Skip to main content

Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries

An Earlier Version of this article was published on 01 December 2006

The Original Version of this article was published on 30 April 2002

Abstract

To be observed and analyzed by the network of gravitational wave detectors on ground (LIGO, VIRGO, etc.) and by the future detectors in space (eLISA, etc.), inspiralling compact binaries — binary star systems composed of neutron stars and/or black holes in their late stage of evolution — require high-accuracy templates predicted by general relativity theory. The gravitational waves emitted by these very relativistic systems can be accurately modelled using a high-order post-Newtonian gravitational wave generation formalism. In this article, we present the current state of the art on post-Newtonian methods as applied to the dynamics and gravitational radiation of general matter sources (including the radiation reaction back onto the source) and inspiralling compact binaries. We describe the post-Newtonian equations of motion of compact binaries and the associated Lagrangian and Hamiltonian formalisms, paying attention to the self-field regularizations at work in the calculations. Several notions of innermost circular orbits are discussed. We estimate the accuracy of the post-Newtonian approximation and make a comparison with numerical computations of the gravitational self-force for compact binaries in the small mass ratio limit. The gravitational waveform and energy flux are obtained to high post-Newtonian order and the binary’s orbital phase evolution is deduced from an energy balance argument. Some landmark results are given in the case of eccentric compact binaries — moving on quasi-elliptical orbits with non-negligible eccentricity. The spins of the two black holes play an important role in the definition of the gravitational wave templates. We investigate their imprint on the equations of motion and gravitational wave phasing up to high post-Newtonian order (restricting to spin-orbit effects which are linear in spins), and analyze the post-Newtonian spin precession equations as well as the induced precession of the orbital plane.

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 XVIIIth century France 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 harmonic-coordinate conditions, take the form of a quasi-linear hyperbolic differential system of equations, involving the famous wave operator or d’Alembertian [140]. The resolution of that system of equations constitutes a problème bien posé in the sense of Hadamard [236, 104], and which is amenable to an analytic solution using approximation methods.

Nowadays, the importance of the field lies in the exciting comparison of the theory with contemporary astrophysical observations, of binary pulsars like the historical Hulse-Taylor pulsar PSR 1913+16 [250], and, in a forthcoming future, of gravitational waves produced by massive and rapidly evolving systems such as inspiralling compact binaries. These should be routinely detected on Earth by the network of large-scale laser interferometers, including the advanced versions of the ground-based interferometers LIGO and VIRGO, with also GEO and the future cryogenic detector KAGRA. The first direct detection of a coalescence of two black holes has been achieved on September 14, 2015 by the advanced LIGO detector [1]. Further ahead, the space-based laser interferometer LISA (actually, the evolved version eLISA) should be able to detect supermassive black-hole binaries at cosmological distances.

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 gravitational waves from the source to the distant detector, as well as their back-reaction onto the source. The solution should then be applied to specific sources like inspiralling compact binaries.

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. Of course the ultimate aim of approximation methods is to extract from the theory some firm predictions to be compared with the outcome of experiments. However, we have to keep in mind that such methods are often missing a clear theoretical framework and are sometimes not related in a very precise mathematical way to the first principles of the theory.

The flagship of approximation methods is the post-Newtonian approximation, which has been developed from the early days of general relativity [303]. This approximation is at the origin of many of the great successes of general relativity, and it gives wonderful answers to the problems of motion and gravitational radiation of systems of compact objects. Three crucial applications are:

  1. 1.

    The motion of N point-like objects at the first post-Newtonian approximation level [184], is taken into account to describe the solar system dynamics (motion of the centers of mass of planets);

  2. 2.

    The gravitational radiation-reaction force, which appears in the equations of motion at the second-and-a-half post-Newtonian (2.5PN) order [148, 147, 143, 142], has been experimentally verified by the observation of the secular acceleration of the orbital motion of the Hulse-Taylor binary pulsar PSR 1913+16 [399, 400, 398];

  3. 3.

    The analysis of gravitational waves emitted by inspiralling compact binaries — two neutron stars or black holes driven into coalescence by emission of gravitational radiation — necessitates the prior knowledge of the equations of motion and radiation field up to very high post-Newtonian order.

This article reviews the current status of the post-Newtonian approach to the problems of the motion of inspiralling compact binaries and their emission of gravitational waves. When the two compact objects approach each other toward merger, the post-Newtonian expansion will lose accuracy and should be taken over by numerical-relativity computations [359, 116, 21]. We shall refer to other review articles like Refs. [233, 187] for discussions of numerical-relativity methods. Despite very intensive developments of numerical relativity, the post-Newtonian approximation remains indispensable for describing the inspiral phase of compact binaries to high accuracy, and for providing a powerful benchmark against which the numerical computations are tested.

Part A of the article deals with general post-Newtonian matter sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the post-Newtonian source in the near-zone the explicit expressions of the source multipole moments are obtained. The relationships between the radiative and source moments involve many non-linear multipole interactions, among them those associated with the tails (and tails-of-tails, etc.) of gravitational waves.

Part B is devoted to the application to compact binary systems, with particular emphasis on black hole binaries with spins. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary’s 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 (in the simple case of circular orbits).

All over the article we try to state the main results in a form that is simple enough to be understood without the full details; however, we also 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. In applications we generally show the most up-to-date results up to the highest known post-Newtonian order.Footnote 1

Analytic approximations and wave generation formalism

The basic problem we face is to relate the asymptotic gravitational-wave form hij generated by some isolated source, at the location of a detector in the wave zone of the source, to the material content of the source, i.e., its stress-energy tensor Tαβ, using approximation methods in general relativity.Footnote 2 Therefore, a general wave-generation formalism must solve the field equations, and the non-linearity therein, by imposing some suitable approximation series in one or several small physical parameters. Some important approximations that we shall use in this article are the post-Newtonian method (or non-linear 1/c-expansion), the post-Minkowskian method or non-linear iteration (G-expansion), the multipole decomposition in irreducible representations of the rotation group (or equivalently a-expansion in the source radius), the far-zone expansion (1/R-expansion in the distance to the source), and the perturbation in the small mass limit (ν-expansion in the mass ratio of a binary system). In particular, the post-Newtonian expansion has provided us in the past with our best insights into the problems of motion and radiation. The most successful wave-generation formalisms make a gourmet cocktail of these approximation methods. For reviews on analytic approximations and applications to the motion and the gravitational wave-generation see Refs. [404, 142, 144, 145, 405, 421, 46, 52, 378]. For reviews on black-hole pertubations and the self-force approach see Refs. [348, 373, 177, 23].

The post-Newtonian 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.Footnote 3 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 the gravitational waves. A serious consequence is the a priori inability of the post-Newtonian expansion to incorporate the boundary conditions at infinity, which determine the radiation reaction force in the source’s local equations of motion.

The post-Minkowskian expansion, by contrast, is uniformly valid, as soon as the source is weakly self-gravitating, over all space-time. In a sense, the post-Minkowskian method is more fundamental than the post-Newtonian one; it can be regarded as an “upstream” approximation with respect to the post-Newtonian expansion, because each coefficient of the post-Minkowskian series can in turn be re-expanded in a post-Newtonian fashion. Therefore, a way to take into account the boundary conditions at infinity in the post-Newtonian series is to control first the post-Minkowskian expansion. Notice that the post-Minkowskian method is also upstream (in the previous sense) with respect to the multipole expansion, when considered outside the source, and with respect to the far-zone expansion, when considered far from the source.

The most “downstream” approximation that we shall use in this article is the post-Newtonian 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 post-Newtonian. For post-Newtonian sources, the parameter defined from the components of the matter stress-energy tensor Tαβ and the source’s Newtonian potential U by

$$\epsilon \equiv \max \left\{{\left| {{{{T^{0i}}} \over {{T^{00}}}}} \right|,{{\left| {{{{T^{ij}}} \over {{T^{00}}}}} \right|}^{1/2}},{{\left| {{U \over {{c^2}}}} \right|}^{1/2}}} \right\},$$
(1)

is much less than one. This parameter represents essentially a slow motion estimate εv/c, where v denotes a typical internal velocity. By a slight abuse of notation, following Chandrasekhar et al. [122, 124, 123], we shall henceforth write formally ε ≡ 1/c, even though ε is dimensionless whereas c has the dimension of a velocity. Thus, 1/c ≪ 1 in the case of post-Newtonian sources. The small post-Newtonian remainders will be denoted \({\mathcal O}{\rm{(1/}}{c^n})\). Furthermore, still following Refs. [122, 124, 123], we shall refer to a small post-Newtonian term with formal order \({\mathcal O}{\rm{(1/}}{c^n})\) relative to the Newtonian acceleration in the equations of motion, as \({n \over 2}{\rm{PN}}\).

We have ∣U/c21/2 ≪ 1/c for sources with negligible self-gravity, and whose dynamics are therefore driven by non-gravitational forces. However, we shall generally assume that the source is self-gravitating; in that case we see that it is necessarily weakly (but not negligibly) self-gravitating, i.e., \(|U/{c^2}{|^{1/2}} = {\mathcal O}{\rm{(1/}}c)\).Footnote 4 Note that the adjective “slow-motion” is a bit clumsy because we shall in fact consider very relativistic sources such as inspiralling compact binaries, for which v/c can be as large as 50% in the last rotations, and whose description necessitates the control of high post-Newtonian approximations.

At the lowest-order in the Newtonian limit 1/c → 0, the gravitational waveform of a post-Newtonian matter source is generated by the time variations of the quadrupole moment of the source. We shall review in Section 1.2 the utterly important “Newtonian” quadrupole moment formalism [183, 285]. Taking into account higher post-Newtonian corrections in a wave generation formalism will mean including into the waveform the contributions of higher multipole moments, beyond the quadrupole. Post-Newtonian corrections of order \({\mathcal O}{\rm{(1/}}{c^n})\) beyond the quadrupole formalism will still be denoted as \({n \over 2}{\rm{PN}}\). Building a post-Newtonian wave generation formalism must be concomitant to understanding the multipole expansion in general relativity.

The multipole expansion is one of the most useful tools of physics, but its use in general relativity is difficult because of the non-linearity 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 [219, 238, 384], while, in the case of non-stationary 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 Einstein quadrupole formula [given by Eq. (4) below] to include the current-quadrupole and mass-octupole moments [332, 333], and obtained the corresponding formulas for linear momentum [332, 333, 30, 358] and angular momentum [339, 134]. The general structure of the infinite multipole series in the linearized theory was investigated by several works [369, 367, 343, 403], from which it emerged that the expansion is characterized by two and only two sets of moments: Mass-type and current-type moments. Below we shall use a particular multipole decomposition of the linearized (vacuum) metric, parametrized by symmetric and trace-free (STF) mass and current moments, as given by Thorne [403]. The expressions of the multipole moments as integrals over the source, valid in the linearized theory but irrespective of a slow motion hypothesis, have been worked out in [309, 119, 118, 154]. In particular, Damour & Iyer [154] obtained the complete closed-form expressions for the time-dependent mass and spin multipole moments (in STF guise) of linearized gravity.

In the full non-linear 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 TR/c = const. The solutions of the field equations in the form of a far-field expansion (power series in 1/R) have been constructed, and their properties elucidated, by Bondi et al. [93] and Sachs [368]. The precise way under which such radiative space-times fall off asymptotically has been formulated geometrically by Penrose [337, 338] in the concept of an asymptotically simple space-time (see also Ref. [220]). The resulting Bondi-Sachs-Penrose 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 matter content and dynamics 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 non-linearities, 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 non-linear theory is that of Blanchet & Damour [57, 41], following pioneering works by Bonnor and collaborators [94, 95, 96, 251] and Thorne [403]. 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 post-Minkowskian expansion of the vacuum field equations (valid in the source’s exterior). Technically, the post-Minkowskian 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.Footnote 5 In this “multipolar-post-Minkowskian” (MPM) formalism, which is physically valid over the entire weak-field 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 non-linear functionals of the more basic source moments. A priori, the method is not limited to post-Newtonian sources; however, we shall see that, in the current situation, the closed-form expressions of the source multipole moments can be established only in the case where the source is post-Newtonian [44, 49]. The reason is that in this case the domain of validity of the post-Newtonian iteration (viz. the near zone) overlaps the exterior weak-field region, so that there exists an intermediate zone in which the post-Newtonian and multipolar expansions can be matched together. This is a standard application of the method of matched asymptotic expansions in general relativity [114, 113, 7, 357].

To be more precise, we shall show how a systematic multipolar and post-Minkowskian iteration scheme for the vacuum Einstein field equations yields the most general physically admissible solution of these equations [57]. The solution is specified once we give two and only two sets of time-varying (source) multipole moments. Some general theorems about the near-zone and far-zone expansions of that general solution will be stated. Notably, we show [41] that the asymptotic behaviour of the solution at future null infinity is in agreement with the findings of the Bondi-Sachs-Penrose [93, 368, 337, 338, 220] 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, as we said, 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 stress-energy tensor Tαβ describing the source? The general expression of the moments was obtained at the level of the second post-Newtonian (2PN) order in Ref. [44], and was subsequently proved to be in fact valid up to any post-Newtonian order in Ref. [49]. The source moments are given by some integrals extending over the post-Newtonian expansion of the total (pseudo) stress-energy tensor ταβ, which is made of a matter part described by Tαβ and a crucial non-linear gravitational source term Λαβ. These moments carry in front a particular operation of taking the finite part (\({\mathcal F}{\mathcal P}\) as we call it below), which makes them mathematically well-defined despite the fact that the gravitational part Λαβ 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 well-definiteness 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. [59] for the mass-type moments [see Eq. (157a) below], and in Ref. [155] for the current-type ones.

The wave-generation formalism resulting from matching the exterior multipolar and post-Minkowskian field [57, 41] to the post-Newtonian source [44, 49] is able to take into account, in principle, any post-Newtonian 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 non-linear 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 non-static moments with the total mass M of the source. The non-linear multipole interactions have been computed within the present wave-generation formalism up to the 3.5PN order in Refs. [60, 50, 48, 74, 197]. Furthermore, the back-reaction of the gravitational-wave 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 [58, 43, 47]. 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, i.e., 4PN absolute order [58]. A systematic post-Newtonian iteration scheme for the near-zone field, formally taking into account all radiation reaction effects, has been obtained, fully consistent with the present formalism [357, 75].

A different wave-generation formalism has been devised by Will & Wiseman [424] (see also Refs. [422, 335, 336]), after earlier attempts by Epstein & Wagoner [185] and Thorne [403]. This formalism has exactly the same scope as the one of Refs. [57, 41, 44, 49], i.e., it applies to any isolated post-Newtonian sources, but it differs in the definition of the source multipole moments and in many technical details when properly implemented [424]. In both formalisms, the moments are generated by the post-Newtonian expansion of the pseudo-tensor ταβ, but in the Will-Wiseman formalism they are defined by some compact-support integrals terminating at some finite radius enclosing the source, e.g., the radius \({\mathcal R}\) of the near zone. By contrast, in Refs. [44, 49], the moments are given by some integrals covering the whole space (ℝ3) and regularized by means of the finite part \({\mathcal F}{\mathcal P}\). Nevertheless, in both formalisms the source multipole moments, which involve a whole series of relativistic corrections, must be coupled together in a complicated way in the true non-linear solution; such non-linear couplings form an integral part of the radiative moments at infinity and thereby of the observed signal. We shall prove in Section 4.3 the complete equivalence, at the most general level, between the two formalisms.

The quadrupole moment formalism

The lowest-order wave generation formalism is the famous quadrupole formalism of Einstein [183] and Landau & Lifshitz [285]. This formalism applies to a general isolated matter source which is post-Newtonian in the sense of existence of the small post-Newtonian parameter ε defined by Eq. (1). However, the quadrupole formalism is valid in the Newtonian limit ε → 0; it can rightly be qualified as “Newtonian” because the quadrupole moment of the matter source is Newtonian and its evolution obeys Newton’s laws of gravity. In this formalism the gravitational field \(h_{ij}^{{\rm{TT}}}\) is expressed in a transverse and traceless (TT) coordinate system covering the far zone of the source at retarded times,Footnote 6 as

$$h_{ij}^{{\rm{TT}}} = {{2G} \over {{c^4}R}}{{\mathcal P}_{ijab}}(N)\left\{{{{{{\rm{d}}^2}{{\rm{Q}}_{ab}}} \over {{\rm{d}}{T^2}}}(T - R/c) + {\mathcal O}\left({{1 \over c}} \right)} \right\} + {\mathcal O}\left({{1 \over {{R^2}}}} \right)\,,$$
(2)

where R = ∣X∣ is the distance to the source, TR/c is the retarded time, N = X/R is the unit direction from the source to the far away observer, and \({{\mathcal P}_{ijab}} = {{\mathcal P}_{ia}}{{\mathcal P}_{jb}} - {1 \over 2}{{\mathcal P}_{ij}}{{\mathcal P}_{ab}}\) is the TT projection operator, with \({\mathcal P}_{ij} = {\delta_{ij}} - {N_i}{N_j}\) being the projector onto the plane orthogonal to N. The source’s quadrupole moment takes the familiar Newtonian form

$${{\rm{Q}}_{ij}}(t) = \int\nolimits_{{\rm{source}}} {{{\rm{d}}^3}{\bf{x}}\,\rho ({\bf{x}},t)\left({{x_i}{x_j} - {1 \over 3}{\delta _{ij}}{{\bf{x}}^2}} \right)\,,}$$
(3)

where ρ is the Newtonian mass density. The total gravitational power emitted by the source in all directions around the source is given by the Einstein quadrupole formula

$${\mathcal F} \equiv {\left({{{{\rm{d}}E} \over {{\rm{d}}T}}} \right)^{{\rm{GW}}}} = {G \over {{c^5}}}\left\{{{1 \over 5}{{{{\rm{d}}^3}{{\rm{Q}}_{ab}}} \over {{\rm{d}}{T^3}}}{{{{\rm{d}}^3}{{\rm{Q}}_{ab}}} \over {{\rm{d}}{T^3}}} + {\mathcal O}\left({{1 \over {{c^2}}}} \right)} \right\}.$$
(4)

Our notation \({\mathcal F}\) stands for the total gravitational energy flux or gravitational “luminosity” of the source. Similarly, the total angular momentum flux is given by

$${{\mathcal G}_i} \equiv {\left({{{{\rm{d}}{{\rm{J}}_i}} \over {{\rm{d}}T}}} \right)^{{\rm{GW}}}} = {G \over {{c^5}}}\left\{{{2 \over 5}{\epsilon _{iab}}{{{{\rm{d}}^2}{{\rm{Q}}_{ac}}} \over {{\rm{d}}{T^2}}}{{{{\rm{d}}^3}{{\rm{Q}}_{bc}}} \over {{\rm{d}}{T^3}}} + {\mathcal O}\left({{1 \over {{c^2}}}} \right)} \right\}\,,$$
(5)

where εabc denotes the standard Levi-Civita symbol with ε123 = 1.

Associated with the latter energy and angular momentum fluxes, there is also a quadrupole formula for the radiation reaction force, which reacts on the source’s dynamics in consequence of the emission of waves. This force will inflect the time evolution of the orbital phase of the binary pulsar and inspiralling compact binaries. At the position (x, t) in a particular coordinate system covering the source, the reaction force density can be written as [114, 113, 319]

$$F_i^{{\rm{reac}}} = {G \over {{c^5}}}\rho \left\{{- {2 \over 5}{x^a}{{{{\rm{d}}^5}{{\rm{Q}}_{ia}}} \over {{\rm{d}}{t^5}}} + {\mathcal O}\left({{1 \over {{c^2}}}} \right)} \right\}\,.$$
(6)

This is the gravitational analogue of the damping force of electromagnetism. However, notice that gravitational radiation reaction is inherently gauge-dependent, so the expression of the force depends on the coordinate system which is used. Consider now the energy and angular momentum of a matter system made of some perfect fluid, say

$$E = \int {{{\rm{d}}^3}} {\bf{x}}\,\rho \left[ {{{{{\bf{v}}^2}} \over 2} + \Pi - {U \over 2}} \right] + {\mathcal O}\left({{1 \over {{c^2}}}} \right)\,,$$
(7a)
$${{\rm{J}}_i} = \int {{{\rm{d}}^3}} {\bf{x}}\,\rho \,{\epsilon _{iab}}\,{x_a}\,{v_b} + {\mathcal O}\left({{1 \over {{c^2}}}} \right)\,.$$
(7b)

The specific internal energy of the fluid is denoted Π, and obeys the usual thermodynamic relation dΠ = − Pd(1/ρ) where P is the pressure; the gravitational potential obeys the Poisson equation ΔU = −4πGρ. We compute the mechanical losses of energy and angular momentum from the time derivatives of E and Ji. We employ the usual Eulerian equation of motion \(\rho \,{\rm{d}}{v^i}/{\rm{d}}t = - {\partial _i}P + \rho {\partial _i}U + F_i^{{\rm{reac}}}\) and continuity equation tρ + i(ρvi) = 0. Note that we add the small dissipative radiation-reaction contribution \(F_i^{{\rm{reac}}}\) in the equation of motion but neglect all conservative post-Newtonian corrections. The result is

$${{{\rm{d}}E} \over {{\rm{d}}t}} = \int {{{\rm{d}}^3}} {\bf{x}}\,{v^i}\,F_i^{{\rm{reac}}} = - {\mathcal F} + {{{\rm{d}}f} \over {{\rm{d}}t}}\,,$$
(8a)
$${{{\rm{d}}{{\rm{J}}_i}} \over {{\rm{d}}t}} = \int {{{\rm{d}}^3}} {\bf{x}}\,{\epsilon _{iab}}\,{x_a}\,F_b^{{\rm{reac}}} = - {{\mathcal G}_i} + {{{\rm{d}}{g_i}} \over {{\rm{d}}t}}\,,$$
(8b)

where one recognizes the fluxes at infinity given by Eqs. (4) and (5), and where the second terms denote some total time derivatives made of quadratic products of derivatives of the quadrupole moment. Looking only for secular effects, we apply an average over time on a typical period of variation of the system; the latter time derivatives will be in average numerically small in the case of quasi-periodic motion (see e.g., [103] for a discussion). Hence we obtain

$$\langle {{{\rm{d}}E} \over {{\rm{d}}t}}\rangle = - \langle {\mathcal F}\rangle \,,$$
(9a)
$$\langle {{{\rm{d}}{{\rm{J}}_i}} \over {{\rm{d}}t}}\rangle = - \langle {{\mathcal G}_i}\rangle \,,$$
(9b)

where the brackets denote the time averaging over an orbit. These balance equations encode the secular decreases of energy and angular momentum by gravitational radiation emission.

The cardinal virtues of the Einstein-Landau-Lifshitz 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 binary pulsar PSR 1913+16 [399, 400, 398]. Indeed let us apply the balance equations (9) to a system of two point masses moving on an eccentric orbit modelling the binary pulsar PSR 1913+16 — the classic references are [340, 339]; see also [186, 415]. We use the binary’s Newtonian energy and angular momentum,

$$E = - {{G{m_1}{m_2}} \over {2a}},$$
(10a)
$${\rm{J}} = {m_1}{m_2}\sqrt {{{Ga(1 - {e^2})} \over {{m_1} + {m_2}}}} \,,$$
(10b)

where a and e are the semi-major axis and eccentricity of the orbit and m1 and m2 are the two masses. From the energy balance equation (9a) we obtain first the secular evolution of a; next changing from a to the orbital period P using Kepler’s third law,Footnote 7 we get the secular evolution of the orbital period P as

$$\langle {{{\rm{d}}P} \over {{\rm{d}}t}}\rangle = - {{192\pi} \over {5{c^5}}}{\left({{{2\pi G} \over P}} \right)^{5/3}}{{{m_1}{m_2}} \over {{{({m_1} + {m_2})}^{1/3}}}}\,{{1 + {{73} \over {24}}{e^2} + {{37} \over {96}}{e^4}} \over {{{(1 - {e^2})}^{7/2}}}}.$$
(11)

The last factor, depending on the eccentricity, comes out from the orbital average and is known as the Peters & Mathews [340] “enhancement” factor, so designated because in the case of the binary pulsar, which has a rather large eccentricity e ≃ 0.617, it enhances the effect by a factor ∼ 12. Numerically, one finds 〈dP/dt〉 = −2.4 × 10−12, a dimensionless number in excellent agreement with the observations of the binary pulsar [399, 400, 398]. On the other hand the secular evolution of the eccentricity e is deduced from the angular momentum balance equation (9b) [together with the previous result (11)], as

$$\langle {{{\rm{d}}e} \over {{\rm{d}}t}}\rangle = - {{608\pi} \over {15{c^5}}}\,{e \over P}{\left({{{2\pi G} \over P}} \right)^{5/3}}{{{m_1}{m_2}} \over {{{({m_1} + {m_2})}^{1/3}}}}\,{{1 + {{121} \over {304}}{e^2}} \over {{{(1 - {e^2})}^{5/2}}}}\,.$$
(12)

Interestingly, the system of equations (11)(12) can be thoroughly integrated in closed analytic form. This yields the evolution of the eccentricity [339]:

$${{{e^2}} \over {{{(1 - {e^2})}^{19/6}}}}{\left({1 + {{121} \over {304}}{e^2}} \right)^{145/121}} = {c_0}\,{P^{19/9}}\,,$$
(13)

where c0 denotes an integration constant to be determined by the initial conditions at the start of the binary evolution. When e ≪ 1 the latter relation gives approximately e2c0 P19/9.

For a long while, it was thought that the various quadrupole formulas would be sufficient for sources of gravitational radiation to be observed directly on Earth — as they had proved to be amply sufficient in the case of the binary pulsar. However, further works [139]Footnote 8 and [87, 138] showed that this is not true, as one has to include post-Newtonian corrections to the quadrupole formalism in order to prepare for the data analysis of future detectors, in the case of inspiralling compact binaries. From the beautiful test of the orbital decay (11) of the binary pulsar, we can say that the post-Newtonian corrections to the “Newtonian” quadrupole formalism — which we shall compute in this article — have already received a strong observational support.

Problem posed by compact binary systems

Inspiralling compact binaries, containing neutron stars and/or black holes, are likely to become the bread-and-butter sources of gravitational waves for the detectors LIGO, VIRGO, GEO and KAGRA on ground, and also e LISA in space. 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 gravitational-wave 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. This effect is due to the emission of angular momentum by gravitational waves, resulting in a secular decrease of the eccentricity of the orbit, which has been computed within the quadrupole formalism in Eq. (12). For instance, suppose that the inspiralling compact binary was long ago (a few hundred million years ago) a system similar to the binary pulsar system, with an orbital frequency Ω0 ≡ 2π/P0 ∼ 10−4 rad/s and a rather large orbital eccentricity e0 ∼ 0.6. When it becomes visible by the detectors on ground, i.e., when the gravitational wave signal frequency reaches about f ≡ Ω/π ∼ 10 Hz, the eccentricity of the orbit should be e ∼ 10−6 according to the formula (13). This is a very small eccentricity, even when compared to high-order relativistic corrections. Only non-isolated binary systems could have a non negligible eccentricity. For instance, the Kozai mechanism [283, 300] is one important scenario that produces eccentric binaries and involves the interaction between a pair of binaries in the dense cores of globular clusters [315]. If the mutual inclination angle of the inner binary is strongly tilted with respect to the outer compact star, then a resonance occurs and can increase the eccentricity of the inner binary to large values. This is one motivation for looking at the waves emitted by inspiralling binaries in non-circular, quasi-elliptical orbits (see Section 10).

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 ma and possibly two spins Sa (with a = 1, 2 labelling the particles), is sufficient in first approximation. Indeed, most of the non-gravitational effects usually plaguing the dynamics of binary star systems, such as the effects of a magnetic field, of an interstellar medium, of the internal structure of extended bodies, are dominated by gravitational effects. The main 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 Qa induced by tidal interaction between two neutron stars. Let aa be the radius of the stars, and r12 be the distance between the two centers of mass. We have, for tidal moments,

$${{\rm{Q}}_1} = {k_1}{{{m_2}\,a_1^5} \over {r_{12}^3}}\quad {\rm{and}}\quad {{\rm{Q}}_2} = {k_2}{{{m_1}\,a_2^5} \over {r_{12}^3}}\,,$$
(14)

where ka are the star’s dimensionless (second) Love numbers [321], 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” parameters, defined by the dimensionless ratios

$${K_{\rm{a}}} \equiv {{G{m_{\rm{a}}}} \over {{a_{\rm{a}}}{c^2}}}\,,$$
(15)

and equal ∼ 0.2 for neutron stars (depending on their equation of state). The quadrupoles Qa will affect the Newtonian binding energy E of the two bodies, and also the emitted total gravitational flux \({\mathcal F}\) as computed using the Newtonian quadrupole formula (4). It is known that for inspiralling compact binaries the neutron stars are not co-rotating because the tidal synchronization time is much larger than the time left till the coalescence. As shown by Kochanek [276] the best models for the fluid motion inside the two neutron stars are the so-called Roche-Riemann 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, using the energy balance equation (9a), we find that within such a model (in the case of two identical neutron stars with same mass m, compactness K and Love number k), the orbital phase reads

$${\phi ^{\text{finite size}}} - {\phi _0} = - {1 \over {8{x^{5/2}}}}\left\{ {1 + \text{const }k{{\left( {{x \over K}} \right)}^5}} \right\},$$
(16)

where “const” denotes a numerical coefficient of order one, ϕ0 is some initial phase, and x ≡ (GmΩ/c3)2/3 is a standard dimensionless post-Newtonian parameter of the order of ∼ 1/c2 (with Ω = 2ρ/P the orbital frequency). The first term in the right-hand side of Eq. (16) corresponds to the gravitational-wave damping of two point masses without internal structure; the second term is the finite-size effect, which appears as a relative correction, proportional to (x/K)5, to the latter radiation damping effect. Because the finite-size effect is purely Newtonian, its relative correction ∼ (x/K)5 should not depend on the speed of light c; and indeed the factors 1/c2 cancel out in the ratio x/K. However, the compactness K of neutron stars is of the order of 0.2 say, and by definition of compact objects we can consider that K is formally of the order of unity or one half; therefore the factor 1/c2 it contains in (15) should not be taken into account when estimating numerically the effect. So the real order of magnitude of the relative contribution of the finite-size effect in Eq. (16) is given by the factor x5 alone. This means that for compact objects the finite-size effect should roughly be comparable, numerically, to a post-Newtonian correction of magnitude x5 ∼ 1/c10 namely 5PN order. This is a much higher post-Newtonian order than the one at which we shall investigate the gravitational effects on the phasing formula. Using k ∼ 1, K ∼ 0.2 and the bandwidth of detectors between 10 Hz and 1000 Hz, we find that the cumulative phase error due to the finite-size effect amounts to less that one orbital rotation over a total of ∼ 16 000 produced by the gravitational-wave damping of two neutron stars. The conclusion is that the finite-size effects can in general be neglected in comparison with purely gravitational-wave damping effects. The internal structure of the two compact bodies is “effaced” and their dynamics and radiation depend only, in first approximation, on the masses (and possibly spins); hence this property has been called the “effacement” principle of general relativity [142]. But note that for non-compact or moderately compact objects (such as white dwarfs for instance) the Newtonian tidal interaction dominates over the radiation damping. The constraints on the nuclear equation of state and the tidal deformability of neutron stars which can be inferred from gravitational wave observations of neutron star binary inspirals have been investigated in Refs. [320, 200, 414]. For numerical computations of the merger of two neutron stars see Refs. [187, 249].

Inspiralling compact binaries are ideally suited for application of a high-order post-Newtonian wave generation formalism. These systems are very relativistic, with orbital velocities as high as 0.5c in the last rotations (as compared to ∼ 10−3c for the binary pulsar), so it is not surprising that the quadrupole-moment formalism (2)(6) constitutes a poor description of the emitted gravitational waves, since many post-Newtonian corrections are expected to play a substantial role. This expectation has been confirmed by measurement-analyses [139, 137, 198, 138, 393, 346, 350, 284, 157], which have demonstrated that the post-Newtonian precision needed to implement successfully the optimal filtering technique for the LIGO/VIRGO detectors corresponds grossly, in the case of neutron-star binaries, to the 3PN approximation, or 1/c6 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, 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 Eq. (9a), on the center-of-mass binding energy E and the total gravitational-wave energy flux \({\mathcal F}\).

In summary, the theoretical problem is two-fold: On the one hand E, and on the other hand \({\mathcal F}\), are to be computed with 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 \({\mathcal F}\) it necessitates the application of a 3PN wave generation formalism. It is remarkable that such high PN approximation is needed in preparation for the LIGO and VIRGO data analyses. As we shall see, the signal from compact binaries contains the signature of several non-linear effects which are specific to general relativity. We thus have the possibility of probing, experimentally, some aspects of the non-linear structure of Einstein’s theory [84, 85, 15, 14].

