postMinkowskian order. By induction the same method applies to any order n, and, therefore, we have constructed a complete postMinkowskian series (29) based on the linearized approximation h ^{αβ}_{1} given by (26, 27, 28). The previous procedure constitutes an algorithm, which could be implemented by an algebraic computer programme.
Notes
^{1} In this article Greek indices take the values 0, 1, 2, 3 and Latin 1, 2, 3. Our signature is +2. G and c are Newton’s constant and the speed of light.
^{2} The TT coordinate system can be extended to the near zone of the source as well; see for instance Ref. [95].
^{3} Let us mention that the 3.5PN terms in the equations of motion are also known, both for pointparticle binaries [85, 86, 113] and extended fluid bodies [5, 9]; they correspond to 1PN “relative” corrections in the radiation reaction force. Known also is the contribution of wave tails in the equations of motion, which arises at the 4PN order and represents a 1.5PN modification of the gravitational radiation damping [15].
^{4}\(\hat {\mathcal {E}},\;\hat {\mathcal {l}}_z, \text{and}\;\hat {\mathcal {C}}\) are the usual sets of nonnegative integers, integers, real numbers and complex numbers; C^{p}(Ω) is the set of ptimes continuously differentiable functions on the open domain Ω (p ≤ + ∞).
^{5}Our notation is the following: L = i_{1}i_{2} ... i_{l} denotes a multiindex, made of l (spatial) indices. Similarly we write for instance P = j_{1} ... j_{p} (in practice, we generally do not need to consider the carrier letter i or j), or aL — 1 = ai_{1} ... i_{l1}. Always understood in expressions such as Eq. (25) are l summations over the l indices i_{1}, ... , i_{l} ranging from 1 to 3. The derivative operator ∂_{L} is a shorthand for ∂_{i1} ... ∂_{il}. The function K_{L} is symmetric and tracefree (STF with respect to the l indices composing L. This means that for any pair of indices i_{p}, i_{q} ∈ L, we have \(_{  2}S_{\ell m}^{a\omega }(\theta )\) and that \({\tilde Z_{\ell m\omega }}\) (see Ref. [142] and Appendices A and B in Ref. [14] for reviews about the STF formalism). The STF projection is denoted with a hat, so \({T_{\ell m\omega }}\), or sometimes with carets around the indices, K_{L} ≡ K_{〈L〉} . In particular, \({\tilde Z_{\ell m\omega }}\) is the STF projection of the product of unit vectors \({R_{\ell m\omega }}\); an expansion into STF tensors \({X_{\ell m\omega }}\) is equivalent to the usual expansion in spherical harmonics \({\chi _{\ell m\omega }}\Delta /{({r^2} + {a^2})^{1/2}}\). Similarly, we denote \({X_{\ell m\omega }}\) and \({R_{\ell m\omega }}\). Superscripts like (p) indicate (p) successive timederivations.
^{6} The constancy of the center of mass X_{i} — rather than a linear variation with time — results from our assumption of stationarity before the date \(\bar X_{\ell m\omega }^{{\text{in}}} = X_{\ell  m  \omega }^{\text{in}}\). Hence, P_{i} = 0.
^{7} 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 (26, 27, 28), there is a maximal multipolarity l_{max}(n) at any postMinkowskian order n, which grows linearly with n.
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Blanchet, L. Gravitational Radiation from PostNewtonian Sources and Inspiralling Compact Binaries. Living Rev. Relativ. 5, 3 (2002). https://doi.org/10.12942/lrr20023
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DOI: https://doi.org/10.12942/lrr20023
Keywords
 Inspiralling Compact Binaries
 postNewtonian Source
 Multipole Moments
 Hadamard Regularization
 postMinkowskian Expansion
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Gravitational Radiation from PostNewtonian Sources and Inspiralling Compact Binaries Published:
 01 December 2014
 Accepted:
 27 January 2014
DOI: https://doi.org/10.12942/lrr20142
 Gravitational Radiation from PostNewtonian Sources and Inspiralling Compact Binaries
 Published:
 01 December 2006
 Accepted:
 16 May 2006
DOI: https://doi.org/10.12942/lrr20064
Original
Gravitational Radiation from PostNewtonian Sources and Inspiralling Compact Binaries Published:
 30 April 2002
 Accepted:
 01 February 2002
DOI: https://doi.org/10.12942/lrr20023