Post-Newtonian equations of motion

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 terms, at the 1PN order, were derived by Lorentz & Droste [303]. Subsequently, Einstein, Infeld & Hoffmann [184] obtained the 1PN corrections for N particles by means of their famous “surface-integral” method, in which the equations of motion are deduced from the vacuum field equations, and are therefore applicable to any compact objects (be they neutron stars, black holes, or, perhaps, naked singularities). The 1PN-accurate equations were also obtained, for the motion of the centers of mass of compact bodies, by Fock [201] (see also Refs. [341, 330]).

The 2PN approximation was tackled by Ohta et al. [324, 327, 326, 325], who considered the post-Newtonian iteration of the Hamiltonian of N point-particles. We refer here to the Hamiltonian as a “Fokker-type” Hamiltonian, which is obtained from the matter-plus-field Arnowitt-Deser-Misner (ADM) Hamiltonian by eliminating the field degrees of freedom. The 2.5PN equations of motion were obtained in harmonic coordinates by Damour & Deruelle [148, 147, 175, 141, 142], building on a non-linear (post-Minkowskian) iteration of the metric of two particles initiated in Ref. [31]. The corresponding result for the ADM-Hamiltonian of two particles at the 2PN order was given in Ref. [169] (see also Refs. [375, 376]). The 2.5PN equations of motion have also been derived in the case of two extended compact objects [280, 234]. The 2.5PN equations of two point masses as well as the near zone gravitational field in harmonic-coordinate were computed in Ref. [76].Footnote 9

Up to the 2PN level the equations of motion are conservative. Only at the 2.5PN order does the first non-conservative effect appear, associated with the gravitational radiation emission. The equations of motion up to that level [148, 147, 175, 141, 142], have been used for the study of the radiation damping of the binary pulsar — its orbital [142, 143, 173]. The result was in agreement with the prediction of the quadrupole formalism given by (11). An important point is that the 2.5PN equations of motion have been proved to hold in the case of binary systems of strongly self-gravitating bodies [142]. This is via the effacing principle for the internal structure of the compact bodies. As a result, the equations depend only on the “Schwarzschild” masses, m1 and m2, of the neutron stars. Notably their compactness parameters K1 and K2, defined by Eq. (15), do not enter the equations of motion. This has also been explicitly verified up to the 2.5PN order by Kopeikin et al. [280, 234], who made a physical computation à la Fock, taking into account the internal structure of two self-gravitating extended compact bodies. The 2.5PN equations of motion have also been obtained by Itoh, Futamase & Asada [256, 257] in harmonic coordinates, using a variant (but, much simpler and more developed) of the surface-integral approach of Einstein et al. [184], that is valid for compact bodies, independently of the strength of the internal gravity.

At the 3PN order the equations of motion have been worked out independently by several groups, by means of different methods, and with equivalent results:

  1. 1.

    Jaranowski & Schäfer [261, 262, 263], and then with Damour [162, 164], employ the ADM-Hamiltonian canonical formalism of general relativity, following the line of research initiated in Refs. [324, 327, 326, 325, 169];

  2. 2.

    Blanchet & Faye [69, 71, 70, 72], and with de Andrade [174] and Iyer [79], founding their approach on the post-Newtonian iteration initiated in Ref. [76], compute directly the equations of motion (instead of a Hamiltonian) in harmonic coordinates;

  3. 3.

    Itoh & Futamase [255, 253] (see [213] for a review), continuing the surface-integral method of Refs. [256, 257], obtain the complete 3PN equations of motion in harmonic coordinates, without need of a self-field regularization;

  4. 4.

    Foffa & Sturani [203] derive the 3PN Lagrangian in harmonic coordinates within the effective field theory approach pioneered by Goldberger & Rothstein [223].

It has been shown [164, 174] that there exists a unique “contact” transformation of the dynamical variables that changes the harmonic-coordinates Lagrangian of Ref. [174] (identical to the ones issued from Refs. [255, 253] and [203]) into a new Lagrangian, whose Legendre transform coincides with the ADM-Hamiltonian of Ref. [162]. The equations of motion are therefore physically equivalent. For a while, however, they depended on one unspecified numerical coefficient, which is due to some incompleteness of the Hadamard self-field regularization method. This coefficient has been fixed by means of a better regularization, dimensional regularization, both within the ADM-Hamiltonian formalism [163], and the harmonic-coordinates equations of motion [61]. These works have demonstrated the power of dimensional regularization and its adequateness to the classical problem of interacting point masses in general relativity. By contrast, notice that, interestingly, the surface-integral method [256, 257, 255, 253] by-passes the need of a regularization. We devote Section 6 to questions related to the use of self-field regularizations.

The effective field theory (EFT) approach to the problems of motion and radiation of compact binaries, has been extensively developed since the initial proposal [223] (see [206] for a review). It borrows techniques from quantum field theory and consists of treating the gravitational interaction between point particles as a classical limit of a quantum field theory, i.e., in the “tree level” approximation. The theory is based on the effective action, defined from a Feynman path integral that integrates over the degrees of freedom that mediate the gravitational interaction.Footnote 10 The phase factor in the path integral is built from the standard Einstein-Hilbert action for gravity, augmented by a harmonic gauge fixing term and by the action of particles. The Feynman diagrams naturally show up as a perturbative technique for solving iteratively the Green’s functions. Like traditional approaches [163, 61] the EFT uses the dimensional regularization.

Computing the equations of motion and radiation field using Feynman diagrams in classical general relativity is not a new idea by itself: Bertotti & Plebanski [35] defined the diagrammatic tree-level perturbative expansion of the Green’s functions in classical general relativity; Hari Dass & Soni [240]Footnote 11 showed how to derive the classical energy-loss formula at Newtonian approximation using tree-level propagating gravitons; Feynman diagrams have been used for the equations of motion up to 2PN order in general relativity [324, 327, 326, 325] and in scalar-tensor theories [151]. Nevertheless, the systematic EFT treatment has proved to be powerful and innovative for the field, e.g., with the introduction of a decomposition of the metric into “Kaluza-Klein type” potentials [277], the interesting link with the renormalization group equation [222], and the systematization of the computation of diagrams [203].

The 3.5PN terms in the equations of motion correspond to the 1PN relative corrections in the radiation reaction force. They were derived by Iyer & Will [258, 259] for point-particle binaries in a general gauge, relying on energy and angular momentum balance equations and the known expressions of the 1PN fluxes. The latter works have been extended to 2PN order [226] and to include the leading spin-orbit effects [428]. The result has been then established from first principles (i.e., not relying on balance equations) in various works at 1PN order [260, 336, 278, 322, 254]. The 1PN radiation reaction force has also been obtained for general extended fluid systems in a particular gauge [43, 47]. Known also is the contribution of gravitational-wave tails in the equations of motion, which arises at the 4PN order and represents a 1.5PN modification of the radiation damping force [58]. This 1.5PN tail-induced correction to the radiation reaction force was also derived within the EFT approach [205, 215].

The state of the art on equations of motion is the 4PN approximation. Partial results on the equations of motion at the 4PN order have been obtained in [264, 265, 266] using the ADM Hamiltonian formalism, and in [204] using the EFT. The first derivation of the complete 4PN dynamics was accomplished in [166] by combining the local contributions [264, 265, 266] with a non-local contribution related to gravitational wave tails [58, 43], with the help of the result of an auxiliary analytical self-force calculation [36]. The non-local dynamics of [166] has been transformed in Ref. [167] into a local Hamiltonian containing an infinite series of even powers of the radial momentum. A second computation of the complete 4PN dynamics (including the same non-local interaction as in [166], but disagreeing on the local interaction) was accomplished in [33] using a Fokker Lagrangian in harmonic coordinates. Further works [39, 248] have given independent confirmations of the results of Refs. [166, 167]. More work is needed to understand the difference between the results of [166] and [33].

An important body of works concerns the effects of spins on the equations of motion of compact binaries. In this case we have in mind black holes rather than neutron stars, since astrophysical stellar-size black holes as well as super-massive galactic black holes have spins which can be close to maximal. The dominant effects are the spin-orbit (SO) coupling which is linear in spin, and the spin-spin (SS) coupling which is quadratic. For maximally spinning objects, and adopting a particular convention in which the spin is regarded as a 0.5PN quantity (see Section 11), the leading SO effect arises at the 1.5PN order while the leading SS effect appears at 2PN order. The leading SO and SS effects in the equations of motion have been determined by Barker & O’Connell [27, 28] and Kidder, Will & Wiseman [275, 271]. The next-to-leading SO effect, i.e., 1PN relative order corresponding to 2.5PN order, was obtained by Tagoshi, Ohashi & Owen [394], then confirmed and completed by Faye, Blanchet & Buonanno [194]. The results were also retrieved by two subsequent calculations, using the ADM Hamiltonian [165] and using EFT methods [292, 352]. The ADM calculation was later generalized to the N-body problem [241] and extended to the next-to-leading spin-spin effects (including both the coupling between different spins and spin square terms) in Refs. [387, 389, 388, 247, 243], and the next-to-next-to-leading SS interactions between different spins at the 4PN order [243]. In the meantime EFT methods progressed concurrently by computing the next-to-leading 3PN SS and spin-squared contributions [354, 356, 355, 293, 299], and the next-to-next-to-leading 4PN SS interactions for different spins [294] and for spin-squared [298]. Finally, the next-to-next-to-leading order SO effects, corresponding to 3.5PN order equivalent to 2PN relative order, were obtained in the ADM-coordinates Hamiltonian [242, 244] and in the harmonic-coordinates equations of motion [307, 90], with complete equivalence between the two approaches. Comparisons between the EFT and ADM Hamiltonian schemes for high-order SO and SS couplings can be found in Refs. [295, 299, 297]. We shall devote Section 11 to spin effects (focusing mainly on spin-orbit effects) in black hole binaries.

So far the status of post-Newtonian equations of motion is very satisfying. There is mutual agreement between all the results obtained by means of many different approaches and techniques, whenever they can be compared: point particles described by Dirac delta-functions or extended post-Newtonian fluids; surface-integrals methods; mixed post-Minkowskian and post-Newtonian expansions; direct post-Newtonian iteration and matching; EFT techniques versus traditional expansions; harmonic coordinates versus ADM-type coordinates; different processes or variants of the self-field regularization for point particles; different ways to including spins within the post-Newtonian approximation. In Part B of this article, we present complete results for the 3.5PN equations of motion (including the 1PN radiation reaction), and discuss the conservative part of the equations in the case of quasi-circular orbits. Notably, the conservative part of the dynamics is compared with numerical results for the gravitational self-force in Section 8.4.

Post-Newtonian gravitational radiation

The second problem, that of the computation of the gravitational waveform and the energy flux \({\mathcal F}\), has to be solved by application of a wave generation formalism (see Section 1.1). The earliest computations at the 1PN level beyond the quadrupole moment formalism were done by Wagoner & Will [416], but based on some ill-defined expressions of the multipole moments [185, 403]. The computations were redone and confirmed by Blanchet & Schäfer [86] applying the rigorous wave generation formalism of Refs. [57, 60]. Remember that at that time the post-Newtonian corrections to the emission of gravitational waves had only a purely academic interest.

The energy flux of inspiralling compact binaries was then completed to the 2PN order by Blanchet, Damour & Iyer [64, 224], and, independently, by Will & Wiseman [424, 422], using their own formalism; see Refs. [66, 82] for joint reports of these calculations. The energy flux has been computed using the EFT approach in Ref. [221] with results agreeing with traditional methods.

At the 1.5PN order in the radiation field, appears the first contribution of “hereditary” terms, which are a priori sensitive to the entire past history of the source, i.e., which depend on all previous times up to t → −∞ in the past [60]. This 1.5PN hereditary term represents the dominant contribution of tails in the wave zone. It has been evaluated for compact binaries in Refs. [426, 87] by application of the formula for tail integrals given in Ref. [60]. Higher-order tail effects at the 2.5PN and 3.5PN orders, as well as a crucial contribution of tails generated by the tails themselves (the so-called “tails of tails”) at the 3PN order, were obtained in Refs [45, 48].

The 3PN approximation also involves, besides the tails of tails, many non-tail contributions coming from the relativistic corrections in the (source) multipole moments of the compact binary. Those have been almost completed in Refs. [81, 73, 80], in the sense that the result still involved one unknown numerical coefficient, due to the use of the Hadamard regularization. We shall review in Section 6 the computation of this parameter by means of dimensional regularization [62, 63], and shall present in Section 9 the most up-to-date results for the 3.5PN energy flux and orbital phase, deduced from the energy balance equation. In recent years all the results have been generalized to non-circular orbits, including both the fluxes of energy and angular momentum, and the associated balance equations [10, 9, 12]. The problem of eccentric orbits will be the subject of Section 10.

Besides the problem of the energy flux there is the problem of the gravitational waveform itself, which includes higher-order amplitude corrections and correlatively higher-order harmonics of the orbital frequency, consistent with the post-Newtonian order. Such full post-Newtonian waveform is to be contrasted with the so-called “restricted” post-Newtonian waveform which retains only the leading-order harmonic of the signal at twice the orbital frequency, and is often used in practical data analysis when searching the signal. However, for parameter estimation the full waveform is to be taken into account. For instance it has been shown that using the full waveform in the data analysis of future space-based detectors like eLISA will yield substantial improvements (with respect to the restricted waveform) of the angular resolution and the estimation of the luminosity distance of super-massive black hole binaries [16, 17, 410].

The full waveform has been obtained up to 2PN order in Ref. [82] by means of two independent wave generations (respectively those of Refs. [57, 44] and [424]), and it was subsequently extended up to the 3PN order in Refs. [11, 273, 272, 74]. At that order the signal contains the contributions of harmonics of the orbital frequency up to the eighth mode. The motivation is not only to build accurate templates for the data analysis of gravitational wave detectors, but also to facilitate the comparison and match of the high post-Newtonian prediction for the inspiral waveform with the numerically-generated waveforms for the merger and ringdown. For the latter application it is important to provide the post-Newtonian results in terms of a spin-weighted spherical harmonic decomposition suitable for a direct comparison with the results of numerical relativity. Recently the dominant quadrupole mode (, m) = (2, 2) in the spin-weighted spherical harmonic decomposition has been obtained at the 3.5PN order [197]. Available results will be provided in Sections 9.4 and 9.5.

At the 2.5PN order in the waveform appears the dominant contribution of another hereditary effect called the “non-linear memory” effect (or sometimes Christodoulou effect) [128, 427, 406, 60, 50]. This effect was actually discovered using approximation methods in Ref. [42] (see [60] for a discussion). It implies a permanent change in the wave amplitude from before to after a burst of gravitational waves, which can be interpreted as the contribution of gravitons in the known formulas for the linear memory for massless particles [99]. Note that the non-linear memory takes the form of a simple anti-derivative of an “instantaneous” term, and therefore becomes instantaneous (i.e., non-hereditary) in the energy flux which is composed of the time-derivative of the waveform. In principle the memory contribution must be computed using some model for the evolution of the binary system in the past. Because of the cumulative effect of integration over the whole past, the memory term, though originating from 2.5PN order, finally contributes in the waveform at the Newtonian level [427, 11]. It represents a part of the waveform whose amplitude steadily grows with time, but which is nearly constant over one orbital period. It is therefore essentially a zero-frequency effect (or DC effect), which has rather poor observational consequences in the case of the LIGO-VIRGO detectors, whose frequency bandwidth is always limited from below by some cut-off frequency fseismic > 0. Non-linear memory contributions in the waveform of inspiralling compact binaries have been thoroughly computed by Favata [189, 192].

The post-Newtonian results for the waveform and energy flux are in complete agreement (up to the 3.5PN order) with the results given by the very different technique of linear black-hole perturbations, valid when the mass of one of the bodies is small compared to the other. This is the test-mass limit ν → 0, in which we define the symmetric mass ratio to be the reduced mass divided by the total mass, νμ/m such that ν =1/4 for equal masses. Linear black-hole perturbations, triggered by the geodesic motion of a small particle around the black hole, have been applied to this problem by Poisson [345] at the 1.5PN order (following the pioneering work [216]), by Tagoshi & Nakamura [393], using a numerical code up to the 4PN order, and by Sasaki, Tagoshi & Tanaka [372, 395, 397] (see also Ref. [316]), analytically up to the 5.5PN order. More recently the method has been improved and extended up to extremely high post-Newtonian orders: 14PN [209] and even 22PN [210] orders — but still for linear black-hole perturbations.

To successfully detect the gravitational waves emitted by spinning black hole binaries and to estimate the binary parameters, it is crucial to include spins effects in the templates, most importantly the spin-orbit effect which is linear in spins. The spins will affect the gravitational waves through a modulation of their amplitude, phase and frequency. Notably the orbital plane will precess in the case where the spins are not aligned or anti-aligned with the orbital angular momentum, see e.g., Ref. [8]. The leading SO and SS contributions in the waveform and flux of compact binaries are known from Refs. [275, 271, 314]; the next-to-leading SO terms at order 2.5PN were obtained in Ref. [53] after a previous attempt in [328]; the 3PN SO contribution is due to tails and was computed in Ref. [54], after intermediate results at the same order (but including SS terms) given in [353]. Finally, the next-to-next-to-leading SO contributions in the multipole moments and the energy flux, corresponding to 3.5PN order, and the next-to-leading SO tail corresponding to 4PN order, have been obtained in Refs. [89, 306]. The next-to-leading 3PN SS and spin-squared contributions in the radiation field were derived in Ref. [88]. In Section 11 we shall give full results for the contributions of spins (at SO linear level) in the energy flux and phase evolution up to 4PN order.

A related topic is the loss of linear momentum by gravitational radiation and the resulting gravitational recoil (or “kick”) of black-hole binary systems. This phenomenon has potentially important astrophysical consequences [313]. In models of formation of massive black holes involving successive mergers of smaller “seed” black holes, a recoil with sufficient velocity could eject the system from the host galaxy and effectively terminate the process. Recoils could eject coalescing black holes from dwarf galaxies or globular clusters. Even in galaxies whose potential wells are deep enough to confine the recoiling system, displacement of the system from the center could have important dynamical consequences for the galactic core.

Post-Newtonian methods are not ideally suited to compute the recoil of binary black holes because most of the recoil is generated in the strong field regime close to the coalescence [199]. Nevertheless, after earlier computations of the dominant Newtonian effect [30, 199]Footnote 12 and the 1PN relative corrections [425], the recoil velocity has been obtained up to 2PN order for point particle binaries without spin [83], and is also known for the dominant spin effects [271]. Various estimations of the magnitude of the kick include a PN calculation for the inspiraling phase together with a treatment of the plunge phase [83], an application of the effective-one-body formalism [152], a close-limit calculation with Bowen-York type initial conditions [385], and a close-limit calculation with initial PN conditions for the ringdown phase [288, 290].

In parallel the problem of gravitational recoil of coalescing binaries has attracted considerable attention from the numerical relativity community. These computations led to increasingly accurate estimates of the kick velocity from the merger along quasicircular orbits of binary black holes without spins [115, 20] and with spins [117]. In particular these numerical simulations revealed the interesting result that very large kick velocities can be obtained in the case of spinning black holes for particular spin configurations.

Part A: Post-Newtonian Sources

Non-linear Iteration of the Vacuum Field Equations

Einstein’s field equations

The field equations of general relativity are obtained by varying the space-time metric gαβ in the famous Einstein-Hilbert action,

$${I_{{\rm{EH}}}} = {{{c^3}} \over {16\pi G}}\int {{{\rm{d}}^4}} x\,\sqrt {- g} \,R + {I_{{\rm{mat}}}}[\Psi, {g_{\alpha \beta}}]\,.$$
(17)

They form a system of ten second-order partial differential equations obeyed by the metric,

$${E^{\alpha \beta}}[g,\partial g,{\partial ^2}g] = {{8\pi G} \over {{c^4}}}{T^{\alpha \beta}}[\Psi, g]\,,$$
(18)

where the Einstein curvature tensor EαβRαβ − ½Rgαβ is generated, through the gravitational coupling constant κ = 8πG/c4, by the stress-energy tensor \({T^{\alpha \beta}} \equiv {2 \over {\sqrt {- g}}}\delta {I_{{\rm{mat}}}}/\delta {g_{\alpha \beta}}\) of the matter fields Ψ. Among these ten equations, four govern, via the contracted Bianchi identity, the evolution of the matter system,

$${\nabla _\mu}{E^{\alpha \mu}} = 0\quad \Rightarrow \quad {\nabla _\mu}{T^{\alpha \mu}} = 0\,.$$
(19)

The matter equations can also be obtained by varying the matter action in (17) with respect to the matter fields Ψ. The space-time geometry is constrained by the six remaining equations, which place six independent constraints on the ten components of the metric gαβ, leaving four of them to be fixed by a choice of the coordinate system.

In most of this paper we adopt the conditions of harmonic coordinates, sometimes also called de Donder coordinates. We define, as a basic variable, the gravitational-field amplitude

$${h^{\alpha \beta}} \equiv \sqrt {- g} \,{g^{\alpha \beta}} - {\eta ^{\alpha \beta}}\,,$$
(20)

where gαβ denotes the contravariant metric (satisfying \({g^{\alpha \mu}}{g_{\mu \beta}} = \delta _\beta ^\alpha\)), where g is the determinant of the covariant metric, g ≡ det(gαβ), and where ηαβ represents an auxiliary Minkowskian metric ηαβ ≡ diag(−1, 1, 1, 1). The harmonic-coordinate condition, which accounts exactly for the four equations (19) corresponding to the conservation of the matter tensor, readsFootnote 13

$${\partial _\mu}{h^{\alpha \mu}} = 0.$$
(21)

Equation (21) introduces into the definition of our coordinate system a preferred Minkowskian structure, with Minkowski metric ηαβ. Of course, this is not contrary to the spirit of general relativity, where there is only one physical metric gαβ without any flat prior geometry, because the coordinates are not governed by geometry (so to speak), but rather can be chosen at convenience, depending on physical phenomena under study. The coordinate condition (21) is especially useful when studying gravitational waves as perturbations of space-time propagating on the fixed background metric ηαβ. This view is perfectly legitimate and represents a fruitful and rigorous way to think of the problem using approximation methods. Indeed, the metric ηαβ, originally introduced in the coordinate condition (21), does exist at any finite order of approximation (neglecting higher-order terms), and plays the role of some physical “prior” flat geometry at any order of approximation.

The Einstein field equations in harmonic coordinates can be written in the form of inhomogeneous flat d’Alembertian equations,

$$\Box{h^{\alpha \beta}} = {{16\pi G} \over {{c^4}}}{\tau ^{\alpha \beta}}\,,$$
(22)

where □ ≡ □η = ημνμν. The source term ταβ can rightly be interpreted as the stress-energy pseudo-tensor (actually, ταβ is a Lorentz-covariant tensor) of the matter fields, described by Tαβ, and the gravitational field, given by the gravitational source term Λαβ, i.e.,

$${\tau ^{\alpha \beta}} = |g|{T^{\alpha \beta}} + {{{c^4}} \over {16\pi G}}{\Lambda ^{\alpha \beta}}\,.$$
(23)

The exact expression of Λαβ in harmonic coordinates, including all non-linearities, readsFootnote 14

$$\begin{array}{*{20}c} {{\Lambda ^{\alpha \beta}} = - {h^{\mu \nu}}\partial _{\mu \nu}^2{h^{\alpha \beta}} + {\partial _\mu}{h^{\alpha \nu}}{\partial _\nu}{h^{\beta \mu}} + {1 \over 2}{g^{\alpha \beta}}{g_{\mu \nu}}{\partial _\lambda}{h^{\mu \tau}}{\partial _\tau}{h^{\nu \lambda}}\quad \quad \quad \quad \quad} \\ {- {g^{\alpha \mu}}{g_{\nu \tau}}{\partial _\lambda}{h^{\beta \tau}}{\partial _\mu}{h^{\nu \lambda}} - {g^{\beta \mu}}{g_{\nu \tau}}{\partial _\lambda}{h^{\alpha \tau}}{\partial _\mu}{h^{\nu \lambda}} + {g_{\mu \nu}}{g^{\lambda \tau}}{\partial _\lambda}{h^{\alpha \mu}}{\partial _\tau}{h^{\beta \nu}}} \\ {+ {1 \over 8}\left({2{g^{\alpha \mu}}{g^{\beta \nu}} - {g^{\alpha \beta}}{g^{\mu \nu}}} \right)\left({2{g_{\lambda \tau}}{g_{\epsilon \pi}} - {g_{\tau \epsilon}}{g_{\lambda \pi}}} \right){\partial _\mu}{h^{\lambda \pi}}{\partial _\nu}{h^{\tau \epsilon}}\,.\quad \quad} \\ \end{array}$$
(24)

As is clear from this expression, Λαβ is made of terms at least quadratic in the gravitational-field strength h and its first and second space-time derivatives. In the following, for the highest post-Newtonian order that we shall consider, we will need the quadratic, cubic and quartic pieces of Λαβ; with obvious notation, we can write them as

$${\Lambda ^{\alpha \beta}} = {N^{\alpha \beta}}[h,h] + {M^{\alpha \beta}}[h,h,h] + {L^{\alpha \beta}}[h,h,h,h] + {\mathcal O}({h^5})\,.$$
(25)

These various terms can be straightforwardly computed from expanding Eq. (24); for instance the leading quadratic piece is explicitly given byFootnote 15

$$\begin{array}{*{20}c} {{N^{\alpha \beta}} = - {h^{\mu \nu}}\partial _{\mu \nu}^2{h^{\alpha \beta}} + {1 \over 2}{\partial ^\alpha}{h_{\mu \nu}}{\partial ^\beta}{h^{\mu \nu}} - {1 \over 4}{\partial ^\alpha}h{\partial ^\beta}h + {\partial _\nu}{h^{\alpha \mu}}\left({{\partial ^\nu}h_\mu ^\beta + {\partial _\mu}{h^{\beta \nu}}} \right)\quad \quad \quad \quad} \\ {- 2{\partial ^{(\alpha}}{h_{\mu \nu}}{\partial ^\mu}{h^{\beta)\nu}} + {\eta ^{\alpha \beta}}\left[ {- {1 \over 4}{\partial _\tau}{h_{\mu \nu}}{\partial ^\tau}{h^{\mu \nu}} + {1 \over 8}{\partial _\mu}h{\partial ^\mu}h + {1 \over 2}{\partial _\mu}{h_{\nu \tau}}{\partial ^\nu}{h^{\mu \tau}}} \right].} \\ \end{array}$$
(26)

As we said, the condition (21) is equivalent to the matter equations of motion, in the sense of the conservation of the total pseudo-tensor ταβ,

$${\partial _\mu}{\tau ^{\alpha \mu}} = 0\quad \Leftrightarrow \quad {\nabla _\mu}{T^{\alpha \mu}} = 0\,.$$
(27)

In this article, we shall look for approximate solutions of the field equations (21)(22) under the following four hypotheses:

  1. 1.

    The matter stress-energy tensor is of spatially compact support, i.e., can be enclosed into some time-like world tube, say ra, where r = ∣x∣ is the harmonic-coordinate radial distance. Outside the domain of the source, when r > a, the gravitational source term, according to Eq. (27), is divergence-free,

    $${\partial _\mu}{\Lambda ^{\alpha \mu}} = 0\qquad ({\rm{when}}\;r > a);$$
    (28)
  2. 2.

    The matter distribution inside the source is smooth: TαβC(ℝ3).Footnote 16 We have in mind a smooth hydrodynamical fluid system, without any singularities nor shocks (a priori), that is described by some Euler-type equations including high relativistic corrections. In particular, we exclude from the start the presence of any black holes; however, we shall return to this question in Part B when we look for a model describing compact objects;

  3. 3.

    The source is post-Newtonian 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 post-Newtonian field and the outer multipolar decomposition in the source’s exterior near zone;

  4. 4.

    The gravitational field has been independent of time (stationary) in some remote past, i.e., before some finite instant \(- {\mathcal T}\) in the past, namely

    $${\partial \over {\partial t}}\left[ {{h^{\alpha \beta}}({\bf{x}},t)} \right] = 0\qquad {\rm{when}}\;t\leqslant - {\mathcal T}\,.$$
    (29)

The latter condition is a means to impose, by brute force, the famous no-incoming 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 no-incoming radiation condition should be imposed at past null infinity. As we shall see, this condition entirely fixes the radiation reaction forces inside the isolated source. We shall later argue (see Section 3.2) that our condition of stationarity in the past (29), although weaker than the ideal no-incoming radiation condition, does not entail any physical restriction on the general validity of the formulas we derive. Even more, the condition (29) is actually better suited in the case of real astrophysical sources like inspiralling compact binaries, for which we do not know the details of the initial formation and remote past evolution. In practice the initial instant \(- {\mathcal T}\) can be set right after the explosions of the two supernovæ yielding the formation of the compact binary system.

Subject to the past-stationarity condition (29), the differential equations (22) can be written equivalently into the form of the integro-differential equations

$${h^{\alpha \beta}} = {{16\pi G} \over {{c^4}}}\Box_{{\rm{ret}}}^{- 1}{\tau ^{\alpha \beta}}\,,$$
(30)

containing the usual retarded inverse d’Alembertian integral operator, given by

$$(\Box_{{\rm{ret}}}^{- 1}\tau)({\bf{x}},t) \equiv - {1 \over {4\pi}}\int {\int {\int {{{{{\rm{d}}^3}{\bf{x\prime}}} \over {|{\bf{x}} - {\bf{x\prime}}|}}\tau}}} \left({{\bf{x\prime}},t - |{\bf{x}} - {\bf{x\prime}}|/c} \right)\,,$$
(31)

extending over the whole three-dimensional space ℝ3.

Linearized vacuum equations

In what follows we solve the field equations (21)(22), in the vacuum region outside the compact-support source, in the form of a formal non-linearity or post-Minkowskian expansion, considering the field variable hαβ as a non-linear metric perturbation of Minkowski space-time. At the linearized level (or first-post-Minkowskian approximation), we write:

$$h_{{\rm{ext}}}^{\alpha \beta} = G\,h_{(1)}^{\alpha \beta} + {\mathcal O}({G^2})\,,$$
(32)

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 book-keeping parameter, enabling one to label conveniently the successive post-Minkowskian approximations. Since hαβ is a dimensionless variable, with our convention the linear coefficient \(h_{(1)}^{\alpha \beta}\) in Eq. (32) has the dimension of the inverse of G (which should be a mass squared in a system of units where ħ = c = 1). In vacuum, the harmonic-coordinate metric coefficient \(h_{(1)}^{\alpha \beta}\) satisfies

$$\Box h_{(1)}^{\alpha \beta} = 0\,,$$
(33a)
$${\partial _\mu}h_{(1)}^{\alpha \mu} = 0\,.$$
(33b)

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 appropriate method for describing the physics of the source as seen from its exterior (r > a). On the other hand, the post-Minkowskian series is physically valid in the weak-field region, which surely includes the exterior of any source, starting at a sufficiently large distance. For post-Newtonian sources the exterior weak-field region, where both multipole and post-Minkowskian expansions are valid, simply coincides with the exterior region r > a. It is therefore quite natural, and even, one would say inescapable when considering general sources, to combine the post-Minkowskian approximation with the multipole decomposition. This is the original idea of the “double-expansion” series of Bonnor and collaborators [94, 95, 96, 251], which combines the G-expansion (or m-expansion in their notation) with the a-expansion (equivalent to the multipole expansion, since the -th order multipole moment scales with the source radius like a).

The multipolar-post-Minkowskian (MPM) method will be implemented systematically, using symmetric-trace-free (STF) harmonics to describe the multipole expansion [403], and looking for a definite algorithm for the approximation scheme [57]. The solution of the system of equations (33) takes the form of a series of retarded multipolar wavesFootnote 17

$$h_{(1)}^{\alpha \beta} = \sum\limits_{\ell = 0}^{+ \infty} {{\partial _L}} \left({{{{\rm{K}}_L^{\alpha \beta}(t - r/c)} \over r}} \right)\,,$$
(34)

where r = ∣x∣, and where the functions \({\rm{K}}_L^{\alpha \beta} \equiv {\rm{K}}_{{i_{1 \cdots {i_\ell}}}}^{\alpha \beta}\) are smooth functions of the retarded time utr/c [i.e., KL(u) ∈ C(ℝ)], which become constant in the past, when \(t\leqslant - {\mathcal T}\), see Eq. (29). Since a monopolar wave satisfies □(KL(u)/r) = 0 and the d’Alembertian commutes with the multi-derivative L, it is evident that Eq. (34) represents the most general solution of the wave equation (33a); but see Section 2 in Ref. [57] for a rigorous proof based on the Euler-Poisson-Darboux equation. The gauge condition (33b), however, is not fulfilled in general, and to satisfy it we must algebraically decompose the set of functions \({\rm{K}}_L^{00},\,{\rm{K}}_L^{0i},\,{\rm{K}}_L^{ij}\) into ten tensors which are STF with respect to all their indices, including the spatial indices i, ij. Imposing the condition (33b) 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 [403, 57].

Theorem 1. The most general solution of the linearized field equations (33) outside some time-like world tube enclosing the source (r > a), and stationary in the past [see Eq. (29)], reads

$$h_{(1)}^{\alpha \beta} = k_{(1)}^{\alpha \beta} + {\partial ^\alpha}\varphi _{(1)}^\beta + {\partial ^\beta}\varphi _{(1)}^\alpha - {\eta ^{\alpha \beta}}{\partial _\mu}\varphi _{(1)}^\mu \,.$$
(35)

The first term depends on two STF-tensorial multipole moments, IL(u) and JL(u), which are arbitrary functions of time except for the laws of conservation of the monopole: I = const, and dipoles: Ii = const, Ji = const. It is given by

$$k_{(1)}^{00} = - {4 \over {{c^2}}}\sum\limits_{\ell \geqslant 0} {{{{{(-)}^\ell}} \over {\ell !}}} {\partial _L}\left({{1 \over r}{{\rm{I}}_L}(u)} \right)\,,$$
(36a)
$$k_{(1)}^{0i} = {4 \over {{c^3}}}\sum\limits_{\ell \geqslant 1} {{{{{(-)}^\ell}} \over {\ell !}}} \left\{{{\partial _{L - 1}}\left({{1 \over r}{\rm{I}}_{iL - 1}^{(1)}(u)} \right) + {\ell \over {\ell + 1}}{\epsilon _{iab}}{\partial _{aL - 1}}\left({{1 \over r}{{\rm{J}}_{bL - 1}}(u)} \right)} \right\}\,,$$
(36b)
$$k_{(1)}^{ij} = - {4 \over {{c^4}}}\sum\limits_{\ell \geqslant2} {{{{{(-)}^\ell}} \over {\ell !}}} \left\{{{\partial _{L - 2}}\left({{1 \over r}{\rm{I}}_{ijL - 2}^{(2)}(u)} \right) + {{2\ell} \over {\ell + 1}}{\partial _{aL - 2}}\left({{1 \over r}{\epsilon _{ab(i}}{\rm{J}}_{j)bL - 2}^{(1)}(u)} \right)} \right\}\,.$$
(36c)

The other terms represent a linearized gauge transformation, with gauge vector \(\varphi _{(1)}^\alpha\) parametrized by four other multipole moments, say WL(u), XL(u), YL(u) and ZL(u) [see Eqs. (37)].

The conservation of the lowest-order moments gives the constancy of the total mass of the source, M ≡ I = const, center-of-mass position, Xi ≡ Ii/I = const, total linear momentum \({{\rm{P}}_i} \equiv {\rm{I}}_i^{(1)} = 0\),Footnote 18 and total angular momentum, Ji = const. It is always possible to achieve Xi = 0 by translating the origin of our coordinates to the center of mass. The total mass M is the ADM mass of the Hamiltonian formulation of general relativity. Note that the quantities M, Xi, Pi and Ji 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 IL(u) and JL(u), which thoroughly encode the physical properties of the source at the linearized level (because the other moments WL, …, ZL, parametrize a gauge transformation), will be referred to as the mass-type and current-type source multipole moments. Beware, however, that at this stage the moments are not specified in terms of the stress-energy tensor Tαβ of the source: Theorem 1 follows merely from the algebraic and differential properties of the vacuum field equations outside the source.

For completeness, we give the components of the gauge-vector \(\varphi _{(1)}^\alpha\) entering Eq. (35):

$$\varphi _{(1)}^0 = {4 \over {{c^3}}}\sum\limits_{\ell \geqslant 0} {{{{{(-)}^\ell}} \over {\ell !}}} {\partial _L}\left({{1 \over r}{{\rm{W}}_L}(u)} \right)\,,$$
(37a)
$$\varphi _{(1)}^i = - {4 \over {{c^4}}}\sum\limits_{\ell \geqslant 0} {{{{{(-)}^\ell}} \over {\ell !}}} {\partial _{iL}}\left({{1 \over r}{{\rm{X}}_L}(u)} \right)$$
(37b)
$$- {4 \over {{c^4}}}\sum\limits_{\ell \geqslant 1} {{{{{(-)}^\ell}} \over {\ell !}}} \left\{{{\partial _{L - 1}}\left({{1 \over r}{{\rm{Y}}_{iL - 1}}(u)} \right) + {\ell \over {\ell + 1}}{\epsilon _{iab}}{\partial _{aL - 1}}\left({{1 \over r}{{\rm{Z}}_{bL - 1}}(u)} \right)} \right\}\,.$$
(37c)

Because the theory is covariant with respect to non-linear diffeomorphisms and not merely with respect to linear gauge transformations, the moments WL, …, ZL do play a physical role starting at the non-linear level, in the following sense. If one takes these moments equal to zero and continues the post-Minkowskian iteration [see Section 2.3] one ends up with a metric depending on IL and JL only, but that metric will not describe the same physical source as the one which would have been constructed starting from the six moments IL, JL, …, ZL altogether. In other words, the two non-linear metrics associated with the sets of multipole moments {IL, JL, 0, …, 0} and {IL, JL, WL, …, ZL} are not physically equivalent — they are not isometric. We shall point out in Section 2.4 below that the full set of moments {IL, JL, WL, …, ZL} is in fact physically equivalent to some other reduced set of moments {ML, SL, 0, …, 0}, but with some moments ML, SL that differ from IL, JL by non-linear corrections [see Eqs. (97)(98)]. The moments ML, SL are called “canonical” moments; they play a useful role in intermediate calculations. All the multipole moments IL, JL, WL, XL, YL, ZL will be computed in Section 4.4.

The multipolar post-Minkowskian solution

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 post-Minkowskian iteration of that solution. We consider the full post-Minkowskian series

$$h_{{\rm{ext}}}^{\alpha \beta} = \sum\limits_{n = 1}^{+ \infty} {{G^n}} \,h_{(n)}^{\alpha \beta}\,,$$
(38)

where the first term is composed of the result given by Eqs. (35)(37). In this article, we shall always understand the infinite sums such as the one in Eq. (38) in the sense of formal power series, i.e., as an ordered collection of coefficients, \({(h_{(n)}^{\alpha \beta})_{n \in \mathbb N}}\). We do not attempt to control the mathematical nature of the series and refer to the mathematical-physics literature for discussion of that point (see, in the present context, Refs. [130, 171, 361, 362, 363]).

We substitute the post-Minkowski ansatz (38) into the vacuum Einstein field equations (21)(22), i.e., with ταβ simply given by the gravitational source term Λαβ, and we equate term by term the factors of the successive powers of our book-keeping parameter G. We get an infinite set of equations for each of the \(h_{(n)}^{\alpha \beta}\)’s: namely, ∀n ⩾ 2,

$$\Box h_{(n)}^{\alpha \beta} = \Lambda _{(n)}^{\alpha \beta}[{h_{(1)}},{h_{(2)}}, \ldots, {h_{(n - 1)}}]\,,$$
(39a)
$${\partial _\mu}h_{(n)}^{\alpha \mu} = 0\,.$$
(39b)

The right-hand side of the wave equation (39a) is obtained from inserting the previous iterations, known up to the order n − 1, into the gravitational source term. In more details, the series of equations (39a) reads

$$\Box h_{(2)}^{\alpha \beta} = {N^{\alpha \beta}}[{h_{(1)}},{h_{(1)}}]\,,$$
(40a)
$$\Box h_{(3)}^{\alpha \beta} = {M^{\alpha \beta}}[{h_{(1)}},{h_{(1)}},{h_{(1)}}] + {N^{\alpha \beta}}[{h_{(1)}},{h_{(2)}}] + {N^{\alpha \beta}}[{h_{(2)}},{h_{(1)}}]\,,$$
(40b)
$$\begin{array}{*{20}c} {\Box h_{(4)}^{\alpha \beta} = {L^{\alpha \beta}}[{h_{(1)}},{h_{(1)}},{h_{(1)}},{h_{(1)}}]\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ \,{M^{\alpha \beta}}[{h_{(1)}},{h_{(1)}},{h_{(2)}}] + {M^{\alpha \beta}}[{h_{(1)}},{h_{(2)}},{h_{(1)}}] + {M^{\alpha \beta}}[{h_{(2)}},{h_{(1)}},{h_{(1)}}]} \\ {+ \,{N^{\alpha \beta}}[{h_{(2)}},{h_{(2)}}] + {N^{\alpha \beta}}[{h_{(1)}},{h_{(3)}}] + {N^{\alpha \beta}}[{h_{(3)}},{h_{(1)}}]\,,\quad \quad \quad \quad \quad} \\ {\vdots \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(40c)

The quadratic, cubic and quartic pieces of Λαβ are defined by Eq. (25)(26).

Let us now proceed by induction. Some n ∈ ℕ being given, we assume that we succeeded in constructing, starting from the linearized solution h(1), the sequence of post-Minkowskian solutions h(2), h(3), …, h(n−1), and from this we want to infer the next solution h(n). The right-hand side of Eq. (39a), \(\Lambda _{(n)}^{\alpha \beta}\), is known by induction hypothesis. Thus the problem is that of solving a flat wave equation whose source is given. The point is that this wave equation, instead of being valid everywhere in ℝ3, is physically correct only outside the matter source (r > a), and it makes no sense to solve it by means of the usual retarded integral. Technically speaking, the right-hand side of Eq. (39a) 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 \(\Lambda _{(n)}^{\alpha \beta}\), 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 method of Ref. [57], by means of a mathematical trick, in which one first “regularizes” the source term \(\Lambda _{(n)}^{\alpha \beta}\) by multiplying it by the factor rB, where r = ∣x∣ is the spatial radial distance and B is a complex number, B ∈ ℂ. Let us assume, for definiteness, that \(\Lambda _{(n)}^{\alpha \beta}\) is composed of multipolar pieces with maximal multipolarity max. This means that we start the iteration from the linearized metric (35)(37) in which the multipolar sums are actually finite.Footnote 19 The divergences when r → 0 of the source term are typically power-like, say 1/rk (there are also powers of the logarithm of r), and with the previous assumption there will exist a maximal order of divergency, say kmax. Thus, when the real part of B is large enough, i.e., ℜ(B) > kmax − 3, the “regularized” source term \({r^B}\Lambda _{(n)}^{\alpha \beta}\) is regular enough when r → 0 so that one can perfectly apply the retarded integral operator. This defines the B-dependent retarded integral, when ℜ(B) is large enough,

$${I^{\alpha \beta}}(B) \equiv \Box_{{\rm{ret}}}^{- 1}\left[ {{{\tilde r}^B}\Lambda _{(n)}^{\alpha \beta}} \right]\,,$$
(41)

where the symbol \(\square_{{\rm{ret}}}^{- 1}\) stands for the retarded integral defined by Eq. (31). It is convenient to introduce inside the regularizing factor some arbitrary constant length scale r0 in order to make it dimensionless. Everywhere in this article we pose

$$\tilde r \equiv {r \over {{r_0}}}\,.$$
(42)

The fate of the constant r0 in a detailed calculation will be interesting to follow, as we shall see. Now the point for our purpose is that the function Iαβ(B) on the complex plane, which was originally defined only when ℜ(B) > kmax − 3, admits a unique analytic continuation to all values of B ∈ ℂ except at some integer values. Furthermore, the analytic continuation of Iαβ(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 Iαβ(B), when B → 0, in the form

$${I^{\alpha \beta}}(B) = \sum\limits_{p = {p_0}}^{+ \infty} {\iota _p^{\alpha \beta}} {B^p}\,,$$
(43)

where p ∈ ℤ, and the various coefficients \(\iota _p^{\alpha \beta}\) are functions of the field point (x, t). When p0 ⩾ −1 there are poles; and −p0, which depends on n, refers to the maximal order of these poles. By applying the d’Alembertian operator onto both sides of Eq. (43), and equating the different powers of B, we arrive at

$${p_0}\leqslant p\leqslant - 1\quad \Rightarrow \quad \Box \iota _p^{\alpha \beta} = 0\,,$$
(44a)
$$p\geqslant 0\quad \Rightarrow \quad \Box \iota _p^{\alpha \beta} = {{{{(\ln r)}^p}} \over {p!}}\Lambda _{(n)}^{\alpha \beta}\,.$$
(44b)

As we see, the case p = 0 shows that the finite-part coefficient in Eq. (43), namely \(\iota _0^{\alpha \beta}\), is a particular solution of the requested equation: \(\square\iota _0^{\alpha \beta} = \Lambda _{(n)}^{\alpha \beta}\). Furthermore, we can prove that this solution, by its very construction, owns the same structure made of a multipolar expansion singular at r = 0 as the corresponding source.

Let us forget about the intermediate name \(\iota _0^{\alpha \beta}\), and denote, from now on, the latter solution by \(u_{(n)}^{\alpha \beta} \equiv \iota _0^{\alpha \beta}\), or, in more explicit terms,

$$u_{(n)}^{\alpha \beta} = {\mathcal F}{{\mathcal P}_{B = 0}}\,\Box_{{\rm{ret}}}^{- 1}\left[ {{{\tilde r}^B}\Lambda _{(n)}^{\alpha \beta}} \right]\,,$$
(45)

where the finite-part symbol \({\mathcal F}{{\mathcal P}_{B = 0}}\) means the previously detailed operations of considering the analytic continuation, taking the Laurent expansion, and picking up the finite-part coefficient when B → 0. The story is not complete, however, because \(u_{(n)}^{\alpha \beta}\) does not fulfill the constraint of harmonic coordinates (39b); its divergence, say \(w_{(n)}^\alpha = {\partial _\mu}u_{(n)}^{\alpha \mu}\), is different from zero in general. From the fact that the source term is divergence-free in vacuum, \({\partial _\mu}\Lambda _{(n)}^{\alpha \mu} = 0\) [see Eq. (28)], we find instead

$$w_{(n)}^\alpha = {\mathcal F}{{\mathcal P}_{B = 0}}\,\Box_{{\rm{ret}}}^{- 1}\left[ {B\,{{\tilde r}^B}{{{n_i}} \over r}\Lambda _{(n)}^{\alpha i}} \right]\,.$$
(46)

The factor B comes from the differentiation of the regularization factor \({\tilde r^B}\). So, \(w_{(n)}^\alpha\) is zero only in the special case where the Laurent expansion of the retarded integral in Eq. (46) 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 an homogeneous solution of the source-free d’Alembertian equation, and, what is more (from its stationarity in the past), that solution is a retarded one. Hence, taking into account the index structure of \(w_{(n)}^\alpha\), there must exist four STF-tensorial functions of u = tr/c, say NL(u), PL(u), QL(u) and RL(u), such that

$$w_{(n)}^0 = \sum\limits_{l = 0}^{+ \infty} {{\partial _L}} \left[ {{r^{- 1}}{N_L}(u)} \right]\,,$$
(47a)
$$w_{(n)}^i = \sum\limits_{l = 0}^{+ \infty} {{\partial _{iL}}} \left[ {{r^{- 1}}{P_L}(u)} \right] + \sum\limits_{l = 1}^{+ \infty} {\left\{{{\partial _{L - 1}}\left[ {{r^{- 1}}{Q_{iL - 1}}(u)} \right] + {\epsilon _{iab}}{\partial _{aL - 1}}\left[ {{r^{- 1}}{R_{bL - 1}}(u)} \right]} \right\}} \,.$$
(47b)

From that expression we are able to find a new object, say \(v_{(n)}^{\alpha \beta}\), which takes the same structure as \(w_{(n)}^\alpha\) (a retarded solution of the source-free wave equation) and, furthermore, whose divergence is exactly the opposite of the divergence of \(u_{(n)}^{\alpha \beta}\), i.e. \({\partial _\mu}v_{(n)}^{\alpha \mu} = - w_{(n)}^\alpha\). Such a \(v_{(n)}^{\alpha \beta}\) is not unique, but we shall see that it is simply necessary to make a choice for (the simplest one) in order to obtain the general solution. The formulas that we adopt are

$$v_{(n)}^{00} = - {r^{- 1}}{N^{(- 1)}} + {\partial _a}\left[ {{r^{- 1}}\left({- N_a^{(- 1)} + C_a^{(- 2)} - 3{P_a}} \right)} \right]\,,$$
(48a)
$$v_{(n)}^{0i} = {r^{- 1}}\left({- Q_i^{(- 1)} + 3P_i^{(1)}} \right) - {\epsilon _{iab}}{\partial _a}\left[ {{r^{- 1}}R_b^{(- 1)}} \right] - \sum\limits_{l = 2}^{+ \infty} {{\partial _{L - 1}}} \left[ {{r^{- 1}}{N_{iL - 1}}} \right]\,,$$
(48b)
$$\begin{array}{*{20}c} {v_{(n)}^{ij} = - {\delta _{ij}}{r^{- 1}}P + \sum\limits_{l = 2}^{+ \infty} {\left\{{2{\delta _{ij}}{\partial _{L - 1}}\left[ {{r^{- 1}}{P_{L - 1}}} \right] - 6{\partial _{L - 2(i}}\left[ {{r^{- 1}}{P_{j)L - 2}}} \right]} \right.} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\left. {+ {\partial _{L - 2}}\left[ {{r^{- 1}}(N_{ijL - 2}^{(1)} + 3P_{ijL - 2}^{(2)} - {Q_{ijL - 2}})} \right] - 2{\partial _{aL - 2}}\left[ {{r^{- 1}}{\epsilon _{ab(i}}{R_{j)bL - 2}}} \right]} \right\}.} \\ \end{array}$$
(48c)

Notice the presence of anti-derivatives, denoted e.g., by \({N^{(- 1)}}(u) = \int\nolimits_{- \infty}^u {{\rm{d}}v\,N(v)}\); there is no problem with the limit v → −∞ since all the corresponding functions are zero when \(t\leqslant - \mathcal T\). The choice made in Eqs. (48) is dictated by the fact that the 00 component involves only some monopolar and dipolar terms, and that the spatial trace ii is monopolar: \(v_{(n)}^{ii} = - 3{r^{- 1}}P\). Finally, if we pose

$$h_{(n)}^{\alpha \beta} = u_{(n)}^{\alpha \beta} + v_{(n)}^{\alpha \beta}\,,$$
(49)

we see that we solve at once the d’Alembertian equation (39a) and the coordinate condition (39b). That is, we have succeeded in finding a solution of the field equations at the n-th post-Minkowskian order. By induction the same method applies to any order n, and, therefore, we have constructed a complete post-Minkowskian series (38) based on the linearized approximation \(h_{(1)}^{\alpha \beta}\) given by Eqs. (35)(37). The previous procedure constitutes an algorithm, which can be (and has recently been [74, 197]) implemented by an algebraic computer programme. Again, note that this algorithm permits solving the full Einstein field equations together with the gauge condition (i.e., not only the relaxed field equations).

Generality of the MPM solution

We have a solution, but is that a general solution? The answer, “yes”, is provided by the following result [57].

Theorem 2. The most general solution of the harmonic-coordinates Einstein field equations in the vacuum region outside an isolated source, admitting some post-Minkowskian and multipolar expansions, is given by the previous construction as

$$h_{ext}^{\alpha \beta} = \sum\limits_{n = 1}^{+ \infty} {{G^n}} h_{(n)}^{\alpha \beta}[{{\rm{I}}_L},{{\rm{J}}_L}, \ldots, {{\rm{Z}}_L}]\,.$$
(50)

It depends on two sets of arbitrary STF-tensorial functions of time IL(u) and JL(u) (satisfying the conservation laws) defined by Eqs. (36), and on four supplementary functions WL(u), …, ZL(u) parametrizing the gauge vector (37).

The proof is quite easy. With Eq. (49) we obtained a particular solution of the system of equations (39). To it we should add the most general solution of the corresponding homogeneous system of equations, which is obtained by setting \(\Lambda _{(n)}^{\alpha \beta} = 0\) into Eqs. (39). But this homogeneous system of equations is nothing but the linearized vacuum field equations (33), to which we know the most general solution \(h_{(1)}^{\alpha \beta}\) given by Eqs. (35)(37). Thus, we must add to our particular solution \(h_{(n)}^{\alpha \beta}\) a general homogeneous solution that is necessarily of the type \(h_{(1)}^{\alpha \beta}[\delta {{\rm{I}}_{L, \cdots,}}\delta {{\rm{Z}}_L}]\), where \(\delta {{\rm{I}}_{L, \cdots,}}\delta {{\rm{Z}}_L}\) denote some corrections to the multipole moments at the n-th post-Minkowskian order (with the monopole δI and dipoles δIi, δJi being constant). It is then clear, since precisely the linearized metric is a linear functional of all these moments, that the previous corrections to the moments can be absorbed into a re-definition of the original ones IL, …, ZL by posing

$${\rm{I}}_L^{{\rm{new}}} = {{\rm{I}}_L} + {G^{n - 1}}\delta {{\rm{I}}_L}\,,$$
(51a)
$$\begin{array}{*{20}c} {\vdots \quad \quad \quad \quad \quad} \\ {{\rm{Z}}_L^{{\rm{new}}} = {{\rm{Z}}_L} + {G^{n - 1}}\delta {{\rm{Z}}_L}\,.} \\ \end{array}$$
(51b)

After re-arranging the metric in terms of these new moments, taking into account the fact that the precision of the metric is limited to the n-th post-Minkowskian order, and dropping the superscript “new”, we find exactly the same solution as the one we had before (indeed, the moments are arbitrary functions of time) — hence the proof.

The six sets of multipole moments IL(u), …, ZL(u) contain the physical information about any isolated source as seen in its exterior. However, as we now discuss, it is always possible to find two, and only two, sets of multipole moments, ML(u) and SL(u), for parametrizing the most general isolated source as well. The route for constructing such a general solution is to get rid of the moments WL, XL, YL, ZL at the linearized level by performing the linearized gauge transformation \(\delta {x^\alpha} = \varphi _{(1)}^\alpha\), where \(\varphi _{(1)}^\alpha\) is the gauge vector given by Eqs. (37). So, at the linearized level, we have only the two types of moments M L and SL, parametrizing \(k_{(1)}^{\alpha \beta}\) by the same formulas as in Eqs. (36). We must be careful to denote these moments with names different from IL and JL because they will ultimately correspond to a different physical source. Then we apply exactly the same post-Minkowskian algorithm, following the formulas (45)(49) as we did above, but starting from the gauge-transformed linear metric \(k_{(1)}^{\alpha \beta}\) instead of \(h_{(1)}^{\alpha \beta}\). The result of the iteration is therefore some

$$k_{{\rm{ext}}}^{\alpha \beta} = \sum\limits_{n = 1}^{+ \infty} {{G^n}} k_{(n)}^{\alpha \beta}[{{\rm{M}}_L},{{\rm{S}}_L}]\,.$$
(52)

Obviously this post-Minkowskian algorithm yields some simpler calculations as we have only two multipole moments to iterate. The point is that one can show that the resulting metric (52) is isometric to the original one (50) if and only if the so-called canonical moments ML, and SL are related to the source moments IL, JL, …, ZL by some (quite involved) non-linear equations. We shall give in Eqs. (97)(98) the most up to date relations we have between these moments. Therefore, the most general solution of the field equations, modulo a coordinate transformation, can be obtained by starting from the linearized metric \(k_{(1)}^{\alpha \beta}[{{\rm{M}}_{L,}}{{\rm{S}}_L}]\) instead of the more complicated \(k_{(1)}^{\alpha \beta}[{{\rm{I}}_L},{{\rm{J}}_L}] + {\partial ^\alpha}\varphi _{(1)}^\beta + {\partial ^\beta}\varphi _{(1)}^\alpha - {\eta ^{\alpha \beta}}{\partial _\mu}\varphi _{(1)}^\mu\), and continuing the post-Minkowskian calculation.

So why not consider from the start that the best description of the isolated source is provided by only the two types of multipole moments, ML and SL, instead of the six types, IL, JL, …, ZL? The reason is that we shall determine in Theorem 6 below the explicit closed-form expressions of the six source moments IL, JL, …, ZL, but that, by contrast, it seems to be impossible to obtain some similar closed-form expressions for the canonical moments ML and SL. The only thing we can do is to write down the explicit non-linear algorithm that computes ML, SL starting from IL, JL, …, ZL. In consequence, it is better to view the moments IL, JL, …, ZL as more “fundamental” than ML and SL, in the sense that they appear to be more tightly related to the description of the source, since they admit closed-form expressions as some explicit integrals over the source. Hence, we choose to refer collectively to the six moments IL, JL, …, ZL as the multipole moments of the source. This being said, the moments ML and SL are generally very useful in practical computations because they yield a simpler post-Minkowskian iteration. Then, one can generally come back to the more fundamental source-rooted moments by using the fact that ML and SL differ from the corresponding IL and JL only by high-order post-Newtonian terms like 2.5PN; see Eqs. (97)(98) below. Indeed, this is to be expected because the physical difference between both types of moments stems only from non-linearities.

Near-zone and far-zone structures

In our presentation of the post-Minkowskian algorithm (45)(49) we have for the moment omitted a crucial recursive hypothesis, which is required in order to prove that at each post-Minkowskian order n, the inverse d’Alembertian operator can be applied in the way we did — notably that the B-dependent retarded integral can be analytically continued down to a neighbourhood of B = 0. This hypothesis is that the “near-zone” expansion, i.e., when r → 0, of each one of the post-Minkowskian coefficients h(n) has a certain structure (here we often omit the space-time indices αβ); this hypothesis is established as a theorem once the mathematical induction succeeds.

Theorem 3. The general structure of the expansion of the post-Minkowskian exterior metric in the near-zone (when r → 0) is of the type: ∀N ∈ ℕ,Footnote 20

$${h_{(n)}}({\bf{x}},t) = \sum {{{\hat n}_L}} {r^m}{(\ln r)^p}{F_{L,m,p,n}}(t) + o({r^N})\,,$$
(53)

where m ∈ ℤ, with m0mN (and m0 becoming more and more negative as n grows), p ∈ ℕ with pn − 1. The functions FL,m,p,n are multilinear functionals of the source multipole moments IL, …, ZL.

For the proof see Ref. [57]. As we see, the near-zone expansion involves, besides the simple powers of r, some powers of the logarithm of r, with a maximal power of n − 1. As a corollary of that theorem, we find, by restoring all the powers of c in Eq. (53) and using the fact that each r goes into the combination r/c, that the general structure of the post-Newtonian expansion (c → +∞) is necessarily of the type

$${h_{(n)}}(c) \simeq \,\,\sum\limits_{p,q \in {\mathbb N}} {{{{{(\ln c)}^p}} \over {{c^q}}}} \,,$$
(54)

where pn − 1 (and q ⩾ 2). The post-Newtonian expansion proceeds not only with the normal powers of 1/c but also with powers of the logarithm of c [57]. It is remarkable that there is no more complicated structure like for instance ln(ln c).

Paralleling the structure of the near-zone expansion, we have a similar result concerning the structure of the far-zone expansion at Minkowskian future null infinity, i.e., when r → +∞ with u = tr/c = const: ∀N ∈ ℕ,

$${h_{(n)}}({\bf{x}},t) = \sum {{{{{\hat n}_L}{{(\ln r)}^p}} \over {{r^k}}}} {G_{L,k,p,n}}(u) + o\left({{1 \over {{r^N}}}} \right)\,,$$
(55)

where k, p ∈ ℕ, with 1 ⩽ kN, and where, likewise in the near-zone expansion (53), some powers of logarithms, such that pn − 1, appear. The appearance of logarithms in the far-zone expansion of the harmonic-coordinates metric has been known since the work of Fock [202]. One knows also that this is a coordinate effect, because the study of the “asymptotic” structure of space-time at future null infinity by Bondi et al. [93], Sachs [368], and Penrose [337, 338], has revealed the existence of other coordinate systems that avoid the appearance of any logarithms: the so-called radiative coordinates, in which the far-zone expansion of the metric proceeds with simple powers of the inverse radial distance. Hence, the logarithms are simply an artifact of the use of harmonic coordinates [252, 304, 41]. The following theorem, proved in Ref. [41], shows that our general construction of the metric in the exterior of the source, when developed at future null infinity, is consistent with the Bondi-Sachs-Penrose [93, 368, 337, 338] approach to gravitational radiation.

Theorem 4. The most general multipolar-post-Minkowskian solution, stationary in the past [see Eq. (29)], admits some radiative coordinates (T, X), for which the expansion at future null infinity, R → +00 with UTR/c = const, takes the form

$${H_{(n)}}({\bf{X}},T) = \sum {{{{{\hat N}_L}} \over {{R^k}}}} {K_{L,k,n}}(U) + {\mathcal O}\left({{1 \over {{R^N}}}} \right)\,.$$
(56)

The functions KL,k,n are computable functionals of the source multipole moments. In radiative coordinates the retarded time U is a null coordinate in the asymptotic limit. The metric \(H_{{\rm{ext}}}^{\alpha \beta} = \sum\nolimits_{n\geqslant1} {{G^n}H_{(n)}^{\alpha \beta}}\) is asymptotically simple in the sense of Penrose [337, 338, 220], perturbatively to any post-Minkowskian order.

Proof. We introduce a linearized “radiative” metric by performing a gauge transformation of the harmonic-coordinates metric defined by Eqs. (35)(37), namely

$$H_{(1)}^{\alpha \beta} = h_{(1)}^{\alpha \beta} + {\partial ^\alpha}\xi _{(1)}^\beta + {\partial ^\beta}\xi _{(1)}^\alpha - {\eta ^{\alpha \beta}}{\partial _\mu}\xi _{(1)}^\mu \,,$$
(57)

where the gauge vector \(\xi _{(1)}^\alpha\) is

$$\xi _{(1)}^\alpha = {{2{\rm{M}}} \over {{c^2}}}\,{\eta ^{0\alpha}}\ln \left({{r \over {{r_0}}}} \right)\,.$$
(58)

This gauge transformation is non-harmonic:

$${\partial _\mu}H_{(1)}^{\alpha \mu} = \Box \xi _{(1)}^\alpha = {{2{\rm{M}}} \over {{c^2}{r^2}}}\,{\eta ^{0\alpha}}.$$
(59)

Its effect is to correct for the well-known logarithmic deviation of the retarded time in harmonic coordinates, with respect to the true space-time characteristic or light cones. After the change of gauge, the coordinate u = tr/c coincides with a null coordinate at the linearized level.Footnote 21 This is the requirement to be satisfied by a linearized metric so that it can constitute the linearized approximation to a full (post-Minkowskian) radiative field [304]. One can easily show that, at the dominant order when r → +∞,

$${k_\mu}{k_\nu}H_{(1)}^{\mu \nu} = {\mathcal O}\left({{1 \over {{r^2}}}} \right)\,,$$
(60)

where kμ = ημνkν = (1, n) is the outgoing Minkowskian null vector. Given any n ⩾ 2, let us recursively assume that we have obtained all the previous radiative post-Minkowskian coefficients \(H_{(m)}^{\alpha \beta}\), i.e. ∀mn − 1, and that all of them satisfy

$${k_\mu}{k_\nu}H_{(m)}^{\mu \nu} = {\mathcal O}\left({{1 \over {{r^2}}}} \right)\,.$$
(61)

From this induction hypothesis one can prove that the n-th post-Minkowskian source term \(\Lambda _{(n)}^{\alpha \beta} = \Lambda _{(n)}^{\alpha \beta}({H_{(1), \cdots,}}{H_{(n - 1)}})\) is such that

$$\Lambda _{(n)}^{\alpha \beta} = {{{k^\alpha}{k^\beta}} \over {{r^2}}}{\sigma _{(n)}}\left({u,{\bf{n}}} \right) + {\mathcal O}\left({{1 \over {{r^3}}}} \right)\,.$$
(62)

To the leading order this term takes the classic form of the stress-energy tensor of massless particles, with σ(n) being proportional to the power in the massless waves. One can show that all the problems with the appearance of logarithms come from the retarded integral of the terms in Eq. (62) that behave like 1/r2: See indeed the integration formula (83), which behaves like lnr/r at infinity. But now, thanks to the particular index structure of the term (62), we can correct for the effect by adjusting the gauge at the n-th post-Minkowskian order. We pose, as a gauge vector,

$$\xi _{(n)}^\alpha = {\mathcal F}{\mathcal P}\,\Box_{{\rm{ret}}}^{- 1}\left[ {{{{k^\alpha}} \over {2{r^2}}}\,\int\nolimits_{- \infty}^u {{\rm{d}}v\,{\sigma _{(n)}}(v,{\bf{n}})}} \right]\,,$$
(63)

where \({\mathcal F}{\mathcal P}\) refers to the same finite part operation as in Eq. (45). This vector is such that the logarithms that will appear in the corresponding gauge terms cancel out the logarithms coming from the retarded integral of the source term (62); see Ref. [41] for the details. Hence, to the n-th post-Minkowskian order, we define the radiative metric as

$$H_{(n)}^{\alpha \beta} = U_{(n)}^{\alpha \beta} + V_{(n)}^{\alpha \beta} + {\partial ^\alpha}\xi _{(n)}^\beta + {\partial ^\beta}\xi _{(n)}^\alpha - {\eta ^{\alpha \beta}}{\partial _\mu}\xi _{(n)}^\mu \,.$$
(64)

Here \(U_{(n)}^{\alpha \beta}\) and \(V_{(n)}^{\alpha \beta}\) denote the quantities that are the analogues of \(u_{(n)}^{\alpha \beta}\) and \(v_{(n)}^{\alpha \beta}\), which were introduced into the harmonic-coordinates algorithm: See Eqs. (45)(48). In particular, these quantities are constructed in such a way that the sum \(U_{(n)}^{\alpha \beta} + V_{(n)}^{\alpha \beta}\) is divergence-free, so we see that the radiative metric does not obey the harmonic-gauge condition, but instead

$${\partial _\mu}H_{(n)}^{\alpha \mu} = \Box \xi _{(n)}^\alpha = {{{k^\alpha}} \over {2{r^2}}}\int\nolimits_{- \infty}^u {{\rm{d}}v\,{\sigma _{(n)}}(v,{\bf{n}})} \,.$$
(65)

The far-zone expansion of the latter metric is of the type (56), i.e., is free of any logarithms, and the retarded time in these coordinates tends asymptotically toward a null coordinate at future null infinity. The property of asymptotic simplicity, in the form given by Geroch & Horowitz [220], is proved by introducing the usual conformal factor Ω = 1/R in radiative coordinates [41]. Finally, it can be checked that the metric so constructed, which is a functional of the source multipole moments IL, …, ZL (from the definition of the algorithm), is as general as the general harmonic-coordinate metric of Theorem 2, since it merely differs from it by a coordinate transformation (t, x) → (T, X), where (t, x) are the harmonic coordinates and (T, X) the radiative ones, together with a re-definition of the multipole moments.

Asymptotic Gravitational Waveform

The radiative multipole moments

The leading-order term 1/R of the metric in radiative coordinates (T, X) as given in Theorem 4, neglecting \({\mathcal O}(1/{R^2})\), 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 UTR/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]

$$\begin{array}{*{20}c} {H_{ij}^{{\rm{TT}}}(U,{\bf{X}}) = {{4G} \over {{c^2}R}}{{\mathcal P}_{ijab}}(N)\sum\limits_{\ell = 2}^{+ \infty} {{1 \over {{c^\ell}\ell !}}} \left\{{{N_{L - 2}}{{\rm{U}}_{abL - 2}}(U) - {{2\ell} \over {c(\ell + 1)}}{N_{cL - 2}}{\epsilon _{cd(a}}{{\rm{V}}_{b)dL - 2}}(U)} \right\}} \\ {+ \,{\mathcal O}\left({{1 \over {{R^2}}}} \right)\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(66)

We have formally re-summed the whole post-Minkowskian series in Eq. (56) from n = 1 up to +∞. As before we denote for instance \({N_{L - 2}} = {N_{{i_1}}} \cdots {N_{{i_{\ell - 2}}}}\) and so on, where Ni = (N)i and N = X/R. The TT algebraic projection operator \({{\mathcal P}_{ijab}}\) 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

$${{\rm{U}}_{ij}}(U) = {\rm{Q}}_{ij}^{(2)}(U) + {\mathcal O}\left({{1 \over {{c^2}}}} \right)\,,$$
(67)

where Qij is the Newtonian quadrupole moment (3). Associated to the asymptotic waveform (66) we can compute by standard methods the total energy flux \({\mathcal F} = {({\rm{d}}E/{\rm{d}}U)^{{\rm{GW}}}}\) and angular momentum flux \({{\mathcal G}_i} = {({\rm{d}}{{\rm{J}}_i}/{\rm{d}}U)^{{\rm{GW}}}}\) in gravitational waves [403]:

$${\mathcal F} = \sum\limits_{\ell = 2}^{+ \infty} {{G \over {{c^{2\ell + 1}}}}} \left\{{{{(\ell + 1)(\ell + 2)} \over {(\ell - 1)\ell \ell !(2\ell + 1)!!}}{\rm{U}}_L^{(1)}{\rm{U}}_L^{(1)} + {{4\ell (\ell + 2)} \over {{c^2}(\ell - 1)(\ell + 1)!(2\ell + 1)!!}}{\rm{V}}_L^{(1)}{\rm{V}}_L^{(1)}} \right\}\,.$$
(68a)
$${{{\mathcal G}_i} = {\epsilon _{iab}}\sum\limits_{\ell = 2}^{+ \infty} {{G \over {{c^{2\ell + 1}}}}} \left\{{{{(\ell + 1)(\ell + 2)} \over {(\ell - 1)\ell !(2\ell + 1)!!}}\,{{\rm{U}}_{aL - 1}}{\rm{U}}_{bL - 1}^{(1)} + {{4{\ell ^2}(\ell + 2)} \over {{c^2}(\ell - 1)(\ell + 1)!(2\ell + 1)!!}}\,{{\rm{V}}_{aL - 1}}{\rm{V}}_{bL - 1}^{(1)}} \right\}}$$
(68b)

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

$${h_ +} = {1 \over 2}\left({{P_i}{P_j} - {Q_i}{Q_j}} \right)H_{ij}^{{\rm{TT}}}\,,$$
(69a)
$${h_ \times} = {1 \over 2}\left({{P_i}{Q_j} + {P_j}{Q_i}} \right)H_{ij}^{{\rm{TT}}}\,.$$
(69b)

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 (, 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

$$Y_{(- 2)}^{\ell m} = \sqrt {{{2\ell + 1} \over {4\pi}}} {d^{\,\ell m}}(\theta)\,{e^{{\rm{i}}\,m\,\phi}}\,,$$
(70a)
$${d^{\ell m}} = \sum\limits_{k = {k_1}}^{{k_2}} {{{{{(-)}^k}} \over {k!}}} e_k^{\,\ell m}{\left({\cos {\theta \over 2}} \right)^{2\ell + m - 2k - 2}}{\left({\sin {\theta \over 2}} \right)^{2k - m + 2}}\,,$$
(70b)
$$e_k^{\,\ell m} = {{\sqrt {(\ell + m)!(\ell - m)!(\ell + 2)!(\ell - 2)!}} \over {(k - m + 2)!(\ell + m - k)!(\ell - k - 2)!}}\,,$$
(70c)

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

$${h_ +} - {\rm{i}}{h_ \times} = \sum\limits_{\ell = 2}^{+ \infty} {\sum\limits_{m = - \ell}^\ell {{h^{\ell m}}}} \,Y_{(- 2)}^{\ell m}(\theta, \phi)\,.$$
(71)

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

$${h^{\ell m}} = \int {\rm{d}} \Omega \,\left[ {{h_ +} - {\rm{i}}{h_ \times}} \right]\,\bar Y_{(- 2)}^{\,\ell m}(\theta, \phi)\,,$$
(72)

where the overline refers to the complex conjugation. On the other hand, we can also relate hℓm to the radiative multipole moments UL and VL. The result is

$${h^{\ell m}} = - {G \over {\sqrt 2 \,R\,{c^{\ell + 2}}}}\left[ {{{\rm{U}}^{\ell m}} - {{\rm{i}} \over c}{{\rm{V}}^{\ell m}}} \right]\,,$$
(73)

where Uℓm and Vℓm denote the radiative mass and current moments in standard (non-STF) guise. These are related to the STF moments by

$${{\rm{U}}^{\ell m}} = {4 \over {\ell !}}\,\sqrt {{{(\ell + 1)(\ell + 2)} \over {2\ell (\ell - 1)}}} \,\alpha _L^{\ell m}\,{{\rm{U}}_L}\,,$$
(74a)
$${{\rm{V}}^{\ell m}} = - {8 \over {\ell !}}\,\sqrt {{{\ell (\ell + 2)} \over {2(\ell + 1)(\ell - 1)}}} \,\alpha _L^{\ell m}\,{{\rm{V}}_L}\,.$$
(74b)

Here \(\alpha _L^{\ell m}\) denotes the STF tensor connecting together the usual basis of spherical harmonics Yℓm to the set of STF tensors \({\hat N_L} = {N_{\langle {i_1}}} \ldots {N_{{i_\ell}}}_\rangle\) (where the brackets indicate the STF projection). Indeed both Yℓm and \({\hat N_L}\) are basis of an irreducible representation of weight of the rotation group; the two basis are related byFootnote 22

$${\hat N_L}(\theta, \phi) = \sum\limits_{m = - \ell}^\ell {\alpha _L^{\ell m}} \,{Y^{\ell m}}(\theta, \phi)\,,$$
(75a)
$${Y^{\ell m}}(\theta, \phi) = {{(2\ell + 1)!!} \over {4\pi l!}}\,\bar \alpha _L^{\ell m}\,{\hat N_L}(\theta, \phi)\,.$$
(75b)

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

$${{\rm{U}}_L}(U) = {\rm{M}}_L^{(\ell)}(U) + {{2G{\rm{M}}} \over {{c^3}}}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_L^{(\ell + 2)}(U - \tau)\left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {\kappa _\ell}} \right] + {\mathcal O}\left({{1 \over {{c^5}}}} \right)} \,,$$
(76a)
$${{\rm{V}}_L}(U) = {\rm{S}}_L^{(\ell)}(U) + {{2G{\rm{M}}} \over {{c^3}}}\int\nolimits_0^{+ \infty} \, \,{\rm{d}}\tau \,{\rm{S}}_L^{(\ell + 2)}(U - \tau)\left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {\pi _\ell}} \right] + {\mathcal O}\left({{1 \over {{c^5}}}} \right)\,,$$
(76b)

where r0 is the length scale introduced in Eq. (42), and the constants κ and π are given by

$${\kappa _\ell} = {{2{\ell ^2} + 5\ell + 4} \over {\ell (\ell + 1)(\ell + 2)}} + \sum\limits_{k = 1}^{\ell - 2} {{1 \over k}} \,,$$
(77a)
$${\pi _\ell} = {{\ell - 1} \over {\ell (\ell + 1)}} + \sum\limits_{k = 1}^{\ell - 1} {{1 \over k}} \,.$$
(77b)

Recall from the gauge vector \(\xi _{(1)}^\alpha\) found in Eq. (58) that the retarded time U = TR/c in radiative coordinates is related to the retarded time u = tr/c in harmonic coordinates by

$$U = u - {{2G{\rm{M}}} \over {{c^3}}}\ln \left({{r \over {{r_0}}}} \right) + {\mathcal O}\left({{G^2}} \right)\,.$$
(78)

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.,

$${{\rm{U}}_L}(U) = {\rm{M}}_L^{(\ell)}(u) + {{2G{\rm{M}}} \over {{c^3}}}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_L^{(\ell + 2)}(u - \tau)\left[ {\ln \left({{{c\tau} \over {2r}}} \right) + {\kappa _\ell}} \right] + {\mathcal O}\left({{1 \over {{c^5}}}} \right)} \,.$$
(79)

The remainder \({\mathcal O}({G^2})\) in Eq. (78) is negligible here. This expression no longer depends on the constant r0, i.e., we find that r0 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 r′ = r + GM/c2 (and u′ = uGM/c3), we get

$${{\rm{U}}_L}(U) = {\rm{M}}_L^{(\ell)}(u\prime) + {{2G{\rm{M}}} \over {{c^3}}}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_L^{(l + 2)}(u\prime - \tau)\left[ {\ln \left({{{c\tau} \over {2r\prime}}} \right) + {{\kappa \prime}_\ell}} \right] + {\mathcal O}\left({{1 \over {{c^5}}}} \right)\,,}$$
(80)

where κ = κ + 1/2. This shows that the constant κ (and π 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 κ2 = 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 \(- {\mathcal T}\). The range of integration of the tails is therefore limited a priori to the time interval \([ - {\mathcal T}{\rm{,}}U]\). 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 \(- {\mathcal T}\).

In this situation let us check, using a simple Newtonian model for the behaviour of the multipole moment ML(Uτ) 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 xi(Uτ) = Viτ + Wi ln τ + Xi + o(1), where Vi, Wi 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

$${{\rm{M}}_L}(U - \tau) = {A_L}{\tau ^\ell} + {B_L}{\tau ^{\ell - 1}}\ln \tau + {C_L}{\tau ^{\ell - 1}} + o({\tau ^{\ell - 1}})\,,$$
(81)

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 -th mass multipole moment QL. The appropriate time derivatives of the moment appearing in Eq. (76a) are therefore dominantly like

$${\rm{M}}_L^{(\ell + 2)}(U - \tau) = {{{D_L}} \over {{\tau ^3}}} + o({\tau ^{- 3}})\,,$$
(82)

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 Nαβ(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 . 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 1/r2:

$$\Box_{{\rm{ret}}}^{- 1}\left[ {{{{{\hat n}_L}} \over {{r^2}}}{\rm{F}}(t - r)} \right] = - {\hat n_L}\int\nolimits_1^{+ \infty} {{\rm{d}}x\,{Q_\ell}(x){\rm{F}}(t - rx)} \,,$$
(83)

where F is any smooth function representing a time derivative of the quadrupole moment, and Q denotes the Legendre function of the second kind.Footnote 23 Note that there is no need to include a finite part operation \({\mathcal F}{\mathcal P}\) 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]Footnote 24

$$\begin{array}{*{20}c} {h_{(2)}^{00} = {{{\rm{M}}{n_{ab}}} \over {{r^4}}}\left[ {- 21{{\rm{M}}_{ab}} - 21r{\rm{M}}_{ab}^{(1)} + 7{r^2}{\rm{M}}_{ab}^{(2)} + 10{r^3}{\rm{M}}_{ab}^{(3)}} \right]} \\ {+ 8{\rm{M}}{n_{ab}}\int\nolimits_1^{+ \infty} {{\rm{d}}x\,{Q_2}(x){\rm{M}}_{ab}^{(4)}(t - rx)\,,} \quad \quad \quad \quad \quad} \\ \end{array}$$
(84a)
$$\begin{array}{*{20}c} {h_{(2)}^{0i} = {{{\rm{M}}{n_{iab}}} \over {{r^3}}}\left[ {- {\rm{M}}_{ab}^{(1)} - r{\rm{M}}_{ab}^{(2)} - {1 \over 3}{r^2}{\rm{M}}_{ab}^{(3)}} \right]\quad \quad \quad \quad \quad \quad} \\ {+ {{{\rm{M}}{n_a}} \over {{r^3}}}\left[ {- 5{\rm{M}}_{ai}^{(1)} - 5r{\rm{M}}_{ai}^{(2)} + {{19} \over 3}{r^2}{\rm{M}}_{ai}^{(3)}} \right]\quad \quad \quad \quad} \\ {+ 8{\rm{M}}{n_a}\int\nolimits_1^{+ \infty} {{\rm{d}}x\,{Q_1}(x){\rm{M}}_{ai}^{(4)}(t - rx)\,,} \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(84b)
$$\begin{array}{*{20}c} {h_{(2)}^{ij} = {{{\rm{M}}{n_{ijab}}} \over {{r^4}}}\left[ {- {{15} \over 2}{{\rm{M}}_{ab}} - {{15} \over 2}r{\rm{M}}_{ab}^{(1)} - 3{r^2}{\rm{M}}_{ab}^{(2)} - {1 \over 2}{r^3}{\rm{M}}_{ab}^{(3)}} \right]} \\ {\quad + {{{\rm{M}}{\delta _{ij}}{n_{ab}}} \over {{r^4}}}\left[ {- {1 \over 2}{{\rm{M}}_{ab}} - {1 \over 2}r{\rm{M}}_{ab}^{(1)} - 2{r^2}{\rm{M}}_{ab}^{(2)} - {{11} \over 6}{r^3}{\rm{M}}_{ab}^{(3)}} \right]} \\ {+ {{{\rm{M}}{n_{a(i}}} \over {{r^4}}}\left[ {6{{\rm{M}}_{j)a}} + 6r{\rm{M}}_{j)a}^{(1)} + 6{r^2}{\rm{M}}_{j)a}^{(2)} + 4{r^3}{\rm{M}}_{j)a}^{(3)}} \right]\quad} \\ {+ {{\rm{M}} \over {{r^4}}}\left[ {- {{\rm{M}}_{ij}} - r{\rm{M}}_{ij}^{(1)} - 4{r^2}{\rm{M}}_{ij}^{(2)} - {{11} \over 3}{r^3}{\rm{M}}_{ij}^{(3)}} \right]\quad \quad \quad} \\ {+ 8{\rm{M}}\int\nolimits_1^{+ \infty} {{\rm{d}}x\,{Q_0}(x){\rm{M}}_{ij}^{(4)}(t - rx)\,.} \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(84c)

The metric is composed of two types of terms: “instantaneous” ones depending on the values of the quadrupole moment at the retarded time u = tr, and “hereditary” tail integrals, depending on all previous instants trx < 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 \({h_{\rm{M}}} \times {h_{\rm{M}}} \times {h_{{{\rm{M}}_{ij}}}}\), and quadratic products between one linear metric and one quadratic, say \({h_{{{\rm{M}}^2}}} \times {h_{{{\rm{M}}_{ij}}}}\) and \({h_{\rm{M}}} \times {h_{{\rm{M}}{{\rm{M}}_{ij}}}}\). The latter case is the most tricky because the tails present in \({h_{{\rm{M}}{{\rm{M}}_{ij}}}}\), which are given explicitly by Eqs. (84), will produce in turn some tails of tails in the cubic metric \({h_{{{\rm{M}}^2}{{\rm{M}}_{ij}}}}\). 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:Footnote 25

$$\begin{array}{*{20}c} {{\mathcal F}{\mathcal P}\Box_{{\rm{ret}}}^{- 1}\left[ {{{{{\hat n}_L}} \over r}\int\nolimits_1^{+ \infty} {{\rm{d}}x\,{Q_m}(x){\rm{F}}(t - rx)}} \right] = {{\hat n}_L}\int\nolimits_1^{+ \infty} {{\rm{d}}y\,{{\rm{F}}^{(- 1)}}(t - ry)} \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\times \left\{{{Q_\ell}(y)\int\nolimits_1^y {\rm{d}} x\,{Q_m}(x){{{\rm{d}}{P_\ell}} \over {{\rm{d}}x}}(x) + {P_\ell}(y)\int\nolimits_y^{+ \infty} {{\rm{d}}x\,{Q_m}(x){{{\rm{d}}{Q_\ell}} \over {{\rm{d}}x}}(x)}} \right\}\,,} \\ \end{array}$$
(85)

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 isFootnote 26

$$\begin{array}{*{20}c} {h_{(3)}^{00} = {{{{\rm{M}}^2}{n_{ab}}} \over r}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{ab}^{(5)}\left[ {- 4{{\ln}^2}\left({{\tau \over {2r}}} \right) - 4\ln \left({{\tau \over {2r}}} \right) + {{116} \over {21}}\ln \left({{\tau \over {2{r_0}}}} \right) - {{7136} \over {2205}}} \right]}} \\ {+ o\left({{1 \over r}} \right)\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(86a)
$$\begin{array}{*{20}c} {h_{(3)}^{0i} = {{{{\rm{M}}^2}{{\hat n}_{iab}}} \over r}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau {\rm{M}}_{ab}^{(5)}\left[ {- {2 \over 3}\ln \left({{\tau \over {2r}}} \right) - {4 \over {105}}\ln \left({{\tau \over {2{r_0}}}} \right) - {{716} \over {1225}}} \right]\quad \quad \quad \quad \quad \quad \quad}} \\ {+ {{{{\rm{M}}^2}{n_a}} \over r}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{ai}^{(5)}\left[ {- 4{{\ln}^2}\left({{\tau \over {2r}}} \right) - {{18} \over 5}\ln \left({{\tau \over {2r}}} \right) + {{416} \over {75}}\ln \left({{\tau \over {2{r_0}}}} \right) - {{22724} \over {7875}}} \right]}} \\ {+ o\left({{1 \over r}} \right)\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,} \\ \end{array}$$
(86b)
$$\begin{array}{*{20}c} {h_{(3)}^{ij} = {{{{\rm{M}}^2}{{\hat n}_{ijab}}} \over r}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{ab}^{(5)}\left[ {- \ln \left({{\tau \over {2r}}} \right) - {{191} \over {210}}} \right]} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{{\rm{M}}^2}{\delta _{ij}}{n_{ab}}} \over r}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{ab}^{(5)}\left[ {- {{80} \over {21}}\ln \left({{\tau \over {2r}}} \right) - {{32} \over {21}}\ln \left({{\tau \over {2{r_0}}}} \right) - {{296} \over {35}}} \right]} \quad \quad \quad} \\ {+ {{{{\rm{M}}^2}{{\hat n}_{a(i}}} \over r}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{j)a}^{(5)}\left[ {{{52} \over 7}\ln \left({{\tau \over {2r}}} \right) + {{104} \over {35}}\ln \left({{\tau \over {2{r_0}}}} \right) + {{8812} \over {525}}} \right]\quad} \quad \quad \,\,\,} \\ {+ {{{{\rm{M}}^2}} \over r}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{ij}^{(5)}\left[ {- 4{{\ln}^2}\left({{\tau \over {2r}}} \right) - {{24} \over 5}\ln \left({{\tau \over {2r}}} \right) + {{76} \over {15}}\ln \left({{\tau \over {2{r_0}}}} \right) - {{198} \over {35}}} \right]} \,} \\ {+ o\left({{1 \over r}} \right)\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,} \\ \end{array}$$
(86c)

where all the moments Mab are evaluated at the instant uτ = trτ. 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 Xα 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: \({X^\alpha} = {x^\alpha} + G\xi _{(1)}^\alpha + {\mathcal O}({G^2})\), where \(\xi _{(1)}^\alpha\) 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

$$\begin{array}{*{20}c} {H_{(3)}^{00} = {{{{\rm{M}}^2}{N_{ab}}} \over R}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{ab}^{(5)}\left[ {- 4{{\ln}^2}\left({{\tau \over {2{r_0}}}} \right) + 3221\ln \left({{\tau \over {2{r_0}}}} \right) - {{7136} \over {2205}}} \right]}} \\ {+ {\mathcal O}\left({{1 \over {{R^2}}}} \right)\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(87a)
$$\begin{array}{*{20}c} {H_{(3)}^{0i} = {{{{\rm{M}}^2}{{\hat N}_{iab}}} \over R}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{ab}^{(5)}\left[ {- {{74} \over {105}}\ln \left({{\tau \over {2{r_0}}}} \right) - {{716} \over {1225}}} \right]\quad \quad \quad \quad \quad \quad \quad}} \\ {+ \,{{{{\rm{M}}^2}{N_a}} \over R}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{ai}^{(5)}\left[ {- 4{{\ln}^2}\left({{\tau \over {2{r_0}}}} \right) + {{146} \over {75}}\ln \left({{\tau \over {2{r_0}}}} \right) - {{22724} \over {7875}}} \right]}} \\ {+ {\mathcal O}\left({{1 \over {{R^2}}}} \right)\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(87b)
$$\begin{array}{*{20}c} {H_{(3)}^{ij} = {{{{\rm{M}}^2}{{\hat N}_{ijab}}} \over R}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{ab}^{(5)}\left[ {- \ln \left({{\tau \over {2{r_0}}}} \right) - {{191} \over {210}}} \right]} \quad \quad \quad \quad \quad} \\ {+ {{{{\rm{M}}^2}{\delta _{ij}}{N_{ab}}} \over R}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{ab}^{(5)}\left[ {- {{16} \over 3}\ln \left({{\tau \over {2{r_0}}}} \right) - {{296} \over {35}}} \right]\quad \quad}} \\ {+ {{{{\rm{M}}^2}{{\hat N}_{a(i}}} \over R}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{j)a}^{(5)}\left[ {{{52} \over 5}\ln \left({{\tau \over {2{r_0}}}} \right) + {{8812} \over {525}}} \right]\quad \quad \quad}} \\ {\,\, + \,{{{{\rm{M}}^2}} \over R}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \,{\rm{M}}_{ij}^{(5)}\left[ {- 4{{\ln}^2}\left({{\tau \over {2{r_0}}}} \right) + {4 \over {15}}\ln \left({{\tau \over {2{r_0}}}} \right) - {{198} \over {35}}} \right]}} \\ {\, + {\mathcal O}\left({{1 \over {{R^2}}}} \right)\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(87c)

where the moments are evaluated at time Uτ = TRτ. 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.

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:

$${{\rm{U}}_{ij}} = {\rm{U}}_{ij}^{{\rm{inst}}} + {\rm{U}}_{ij}^{{\rm{tail}}} + {\rm{U}}_{ij}^{{\rm{tail - tail}}} + {\rm{U}}_{ij}^{{\rm{mem}}} + {\mathcal O}\left({{1 \over {{c^8}}}} \right)\,.$$
(88)
  1. 1.

    The instantaneous (i.e., non-hereditary) piece \({\rm{U}}_{ij}^{{\rm{inst}}}\) up to 3.5PN order reads

    $$\begin{array}{*{20}c} {{\rm{U}}_{ij}^{{\rm{inst}}} = {\rm{M}}_{ij}^{(2)}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {G \over {{c^5}}}\left[ {{1 \over 7}{\rm{M}}_{a\langle i}^{(5)}{{\rm{M}}_{j\rangle a}} - {5 \over 7}{\rm{M}}_{a\langle i}^{(4)}{\rm{M}}_{j\rangle a}^{(1)} - {2 \over 7}{\rm{M}}_{a\langle i}^{(3)}{\rm{M}}_{j\rangle a}^{(2)} + {1 \over 3}{\epsilon _{ab\langle i}}{\rm{M}}_{j\rangle a}^{(4)}{{\rm{S}}_b}} \right]\quad \quad \quad \quad} \\ {+ {G \over {{c^7}}}\left[ {- {{64} \over {63}}{\rm{S}}_{a\langle i}^{(2)}{\rm{S}}_{j\rangle a}^{(3)} + {{1957} \over {3024}}{\rm{M}}_{ijab}^{(3)}{\rm{M}}_{ab}^{(4)} + {5 \over {2268}}{\rm{M}}_{ab\langle i}^{(3)}{\rm{M}}_{j\rangle ab}^{(4)} + {{19} \over {648}}{\rm{M}}_{ab}^{(3)}{\rm{M}}_{ijab}^{(4)}} \right.} \\ {+ {{16} \over {63}}{\rm{S}}_{a\langle i}^{(1)}{\rm{S}}_{j\rangle a}^{(4)} + {{1685} \over {1008}}{\rm{M}}_{ijab}^{(2)}{\rm{M}}_{ab}^{(5)} + {5 \over {126}}{\rm{M}}_{ab\langle i}^{(2)}{\rm{M}}_{j\rangle ab}^{(5)} - {5 \over {756}}{\rm{M}}_{ab}^{(2)}{\rm{M}}_{ijab}^{(5)}\quad \quad \quad} \\ {+ {{80} \over {63}}{{\rm{S}}_{a\langle i}}{\rm{S}}_{j\rangle a}^{(5)} + {5 \over {42}}{{\rm{S}}_a}{\rm{S}}_{ija}^{(5)} + {{41} \over {28}}{\rm{M}}_{ijab}^{(1)}{\rm{M}}_{ab}^{(6)} + {5 \over {189}}{\rm{M}}_{ab\langle i}^{(1)}{\rm{M}}_{j\rangle ab}^{(6)}\quad \quad \quad \quad \quad \quad} \\ {+ {1 \over {432}}{\rm{M}}_{ab}^{(1)}{\rm{M}}_{ijab}^{(6)} + {{91} \over {216}}{{\rm{M}}_{ijab}}{\rm{M}}_{ab}^{(7)} - {5 \over {252}}{{\rm{M}}_{ab\langle i}}{\rm{M}}_{j\rangle ab}^{(7)} - {1 \over {432}}{{\rm{M}}_{ab}}{\rm{M}}_{ijab}^{(7)}\quad \quad} \\ {+ {\epsilon _{ac\langle i}}\left({{{32} \over {189}}{\rm{M}}_{j\rangle bc}^{(3)}{\rm{S}}_{ab}^{(3)} - {1 \over 6}{\rm{M}}_{ab}^{(3)}{\rm{S}}_{j\rangle bc}^{(3)} + {3 \over {56}}{\rm{S}}_{j\rangle bc}^{(2)}{\rm{M}}_{ab}^{(4)} + {{10} \over {189}}{\rm{S}}_{ab}^{(2)}{\rm{M}}_{j\rangle bc}^{(4)}} \right.\quad \quad \quad} \\ {+ {{65} \over {189}}{\rm{M}}_{j\rangle bc}^{(2)}{\rm{S}}_{ab}^{(4)} + {1 \over {28}}{\rm{M}}_{ab}^{(2)}{\rm{S}}_{j\rangle bc}^{(4)} + {{187} \over {168}}{\rm{S}}_{j\rangle bc}^{(1)}{\rm{M}}_{ab}^{(5)} - {1 \over {189}}{\rm{S}}_{ab}^{(1)}{\rm{M}}_{j\rangle bc}^{(5)}\quad \quad \quad \quad} \\ {- {5 \over {189}}{\rm{M}}_{j\rangle bc}^{(1)}{\rm{S}}_{ab}^{(5)} + {1 \over {24}}{\rm{M}}_{ab}^{(1)}{\rm{S}}_{j\rangle bc}^{(5)} + {{65} \over {84}}{{\rm{S}}_{j\rangle bc}}{\rm{M}}_{ab}^{(6)} + {1 \over {189}}{{\rm{S}}_{ab}}{\rm{M}}_{j\rangle bc}^{(6)}\quad \quad \quad \quad \quad} \\ {\left. {\left. {- {{10} \over {63}}{{\rm{M}}_{j\rangle bc}}{\rm{S}}_{ab}^{(6)} + {1 \over {168}}{{\rm{M}}_{ab}}{\rm{S}}_{j\rangle bc}^{(6)}} \right)} \right].\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
    (89)

    The Newtonian term in this expression contains the Newtonian quadrupole moment Qij and recovers the standard quadrupole formalism [see Eq. (67)];

  2. 2.

    The hereditary tail integral \({\rm{U}}_{ij}^{{\rm{tail}}}\) is made of the dominant tail term at 1.5PN order in agreement with Eq. (76a) above:

    $${\rm{U}}_{ij}^{{\rm{tail}}} = {{2G{\rm{M}}} \over {{c^3}}}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {{11} \over {12}}} \right]{\rm{M}}_{ij}^{(4)}(U - \tau)\,.}$$
    (90)

    The length scale r0 is the one that enters our definition of the finite-part operation \({\mathcal F}{\mathcal P}\) [see Eq. (42)] and it enters also the relation between the radiative and harmonic retarded times given by Eq. (78);

  3. 3.

    The hereditary tail-of-tail term appears dominantly at 3PN order [48] and is issued from the radiative metric computed in Eqs. (87):

    $${\rm{U}}_{ij}^{{\rm{tail - tail}}} = 2{\left({{{G{\rm{M}}} \over {{c^3}}}} \right)^2}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \left[ {{{\ln}^2}\left({{{c\tau} \over {2{r_0}}}} \right) + {{57} \over {70}}\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {{124627} \over {44100}}} \right]{\rm{M}}_{ij}^{(5)}(U - \tau)\,;}$$
    (91)
  4. 4.

    Finally the memory-type hereditary piece \({\rm{U}}_{ij}^{{\rm{mem}}}\) contributes at orders 2.5PN and 3.5PN and is given by

    $$\begin{array}{*{20}c} {{\rm{U}}_{ij}^{{\rm{mem}}} = {G \over {{c^5}}}\left[ {- {2 \over 7}\int\nolimits_0^{+ \infty} {\rm{d}} \tau \,{\rm{M}}_{a\langle i}^{(3)}\,{\rm{M}}_{j\rangle a}^{(3)}(U - \tau)} \right]\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {G \over {{c^7}}}\left[ {- {{32} \over {63}}\int\nolimits_0^{+ \infty} {\rm{d}} \tau \,{\rm{S}}_{a\langle i}^{(3)}\,{\rm{S}}_{j\rangle a}^{(3)}(U - \tau) - {5 \over {756}}\int\nolimits_0^{+ \infty} {\rm{d}} \tau \,{\rm{M}}_{ab}^{(4)}\,{\rm{M}}_{ijab}^{(4)}(U - \tau)} \right.} \\ {\quad \quad \quad \left. {- {{20} \over {189}}\,{\epsilon _{ab\langle i}}\int\nolimits_0^{+ \infty} {\rm{d}} \tau \,{\rm{S}}_{ac}^{(3)}\,{\rm{M}}_{j\rangle bc}^{(4)}(U - \tau) + {5 \over {42}}\,{\epsilon _{ab\langle i}}\int\nolimits_0^{+ \infty} {\rm{d}} \tau \,{\rm{M}}_{ac}^{(3)}\,{\rm{S}}_{j\rangle bc}^{(4)}(U - \tau)} \right].\quad} \\ \end{array}$$
    (92)

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 \(U = u - {{2G{\rm{M}}} \over {{c^3}}}\ln (r/{r_0})\) 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

$${U_{ij}} = M_{ij}^{(2)}(u) - {{214} \over {105}}\ln \left( {{r \over {{r_0}}}} \right){\left( {{{GM} \over {{c^3}}}} \right)^2}M_{ij}^{(4)}(u) + \text{terms of independent of}\; {r_0}.$$
(93)

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 Iij of compact binaries at the 3PN order, and we do observe on the result (300)(301) below the requested terms depending on r0, namelyFootnote 27

$${{\rm{M}}_{ij}} = {{\rm{Q}}_{ij}} + {{214} \over {105}}\ln \left({{{{r_{12}}} \over {{r_0}}}} \right){\left({{{Gm} \over {{c^3}}}} \right)^2}{\rm{Q}}_{ij}^{(2)} + {\rm{terms}}\,{\rm{independent}}\,{\rm{of}}\,{r_0}.$$
(94)

where \({{\rm{Q}}_{ij}} = \mu {\hat x_{ij}}\) 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]:Footnote 28

$$\begin{array}{*{20}c} {{{\rm{U}}_{ijk}}(U) = {\rm{M}}_{ijk}^{(3)}(U) + {{2GM} \over {{c^3}}}\int\nolimits_0^{+ \infty} {\rm{d}} \tau \left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {{97} \over {60}}} \right]{\rm{M}}_{ijk}^{(5)}(U - \tau)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {G \over {{c^5}}}\left\{{\int\nolimits_0^{+ \infty} {\rm{d}} \tau \left[ {- {1 \over 3}{\rm{M}}_{a\langle i}^{(3)}{\rm{M}}_{jk\rangle a}^{(4)} - {4 \over 5}{\epsilon _{ab\langle i}}{\rm{M}}_{ja}^{(3)}{\rm{S}}_{k\rangle b}^{(3)}} \right](U - \tau)\quad \quad \quad \quad \quad \quad \quad} \right.} \\ {- {4 \over 3}{\rm{M}}_{a\langle i}^{(3)}{\rm{M}}_{jk\rangle a}^{(3)} - {9 \over 4}{\rm{M}}_{a\langle i}^{(4)}{\rm{M}}_{jk\rangle a}^{(2)} + {1 \over 4}{\rm{M}}_{a\langle i}^{(2)}{\rm{M}}_{jk\rangle a}^{(4)} - {3 \over 4}{\rm{M}}_{a\langle i}^{(5)}{\rm{M}}_{jk\rangle a}^{(1)} + {1 \over 4}{\rm{M}}_{a\langle i}^{(1)}{\rm{M}}_{jk\rangle a}^{(5)}} \\ {\quad \, + {1 \over {12}}{\rm{M}}_{a\langle i}^{(6)}{{\rm{M}}_{jk\rangle a}} + {1 \over 4}{{\rm{M}}_{a\langle i}}{\rm{M}}_{jk\rangle a}^{(6)} + {1 \over 5}{\epsilon _{ab\langle i}}\left[ {- 12{\rm{S}}_{ja}^{(2)}{\rm{M}}_{k\rangle b}^{(3)} - 8{\rm{M}}_{ja}^{(2)}{\rm{S}}_{k\rangle b}^{(3)} - 3{\rm{S}}_{ja}^{(1)}{\rm{M}}_{k\rangle b}^{(4)}} \right.} \\ {\left. {\left. {- 27{\rm{M}}_{ja}^{(1)}{\rm{S}}_{k\rangle b}^{(4)} - {{\rm{S}}_{ja}}{\rm{M}}_{k\rangle b}^{(5)} - 9{{\rm{M}}_{ja}}{\rm{S}}_{k\rangle b}^{(5)} - {9 \over 4}{{\rm{S}}_a}{\rm{M}}_{jk\rangle b}^{(5)}} \right] + {{12} \over 5}{{\rm{S}}_{\langle i}}{\rm{S}}_{jk\rangle}^{(4)}} \right\}\quad \quad \quad \quad \quad} \\ {+ {\mathcal O}\left({{1 \over {{c^6}}}} \right)\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(95a)
$$\begin{array}{*{20}c} {{{\rm{V}}_{ij}}(U) = {\rm{S}}_{ij}^{(2)}(U) + {{2GM} \over {{c^3}}}\int\nolimits_0^{+ \infty} {\rm{d}} \tau \left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {7 \over 6}} \right]{\rm{S}}_{ij}^{(4)}(U - \tau)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {G \over {7\,{c^5}}}\left\{{4{\rm{S}}_{a\langle i}^{(2)}{\rm{M}}_{j\rangle a}^{(3)} + 8{\rm{M}}_{a\langle i}^{(2)}{\rm{S}}_{j\rangle a}^{(3)} + 17{\rm{S}}_{a\langle i}^{(1)}{\rm{M}}_{j\rangle a}^{(4)} - 3{\rm{M}}_{a\langle i}^{(1)}{\rm{S}}_{j\rangle a}^{(4)} + 9{{\rm{S}}_{a\langle i}}{\rm{M}}_{j\rangle a}^{(5)}} \right.\quad \quad \quad \quad \quad} \\ {- 3{{\rm{M}}_{a\langle i}}{\rm{S}}_{j\rangle a}^{(5)} - {1 \over 4}{{\rm{S}}_a}{\rm{M}}_{ija}^{(5)} - 7{\epsilon _{ab\langle i}}{{\rm{S}}_a}{\rm{S}}_{j\rangle b}^{(4)} + {1 \over 2}{\epsilon _{ac\langle i}}\left[ {3{\rm{M}}_{ab}^{(3)}{\rm{M}}_{j\rangle bc}^{(3)} + {{353} \over {24}}{\rm{M}}_{j\rangle bc}^{(2)}{\rm{M}}_{ab}^{(4)}} \right.\quad \,\,} \\ {\left. {\left. {- {5 \over {12}}{\rm{M}}_{ab}^{(2)}{\rm{M}}_{j\rangle bc}^{(4)} + {{113} \over 8}{\rm{M}}_{j\rangle bc}^{(1)}{\rm{M}}_{ab}^{(5)} - {3 \over 8}{\rm{M}}_{ab}^{(1)}{\rm{M}}_{j\rangle bc}^{(5)} + {{15} \over 4}{{\rm{M}}_{j\rangle bc}}{\rm{M}}_{ab}^{(6)} + {3 \over 8}{{\rm{M}}_{ab}}{\rm{M}}_{j\rangle bc}^{(6)}} \right]} \right\}} \\ {+ {\mathcal O}\left({{1 \over {{c^6}}}} \right)\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(95b)
$$\begin{array}{*{20}c} {{{\rm{U}}_{ijkl}}(U) = {\rm{M}}_{ijkl}^{(4)}(U) + {G \over {{c^3}}}\left\{{2M\int\nolimits_0^{+ \infty} {\rm{d}} \tau \left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {{59} \over {30}}} \right]{\rm{M}}_{ijkl}^{(6)}(U - \tau)} \right.\quad \quad \quad \quad \quad \quad \quad \quad} \\ {\quad \left. {+ {2 \over 5}\int\nolimits_0^{+ \infty} {\rm{d}} \tau {\rm{M}}_{\langle ij}^{(3)}{\rm{M}}_{kl\rangle}^{(3)}(U - \tau) - {{21} \over 5}{\rm{M}}_{\langle ij}^{(5)}{{\rm{M}}_{kl\rangle}} - {{63} \over 5}{\rm{M}}_{\langle ij}^{(4)}{\rm{M}}_{kl\rangle}^{(1)} - {{102} \over 5}{\rm{M}}_{\langle ij}^{(3)}{\rm{M}}_{kl\rangle}^{(2)}} \right\}} \\ {+ \,{\mathcal O}\left({{1 \over {{c^5}}}} \right)\,\,,\,\,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,} \\ \end{array}$$
(95c)
$$\begin{array}{*{20}c} {{{\rm{V}}_{ijk}}(U) = {\rm{S}}_{ijk}^{(3)}(U) + {G \over {{c^3}}}\left\{{2M\int\nolimits_0^{+ \infty} {\rm{d}} \tau \left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {5 \over 3}} \right]{\rm{S}}_{ijk}^{(5)}(U - \tau)} \right.\quad} \\ {\left. {+ {1 \over {10}}{\epsilon _{ab\langle i}}{\rm{M}}_{ja}^{(5)}{{\rm{M}}_{k\rangle b}} - {1 \over 2}{\epsilon _{ab\langle i}}{\rm{M}}_{ja}^{(4)}{\rm{M}}_{k\rangle b}^{(1)} - 2{{\rm{S}}_{\langle i}}{\rm{M}}_{jk\rangle}^{(4)}} \right\}} \\ {+ {\mathcal O}\left({{1 \over {{c^5}}}} \right)\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(95d)
$$\begin{array}{*{20}c} {{{\rm{U}}_{ijklm}}(U) = {\rm{M}}_{ijklm}^{(5)}(U) + {G \over {{c^3}}}\left\{{2M\int\nolimits_0^{+ \infty} {\rm{d}} \tau \left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {{232} \over {105}}} \right]{\rm{M}}_{ijklm}^{(7)}(U - \tau)} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{20} \over {21}}\int\nolimits_0^{+ \infty} {\rm{d}} \tau {\rm{M}}_{\langle ij}^{(3)}{\rm{M}}_{klm\rangle}^{(4)}(U - \tau) - {{710} \over {21}}{\rm{M}}_{\langle ij}^{(3)}{\rm{M}}_{klm\rangle}^{(3)} - {{265} \over 7}{\rm{M}}_{\langle ijk}^{(2)}{\rm{M}}_{lm\rangle}^{(4)} - {{120} \over 7}{\rm{M}}_{\langle ij}^{(2)}{\rm{M}}_{klm\rangle}^{(4)}} \\ {\left. {- {{155} \over 7}{\rm{M}}_{\langle ijk}^{(1)}{\rm{M}}_{lm\rangle}^{(5)} - {{41} \over 7}{\rm{M}}_{\langle ij}^{(1)}{\rm{M}}_{klm\rangle}^{(5)} - {{34} \over 7}{{\rm{M}}_{\langle ijk}}{\rm{M}}_{lm\rangle}^{(6)} - {{15} \over 7}{{\rm{M}}_{\langle ij}}{\rm{M}}_{klm\rangle}^{(6)}} \right\}\quad \quad \quad \quad \quad \quad} \\ {+ {\mathcal O}\left({{1 \over {{c^4}}}} \right)\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(95e)
$$\begin{array}{*{20}c} {{{\rm{V}}_{ijkl}}(U) = {\rm{S}}_{ijkl}^{(4)}(U) + {G \over {{c^3}}}\left\{{2M\int\nolimits_0^{+ \infty} {\rm{d}} \tau \left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {{119} \over {60}}} \right]{\rm{S}}_{ijkl}^{(6)}(U - \tau)} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {- {{35} \over 3}{\rm{S}}_{\langle ij}^{(2)}{\rm{M}}_{kl\rangle}^{(3)} - {{25} \over 3}{\rm{M}}_{\langle ij}^{(2)}{\rm{S}}_{kl\rangle}^{(3)} - {{65} \over 6}{\rm{S}}_{\langle ij}^{(1)}{\rm{M}}_{kl\rangle}^{(4)} - {{25} \over 6}{\rm{M}}_{\langle ij}^{(1)}{\rm{S}}_{kl\rangle}^{(4)} - {{19} \over 6}{{\rm{S}}_{\langle ij}}{\rm{M}}_{kl\rangle}^{(5)}\quad \quad \quad \quad} \\ {- {{11} \over 6}{{\rm{M}}_{\langle ij}}{\rm{S}}_{kl\rangle}^{(5)} - {{11} \over {12}}{{\rm{S}}_{\langle i}}{\rm{M}}_{jkl\rangle}^{(5)} + {1 \over 6}{\epsilon _{ab\langle i}}\left[ {- 5{\rm{M}}_{ja}^{(3)}{\rm{M}}_{kl\rangle b}^{(3)} - {{11} \over 2}{\rm{M}}_{ja}^{(4)}{\rm{M}}_{kl\rangle b}^{(2)} - {5 \over 2}{\rm{M}}_{ja}^{(2)}{\rm{M}}_{kl\rangle b}^{(4)}} \right.} \\ {\left. {\left. {- {1 \over 2}{\rm{M}}_{ja}^{(5)}{\rm{M}}_{kl\rangle b}^{(1)} + {{37} \over {10}}{\rm{M}}_{ja}^{(1)}{\rm{M}}_{kl\rangle b}^{(5)} + {3 \over {10}}{\rm{M}}_{ja}^{(6)}{{\rm{M}}_{kl\rangle b}} + {1 \over 2}{{\rm{M}}_{ja}}{\rm{M}}_{kl\rangle b}^{(6)}} \right]} \right\}\quad \quad \quad \quad \quad \quad} \\ {+ {\mathcal O}\left({{1 \over {{c^4}}}} \right)\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(95f)

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

$${{\rm{U}}_L}(U) = {\rm{M}}_L^{(\ell)}(U) + {\mathcal O}\left({{1 \over {{c^3}}}} \right)\,,$$
(96a)
$${{\rm{V}}_L}(U) = {\rm{S}}_L^{(\ell)}(U) + {\mathcal O}\left({{1 \over {{c^3}}}} \right)\,.$$
(96b)

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 {IL, JL, WL, XL, YL, ZL}. 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]

$$\begin{array}{*{20}c} {{{\rm{M}}_{ij}} = {{\rm{I}}_{ij}} + {{4G} \over {{c^5}}}\left[ {{{\rm{W}}^{(2)}}{{\rm{I}}_{ij}} - {{\rm{W}}^{(1)}}{\rm{I}}_{ij}^{(1)}} \right]\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{4G} \over {{c^7}}}\left[ {{4 \over 7}{\rm{W}}_{a\langle i}^{(1)}{\rm{I}}_{j\rangle a}^{(3)} + {6 \over 7}{{\rm{W}}_{a\langle i}}{\rm{I}}_{j\rangle a}^{(4)} - {1 \over 7}{{\rm{I}}_{a\langle i}}{\rm{Y}}_{j\rangle a}^{(3)} - {{\rm{Y}}_{a\langle i}}{\rm{I}}_{j\rangle a}^{(3)} - 2{\rm{XI}}_{ij}^{(3)} - {5 \over {21}}{{\rm{I}}_{ija}}{\rm{W}}_a^{(4)}} \right.\quad \quad} \\ {\quad \quad \quad \quad + {1 \over {63}}{\rm{I}}_{ija}^{(1)}{\rm{W}}_a^{(3)} - {{25} \over {21}}{{\rm{I}}_{ija}}{\rm{Y}}_a^{(3)} - {{22} \over {63}}{\rm{I}}_{ija}^{(1)}{\rm{Y}}_a^{(2)} + {5 \over {63}}{\rm{Y}}_a^{(1)}{\rm{I}}_{ija}^{(2)} + 2{{\rm{W}}_{ij}}{{\rm{W}}^{(3)}}\quad \quad \quad \quad} \\ {+ 2{\rm{W}}_{ij}^{(1)}{{\rm{W}}^{(2)}} - {4 \over 3}{{\rm{W}}_{\langle i}}{\rm{W}}_{j\rangle}^{(3)} + 2{{\rm{Y}}_{ij}}{{\rm{W}}^{(2)}} - 4{{\rm{W}}_{\langle i}}{\rm{Y}}_{j\rangle}^{(2)}\quad \quad \quad \quad \quad \quad} \\ {\quad \quad \quad \quad \quad + {\epsilon _{ab\langle i}}\left. {\left({{1 \over 3}{{\rm{I}}_{j\rangle a}}{\rm{Z}}_b^{(3)} + {4 \over 9}{{\rm{J}}_{j\rangle a}}{\rm{W}}_b^{(3)} - {4 \over 9}{{\rm{J}}_{j\rangle a}}{\rm{Y}}_b^{(2)} + {8 \over 9}{\rm{J}}_{j\rangle a}^{(1)}{\rm{Y}}_b^{(1)} + {{\rm{Z}}_a}{\rm{I}}_{j\rangle b}^{(3)}} \right)} \right] + {\mathcal O}\left({{1 \over {{c^8}}}} \right)\,.} \\ \end{array}$$
(97)

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:

$${{\rm{M}}_{ijk}} = {{\rm{I}}_{ijk}} + {{4G} \over {{c^5}}}\left[ {{{\rm{W}}^{(2)}}{{\rm{I}}_{ijk}} - {{\rm{W}}^{(1)}}{\rm{I}}_{ijk}^{(1)} + 3\,{{\rm{I}}_{\langle ij}}{\rm{Y}}_{k\rangle}^{(1)}} \right] + {\mathcal O}\left({{1 \over {{c^6}}}} \right)\,,$$
(98a)
$$\begin{array}{*{20}c} {{{\rm{S}}_{ij}} = {{\rm{J}}_{ij}} + {{2G} \over {{c^5}}}\left[ {{\epsilon _{ab\langle i}}\left({- {\rm{I}}_{j\rangle b}^{(3)}{{\rm{W}}_a} - 2{{\rm{I}}_{j\rangle b}}{\rm{Y}}_a^{(2)} + {\rm{I}}_{j\rangle b}^{(1)}{\rm{Y}}_a^{(1)}} \right) + 3{{\rm{J}}_{\langle i}}{\rm{Y}}_{j\rangle}^{(1)} - 2{\rm{J}}_{ij}^{(1)}{{\rm{W}}^{(1)}}} \right]} \\ {+ {\mathcal O}\left({{1 \over {{c^6}}}} \right)\,.\,\,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(98b)

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 IL, JL:

$${{\rm{M}}_L} = {{\rm{I}}_L} + {\mathcal O}\left({{1 \over {{c^5}}}} \right)\,,$$
(99a)
$${{\rm{S}}_L} = {{\rm{J}}_L} + {\mathcal O}\left({{1 \over {{c^5}}}} \right)\,.$$
(99b)

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.

Matching to a Post-Newtonian Source

By Theorem 2 we control the most general class of solutions of the vacuum equations outside the source, in the form of non-linear functionals of the source multipole moments. For instance, these solutions include the Schwarzschild and Kerr solutions for black holes, as well as all their perturbations. By Theorem 4 we learned how to construct the radiative moments at infinity, which constitute the observables of the radiation field at large distances from the source, and we obtained in Section 3.3 explicit relationships between radiative and source moments. We now want to understand how a specific choice of matter stress-energy tensor Tαβ, i.e., a specific choice of some physical model describing the material source, selects a particular physical exterior solution among our general class, and therefore a given set of multipole moments for the source.

The matching equation

We shall provide the answer to that problem in the case of a post-Newtonian source for which the post-Newtonian parameter ε ∼ 1/c defined by Eq. (1) is small. The fundamental fact that permits the connection of the exterior field to the inner field of the source is the existence of a “matching” region, in which both the multipole expansion and the post-Newtonian expansion are valid. This region is nothing but the exterior part of the near zone, such that r > a (exterior) and rλ (near zone); it always exists around post-Newtonian sources whose radius is much less than the emitted wavelength, \({\alpha \over \lambda} \sim \epsilon \ll 1\). In our formalism the multipole expansion is defined by the multipolar-post-Minkowskian (MPM) solution; see Section 2. Matching together the post-Newtonian and MPM solutions in this overlapping region is an application of the method of matched asymptotic expansions, which has frequently been applied in the present context, both for radiation-reaction [114, 113, 7, 58, 43] and wave-generation [59, 155, 44, 49] problems.

Let us denote by \({\mathcal M}{\rm{(}}h{\rm{)}}\) the multipole expansion of h (for simplicity, we suppress the space-time indices). By \({\mathcal M}{\rm{(}}h{\rm{)}}\) we really mean the MPM exterior metric that we have constructed in Sections 2.2 and 2.3:

$${\mathcal M}(h) \equiv {h_{{\rm{ext}}}} = \sum\limits_{n = 1}^{+ \infty} {{G^n}} {h_{(n)}}[{{\rm{I}}_L}, \ldots, {{\rm{Z}}_L}]\,.$$
(100)

This solution is formally defined for any radius r > 0. Of course, the true solution h agrees with its own multipole expansion in the exterior of the source, i.e.

$$r > a\quad \Rightarrow \quad {\mathcal M}(h) = h\,.$$
(101)

By contrast, inside the source, h and \({\mathcal M}{\rm{(}}h{\rm{)}}\) disagree with each other because h is a fully-fledged solution of the field equations within the matter source, while \({\mathcal M}{\rm{(}}h{\rm{)}}\) is a vacuum solution becoming singular at r = 0. Now let us denote by h the post-Newtonian expansion of h. We have already anticipated the general structure of this expansion which is given in Eq. (54). In the matching region, where both the multipolar and post-Newtonian expansions are valid, we write the numerical equality

$$a < r \ll \lambda \quad \Rightarrow \quad {\mathcal M}(h) = \bar h\,.$$
(102)

This “numerical” equality is viewed here in a sense of formal expansions, as we do not control the convergence of the series. In fact, we should be aware that such an equality, though quite natural and even physically obvious, is probably not really justified within the approximation scheme (mathematically speaking), and we simply take it here as part of our fundamental assumptions.

We now transform Eq. (102) into a matching equation, by replacing in the left-hand side \({\mathcal M}{\rm{(}}h{\rm{)}}\) by its near-zone re-expansion \(\overline {{\mathcal M}(h)}\), and in the right-hand side \(\overline h\) by its multipole expansion \({\mathcal M}(\overline h)\). The structure of the near-zone expansion (r → 0) of the exterior multipolar field has been found in Theorem 3, see Eq. (53). We denote the corresponding infinite series \(\overline {{\mathcal M}(h)}\) with the same overbar as for the post-Newtonian expansion because it is really an expansion when r/c → 0, equivalent to an expansion when c → ∞. Concerning the multipole expansion of the post-Newtonian metric, \({\mathcal M}(\overline h)\), we simply postulate for the moment its existence, but we shall show later how to construct it explicitly. Therefore, the matching equation is the statement that

$$\overline {{\mathcal M}(h)} = {\mathcal M}(\bar h)\,,$$
(103)

by which we really mean an infinite set of functional identities, valid \(\forall ({\rm{x,}}\,t) \in \mathbb R_*^3 \times \mathbb R\), between the coefficients of the series in both sides of the equation. Note that such a meaning is somewhat different from that of a numerical equality like Eq. (102), which is valid only when x belongs to some limited spatial domain. The matching equation (103) tells us that the formal near-zone expansion of the multipole decomposition is identical, term by term, to the multipole expansion of the post-Newtonian solution. However, the former expansion is nothing but the formal far-zone expansion, when r → ∞, of each of the post-Newtonian coefficients. Most importantly, it is possible to write down, within the present formalism, the general structure of these identical expansions as a consequence of Eq. (53):

$$\overline {{\mathcal M}(h)} = \,\sum {{{\hat n}_L}} {r^m}{(\ln r)^p}{F_{L,m,p}}(t) = {\mathcal M}(\bar h)\,,$$
(104)

where the functions FL,m,p = ∑n⩾1 GnFL,m,p,n. The latter expansion can be interpreted either as the singular re-expansion of the multipole decomposition when r → 0 — i.e., the first equality in Eq. (104) —, or the singular re-expansion of the post-Newtonian series when r → +∞ — the second equality

We recognize the beauty of singular perturbation theory, where two asymptotic expansions, taken formally outside their respective domains of validity, are matched together. Of course, the method works because there exists, physically, an overlapping region in which the two approximation series are expected to be numerically close to the exact solution. As we shall detail in Sections 4.2 and 5.2, the matching equation (103), supplemented by the condition of no-incoming radiation [say in the form of Eq. (29)], permits determining all the unknowns of the problem: On the one hand, the external multipolar decomposition \({\mathcal M}(h)\), i.e., the explicit expressions of the multipole moments therein (see Sections 4.2 and 4.4); on the other hand, the terms in the inner post-Newtonian expansion \(\overline h\) that are associated with radiation-reaction effects, i.e., those terms which depend on the boundary conditions of the radiative field at infinity, and which correspond in the present case to a post-Newtonian source which is isolated from other sources in the Universe; see Section 5.2.

General expression of the multipole expansion

Theorem 5. Under the hypothesis of matching, Eq. (103), the multipole expansion of the solution of the Einstein field equation outside a post-Newtonian source reads

$${\mathcal M}({h^{\alpha \beta}}) = {\mathcal F}{{\mathcal P}_{B = 0}}\,\Box _{{\rm{ret}}}^{- 1}\left[ {{{\tilde r}^B}{\mathcal M}({\Lambda ^{\alpha \beta}})} \right] - {{4G} \over {{c^4}}}\sum\limits_{\ell = 0}^{+ \infty} {{{{{(-)}^\ell}} \over {\ell !}}} {\partial _L}\left\{{{1 \over r}{\mathcal H}_L^{\alpha \beta}(t - r/c)} \right\}\,,$$
(105)

where the “multipole moments” are given by

$${\mathcal H}_L^{\alpha \beta}(u) = {\mathcal F}{{\mathcal P}_{B = 0}}\int {{{\rm{d}}^3}} {\bf{x}}\,{\tilde r^B}{x_L}\,{\bar \tau ^{\alpha \beta}}({\bf{x}},u)\,.$$
(106)

Here, \({\overline \tau ^{\alpha \beta}}\) denotes the post-Newtonian expansion of the stress-energy pseudo-tensor in harmonic coordinates as defined by Eq. (23).

Proof (see Refs. [44, 49]): First notice where the physical restriction of considering a post-Newtonian source enters this theorem: The multipole moments (106) depend on the post-Newtonian expansion \({\overline \tau ^{\alpha \beta}}\) of the pseudo-tensor, rather than on ταβ itself. Consider Δαβ, namely the difference between hαβ, which is a solution of the field equations everywhere inside and outside the source, and the first term in Eq. (105), namely the finite part of the retarded integral of the multipole expansion \({\mathcal M}({\Lambda ^{\alpha \beta}})\):

$${\Delta ^{\alpha \beta}} \equiv {h^{\alpha \beta}} - {\mathcal F}{\mathcal P}\,\Box_{{\rm{ret}}}^{- 1}[{\mathcal M}({\Lambda ^{\alpha \beta}})]\,.$$
(107)

From now on we shall generally abbreviate the symbols concerning the finite-part operation at B = 0 by a mere \({\mathcal F}{\mathcal P}\). According to Eq. (30), hαβ is given by the retarded integral of the pseudotensor ταβ. So,

$${\Delta ^{\alpha \beta}} = {{16\pi G} \over {{c^4}}}\Box_{{\rm{ret}}}^{- 1}{\tau ^{\alpha \beta}} - {\mathcal F}{\mathcal P}\,\Box_{{\rm{ret}}}^{- 1}\left[ {{\mathcal M}({\Lambda ^{\alpha \beta}})} \right]\,.$$
(108)

In the second term the finite part plays a crucial role because the multipole expansion \({\mathcal M}({\Lambda ^{\alpha \beta}})\) is singular at r = 0. By contrast, the first term in Eq. (108), as it stands, is well-defined because we are considering only some smooth field distributions: ταβC(ℝ4). There is no need to include a finite part \({\mathcal F}{\mathcal P}\) in the first term, but a contrario there is no harm to add one in front of it, because for convergent integrals the finite part simply gives back the value of the integral. The advantage of adding artificially the \({\mathcal F}{\mathcal P}\) in the first term is that we can re-write Eq. (108) into the more interesting form

$${\Delta ^{\alpha \beta}} = {{16\pi G} \over {{c^4}}}{\mathcal F}{\mathcal P}\,\Box_{{\rm{ret}}}^{- 1}\left[ {{\tau ^{\alpha \beta}} - {\mathcal M}({\tau ^{\alpha \beta}})} \right]\,,$$
(109)

in which we have also used the fact that \({\mathcal M}({\Lambda ^{\alpha \beta}}) = {{16\pi G} \over {{c^4}}}{\mathcal M}{\rm{(}}{\tau ^{\alpha \beta}}{\rm{)}}\) because Tαβ has a compact support. The interesting point about Eq. (109) is that Δαβ appears now to be the (finite part of a) retarded integral of a source with spatially compact support. This follows from the fact that the pseudo-tensor agrees numerically with its own multipole expansion when r > a [by the same equation as Eq. (102)]. Therefore, \({\mathcal M}({\Delta ^{\alpha \beta}})\) can be obtained from the known formula for the multipole expansion of the retarded solution of a wave equation with compact-support source. This formula, given in Appendix B of Ref. [59], yields the second term in Eq. (105),

$${\mathcal M}({\Delta ^{\alpha \beta}}) = - {{4G} \over {{c^4}}}\sum\limits_{\ell = 0}^{+ \infty} {{{{{(-)}^\ell}} \over {\ell !}}} {\partial _L}\left\{{{1 \over r}{\mathcal H}_L^{\alpha \beta}(u)} \right\}\,,$$
(110)

but in which the moments do not yet match the result (106); instead,Footnote 29

$${\mathcal H}_L^{\alpha \beta} = {\mathcal F}{\mathcal P}\int {{{\rm{d}}^3}} {\bf{x}}\,{x_L}\left[ {{\tau ^{\alpha \beta}} - {\mathcal M}({\tau ^{\alpha \beta}})} \right]\,.$$
(111)

The reason is that we have not yet applied the assumption of a post-Newtonian source. Such sources are entirely covered by their own near zone (i.e., aλ), and, in addition, for them the integral (111) has a compact support limited to the domain of the source. In consequence, we can replace the integrand in Eq. (111) by its post-Newtonian expansion, valid over all the near zone:

$${\mathcal H}_L^{\alpha \beta} = {\mathcal F}{\mathcal P}\int {{{\rm{d}}^3}} {\bf{x}}\,{x_L}[{\bar \tau ^{\alpha \beta}} - \overline {{\mathcal M}({\tau ^{\alpha \beta}})} ]\,.$$
(112)

Strangely enough, we do not get the expected result because of the presence of the second term in Eq. (112). Actually, this term is a bit curious, because the object \(\overline {{\mathcal M}{\rm{(}}{\tau ^{\alpha \beta}}{\rm{)}}}\) it contains is only known in the form of the formal series whose structure is given by the first equality in Eq. (104) (indeed τ and h have the same type of structure). Happily — because we would not know what to do with this term in applications — we are now going to prove that the second term in Eq. (112) is in fact identically zero. The proof is based on the properties of the analytic continuation as applied to the formal structure (104) of \(\overline {{\mathcal M}{\rm{(}}{\tau ^{\alpha \beta}}{\rm{)}}}\). Each term of this series yields a contribution to Eq. (112) that takes the form, after performing the angular integration, of the integral \({\mathcal F}{{\mathcal P}_{{\rm{B = 0}}}}\int\nolimits_0^{+ \infty} {{\rm{d}}r\,{r^{B + b}}{{(\ln \,r)}^p}}\), and multiplied by some function of time. We want to prove that the radial integral \(\int\nolimits_0^{+ \infty} {{\rm{d}}r\,{r^{B + b}}{{(\ln \,r)}^p}}\) is zero by analytic continuation (∀B ∈ ℂ). First we can get rid of the logarithms by considering some repeated differentiations with respect to B; thus we need only to consider the simpler integral \(\int\nolimits_0^{+ \infty} {{\rm{d}}r\,{r^{B + b}}}\). We split the integral into a “near-zone” integral \(\int\nolimits_0^{\mathcal R} {{\rm{d}}r\,{r^{B + b}}}\) and a “far-zone” one \(\int\nolimits_{\mathcal R}^{+ \infty} {{\rm{d}}r\,{r^{B + b}}}\), where \({\mathcal R}\) is some constant radius. When ℜ(B) is a large enough positive number, the value of the near-zone integral is \({{\mathcal R}^{B + b + 1}}/(B + b + 1)\), while when ℜ(B) is a large negative number, the far-zone integral reads the opposite, \(- {{\mathcal R}^{B + b + 1}}/(B + b + 1)\). Both obtained values represent the unique analytic continuations of the near-zone and far-zone integrals for any B ∈ ℂ except − b − 1. The complete integral \(\int\nolimits_0^{+ \infty} {{\rm{d}}r\,{r^{B + b}}}\) is equal to the sum of these analytic continuations, and is therefore identically zero (∀B ∈ ℂ, including the value −b − 1). At last we have completed the proof of Theorem 5:

$${\mathcal H}_L^{\alpha \beta} = {\mathcal F}{\mathcal P}\int {{{\rm{d}}^3}} {\bf{x}}\,{x_L}{\bar \tau ^{\alpha \beta}}\,.$$
(113)

The latter proof makes it clear how crucial the analytic-continuation finite part \({\mathcal F}{\mathcal P}\) is, which we recall is the same as in our iteration of the exterior post-Minkowskian field [see Eq. (45)]. Without a finite part, the multipole moment (113) would be strongly divergent, because the pseudo-tensor \({\overline \tau ^{\alpha \beta}}\) has a non-compact support owing to the contribution of the gravitational field, and the multipolar factor xL behaves like r when r → +∞. The latter divergence has plagued the field of post-Newtonian expansions of gravitational radiation for many years. In applications such as in Part B of this article, we must carefully follow the rules for handling the \({\mathcal F}{\mathcal P}\) operator.

The two terms in the right-hand side of Eq. (105) depend separately on the length scale r0 that we have introduced into the definition of the finite part, through the analytic-continuation factor \({\tilde r^B} = {(r/{r_0})^B}\) introduced in Eq. (42). However, the sum of these two terms, i.e., the exterior multipolar field \({\mathcal M}(h)\) itself, is independent of r0. To see this, the simplest way is to differentiate formally \({\mathcal M}(h)\) with respect to r0; the differentiations of the two terms of Eq. (105) cancel each other. The independence of the field upon r0 is quite useful in applications, since in general many intermediate calculations do depend on r0, and only in the final stage does the cancellation of the r0’s occur. For instance, we have already seen in Eqs. (93)(94) that the source quadrupole moment Iij depends on r0 starting from the 3PN level, but that this r0 is compensated by another r0 coming from the non-linear “tails of tails” at the 3PN order.

Equivalence with the Will-Wiseman formalism

Will & Wiseman [424] (see also Refs. [422, 335]), extending previous work of Epstein & Wagoner [185] and Thorne [403], have obtained a different-looking multipole decomposition, with different definitions for the multipole moments of a post-Newtonian source. They find, instead of our multipole decomposition given by Eq. (105),

$${\mathcal M}({h^{\alpha \beta}}) = \Box_{{\rm{ret}}}^{- 1}{\left[ {{\mathcal M}({\Lambda ^{\alpha \beta}})} \right]_{\mathcal R}} - {{4G} \over {{c^4}}}\sum\limits_{\ell = 0}^{+ \infty} {{{{{(-)}^\ell}} \over {\ell !}}} {\partial _L}\left\{{{1 \over r}{\mathcal W}_L^{\alpha \beta}(t - r/c)} \right\}\,.$$
(114)

There is no \({\mathcal F}{\mathcal P}\) operation in the first term, but instead the retarded integral is truncated, as indicated by the subscript \({\mathcal R}\), to extend only in the “far zone”: i.e., \(|{\rm{x\prime| >}}{\mathcal R}\) in the notation of Eq. (31), where \({\mathcal R}\) is a constant radius enclosing the source \({\rm{(}}{\mathcal R}{\rm{>}}a{\rm{)}}\). The near-zone part of the retarded integral is thereby removed, and there is no problem with the singularity of the multipole expansion \({\mathcal M}({\Lambda ^{\alpha \beta}})\) at the origin. The multipole moments \({{\mathcal W}_L}\) are then given, in contrast with our result (106), by an integral extending over the “near zone” only:

$${\mathcal W}_L^{\alpha \beta}(u) = \int\nolimits_{|{\bf{x}}| < {\mathcal R}} {{{\rm{d}}^3}} {\bf{x}}\,{x_L}\,{\bar \tau ^{\alpha \beta}}({\bf{x}},u)\,.$$
(115)

Since the integrand is compact-supported there is no problem with the bound at infinity and the integral is well-defined (no need of a \({\mathcal F}{\mathcal P}\)).

Let us show that the two different formalisms are equivalent. We compute the difference between our moment \({{\mathcal H}_L}\) defined by Eq. (106), and the moment \({{\mathcal W}_L}\) given by Eq. (115). For the comparison we split \({{\mathcal H}_L}\) into far-zone and near-zone pieces corresponding to the radius \({\mathcal R}\). Since the finite part \({\mathcal F}{\mathcal P}\) present in \({{\mathcal H}_L}\) deals only with the bound at infinity, it can be removed from the near-zone piece, which is then seen to reproduce \({{\mathcal W}_L}\) exactly. So the difference between the two moments is simply given by the far-zone piece:

$${\mathcal H}_L^{\alpha \beta}(u) - {\mathcal W}_L^{\alpha \beta}(u) = {\mathcal F}{\mathcal P}\int\nolimits_{|{\bf{x}}| > {\mathcal R}} {{{\rm{d}}^3}} {\bf{x}}\,{x_L}{\bar \tau ^{\alpha \beta}}({\bf{x}},u)\,.$$
(116)

We transform next this expression. Successively we write \({\overline \tau ^{\alpha \beta}} = {\mathcal M}({\overline \tau ^{\alpha \beta}})\) because we are outside the source, and \({\mathcal M}({\overline \tau ^{\alpha \beta}}) = \overline {{\mathcal M}({\tau ^{\alpha \beta}})}\) thanks to the matching equation (103). At this stage, we recall from our reasoning right after Eq. (112) that the finite part of an integral over the whole space ℓ3 of a quantity having the same structure as \(\overline {{\mathcal M}({\tau ^{\alpha \beta}})}\) is identically zero by analytic continuation. The main ingredient of the present proof is made possible by this fact, as it allows us to transform the far-zone integration \(|{\rm{x| >}}{\mathcal R}\) in Eq. (116) into a near-zone one \(|{\rm{x| <}}{\mathcal R}\), at the price of changing the overall sign in front of the integral. So,

$${\mathcal H}_L^{\alpha \beta}(u) - {\mathcal W}_L^{\alpha \beta}(u) = - {\mathcal F}{\mathcal P}\int\nolimits_{|{\bf{x}}| < {\mathcal R}} {{{\rm{d}}^3}} {\bf{x}}\,{x_L}\overline {{\mathcal M}({\tau ^{\alpha \beta}})} ({\bf{x}},u)\,.$$
(117)

Finally, it is straightforward to check that the right-hand side of this equation, when summed up over all multipolarities , accounts exactly for the near-zone part that was removed from the retarded integral of \({\mathcal M}({\Lambda ^{\alpha \beta}})\) in the first term in Eq. (114), so that the “complete” retarded integral as given by the first term in our own definition (105) is exactly reconstituted. In conclusion, the formalism of Ref. [424] is equivalent to the one of Refs. [44, 49].

The source multipole moments

In principle, the bridge between the exterior gravitational field generated by the post-Newtonian source and its inner field is provided by Theorem 5; however, we still have to make the connection with the explicit construction of the general multipolar and post-Minkowskian metric in Section 2. Namely, we must find the expressions of the six STF source multipole moments IL, JL,…, ZL parametrizing the linearized metric (35)(37) at the basis of that construction.Footnote 30

To do this we first find the equivalent of the multipole expansion given in Theorem 5, which was parametrized by non-trace-free multipole functions \({\mathcal H}_L^{\alpha \beta}\), in terms of new multipole functions \({\mathcal F}_L^{\alpha \beta}\) that are STF in all their indices L. The result is

$${\mathcal M}({h^{\alpha \beta}}) = {\mathcal F}{\mathcal P}\,\Box_{{\rm{ret}}}^{- 1}\left[ {{\mathcal M}({\Lambda ^{\alpha \beta}})} \right] - {{4G} \over {{c^4}}}\sum\limits_{\ell = 0}^{+ \infty} {{{{{(-)}^\ell}} \over {\ell !}}} {\partial _L}\left\{{{1 \over r}{\mathcal F}_L^{\alpha \beta}(t - r/c)} \right\}\,,$$
(118)

where the STF multipole functions (witness the multipolar factor \({\hat x_L} \equiv {\rm STF}[{x_L}]\)) read

$${\mathcal F}_L^{\alpha \beta}(u) = {\mathcal F}{\mathcal P}\int {{{\rm{d}}^3}} {\bf{x}}\,{\hat x_L}\,\int\nolimits_{- 1}^1 {\rm{d}} z\,{\delta _\ell}(z)\,{\bar \tau ^{\alpha \beta}}({\bf{x}},u + zr/c)\,.$$
(119)

Notice the presence of an extra integration variable z, ranging from −1 to 1. The z-integration involves the weighting function

$${\delta _\ell}(z) = {{(2\ell + 1)!!} \over {{2^{\ell + 1}}\ell !}}{(1 - {z^2})^\ell}\,,$$
(120)

which approaches the Dirac delta-function (hence its name) in the limit of large multipolarities, lim→+∞ δ(z) = δ(z), and is normalized in such a way that

$$\int\nolimits_{- 1}^1 d z\,{\delta _\ell}(z) = 1\,.$$
(121)

The next step is to impose the harmonic-gauge conditions (21) onto the multipole decomposition (118), and to decompose the multipole functions \({\mathcal F}_L^{\alpha \beta}(u)\) into STF irreducible pieces with respect to both L and their spatial indices contained into αβ = 00, 0i, ij. This technical part of the calculation is identical to the one of the STF irreducible multipole moments of linearized gravity [154]. The formulas needed in this decomposition read

$${\mathcal F}_L^{00} = {R_L}\,,$$
(122a)
$${\mathcal F}_L^{0i}{= ^{(+)}}{T_{iL}} + {\epsilon _{ai < {i_\ell}}}^{(0)}{T_{L - 1 > a}} + {\delta _{i < {i_\ell}}}^{(-)}{T_{L - 1 >}}\,,$$
(122b)
$${\mathcal F}_L^{ij}{= ^{(+ 2)}}{U_{ijL}} + \underset L {STF} \;\underset {ij} {STF} \,[{\epsilon _{ai{i_\ell}}}^{(+ 1)}{U_{ajL - 1}} + {\delta _{i{i_\ell}}}^{(0)}{U_{jL - 1}}$$
(122c)
$$+ {\delta _{i{i_\ell}}}{\epsilon _{aj{i_{\ell - 1}}}}^{(- 1)}{U_{aL - 2}} + {\delta _{i{i_\ell}}}{\delta _{j{i_{\ell - 1}}}}^{(- 2)}{U_{L - 2}}] + {\delta _{ij}}{V_L}\,,$$
(122d)

where the ten tensors RL, (+)TL+1, …,(−2)UL−2, VL are STF, and are uniquely given in terms of the \({\mathcal F}_L^{\alpha \beta}\)’s by some inverse formulas. Finally, the latter decompositions yield the following.

Theorem 6. The STF multipole moments IL and JL of a post-Newtonian source are given, formally up to any post-Newtonian order, by (ℓ ⩾ 2)

$$\begin{array}{*{20}c} {{{\rm{I}}_L}(u) = {\mathcal F}{\mathcal P}\int {{{\rm{d}}^3}} {\bf{x}}\int\nolimits_{- 1}^1 {\rm{d}} z\left\{{{\delta _\ell}{{\hat x}_L}\Sigma - {{4(2\ell + 1)} \over {{c^2}(\ell + 1)(2\ell + 3)}}{\delta _{\ell + 1}}{{\hat x}_{iL}}\Sigma _i^{(1)}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \right.} \\ {\left. {+ {{2(2\ell + 1)} \over {{c^4}(\ell + 1)(\ell + 2)(2\ell + 5)}}{\delta _{\ell + 2}}{{\hat x}_{ijL}}\Sigma _{ij}^{(2)}} \right\}({\bf{x}},u + zr/c)\,,} \\ \end{array}$$
(123a)
$${{\rm{J}}_L}(u) = {\mathcal F}{\mathcal P}\int {{{\rm{d}}^3}} {\bf{x}}\int\nolimits_{- 1}^1 {\rm{d}} z\,{\epsilon _{ab\langle {i_\ell}}}\left\{{{\delta _\ell}{{\hat x}_{L - 1\rangle a}}{\Sigma _b} - {{2\ell + 1} \over {{c^2}(\ell + 2)(2\ell + 3)}}{\delta _{\ell + 1}}{{\hat x}_{L - 1\rangle ac}}\Sigma _{bc}^{(1)}} \right\}({\bf{x}},u + zr/c)\,.$$
(123b)

These moments are the ones that are to be inserted into the linearized metric \(h_{(1)}^{\alpha \beta}\) that represents the lowest approximation to the post-Minkowskian field \(h_{{\rm{ext}}}^{\alpha \beta} = \sum\nolimits_{n\geqslant1} {{G^n}h_{(n)}^{\alpha \beta}}\) defined in Eq. (50).

In these formulas the notation is as follows: Some convenient source densities are defined from the post-Newtonian expansion (denoted by an overbar) of the pseudo-tensor ταβ by

$$\Sigma = {{{{\bar \tau}^{00}} + {{\bar \tau}^{ii}}} \over {{c^2}}}\,,$$
(124a)
$${\Sigma _i} = {{{{\bar \tau}^{0i}}} \over c}\,,$$
(124b)
$${\Sigma _{ij}} = {\bar \tau ^{ij}}\,,$$
(124c)

(where \({\overline \tau ^{ii}} \equiv {\delta _{ij}}{\overline \tau ^{ij}}\)). As indicated in Eqs. (123) all these quantities are to be evaluated at the spatial point x and at time u + zr/c.

For completeness, we give also the formulas for the four auxiliary source moments WL, …, ZL, which parametrize the gauge vector \(\varphi _1^\alpha\) as defined in Eqs. (37):

$$\begin{array}{*{20}c} {{{\rm{W}}_L}(u) = {\mathcal F}{\mathcal P}\int {{{\rm{d}}^3}} {\bf{x}}\int\nolimits_{- 1}^1 {\rm{d}} z\left\{{{{2\ell + 1} \over {(\ell + 1)(2\ell + 3)}}{\delta _{\ell + 1}}{{\hat x}_{iL}}{\Sigma _i}} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\left. {- {{2\ell + 1} \over {2{c^2}(\ell + 1)(\ell + 2)(2\ell + 5)}}{\delta _{\ell + 2}}{{\hat x}_{ijL}}\Sigma _{ij}^{(1)}} \right\},} \\ \end{array}$$
(125a)
$${{\rm{X}}_L}(u) = {\mathcal F}{\mathcal P}\int {{{\rm{d}}^3}} {\bf{x}}\int\nolimits_{- 1}^1 {\rm{d}} z\left\{{{{2\ell + 1} \over {2(\ell + 1)(\ell + 2)(2\ell + 5)}}{\delta _{\ell + 2}}{{\hat x}_{ijL}}{\Sigma _{ij}}} \right\}\,,$$
(125b)
$$\begin{array}{*{20}c} {{{\rm{Y}}_L}(u) = {\mathcal F}{\mathcal P}\int {{{\rm{d}}^3}} {\bf{x}}\int\nolimits_{- 1}^1 {\rm{d}} z\left\{{- {\delta _\ell}{{\hat x}_L}{\Sigma _{ii}} + {{3(2\ell + 1)} \over {(\ell + 1)(2\ell + 3)}}{\delta _{\ell + 1}}{{\hat x}_{iL}}\Sigma _i^{(1)}} \right.\quad \quad \quad \quad \quad \quad \quad} \\ {\left. {- {{2(2\ell + 1)} \over {{c^2}(\ell + 1)(\ell + 2)(2\ell + 5)}}{\delta _{\ell + 2}}{{\hat x}_{ijL}}\Sigma _{ij}^{(2)}} \right\}\,,} \\ \end{array}$$
(125c)
$${{\rm{Z}}_L}(u) = {\mathcal F}{\mathcal P}\int {{{\rm{d}}^3}} {\bf{x}}\int\nolimits_{- 1}^1 {\rm{d}} z\,{\epsilon _{ab\langle {i_\ell}}}\left\{{- {{2\ell + 1} \over {(\ell + 2)(2\ell + 3)}}{\delta _{\ell + 1}}{{\hat x}_{L - 1\rangle bc}}{\Sigma _{ac}}} \right\}\,.$$
(125d)

As discussed in Section 2, one can always find two intermediate “packages” of multipole moments, namely the canonical moments ML and SL, which are some non-linear functionals of the source moments (123) and (125), and such that the exterior field depends only on them, modulo a change of coordinates. However, the canonical moments ML, SL do not admit general closed-form expressions like (123)(125).Footnote 31

These source moments are physically valid for post-Newtonian sources and make sense only in the form of a post-Newtonian expansion, so in practice we need to know how to expand the z-integrals as series when c → +∞. Here is the appropriate formula:

$$\int\nolimits_{- 1}^1 {\rm{d}} z\,{\delta _\ell}(z)\,\Sigma ({\bf{x}},u + zr/c) = \sum\limits_{k = 0}^{+ \infty} {{{(2\ell + 1)!!} \over {{2^k}k!(2\ell + 2k + 1)!!}}} {\left({{r \over c}{\partial \over {\partial u}}} \right)^{\, 2k}}\,\Sigma ({\bf{x}},u)\,.$$
(126)

Since the right-hand side involves only even powers of 1/c, the same result holds equally well for the advanced variable u + zr/c or the retarded one uzr/c. Of course, in the Newtonian limit, the moments IL and JL (and also ML and SL) reduce to the standard Newtonian expressions. For instance, \({I_{ij}}(u) = {{\rm{Q}}_{ij}}(u) + {\mathcal O}(1/{c^2})\) recovers the Newtonian quadrupole moment (3).Footnote 32

Needless to say, the formalism becomes prohibitively difficult to apply at very high post-Newtonian approximations. Some post-Newtonian order being given, we must first compute the relevant relativistic corrections to the pseudo stress-energy-tensor \({\overline \tau ^{a\beta}}\); this necessitates solving the field equations inside the matter, which we shall investigate in the next Section 5. Then \({\overline \tau ^{a\beta}}\) is to be inserted into the source moments (123) and (125), where the formula (126) permits expressing all the terms up to that post-Newtonian order by means of more tractable integrals extending over ℝ3. Given a specific model for the matter source we then have to find a way to compute all these spatial integrals; this is done in Section 9.1 for the case of point-mass binaries. Next, we must substitute the source multipole moments into the linearized metric (35)(37), and iterate them until all the necessary multipole interactions taking place in the radiative moments UL and VL are under control. In fact, we have already worked out these multipole interactions for general sources in Section 3.3 up to the 3PN order in the full waveform, and 3.5PN order for the dominant (2, 2) mode. Only at this point does one have the physical radiation field at infinity, from which we can build the templates for the detection and analysis of gravitational waves. We advocate here that the complexity of the formalism simply reflects the complexity of the Einstein field equations. It is probably impossible to devise a different formalism, valid for general sources devoid of symmetries, that would be substantially simpler.

Interior Field of a Post-Newtonian Source

Theorem 6 solves in principle the question of the generation of gravitational waves by extended post-Newtonian matter sources. However, notice that this result has still to be completed by the precise procedure, i.e., an explicit “algorithm”, for the post-Newtonian iteration of the near-zone field, analogous to the multipolar-post-Minkowskian algorithm we defined in Section 2. Such procedure will permit the systematic computation of the source multipole moments, which contain the full post-Newtonian expansion of the pseudo-tensor \({\overline \tau ^{a\beta}}\), and of the radiation reaction effects occurring within the matter source.

Before proceeding, let us recall that the “standard” post-Newtonian approximation, as it was used until, say, the early 1980’s (see for instance Refs. [6, 181, 269, 270, 334] and also the earlier works [344, 122, 124, 123]), was plagued with some apparently inherent difficulties, which croped up at some high post-Newtonian order. Historically these difficulties, even appearing at higher approximations, have cast a doubt on the actual soundness, from a theoretical point of view, of the post-Newtonian expansion. Practically speaking, they posed the question of the reliability of the approximation, when comparing the theory’s predictions with very precise experimental results. This was one of the main reason for the famous radiation-reaction controversy raging at the time of the binary pulsar data [182, 418]. In this section we assess the nature of these difficulties — are they purely technical or linked with some fundamental drawback of the approximation scheme? — and eventually resolve them.

  1. 1.

    The first problem we face is that in higher approximations some divergent Poisson-type integrals appear. Indeed the post-Newtonian expansion replaces the resolution of a hyperbolic-like d’Alembertian equation by a perturbatively equivalent hierarchy of elliptic-like Poisson equations. Rapidly it is found during the post-Newtonian iteration that the right-hand side of the Poisson equations acquires a non-compact support (it is distributed all over space ℝ3), and that as a result the standard Poisson integral diverges at the bound of the integral at spatial infinity, i.e., when r ≡ ∣x∣ → +∞, with t = const.

  2. 2.

    The second problem is related with the limitation of the post-Newtonian approximation to the near zone — the region surrounding the source of small extent with respect to the wavelength of the emitted radiation: rλ. As we have seen, the post-Newtonian expansion assumes from the start that all retardations r/c are small, so it can rightly be viewed as a formal near-zone expansion, when r → 0. Note that the fact which makes the Poisson integrals to become typically divergent, namely that the coefficients of the post-Newtonian series blow up at spatial infinity, when r → +∞, has nothing to do with the actual behaviour of the field at infinity. However, the serious consequence is that it is a priori impossible to implement within the post-Newtonian scheme alone the physical information that the matter system is isolated from the rest of the Universe. Most importantly, the no-incoming radiation condition, imposed at past null infinity, cannot be taken directly into account, a priori, into the post-Newtonian scheme. In this sense the post-Newtonian approximation is not “self-supporting”, because it necessitates some information taken from outside its own domain of validity.

The divergencies are linked to the fact that the post-Newtonian expansion is actually a singular perturbation, in the sense that the coefficients of the successive powers of 1/c are not uniformly valid in space, since they typically blow up at spatial infinity like some powers of r. We know for instance that the post-Newtonian expansion cannot be “asymptotically flat” starting at the 2PN or 3PN level, depending on the adopted coordinate system [362]. The result is that the standard Poisson integrals are in general badly-behaving at infinity. Trying to solve the post-Newtonian equations by means of the Poisson integral does not make sense. However, this does not mean that there are no solutions to the problem, but simply that the Poisson integral does not constitute the appropriate solution of the Poisson equation in the context of post-Newtonian expansions.

Here we present, following Refs. [357, 75], a solution of both problems, in the form of a general expression for the near-zone gravitational field, developed to any post-Newtonian order, which has been determined from implementing the matching equation (103). This solution is free of the divergences of Poisson-type integrals we mentioned above, and yields, in particular, some general expression, valid up to any order, of the terms associated with the gravitational radiation reaction force inside the post-Newtonian source.

Though we shall focus our attention on the particular approach advocated in [357, 75], there are other ways to resolve the problems of the post-Newtonian approximation. Notably, an alternative solution to the problem of divergencies, proposed in Refs. [214, 211], is based on an initial-value formulation. In this method the problem of the appearance of divergencies is avoided because of the finiteness of the causal region of integration, between the initial Cauchy hypersurface and the considered field point. On the other hand, a different approach to the problem of radiation reaction, which does not use a matching procedure, is to work only within a post-Minkowskian iteration scheme without expanding the retardations, see e.g., Ref. [126].

Post-Newtonian iteration in the near zone

We perform the post-Newtonian iteration of the field equations in harmonic coordinates in the near zone of an isolated matter distribution. We deal with a general hydrodynamical fluid, whose stress-energy tensor is smooth, i.e., TαβC(ℝ4). Thus the scheme a priori excludes the presence of singularities and black holes; these will be dealt with in Part B of this article.

We shall now prove [357] that the post-Newtonian expansion can be indefinitely iterated without divergences. Like in Eq. (106) we denote by means of an overline the formal (infinite) post-Newtonian expansion of the field inside the source’s near-zone. The general structure of the post-Newtonian expansion is denoted (skipping the space-time indices αβ) as

$$\bar h({\bf{x}},t,c) = \sum\limits_{m = 2}^{ + \infty } {{1 \over {{c^m}}}{\mkern 1mu} \mathop {\bar h}\limits_m ({\bf{x}},t;\ln c){\mkern 1mu} } .$$
(127)

The m-th post-Newtonian coefficient is naturally the factor of the m-th power of 1/c. However, we know from restoring the factors c’s in Theorem 3 [see Eq. (53)], that the post-Newtonian expansion also involves powers of the logarithm of c; these are included for convenience here into the definition of the coefficients \({\overline h _m}\).Footnote 33 For the stress-energy pseudo-tensor appearing in Eq. (106) we have the same type of expansion,

$$\bar \tau ({\bf{x}},t,c) = \sum\limits_{m = - 2}^{ + \infty } {{1 \over {{c^m}}}{\mkern 1mu} \mathop {\bar \tau }\limits_m ({\bf{x}},t;\ln c){\mkern 1mu} .}$$
(128)

Note that the expansion starts with a term of order c2 corresponding to the rest mass-energy (\(\overline \tau\) has the dimension of an energy density). As usual we shall understand the infinite sums such as (127)(128) in the sense of formal series, i.e., merely as an ordered collection of coefficients. Because of our consideration of regular extended matter distributions the post-Newtonian coefficients are smooth functions of space-time, i.e., \({\bar h_m}({\rm{x}},t) \in {C^\infty}({\mathbb R^4})\).

Inserting the post-Newtonian ansatz (127) into the harmonic-coordinates Einstein field equation (21)(22) and equating together the powers of 1/c, results is an infinite set of Poisson-type equations (∀m ⩾ 2),

$$\Delta \mathop {\bar h}\limits_m = 16\pi G{\mkern 1mu} {\mkern 1mu} \mathop {\bar \tau }\limits_{m - 4} + {\mkern 1mu} \partial _t^2{\mkern 1mu} {\mkern 1mu} \mathop {\bar h}\limits_{m - 2} {\mkern 1mu} ,$$
(129)

where the second term comes from the split of the d’Alembertian operator into a Laplacian and a second time derivative: \(\square = \Delta - {1 \over {{c^2}}}\partial _t^2\) (this term is zero when m = 2 and 3). We proceed by induction, i.e., we work at some given but arbitrary post-Newtonian order m, assume that we succeeded in constructing the sequence of previous coefficients \({\overline h _p}(\forall p\geqslant {m - 1})\), and from that show how to infer the next-order coefficient \({\overline h _m}\).

To cure the problem of divergencies we introduce a generalized solution of the Poisson equation with non-compact support source, in the form of an appropriate finite part of the usual Poisson integral obtained by regularization of the bound at infinity by means of a specific process of analytic continuation. For any source term like \({\overline \tau _m}\), we multiply it by the regularization factor \({\tilde r^B}\) already extensively used in the construction of the exterior field, thus B ∈ ℂ and \(\tilde r = r/{r_0}\) is given by Eq. (42). Only then do we apply the usual Poisson integral, which therefore defines a certain function of B. The well-definedness of that integral heavily relies on the behaviour of the integrand at the bound at infinity. There is no problem with the vicinity of the origin inside the source because of the smoothness of the pseudo-tensor. Then one can prove [357] that the latter function of B generates a (unique) analytic continuation down to a neighbourhood of the value of interest B = 0, except at B = 0 itself, at which value it admits a Laurent expansion with multiple poles up to some finite order (but growing with the post-Newtonian order m). Then, we consider the Laurent expansion of that function when B → 0 and pick up the finite part, or coefficient of the zero-th power of B, of that expansion. This defines our generalized Poisson integral:

$$\Delta^{-1}[\mathop{\bar\tau}\limits_m](x,t)\equiv-\frac{1}{4\pi}\mathcal{FP}_{B=0}\int\frac{d^{3}x'}{|x-x'|}{\tilde r}'^{B} \mathop{\bar\tau}\limits_{m}(x',t).$$
(130)

The integral extends over all three-dimensional space but with the latter finite-part regularization at infinity denoted \({\mathcal F}{{\mathcal P}_{B = 0}}\) or \({\mathcal F}{\mathcal P}\) for short. The main properties of this generalized Poisson operator is that it solves the Poisson equation,

$$\Delta \left({{\Delta ^{- 1}}\left[ {\underset m {\bar \tau}} \right]} \right) = \underset m {\bar \tau} \,,$$
(131)

and that the solution \({\Delta ^{- 1}}{\overline \tau _m}\) owns the same properties as its source \({\overline \tau _m}\), i.e., the smoothness and the same type of behaviour at infinity, as given by Eq. (104). Similarly, we define the generalized iterated Poisson integral as

$${\Delta ^{- k - 1}}\left[ {\underset m {\bar \tau}} \right]({\bf{x}},t) \equiv - {1 \over {4\pi}}\,{\mathcal F}{{\mathcal P}_{B = 0}}\,\int {{{\rm{d}}^3}} {\bf{x\prime}}\,{{|{\bf{x}} - {\bf{x\prime}}{|^{2k - 1}}} \over {(2k)!}}\,{\tilde r\prime^B}\,\underset m {\bar \tau} ({\bf{x\prime}},t)\,.$$
(132)

The most general solution of the Poisson equation will be obtained by application of the previous generalized Poisson operator to the right-hand side of Eq. (129), and augmented by the most general homogeneous solution of the Poisson equation. Thus, we can write

$$\underset m {\bar h} = 16\pi G\,{\Delta ^{- 1}}\left[ {\underset {m - 4} {\bar \tau}} \right] + \partial _t^2\,{\Delta ^{- 1}}\left[ {\underset {m - 2} {\bar h}} \right] + \sum\limits_{\ell = 0}^{+ \infty} {\underset m {\mathcal B} {\,_L}(t)\,{{\hat x}_L}} \,.$$
(133)

The last term represents the most general solution of the Laplace equation that is regular at the origin r = 0. It can be written in STF guise as a multipolar series of terms of the type \({\hat x_L}\), and multiplied by arbitrary STF-tensorial functions of time \({}_m{{\mathcal B}_L}(t)\). These functions will be associated with the radiation reaction of the field onto the source; they will depend on which boundary conditions are to be imposed on the gravitational field at infinity from the source.

It is now trivial to iterate the process. We substitute for \({\overline h _{m - 2}}\) in the right-hand side of Eq. (133) the same expression but with m replaced by m − 2, and similarly come down until we stop at either one of the coefficients \({\overline h _0} = 0\) or \({\overline h _1} = 0\). At this point \({\overline h _m}\) is expressed in terms of the previous \({\overline \tau _p}\)’s and \({}_p{{\mathcal B}_L}\)’s with pm − 2. To finalize the process we introduce what we call the operator of the “instantaneous” potentials and denote \(\square_{{\rm{inst}}}^{- 1}\). Our notation is chosen to contrast with the standard operator of the retarded potentials \(\square_{{\rm{ret}}}^{- 1}\) defined by Eq. (31). However, beware of the fact that unlike \(\square_{{\rm{ret}}}^{- 1}\) the operator \(\square_{{\rm{inst}}}^{- 1}\) will be defined only when acting on a post-Newtonian series such as \(\overline \tau\). Indeed, we pose

$$\Box_{{\rm{inst}}}^{- 1}\left[ {\bar \tau} \right] \equiv \sum\limits_{k = 0}^{+ \infty} {{{\left({{\partial \over {c\partial t}}} \right)}^{2k}}} {\Delta ^{- k - 1}}\left[ {\bar \tau} \right]\,,$$
(134)

where the k-th iteration of the generalized Poisson operator is defined by Eq. (132). This operator is instantaneous in the sense that it does not involve any integration over time. It is readily checked that in this way we have a solution of the source-free d’Alembertian equation,

$$\Box\left({\Box_{{\rm{inst}}}^{- 1}\left[ {\bar \tau} \right]} \right) = \bar \tau \,.$$
(135)

On the other hand, the homogeneous solution in Eq. (133) will yield by iteration an homogeneous solution of the d’Alembertian equation that is necessarily regular at the origin. Hence it should be of the anti-symmetric type, i.e., be made of the difference between a retarded multipolar wave and the corresponding advanced wave. We shall therefore introduce a new definition for some STF-tensorial functions \({{\mathcal A}_L}(t)\) parametrizing those advanced-minus-retarded free waves. It is very easy to relate if necessary the post-Newtonian expansion of \({{\mathcal A}_L}(t)\) to the functions \({}_m{{\mathcal B}_L}(t)\) previously introduced in Eq. (133). Finally the most general post-Newtonian solution, iterated ad infinitum and without any divergences, is obtained into the form

$$\bar h = {{16\pi G} \over {{c^4}}}\,\Box_{{\rm{inst}}}^{- 1}\left[ {\bar \tau} \right] - {{4G} \over {{c^4}}}\sum\limits_{\ell = 0}^{+ \infty} {{{{{(-)}^\ell}} \over {\ell !}}} {\hat \partial _L}\left\{{{{{{\mathcal A}_L}(t - r/c) - {{\mathcal A}_L}(t + r/c)} \over {2r}}} \right\}\,.$$
(136)

We shall refer to the \({{\mathcal A}_L}(t)\)’s as the radiation-reaction functions. If we stay at the level of the post-Newtonian iteration which is confined into the near zone we cannot do more than Eq. (136): There is no means to compute the radiation-reaction functions \({{\mathcal A}_L}(t)\). We are here touching the second problem faced by the standard post-Newtonian approximation.

Post-Newtonian metric and radiation reaction effects

As we have understood this problem is that of the limitation to the near zone. Such limitation can be circumvented to the lowest post-Newtonian orders by considering retarded integrals that are formally expanded when c → +∞ as series of “instantaneous” Poisson-like integrals, see e.g., [6]. This procedure works well up to the 2.5PN level and has been shown to correctly fix the dominant radiation reaction term at the 2.5PN order [181, 269, 270, 334]. Unfortunately such a procedure assumes fundamentally that the gravitational field, after expansion of all retardations r/c → 0, depends on the state of the source at a single time t, in keeping with the instantaneous character of the Newtonian interaction. However, we know that the post-Newtonian field (as well as the source’s dynamics) will cease at some stage to be given by a functional of the source parameters at a single time, because of the imprint of gravitational-wave tails in the near zone field, in the form of the hereditary modification of the radiation reaction force at the 1.5PN relative order [58, 60, 43]. Since the reaction force is itself of order 2.5PN this means that the formal post-Newtonian expansion of retarded Green functions is no longer valid starting at the 4PN order.

The solution of the problem resides in the matching of the near-zone field to the exterior field. We have already seen in Theorems 5 and 6 that the matching equation (103) yields the expression of the multipole expansion in the exterior domain. Now we prove that it also permits the full determinantion of the post-Newtonian metric in the near-zone, i.e., the radiation-reaction functions \({{\mathcal A}_L}\) which have been left unspecified in Eq. (136).

We find [357] that the radiation-reaction functions \({{\mathcal A}_L}\) are composed of the multipole moment functions \({{\mathcal F}_L}\) defined by Eq. (119), which will here characterize “linear-order” radiation reaction effects starting at 2.5PN order, and of an extra piece \({{\mathcal R}_L}\), which will be due to non-linear effets in the radiation reaction and turn out to arise at the 4PN order. Thus,

$${{\mathcal A}_L}(t) = {{\mathcal F}_L}(t) + {{\mathcal R}_L}(t)\,.$$
(137)

The extra piece \({{\mathcal R}_L}\) is obtained from the multipole expansion of the pseudo-tensor \({\mathcal M}{\rm{(}}\tau {\rm{)}}\).Footnote 34 Hence the radiation-reaction functions do depend on the behaviour of the field far away from the matter source (as physical intuition already told us). The explicit expression reads

$${{\mathcal R}_L}(t) = {\mathcal F}{\mathcal P}\,\int \, {{\rm{d}}^3}{\bf{x}}\,{\hat x_L}\int\nolimits_1^{+ \infty} {\rm{d}} z\,{\gamma _\ell}(z)\,{\mathcal M}(\tau)\,\left({{\bf{x}},t - zr/c} \right)\,.$$
(138)

The fact that the multipolar expansion \({\mathcal M}{\rm{(}}\tau {\rm{)}}\) is the source term for the function \({{\mathcal R}_L}\) is the consequence of the matching equation (103). The specific contributions due to \({{\mathcal R}_L}\) in the post-Newtonian metric (136) are associated with tails of waves [58, 43]. Notice that, remarkably, the \({\mathcal F}{\mathcal P}\) regularization deals with the bound of the integral at r = 0, in contrast with Eq. (119) where it deals with the bound at r = +∞. The weighting function γ(z) therein, where z extends up to infinity in contrast to the analogous function δ(z) in Eq. (119), is simply related to it by γ(z) ≡ −2δ(z); such definition is motivated by the fact that the integral of that function is normalized to one:Footnote 35

$$\int\nolimits_1^{+ \infty} {\rm{d}} z\,{\gamma _\ell}(z) = 1\,.$$
(139)

The post-Newtonian metric (136) is now fully determined. However, let us now prove a more interesting alternative formulation of it, derived in Ref. [75].

Theorem 7. The expression of the post-Newtonian field in the near zone of a post-Newtonian source, satisfying correct boundary conditions at infinity (no incoming radiation), reads

$${\bar h^{\alpha \beta}} = {{16\pi G} \over {{c^4}}}\,\Box_{{\rm{ret}}}^{- 1}\left[ {{{\bar \tau}^{\alpha \beta}}} \right] - {{4G} \over {{c^4}}}\sum\limits_{\ell = 0}^{+ \infty} {{{{{(-)}^\ell}} \over {\ell !}}} {\hat \partial _L}\left\{{{{{\mathcal R}_L^{\alpha \beta}(t - r/c) - {\mathcal R}_L^{\alpha \beta}(t + r/c)} \over {2r}}} \right\}\,.$$
(140)

The first term represents a particular solution of the hierarchy of post-Newtonian equations, while the second one is a homogeneous multipolar solution of the wave equation, of the “anti-symmetric” type that is regular at the origin r = 0 located inside the source, and parametrized by the multipole-moment functions (138).

Let us be more precise about the meaning of the first term in Eq. (140). Indeed such term is made of the formal expansion of the standard retarded integral (31) when c → ∞, but acting on a post-Newtonian source term \(\bar \tau\),

$$\Box_{{\rm{ret}}}^{- 1}\left[ {\,{{\bar \tau}^{\alpha \beta}}} \right]({\bf{x}},t) \equiv - {1 \over {4\pi}}\sum\limits_{m = 0}^{+ \infty} {{{{{(-)}^m}} \over {m!}}} {\left({{\partial \over {c\,\partial t}}} \right)^{\,m}}\,{\mathcal F}{\mathcal P}\,\int {{{\rm{d}}^3}} {\bf{x\prime}}\,|{\bf{x}} - {\bf{x\prime}}{|^{m - 1}}\,{\bar \tau ^{\alpha \beta}}({\bf{x\prime}},t)\,.$$
(141)

We emphasize that (141) constitutes the definition of a (formal) post-Newtonian expansion, each term of which being built from the post-Newtonian expansion of the pseudo-tensor. Crucial in the present formalism, is that each of the terms is regularized by means of the \({\mathcal F}{\mathcal P}\) operation in order to deal with the bound at infinity at which the post-Newtonian expansion is singular. Because of the presence of this regularization, the object (141) should carefully be distinguished from the “global” solution \(\square_{{\rm{ret}}}^{- 1}[\tau ]\) defined by Eq. (31), with global non-expanded pseudo-tensor τ.

The definition (141) is of interest because it corresponds to what one would intuitively think as the natural way of performing the post-Newtonian iteration, i.e., by formally Taylor expanding the retardations in Eq. (31), as was advocated by Anderson & DeCanio [6]. Moreover, each of the terms of the series (141) is mathematically well-defined thanks to the finite part operation, and can therefore be implemented in practical computations. The point is that Eq. (141) solves the wave equation in a perturbative post-Newtonian sense,

$$\Box\left({\Box_{{\rm{ret}}}^{- 1}\left[ {{{\bar \tau}^{\alpha \beta}}} \right]} \right) = {\bar \tau ^{\alpha \beta}}\,,$$
(142)

so constitutes a good prescription for a particular solution of the wave equation — as legitimate as the solution (134). Therefore the two solutions should differ by an homogeneous solution of the wave equation which is necessarily of the anti-symmetric type (regular inside the source). Detailed investigations [357, 75] yield

$$\Box_{{\rm{ret}}}^{- 1}\left[ {{{\bar \tau}^{\alpha \beta}}} \right] = \Box_{{\rm{inst}}}^{- 1}\left[ {{{\bar \tau}^{\alpha \beta}}} \right] - {1 \over {4\pi}}\sum\limits_{\ell = 0}^{+ \infty} {{{{{(-)}^\ell}} \over {\ell !}}} {\hat \partial _L}\left\{{{{{\mathcal F}_L^{\alpha \beta}(t - r/c) - {\mathcal F}_L^{\alpha \beta}(t + r/c)} \over {2r}}} \right\}\,,$$
(143)

where the homogeneous solution is parametrized by the multipole-moments \({{\mathcal F}_L}(t)\). By combining Eqs. (140) and (143), we indeed become reconciled with the previous expression of the post-Newtonian field found in Eq. (136).

For computations limited to the 3.5PN order (level of the 1PN correction to the radiation reaction force), the first term in Eq. (140) with the “intuitive” prescription (141) is sufficient. But because of the second term in (140) there is a fundamental breakdown of this scheme at the 4PN order where it becomes necessary to take into account non-linear radiation reaction effects associated with tails. The second term in (140) constitutes a generalization of the tail-transported radiation reaction arising at the 4PN order, i.e., 1.5PN order relative to the dominant radiation reaction order, as determined in Ref. [58]. The tail-transported radiation reaction is required by energy conservation and the presence of tails in the wave zone. The usual radiation reaction terms, up to 3.5PN order, are contained in the first term of Eq. (140), and are parametrized by the same multipole-moment functions \({{\mathcal F}_L}\) as the exterior multipolar field, as Eq. (143) explicitly shows. In Section 5.4 we shall give an explicit expression of the radiation reaction force showing the usual radiation reaction terms to 3.5PN order, issued from \({{\mathcal F}_L}\), and exhibiting the above tail-induced 4PN effect, issued from \({{\mathcal R}_L}\).

Finally note that the post-Newtonian solution, in either form (136) or (140), has been obtained without imposing the condition of harmonic coordinates (21) in an explicit way. We have simply matched together the post-Newtonian and multipolar expansions, satisfying the “relaxed” Einstein field equations (22) in their respective domains, and found that the matching determines uniquely the solution. An important check done in [357, 75], is therefore to verify that the harmonic coordinate condition (21) is indeed satisfied as a consequence of the conservation of the pseudo-tensor (27), so that we really grasp a solution of the full Einstein field equations.

The 3.5PN metric for general matter systems

The detailed calculations that are called for in applications necessitate having at one’s disposal some explicit expressions of the metric coefficients gαβ, in harmonic coordinates, at the highest possible post-Newtonian order. The 3.5PN metric that we present below can be viewed as an application of the formalism of the previous section. It is expressed by means of some particular retarded-type potentials, V, Vi, \({\hat W_{ij}}\), …, whose main advantages are to somewhat minimize the number of terms, so that even at the 3.5PN order the metric is still tractable, and to delineate the different problems associated with the computation of different categories of terms. Of course, these potentials have no direct physical significance by themselves, but they offer a convenient parametrization of the 3.5PN metric.

The basic idea in our post-Newtonian iteration scheme is to use wherever possible a “direct” integration, with the help of some formulas like \(\square_{{\rm{ret}}}^{- 1}({\partial _\mu}V{\partial ^\mu}V + V\square{V}) = {V^2}/2\). The 3.5PN harmonic-coordinates metric reads [71]

$$\begin{array}{*{20}c} {{g_{00}} = - 1 + {2 \over {{c^2}}}V - {2 \over {{c^4}}}{V^2} + {8 \over {{c^6}}}\left({\hat X + {V_i}{V_i} + {{{V^3}} \over 6}} \right)\quad \quad \quad \quad \quad \quad} \\ {+ {{32} \over {{c^8}}}\left({\hat T - {1 \over 2}V\hat X + {{\hat R}_i}{V_i} - {1 \over 2}V{V_i}{V_i} - {1 \over {48}}{V^4}} \right) + {\mathcal O}\left({{1 \over {{c^{10}}}}} \right)\,,} \\ \end{array}$$
(144a)
$${g_{0i}} = - {4 \over {{c^3}}}{V_i} - {8 \over {{c^5}}}{\hat R_i} - {{16} \over {{c^7}}}\left({{{\hat Y}_i} + {1 \over 2}{{\hat W}_{ij}}{V_j} + {1 \over 2}{V^2}{V_i}} \right) + {\mathcal O}\left({{1 \over {{c^9}}}} \right)\,,$$
(144b)
$$\begin{array}{*{20}c} {{g_{ij}} = {\delta _{ij}}\left[ {1 + {2 \over {{c^2}}}V + {2 \over {{c^4}}}{V^2} + {8 \over {{c^6}}}\left({\hat X + {V_k}{V_k} + {{{V^3}} \over 6}} \right)} \right] + {4 \over {{c^4}}}{{\hat W}_{ij}}} \\ {+ {{16} \over {{c^6}}}\left({{{\hat Z}_{ij}} + {1 \over 2}V{{\hat W}_{ij}} - {V_i}{V_j}} \right) + {\mathcal O}\left({{1 \over {{c^8}}}} \right)\,.\quad \quad \quad \quad \quad} \\ \end{array}$$
(144c)

All the potentials are generated by the matter stress-energy tensor Tαβ through some convenient definitions recalling Eqs. (124),

$$\sigma = {{{T^{00}} + {T^{ii}}} \over {{c^2}}}\,,$$
(145a)
$${\sigma _i} = {{{T^{0i}}} \over c}\,,$$
(145b)
$${\sigma _{ij}} = {T^{ij}}\,.$$
(145c)

Starting at Newtonian and 1PN orders, V and Vi represent some retarded versions of the usual Newtonian and gravitomagnetic potentials,

$$V = \Box_{{\rm{ret}}}^{- 1}\left[ {- 4\pi G\sigma} \right]\,,$$
(146a)
$${V_i} = \Box_{{\rm{ret}}}^{- 1}\left[ {- 4\pi G{\sigma _i}} \right]\,.$$
(146b)

From the 2PN order we have the potentials

$$\hat X = \Box_{{\rm{ret}}}^{- 1}\left[ {- 4\pi GV{\sigma _{ii}} + {{\hat W}_{ij}}{\partial _{ij}}V + 2{V_i}{\partial _t}{\partial _i}V + V\partial _t^2V + {3 \over 2}{{({\partial _t}V)}^2} - 2{\partial _i}{V_j}{\partial _j}{V_i}} \right]\,,$$
(147a)
$${\hat R_i} = \Box_{{\rm{ret}}}^{- 1}\left[ {- 4\pi G(V{\sigma _i} - {V_i}\sigma) - 2{\partial _k}V{\partial _i}{V_k} - {3 \over 2}{\partial _t}V{\partial _i}V} \right],$$
(147b)
$${\hat W_{ij}} = \Box_{{\rm{ret}}}^{- 1}\left[ {- 4\pi G({\sigma _{ij}} - {\delta _{ij}}{\sigma _{kk}}) - {\partial _i}V{\partial _j}V} \right]\,.$$
(147c)

Some parts of these potentials are directly generated by compact-support matter terms, while other parts are made of non-compact-support products of V-type potentials. There exists also an important cubically non-linear term generated by the coupling between \({\hat W_{ij}}\) and V, see the second term in the \(\hat X\)-potential. Note the important point that here and below the retarded integral operator \(\square_{{\rm{ret}}}^{- 1}\) is really meant to be the one given by Eq. (141); thus it involves in principle the finite part regularization \({\mathcal F}{\mathcal P}\) to deal with (IR-type) divergences occurring at high post-Newtonian orders for non-compact-support integrals. For instance, such finite part regularization is important to take into account in the computation of the near zone metric at the 3PN order [68].

At the next level, 3PN, we have even more complicated potentials, namely

$$\begin{array}{*{20}c} {\hat T = \Box_{{\rm{ret}}}^{- 1}\left[ {- 4\pi G\left({{1 \over 4}{\sigma _{ij}}{{\hat W}_{ij}} + {1 \over 2}{V^2}{\sigma _{ii}} + \sigma {V_i}{V_i}} \right) + {{\hat Z}_{ij}}{\partial _{ij}}V + {{\hat R}_i}{\partial _t}{\partial _i}V - 2{\partial _i}{V_j}{\partial _j}{{\hat R}_i} - {\partial _i}{V_j}{\partial _t}{{\hat W}_{ij}}} \right.} \\ {\left. {+ V{V_i}{\partial _t}{\partial _i}V + 2{V_i}{\partial _j}{V_i}{\partial _j}V + {3 \over 2}{V_i}{\partial _t}V{\partial _i}V + {1 \over 2}{V^2}\partial _t^2V + {3 \over 2}V{{({\partial _t}V)}^2} - {1 \over 2}{{({\partial _t}{V_i})}^2}} \right],} \\ \end{array}$$
(148a)
$$\begin{array}{*{20}c} {{{\hat Y}_i} = \Box_{{\rm{ret}}}^{- 1}\left[ {- 4\pi G\left({- \sigma {{\hat R}_i} - \sigma V{V_i} + {1 \over 2}{\sigma _k}{{\hat W}_{ik}} + {1 \over 2}{\sigma _{ik}}{V_k} + {1 \over 2}{\sigma _{kk}}{V_i}} \right)} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ \,{{\hat W}_{kl}}{\partial _{kl}}{V_i} - {\partial _t}{{\hat W}_{ik}}{\partial _k}V + {\partial _i}{{\hat W}_{kl}}{\partial _k}{V_l} - {\partial _k}{{\hat W}_{il}}{\partial _l}{V_k} - 2{\partial _k}V{\partial _i}{{\hat R}_k} - {3 \over 2}{V_k}{\partial _i}V{\partial _k}V} \\ {\left. {- \,{3 \over 2}V{\partial _t}V{\partial _i}V - 2V{\partial _k}V{\partial _k}{V_i} + V\partial _t^2{V_i} + 2{V_k}{\partial _k}{\partial _t}{V_i}} \right],\quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(148b)
$$\begin{array}{*{20}c} {{{\hat Z}_{ij}} = \Box_{{\rm{ret}}}^{- 1}\left[ {- 4\pi GV\left({{\sigma _{ij}} - {\delta _{ij}}{\sigma _{kk}}} \right) - 2{\partial _{(i}}V{\partial _t}{V_{j)}} + {\partial _i}{V_k}{\partial _j}{V_k} + {\partial _k}{V_i}{\partial _k}{V_j} - 2{\partial _{(i}}{V_k}{\partial _k}{V_{j)}}} \right.} \\ {\left. {- {3 \over 4}{\delta _{ij}}{{({\partial _t}V)}^2} - {\delta _{ij}}{\partial _k}{V_m}({\partial _k}{V_m} - {\partial _m}{V_k})} \right].\quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(148c)

These involve many types of compact-support contributions, as well as quadratic-order and cubic-order parts; but, surprisingly, there are no quartically non-linear terms. Indeed it has been possible to “integrate directly” all the quartic contributions in the 3PN metric; see the terms composed of V4 and \(V\,\hat X\) in the first of Eqs. (144).

Note that the 3PN metric (144) does represent the inner post-Newtonian field of an isolated system, because it contains, to this order, the correct radiation-reaction terms corresponding to outgoing radiation. These terms come from the expansions of the retardations in the retarded potentials (146)(148); we elaborate more on radiation-reaction effects in the next Section 5.4.

The above potentials are not independent: They are linked together by some differential identities issued from the harmonic gauge conditions, which are equivalent, via the Bianchi identities, to the equations of motion of the matter fields; see Eq. (27). These identities read

$$\begin{array}{*{20}c} {{\partial _t}\left\{{V + {1 \over {{c^2}}}\left[ {{1 \over 2}{{\hat W}_{kk}} + 2{V^2}} \right] + {4 \over {{c^4}}}\left[ {\hat X + {1 \over 2}{{\hat Z}_{kk}} + {1 \over 2}V{{\hat W}_{kk}} + {2 \over 3}{V^3}} \right]} \right\}\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {\partial _i}\left\{{{V_i} + {2 \over {{c^2}}}\left[ {{{\hat R}_i} + V{V_i}} \right] + {4 \over {{c^4}}}\left[ {{{\hat Y}_i} - {1 \over 2}{{\hat W}_{ij}}{V_j} + {1 \over 2}{{\hat W}_{kk}}{V_i} + V{{\hat R}_i} + {V^2}{V_i}} \right]} \right\} = {\mathcal O}\left({{1 \over {{c^6}}}} \right)\,,} \\ \end{array}$$
(149a)
$${\partial _t}\left\{{{V_i} + {2 \over {{c^2}}}\left[ {{{\hat R}_i} + V{V_i}} \right]} \right\} + {\partial _j}\left\{{{{\hat W}_{ij}} - {1 \over 2}{{\hat W}_{kk}}{\delta _{ij}} + {4 \over {{c^2}}}\left[ {{{\hat Z}_{ij}} - {1 \over 2}{{\hat Z}_{kk}}{\delta _{ij}}} \right]} \right\} = {\mathcal O}\left({{1 \over {{c^4}}}} \right)\,.$$
(149b)

For latter applications to systems of compact objects, let us give the geodesic equations of a particle moving in the 3.5PN metric (144).Footnote 36 It is convenient to write these equations as

$${{{\rm{d}}{P^i}} \over {{\rm{d}}t}} = {F^i}\,,$$
(150)

where the “linear momentum density” Pi and the “force density” Fi of the particle are given by

$${P^i} = c{{{g_{i\mu}}{v^\mu}} \over {\sqrt {- {g_{\rho \sigma}}{v^\rho}{v^\sigma}}}}\,,$$
(151a)
$${F^i} = {c \over 2}{{{\partial _i}{g_{\mu \nu}}{v^\mu}{v^\nu}} \over {\sqrt {- {g_{\rho \sigma}}{v^\rho}{v^\sigma}}}}\,,$$
(151b)

where vμ = (c, vi) with vi = dxi/dt being the particle’s ordinary coordinate velocity, and where the metric components are taken at the location of the particle. Notice that we are here viewing the particle as moving in the fixed background metric (144). In Part B of this article, the metric will be generated by the system of particles itself, and we shall have to supplement the computation of the metric at the location of one of these particles by a suitable self-field regularization.

The expressions of both Pi and Fi in terms of the non-linear potentials follow from insertion of the 3.5PN metric coefficients (144). We obtain some complicated-looking (but useful in applications) sums of products of potentials given by

$$\begin{array}{*{20}c} {{P^i} = {v^i} + {1 \over {{c^2}}}\left({{1 \over 2}{v^2}{v^i} + 3V{v^i} - 4{V_i}} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {1 \over {{c^4}}}\left({{3 \over 8}{v^4}{v^i} + {7 \over 2}V{v^2}{v^i} - 4{V_j}{v^i}{v^j} - 2{V_i}{v^2} + {9 \over 2}{V^2}{v^i} - 4V{V_i} + 4{{\hat W}_{ij}}{v^j} - 8{{\hat R}_i}} \right)} \\ {+ {1 \over {{c^6}}}\left({{5 \over {16}}{v^6}{v^i} + {{33} \over 8}V{v^4}{v^i} - {3 \over 2}{V_i}{v^4} - 6{V_j}{v^i}{v^j}{v^2} + {{49} \over 4}{V^2}{v^2}{v^i} + 2{{\hat W}_{ij}}{v^j}{v^2}\quad \,\,} \right.} \\ {\qquad + 2{{\hat W}_{jk}}{v^i}{v^j}{v^k} - 10V{V_i}{v^2} - 20V{V_j}{v^i}{v^j} - 4{{\hat R}_i}{v^2} - 8{{\hat R}_j}{v^i}{v^j} + {9 \over 2}{V^3}{v^i} + 12{V_j}{V_j}{v^i}\quad} \\ {\left. {\qquad + 12{{\hat W}_{ij}}V{v^j} + 12\hat X{v^i} + 16{{\hat Z}_{ij}}{v^j} - 10{V^2}{V_i} - 8{{\hat W}_{ij}}{V_j} - 8V{{\hat R}_i} - 16{{\hat Y}_i}} \right) + {\mathcal O}\left({{1 \over {{c^8}}}} \right)\,,\quad \quad} \\ \end{array}$$
(152a)
$$\begin{array}{*{20}c} {{F^i} = {\partial _i}V + {1 \over {{c^2}}}\left({- V{\partial _i}V + {3 \over 2}{\partial _i}V{v^2} - 4{\partial _i}{V_j}{v^j}} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {1 \over {{c^4}}}\left({{7 \over 8}{\partial _i}V{v^4} - 2{\partial _i}{V_j}{v^j}{v^2} + {9 \over 2}V{\partial _i}V{v^2} + 2{\partial _i}{{\hat W}_{jk}}{v^j}{v^k}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \right.} \\ {\left. {- 4{V_j}{\partial _i}V{v^j} - 4V\partial {V_j}{v^j} - 8{\partial _i}{{\hat R}_j}{v^j} + {1 \over 2}{V^2}{\partial _i}V + 8{V_j}{\partial _i}{V_j} + 4{\partial _i}\hat X} \right)\quad \quad \quad \quad \quad} \\ {\, + {1 \over {{c^6}}}\left({{{11} \over {16}}{v^6}{\partial _i}V - {3 \over 2}{\partial _i}{V_j}{v^j}{v^4} + {{49} \over 8}V{\partial _i}V{v^4} + {\partial _i}{{\hat W}_{jk}}{v^2}{v^j}{v^k} - 10{V_j}{\partial _i}V{v^2}{v^j} - 10V{\partial _i}{V_j}{v^2}{v^j}} \right.} \\ {\qquad - 4{\partial _i}{{\hat R}_j}{v^2}{v^j} + {{27} \over 4}{V^2}{\partial _i}V{v^2} + 12{V_j}{\partial _i}{V_j}{v^2} + 6{{\hat W}_{jk}}{\partial _i}V{v^j}{v^k} + 6V{\partial _i}{{\hat W}_{jk}}{v^j}{v^k}\quad \quad \quad \quad \quad} \\ {\qquad + 6{\partial _i}\hat X{v^2} + 8{\partial _i}{{\hat Z}_{jk}}{v^j}{v^k} - 20{V_j}V{\partial _i}V{v^j} - 10{V^2}{\partial _i}{V_j}{v^j} - 8{V_k}{\partial _i}{{\hat W}_{jk}}{v^j} - 8{{\hat W}_{jk}}{\partial _i}{V_k}{v^j}\quad \quad} \\ {- 8{{\hat R}_j}{\partial _i}V{v^j} - 8V{\partial _i}{{\hat R}_j}{v^j} - 16{\partial _i}{{\hat Y}_j}{v^j} - {1 \over 6}{V^3}{\partial _i}V - 4{V_j}{V_j}{\partial _i}V + 16{{\hat R}_j}{\partial _i}{V_j} + 16{V_j}{\partial _i}{{\hat R}_j}} \\ {\left. {- 8V{V_j}{\partial _i}{V_j} - 4\hat X{\partial _i}V - 4V{\partial _i}\hat X + 16{\partial _i}\hat T} \right) + {\mathcal O}\left({{1 \over {{c^8}}}} \right)\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(152b)

Note that it will be supposed that all the accelerations appearing in the potentials and in the final expression of the equations of motion are order-reduced by means of the equations of motion themselves. For instance, we see from Eq. (152a) that when computing the time-derivative of Pi we shall meet an acceleration at 1PN order which is therefore to be replaced by the explicit 2.5PN equations of motion. The order-reduction is a crucial aspect of the post-Newtonian method. It is justified by the fact that the matter equations of motion, say ΔμTαμ = 0, represent four out of the ten Einstein field equations, see Section 2.1 for discussion. In the harmonic-coordinate approach the equations of motion are equivalent to the harmonic gauge conditions μhαμ = 0. Thus, each time we get an acceleration in some PN expression (including the PN expression of the acceleration itself), we have also another equation (or the same equation) which tells that the acceleration is given by another PN expression. The post-Newtonian method assumes that it is legitimate to replace that acceleration and to re-expand consistently with the PN order. Post-Newtonian predictions based on such consistent PN order-reduction have been very successful.Footnote 37

Radiation reaction potentials to 4PN order

We said that the metric (144) contains the correct radiation-reaction terms appropriate for an isolated system up to the 3.5PN level included. The metric can even be generalized to include the radiation-reaction terms up to 4PN order. To show this we shall use a particular non-harmonic coordinate system to describe the radiation reaction terms up to 4PN order, which constitutes a natural generalization of the Burke & Thorne [114, 113] coordinate system at 2.5PN order. Recall that at the lowest 2.5PN order the radiation reaction force takes the simple form of Eq. (6), in which the force \(F_i^{{\rm{reac}}} = \rho \,{\partial _i}{V^{{\rm{reac}}}}\) involves only a scalar potential given by

$${V^{{\rm{reac}}}}({\bf{x}},t) = - {G \over {5{c^5}}}\,{x^i}{x^j}\,{\rm{Q}}_{ij}^{(5)}(t) + {\mathcal O}\left({{1 \over {{c^7}}}} \right)\,.$$
(153)

At such dominant 2.5PN level (“Newtonian” radiation reaction) the source quadrupole moment Qij is simply given by the usual Newtonian expression (3).

The novel feature when one extends the Newtonian radiation reaction to include the 1PN corrections is that the reaction force is no longer composed of a single scalar depending on the mass-type multipole moments, but involves also a vectorial component depending in particular on the current-type quadrupole moment. This was noticed in the physically restricted case where the dominant quadrupolar radiation from the source is suppressed [56]. The vectorial component of the reaction force could be important in some astrophysical situations like rotating neutron stars undergoing gravitational instabilities. Here we report the results of the extension to 1.5PN order of the lowest-order Burke & Thorne scalar radiation reaction potential (153), in some appropriate coordinate system, following Refs. [43, 47].

At that level (corresponding to 4PN order in the metric), and in this particular coordinate system, it suffices to incorporate some radiation-reaction contributions into the scalar and vectorial potentials V and Vi which parametrize the metric in Eq. (144). We thus pose

$${\mathcal V} = {V^{{\rm{inst}}}} + {V^{{\rm{reac}}}}\,,$$
(154a)
$${{\mathcal V}_i} = V_i^{{\rm{inst}}} + V_i^{{\rm{reac}}}\,.$$
(154b)

then the metric, accurate to 4PN order regarding the radiation-reaction contributions — we indicate this by using the symbol \({{\mathcal O}^{{\rm{reac}}}}\) for the remainders — reads

$${g_{00}} = - 1 + {2 \over {{c^2}}}{\mathcal V} - {2 \over {{c^4}}}{{\mathcal V}^2} + {{\mathcal O}^{{\rm{reac}}}}\left({{1 \over {{c^{11}}}}} \right)\,,$$
(155a)
$${g_{0i}} = - {4 \over {{c^3}}}{{\mathcal V}_i} + {{\mathcal O}^{{\rm{reac}}}}\left({{1 \over {{c^{10}}}}} \right)\,,$$
(155b)
$${g_{ij}} = {\delta _{ij}}\left[ {1 + {2 \over {{c^2}}}{\mathcal V}} \right] + {{\mathcal O}^{{\rm{reac}}}}\left({{1 \over {{c^9}}}} \right)\,.$$
(155c)

The other contributions, which are conservative (i.e., non radiative), are given up to 3PN order by the metric (144) in which all the potentials take the same form as in Eqs. (146)(148), but where one neglects all the retardations, which means that the retarded integral operator should be replaced by the operator of the instantaneous potentials \(\square_{{\rm{inst}}}^{- 1}\) defined by Eq. (134). This is for instance what we have indicated in Eqs. (154) by writing Vinst and \(V_i^{{\rm{inst}}}\). Up to 3.5PN order, in this particular coordinate system, the effect of all these retardations gets replaced by the effect of the radiation-reaction potentials Vreac and \(V_i^{{\rm{reac}}}\); furthermore, at the 4PN order there is a modification of the scalar radiation-reaction potential that is imposed by gravitational-wave tails propagating in the wave zone [58]. The explicit form of these potentials is [43, 47]Footnote 38

$$\begin{array}{*{20}c} {{V^{{\rm{reac}}}}({\bf{x}},t) = - {G \over {5{c^5}}}{x^{ij}}{\rm{I}}_{ij}^{(5)}(t) + {G \over {{c^7}}}\left[ {{1 \over {189}}{x^{ijk}}\,{\rm{I}}_{ijk}^{(7)}(t) - {1 \over {70}}{{\bf{x}}^2}{x^{ij}}\,{\rm{I}}_{ij}^{(7)}(t)} \right]\quad \quad \quad \quad \quad} \\ {- {{4{G^2}{\rm{M}}} \over {5{c^8}}}\,{x^{ij}}\int\nolimits_0^{+ \infty} {\rm{d}} \tau \,{\rm{I}}_{ij}^{(7)}(t - \tau)\left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {{11} \over {12}}} \right] + {\mathcal O}\left({{1 \over {{c^9}}}} \right)\,,} \\ \end{array}$$
(156a)
$$V_i^{{\rm{reac}}}({\bf{x}},t) = {G \over {{c^5}}}\left[ {{1 \over {21}}{{\hat x}^{ijk}}\,{\rm{I}}_{jk}^{(6)}(t) - {4 \over {45}}\,{\epsilon _{ijk}}\,{x^{jl}}\,{\rm{J}}_{kl}^{(5)}(t)} \right] + {\mathcal O}\left({{1 \over {{c^7}}}} \right)\,,$$
(156b)

where the multipole moments IL and JL denote the source multipole moments defined in Eqs. (123). Witness the tail integral at 4PN order characterized by a logarithmic kernel; see Section 3.2.

The scalar potential Vreac will obviously reproduce Eq. (153) at the dominant order. However, note that it is crucial to include in Eq. (156a) the 1PN correction in the source quadrupole moment Iij. The mass-type moments IL to 1PN order (and the current-type JL to Newtonian order), read

$${{\rm{I}}_L} = \int {{{\rm{d}}^3}} {\bf{x}}\left\{{{{\hat x}_L}\sigma + {1 \over {2{c^2}(2\ell + 3)}}{{\bf{x}}^2}\,{{\hat x}_L}\,\partial _t^2\sigma - {{4(2\ell + 1)} \over {{c^2}(\ell + 1)(2\ell + 3)}}\,{{\hat x}_{iL}}\,{\partial _t}{\sigma _i}} \right\} + {\mathcal O}\left({{1 \over {{c^4}}}} \right)\,,$$
(157a)
$${{\rm{J}}_L} = \int {{{\rm{d}}^3}} {\bf{x}}\,{\epsilon _{ab < {i_\ell}}}\,{\hat x_{L - 1 > a}}\,{\sigma _b} + {\mathcal O}\left({{1 \over {{c^2}}}} \right)\,.$$
(157b)

The matter source densities σ and σi are given in Eqs. (145). Note that the mass multipole moments IL extend only over the compact support of the source even at the 1PN order. Only at the 2PN order will they involve some non-compact supported contributions — i.e., some integrals extending up to infinity [44].

The 3.5PN radiation reaction force in the equations of motion of compact binary systems has been derived by Iyer & Will [258, 259] in an arbitrary gauge, based on the energy and angular momentum balance equations at the relative 1PN order. As demonstrated in Ref. [259] the expressions of the radiation scalar and vector radiation-reaction potentials (156), which are valid in a particular gauge but are here derived from first principles, are fully consistent with the works [258, 259].

With the radiation-reaction potentials (156) in hands, one can prove [47] the energy balance equation up to 1.5PN order, namely

$$\begin{array}{*{20}c} {{{{\rm{d}}{E^{{\rm{4PN}}}}} \over {{\rm{d}}t}} = - {G \over {5{c^5}}}{{\left({{\rm{I}}_{ij}^{(3)} + {{2G{\rm{M}}} \over {{c^3}}}\int\nolimits_0^{+ \infty} {\rm{d}} \tau \,{\rm{I}}_{ij}^{(5)}(t - \tau)\left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {{11} \over {12}}} \right]} \right)}^2}} \\ {- {G \over {{c^7}}}\left[ {{1 \over {189}}{\rm{I}}_{ijk}^{(4)}{\rm{I}}_{ijk}^{(4)} + {{16} \over {45}}{\rm{J}}_{ij}^{(3)}{\rm{J}}_{ij}^{(3)}} \right] + {\mathcal O}\left({{1 \over {{c^9}}}} \right)\,.\quad} \\ \end{array}$$
(158)

One recognizes in the right-hand side the known positive-definite expression for the energy flux at 1.5PN order. Indeed the effective quadrupole moment which appears in the parenthesis of (158) agrees with the tail-modified radiative quadrupole moment Uij parametrizing the field in the far zone; see Eq. (90) where we recall that ML and IL are identical up to 2.5PN order.

Part B: Compact Binary Systems

The problem of the motion and gravitational radiation of compact objects in post-Newtonian approximations is of crucial importance, for at least three reasons listed in the Introduction of this article: Motion of N planets in the solar system; gravitational radiation reaction force in binary pulsars; direct detection of gravitational waves from inspiralling compact binaries. As discussed in Section 1.3, the appropriate theoretical description of inspiralling compact binaries is by two structureless point-particles, characterized solely by their masses m1 and m2 (and possibly their spins), and moving on a quasi-circular orbit.

Strategies to detect and analyze the very weak signals from compact binary inspiral involve matched filtering of a set of accurate theoretical template waveforms against the output of the detectors. Many analyses [139, 137, 198, 138, 393, 346, 350, 284, 157, 158, 159, 156, 105, 106, 3, 18, 111] have shown that, in order to get sufficiently accurate theoretical templates, one must include post-Newtonian effects up to the 3PN level or higher. Recall that in practice, the post-Newtonian templates for the inspiral phase have to be matched to numerical-relativity results for the subsequent merger and ringdown phases. The match proceeds essentially through two routes: Either the so-called Hybrid templates obtained by direct matching between the PN expanded waveform and the numerical computations [4, 371], or the Effective-One-Body (EOB) templates [108, 109, 161, 168] that build on post-Newtonian results and extend their realm of validity to facilitate the analytical comparison with numerical relativity [112, 329]. Note also that various post-Newtonian resummation techniques, based on Padé approximants, have been proposed to improve the efficiency of PN templates [157, 158, 161].

Regularization of the Field of Point Particles

Our aim is to compute the metric (and its gradient needed in the equations of motion) at the 3PN order (say) for a system of two point-like particles. A priori one is not allowed to use directly some metric expressions like Eqs. (144) above, which have been derived under the assumption of a continuous (smooth) matter distribution. Applying them to a system of point particles, we find that most of the integrals become divergent at the location of the particles, i.e., when xy1(t) or y2(t), where y1(t) and y2(t) denote the two trajectories. Consequently, we must supplement the calculation by a prescription for how to remove the infinite part of these integrals. At this stage different choices for a “self-field” regularization (which will take care of the infinite self-field of point particles) are possible. In this section we review the:

  1. 1.

    Hadamard self-field regularization, which has proved to be very convenient for doing practical computations (in particular, by computer), but suffers from the important drawback of yielding some ambiguity parameters, which cannot be determined within this regularization, starting essentially at the 3PN order;

  2. 2.

    Dimensional self-field regularization, an extremely powerful regularization which is free of any ambiguities (at least up to the 3PN level), and therefore permits to uniquely fix the values of the ambiguity parameters coming from Hadamard’s regularization. However, dimensional regularization has not yet been implemented to the present problem in the general case (i.e., for an arbitrary space dimension d ∈ ℂ).

The why and how the final results are unique and independent of the employed self-field regularization (in agreement with the physical expectation) stems from the effacing principle of general relativity [142] — namely that the internal structure of the compact bodies makes a contribution only at the formal 5PN approximation. However, we shall review several alternative computations, independent of the self-field regularization, which confirm the end results.

Hadamard self-field regularization

In most practical computations we employ the Hadamard regularization [236, 381] (see Ref. [382] for an entry to the mathematical literature). Let us present here an account of this regularization, as well as a theory of generalized functions (or pseudo-functions) associated with it, following the detailed investigations in Refs. [70, 72].

Consider the class \({\mathcal F}\) of functions F(x) which are smooth (C) on ℝ3 except for the two points y1 and y2, around which they admit a power-like singular expansion of the type:Footnote 39

$$\forall {\mathcal N} \in {\mathbb N},\qquad F({\bf{x}}) = \sum\limits_{{a_0}\leqslant a\leqslant{\mathcal N}} \, \,\,r_1^a\,\underset 1 f \,{}_a({{\bf{n}}_1}) + o(r_1^{\mathcal N})\,,$$
(159)

and similarly for the other point 2. Here r1 = ∣xy1∣ → 0, and the coefficients 1fa of the various powers of r1 depend on the unit direction n1 = (xy1)/r1 of approach to the singular point. The powers a of r1 are real, range in discrete steps [i.e., a ∈ (ai)i∈ℕ], and are bounded from below (a0a). The coefficients 1fa (and 2fa) for which a < 0 can be referred to as the singular coefficients of F. If F and G belong to \({\mathcal F}\) so does the ordinary product FG, as well as the ordinary gradient iF. We define the Hadamard partie finie of F at the location of the point 1 where it is singular as

$${(F)_1} = \int {{{{\rm{d}}{\Omega _1}} \over {4\pi}}} \,\underset 1 f\,{}_0({{\bf{n}}_1})\,,$$
(160)

where dΩ1 = dQ(n1) denotes the solid angle element centered on y1 and of direction n1. Notice that because of the angular integration in Eq. (160), the Hadamard partie finie is “non-distributive” in the sense that

$${(FG)_1}\not = {(F)_1}{(G)_1}\quad {\rm{in}}\,{\rm{general}}\,.$$
(161)

The non-distributivity of Hadamard’s partie finie is the main source of the appearance of ambiguity parameters at the 3PN order, as discussed in Section 6.2.

The second notion of Hadamard partie finie (Pf) concerns that of the integral ∫ d3x F, which is generically divergent at the location of the two singular points y1 and y2 (we assume that the integral converges at infinity). It is defined by

$${\rm{P}}{{\rm{f}}_{{s_1}{s_2}}}\int {{{\rm{d}}^3}} {\bf{x}}\,F = \underset {s \to 0} {\lim} \,\left\{{\int\nolimits_{{\mathcal S}(s)} \, \,\,{{\rm{d}}^3}{\bf{x}}\,F + 4\pi \sum\limits_{a + 3 < 0} {{{{s^{a + 3}}} \over {a + 3}}} {{\left({{F \over {r_1^a}}} \right)}_1} + 4\pi \ln \left({{s \over {{s_1}}}} \right){{\left({r_1^3F} \right)}_1} + 1 \leftrightarrow 2} \right\}\,.$$
(162)

The first term integrates over a domain \({\mathcal S}{\rm{(}}s{\rm{)}}\) defined as ℝ3 from which the two spherical balls r1s and r2s of radius s and centered on the two singularities, denoted \({\mathcal B}{\rm{(}}{y_1},s)\) and \({\mathcal B}{\rm{(}}{y_2},s)\), are excised: \({\mathcal S}{\rm{(}}s{\rm{)}} \equiv {{\rm{\mathbb R}}^3}\backslash {\mathcal B}({y_1},s) \cup {\mathcal B}({y_2},s)\). The other terms, where the value of a function at point 1 takes the meaning (160), are precisely such that they cancel out the divergent part of the first term in the limit where s → 0 (the symbol 1 ↔ 2 means the same terms but corresponding to the other point 2). The Hadamard partie-finie integral depends on two strictly positive constants s1 and s2, associated with the logarithms present in Eq. (162). We shall look for the fate of these constants in the final equations of motion and radiation field. See Ref. [70] for alternative expressions of the partie-finie integral.

We now come to a specific variant of Hadamard’s regularization called the extended Hadamard regularization (EHR) and defined in Refs. [70, 72]. The basic idea is to associate to any \(F\, \in \,{\mathcal F}\) a pseudo-function, called the partie finie pseudo-function PfF, namely a linear form acting on functions G of \({\mathcal F}\), and which is defined by the duality bracket

$$\forall G \in {\mathcal F},\qquad \langle {\rm{Pf}}F,G\rangle = {\rm{Pf}}\int {{{\rm{d}}^3}} {\bf{x}}\,FG\,.$$
(163)

When restricted to the set \({\mathcal D}\) of smooth functions, i.e., C(ℝ4), with compact support (obviously we have \({\mathcal D}\, \subset \,{\mathcal F}\)), the pseudo-function PfF is a distribution in the sense of Schwartz [381]. The product of pseudo-functions coincides, by definition, with the ordinary point-wise product, namely PfF·PfG = Pf(FG). In practical computations, we use an interesting pseudo-function, constructed on the basis of the Riesz delta function [365], which plays a role analogous to the Dirac measure in distribution theory, δ1(x) ≡ δ(xy1). This is the delta-pseudo-function Pfδ1 defined by

$$\forall F \in {\mathcal F},\qquad \langle {\rm{Pf}}{\delta _1},F\rangle = {\rm{Pf}}\int {{{\rm{d}}^3}} {\bf{x}}\,{\delta _1}F = {(F)_1}\,,$$
(164)

where (F)1 is the partie finie of F as given by Eq. (160). From the product of Pfδ1 with any PfF we obtain the new pseudo-function Pf(1), that is such that

$$\forall G \in {\mathcal F},\qquad \langle {\rm{Pf}}(F{\delta _1}),G\rangle = {(FG)_1}\,.$$
(165)

As a general rule, we are not allowed, in consequence of the “non-distributivity” of the Hadamard partie finie, Eq. (161), to replace F within the pseudo-function Pf(1) by its regularized value: Pf(Fδ1) ≠ (F)1 Pfδ1 in general. It should be noticed that the object Pf(1) has no equivalent in distribution theory.

Next, we treat the spatial derivative of a pseudo-function of the type PfF, namely ∂i(PfF). Essentially, we require [70] that the rule of integration by parts holds. By this we mean that we are allowed to freely operate by parts any duality bracket, with the all-integrated (“surface”) terms always being zero, as in the case of non-singular functions. This requirement is motivated by our will that a computation involving singular functions be as much as possible the same as if we were dealing with regular functions. Thus, by definition,

$$\forall F,G \in {\mathcal F},\quad \langle {\partial _i}({\rm{Pf}}F),G\rangle = - \langle {\partial _i}({\rm{Pf}}G),F\rangle \,.$$
(166)

Furthermore, we assume that when all the singular coefficients of F vanish, the derivative of PfF reduces to the ordinary derivative, i.e., i(PfF) = Pf(iF). Then it is trivial to check that the rule (166) contains as a particular case the standard definition of the distributional derivative [381]. Notably, we see that the integral of a gradient is always zero: 〈i(PfF), 1〉 = 0. This should certainly be the case if we want to compute a quantity like a Hamiltonian density which is defined only modulo a total divergence. We pose

$${\partial _i}({\rm{Pf}}F) = {\rm{Pf}}({\partial _i}F) + {{\rm{D}}_i}[F],$$
(167)

where Pf(iF) represents the “ordinary” derivative and Di[F] is the distributional term. The following solution of the basic relation (166) was obtained in Ref. [70]:

$${{\rm{D}}_i}[F] = 4\pi \,{\rm{Pf}}\left({n_1^i\left[ {{1 \over 2}\,{r_1}\,{f_1}_{\,\, - 1} + \sum\limits_{k\geqslant 0} {{1 \over {r_1^k}}} \,{f_1}_{\,\, - 2 - k}} \right]{\delta _1}} \right) + 1 \leftrightarrow 2\,,$$
(168)

where for simplicity we assume that the powers in the expansion (159) of F are relative integers. The distributional term (168) is of the form Pf(1) plus 1 ↔ 2; it is generated solely by the singular coefficients of F.Footnote 40 The formula for the distributional term associated with the -th distributional derivative, i.e. DL[F] = LPfF − PfLF, where L = i1i2i, reads

$${{\rm{D}}_L}[F] = \sum\limits_{k = 1}^\ell {{\partial _{{i_1} \ldots {i_{k - 1}}}}} {{\rm{D}}_{{i_k}}}[{\partial _{{i_{k + 1}} \ldots {i_\ell}}}F]\,.$$
(169)

We refer to Theorem 4 in Ref. [70] for the definition of another derivative operator, representing the most general derivative satisfying the same properties as the one defined by Eq. (168), and, in addition, the commutation of successive derivatives (or Schwarz lemma).Footnote 41

The distributional derivative defined by (167)(168) does not satisfy the Leibniz rule for the derivation of a product, in accordance with a general result of Schwartz [380]. Rather, the investigation of Ref. [70] suggests that, in order to construct a consistent theory (using the ordinary point-wise product for pseudo-functions), the Leibniz rule should be weakened, and replaced by the rule of integration by part, Eq. (166), which is in fact nothing but an integrated version of the Leibniz rule. However, the loss of the Leibniz rule stricto sensu constitutes one of the reasons for the appearance of the ambiguity parameters at 3PN order, see Section 6.2.

The Hadamard regularization (F)1 is defined by Eq. (160) in a preferred spatial hypersurface t = const of a coordinate system, and consequently is not a priori compatible with the Lorentz invariance. Thus we expect that the equations of motion in harmonic coordinates (which manifestly preserve the global Lorentz invariance) should exhibit at some stage a violation of the Lorentz invariance due to the latter regularization. In fact this occurs exactly at the 3PN order. Up to the 2.5PN level, the use of the regularization (F)1 is sufficient to get some unambiguous equations of motion which are Lorentz invariant [76]. This problem can be dealt with within Hadamard’s regularization, by introducing a Lorentz-invariant variant of this regularization, denoted [F]1 in Ref. [72]. It consists of performing the Hadamard regularization within the spatial hypersurface that is geometrically orthogonal (in a Minkowskian sense) to the four-velocity of the particle. The regularization [F]1 differs from the simpler regularization (F)1 by relativistic corrections of order 1/c2 at least. See [72] for the formulas defining this regularization in the form of some infinite power series in 1/c2. The regularization [F]1 plays a crucial role in obtaining the equations of motion at the 3PN order in Refs. [69, 71]. In particular, the use of the Lorentz-invariant regularization [F]1permits to obtain the value of the ambiguity parameter ωkinetic in Eq. (170a) below.

Hadamard regularization ambiguities

The standard Hadamard regularization yields some ambiguous results for the computation of certain integrals at the 3PN order, as noticed by Jaranowski & Schäfer [261, 262, 263] in their computation of the equations of motion within the ADM-Hamiltonian formulation of general relativity. By standard Hadamard regularization we mean the regularization based solely on the definitions of the partie finie of a singular function, Eq. (160), and the partie finie of a divergent integral, Eq. (162), and without using a theory of pseudo-functions and generalized distributional derivatives as in Refs. [70, 72]. It was shown in Refs. [261, 262, 263] that there are two and only two types of ambiguous terms in the 3PN Hamiltonian, which were then parametrized by two unknown numerical coefficients called ωstatic and ωkinetic.

Progressing concurrently, Blanchet & Faye [70, 72] introduced the “extended” Hadamard regularization — the one we outlined in Section 6.1 — and obtained [69, 71] the 3PN equations of motion complete except for one and only one unknown numerical constant, called λ. The new extended Hadamard regularization is mathematically well-defined and yields unique results for the computation of any integral in the problem; however, it turned out to be in a sense “incomplete” as it could not determine the value of this constant. The comparison of the result with the work [261, 262], on the basis of the computation of the invariant energy of compact binaries moving on circular orbits, revealed [69] that

$${\omega _{{\rm{kinetic}}}} = {{41} \over {24}}\,,$$
(170a)
$${\omega _{{\rm{static}}}} = - {{11} \over 3}\lambda - {{1987} \over {840}}\,.$$
(170b)

Therefore, the ambiguity ωkinetic is fixed, while λ is equivalent to the other ambiguity ωstatic. Notice that the value (170a) for the kinetic ambiguity parameter ωkinetic, which is in factor of some velocity dependent terms, is the only one for which the 3PN equations of motion are Lorentz invariant. Fixing up this value was possible because the extended Hadamard regularization [70, 72] was defined in such a way that it keeps the Lorentz invariance.

The value of ωkinetic given by Eq. (170a) was recovered in Ref. [162] by directly proving that such value is the unique one for which the global Poincaré invariance of the ADM-Hamiltonian formalism is verified. Since the coordinate conditions associated with the ADM formalism do not manifestly respect the Poincaré symmetry, it was necessary to prove that the 3PN Hamiltonian is compatible with the existence of generators for the Poincaré algebra. By contrast, the harmonic-coordinate conditions preserve the Poincaré invariance, and therefore the associated equations of motion at 3PN order are manifestly Lorentz-invariant, as was found to be the case in Refs. [69, 71].

The appearance of one and only one physical unknown coefficient λ in the equations of motion constitutes a quite striking fact, that is related specifically with the use of a Hadamard-type regularization.Footnote 42 Technically speaking, the presence of the ambiguity parameter is associated with the non-distributivity of Hadamard’s regularization, in the sense of Eq. (161). Mathematically speaking, λ is probably related to the fact that it is impossible to construct a distributional derivative operator, such as Eqs. (167)(168), satisfying the Leibniz rule for the derivation of the product [380]. The Einstein field equations can be written in many different forms, by shifting the derivatives and operating some terms by parts with the help of the Leibniz rule. All these forms are equivalent in the case of regular sources, but since the derivative operator (167)(168) violates the Leibniz rule they become inequivalent for point particles.

Physically speaking, let us also argue that has its root in the fact that in a complete computation of the equations of motion valid for two regular extended weakly self-gravitating bodies, many non-linear integrals, when taken individually, start depending, from the 3PN order, on the internal structure of the bodies, even in the “compact-body” limit where the radii tend to zero. However, when considering the full equations of motion, one expects that all the terms depending on the internal structure can be removed, in the compact-body limit, by a coordinate transformation (or by some appropriate shifts of the central world lines of the bodies), and that finally λ is given by a pure number, for instance a rational fraction, independent of the details of the internal structure of the compact bodies. From this argument (which could be justified by the effacing principle in general relativity) the value of λ is necessarily the one we compute below, Eq. (172), and will be valid for any compact objects, for instance black holes.

The ambiguity parameter ωstatic, which is in factor of some static, velocity-independent term, and hence cannot be derived by invoking Lorentz invariance, was computed by Damour, Jaranowski & Schäfer [163] by means of dimensional regularization, instead of some Hadamard-type one, within the ADM-Hamiltonian formalism. Their result is

$${\omega _{{\rm{static}}}} = 0\,.$$
(171)

As argued in [163], clearing up the static ambiguity is made possible by the fact that dimensional regularization, contrary to Hadamard’s regularization, respects all the basic properties of the algebraic and differential calculus of ordinary functions: Associativity, commutativity and distributivity of point-wise addition and multiplication, Leibniz’s rule, and the Schwarz lemma. In this respect, dimensional regularization is certainly superior to Hadamard’s one, which does not respect the distributivity of the product [recall Eq. (161)] and unavoidably violates at some stage the Leibniz rule for the differentiation of a product.

The ambiguity parameter λ is fixed from the result (171) and the necessary link (170b) provided by the equivalence between the harmonic-coordinates and ADM-Hamiltonian formalisms [69, 164]. However, λ has also been computed directly by Blanchet, Damour & Esposito-Farèse [61] applying dimensional regularization to the 3PN equations of motion in harmonic coordinates (in the line of Refs. [69, 71]). The end result,

$$\lambda = - {{1987} \over {3080}}\,,$$
(172)

is in full agreement with Eq. (171).Footnote 43 Besides the independent confirmation of the value of ωstatic or λ, the work [61] provides also a confirmation of the consistency of dimensional regularization, since the explicit calculations are entirely different from the ones of Ref. [163]: Harmonic coordinates instead of ADM-type ones, work at the level of the equations of motion instead of the Hamiltonian, a different form of Einstein’s field equations which is solved by a different iteration scheme.

Let us comment that the use of a self-field regularization, be it dimensional or based on Hadamard’s partie finie, signals a somewhat unsatisfactory situation on the physical point of view, because, ideally, we would like to perform a complete calculation valid for extended bodies, taking into account the details of the internal structure of the bodies (energy density, pressure, internal velocity field, etc.). By considering the limit where the radii of the objects tend to zero, one should recover the same result as obtained by means of the point-mass regularization. This would demonstrate the suitability of the regularization. This program was undertaken at the 2PN order in Refs. [280, 234] which derived the equations of motion of two extended fluid balls, and obtained equations of motion depending only on the two masses m1 and m2 of the compact bodies.Footnote 44 At the 3PN order we expect that the extended-body program should give the value of the regularization parameter λ — probably after a coordinate transformation to remove the terms depending on the internal structure. Ideally, its value should also be confirmed by independent and more physical methods like those of Refs. [407, 281, 172].

An important work, in several aspects more physical than the formal use of regularizations, is the one of Itoh & Futamase [255, 253, 254], following previous investigations in Refs. [256, 257]. These authors derived the 3PN equations of motion in harmonic coordinates by means of a particular variant of the famous “surface-integral” method à la Einstein, Infeld & Hoffmann [184]. The aim is to describe extended relativistic compact binary systems in the so-called strong-field point particle limit which has been defined in Ref. [212]. This approach is interesting because it is based on the physical notion of extended compact bodies in general relativity, and is free of the problems of ambiguities. The end result of Refs. [255, 253] is in agreement with the 3PN harmonic coordinates equations of motion [69, 71] and is unambiguous, as it does directly determine the ambiguity parameter λ to exactly the value (172).

The 3PN equations of motion in harmonic coordinates or, more precisely, the associated 3PN Lagrangian, were also derived by Foffa & Sturani [203] using another important approach, coined the effective field theory (EFT) [223]. Their result is fully compatible with the value (172) for the ambiguity parameter λ; however, in contrast with the surface-integral method of Refs. [255, 253], this does not check the method of regularization because the EFT approach is also based on dimensional self-field regularization.

We next consider the problem of the binary’s radiation field, where the same phenomenon occurs, with the appearance of some Hadamard regularization ambiguity parameters at 3PN order. More precisely, Blanchet, Iyer & Joguet [81], computing the 3PN compact binary’s mass quadrupole moment Iij, found it necessary to introduce three Hadamard regularization constants ξ, κ, and ζ, which are independent of the equation-of-motion related constant λ. The total gravitational-wave flux at 3PN order, in the case of circular orbits, was found to depend on a single combination of the latter constants, θ = ξ + 2κ + ζ, and the binary’s orbital phase, for circular orbits, involved only the linear combination of θ and λ given by \(\hat \theta = \theta - {7 \over 3}\lambda\), as shown in [73].

Dimensional regularization (instead of Hadamard’s) has next been applied in Refs. [62, 63] to the computation of the 3PN radiation field of compact binaries, leading to the following unique determination of the ambiguity parameters:Footnote 45

$$\xi = - {{9871} \over {9240}}\,,$$
(173a)
$$\kappa = 0\,,$$
(173b)
$$\zeta = - {7 \over {33}}\,.$$
(173c)

These values represent the end result of dimensional regularization. However, several alternative calculations provide a check, independent of dimensional regularization, for all the parameters (173). One computes [80] the 3PN binary’s mass dipole moment Ii using Hadamard’s regularization, and identifies Ii with the 3PN center of mass vector position Gi, already known as a conserved integral associated with the Poincaré invariance of the 3PN equations of motion in harmonic coordinates [174]. This yields ξ + κ = −9871/9240 in agreement with Eqs. (173). Next, one considers [65] the limiting physical situation where the mass of one of the particles is exactly zero (say, m2 = 0), and the other particle moves with uniform velocity. Technically, the 3PN quadrupole moment of a boosted Schwarzschild black hole is computed and compared with the result for Iij in the limit m2 = 0. The result is ζ = −7/33, and represents a direct verification of the global Poincaré invariance of the wave generation formalism (the parameter ζ representing the analogue for the radiation field of the parameter ωkinetic). Finally, one proves [63] that κ = 0 by showing that there are no dangerously divergent diagrams corresponding to non-zero κ-values, where a diagram is meant here in the sense of Ref. [151].

The determination of the parameters (173) completes the problem of the general relativistic prediction for the templates of inspiralling compact binaries up to 3.5PN order. The numerical values of these parameters indicate, following measurement-accuracy analyses [105, 106, 159, 156], that the 3.5PN order should provide an excellent approximation for both the on-line search and the subsequent off-line analysis of gravitational wave signals from inspiralling compact binaries in the LIGO and VIRGO detectors.

Dimensional regularization of the equations of motion

As reviewed in Section 6.2, work at 3PN order using Hadamard’s self-field regularization showed the appearance of ambiguity parameters, due to an incompleteness of the Hadamard regularization employed for curing the infinite self field of point particles. We give here more details on the determination using dimensional regularization of the ambiguity parameter λ [or equivalently ωstatic, see Eq. (170b)] which appeared in the 3PN equations of motion.

Dimensional regularization was invented as a means to preserve the gauge symmetry of perturbative quantum field theories [391, 91, 100, 131]. Our basic problem here is to respect the gauge symmetry associated with the diffeomorphism invariance of the classical general relativistic description of interacting point masses. Hence, we use dimensional regularization not merely as a trick to compute some particular integrals which would otherwise be divergent, but as a powerful tool for solving in a consistent way the Einstein field equations with singular point-mass sources, while preserving its crucial symmetries. In particular, we shall prove that dimensional regularization determines the kinetic ambiguity parameter ωkinetic (and its radiation-field analogue ζ), and is therefore able to correctly keep track of the global Lorentz-Poincaré invariance of the gravitational field of isolated systems. The dimensional regularization is also an important ingredient of the EFT approach to equations of motion and gravitational radiation [223].

The Einstein field equations in d+1 space-time dimensions, relaxed by the condition of harmonic coordinates dμhαμ = 0, take exactly the same form as given in Eqs. (18)(23). In particular the box operator □ now denotes the flat space-time d’Alembertian operator in d + 1 dimensions with signature (−1, 1, 1, ⋯). The gravitational constant G is related to the usual three-dimensional Newton’s constant GN by

$$G = {G_{\rm{N}}}\,\ell _0^{d - 3}\,,$$
(174)

where 0 denotes an arbitrary length scale. The explicit expression of the gravitational source term Λαβ involves some d-dependent coefficients, and is given by

$$\begin{array}{*{20}c} {{\Lambda ^{\alpha \beta}} = - {h^{\mu \nu}}\partial _{\mu \nu}^2{h^{\alpha \beta}} + {\partial _\mu}{h^{\alpha \nu}}{\partial _\nu}{h^{\beta \mu}} + {1 \over 2}{g^{\alpha \beta}}{g_{\mu \nu}}{\partial _\lambda}{h^{\mu \tau}}{\partial _\tau}{h^{\nu \lambda}}\quad \quad \quad \quad \quad} \\ {- {g^{\alpha \mu}}{g_{\nu \tau}}{\partial _\lambda}{h^{\beta \tau}}{\partial _\mu}{h^{\nu \lambda}} - {g^{\beta \mu}}{g_{\nu \tau}}{\partial _\lambda}{h^{\alpha \tau}}{\partial _\mu}{h^{\nu \lambda}} + {g_{\mu \nu}}{g^{\lambda \tau}}{\partial _\lambda}{h^{\alpha \mu}}{\partial _\tau}{h^{\beta \nu}}} \\ {+ {1 \over 4}(2{g^{\alpha \mu}}{g^{\beta \nu}} - {g^{\alpha \beta}}{g^{\mu \nu}})\left({{g_{\lambda \tau}}{g_{\epsilon \pi}} - {1 \over {d - 1}}{g_{\tau \epsilon}}{g_{\lambda \pi}}} \right){\partial _\mu}{h^{\lambda \pi}}{\partial _\nu}{h^{\tau \epsilon}}\,.} \\ \end{array}$$
(175)

When d = 3 we recover Eq. (24). In the following we assume, as usual in dimensional regularization, that the dimension of space is a complex number, d ∈ ℂ, and prove many results by invoking complex analytic continuation in d. We shall often pose εd − 3.

We parametrize the 3PN metric in d dimensions by means of some retarded potentials V, Vi, \({\hat W_{ij}}\), …, which are straightforward d-dimensional generalizations of the potentials used in three dimensions and which were defined in Section 5.3. Those are obtained by post-Newtonian iteration of the d-dimensional field equations, starting from appropriate definitions of matter source densities generalizing Eqs. (145):

$$\sigma = {2 \over {d - 1}}{{(d - 2){T^{00}} + {T^{ii}}} \over {{c^2}}}\,,$$
(176a)
$${\sigma _i} = {{{T^{0i}}} \over c}\,,$$
(176b)
$${\sigma _{ij}} = {T^{ij}}\,.$$
(176c)

As a result all the expressions of Section 5.3 acquire some explicit d-dependent coefficients. For instance we find [61]

$$V = \Box_{{\rm{ret}}}^{- 1}\left[ {- 4\pi G\sigma} \right]\,,$$
(177a)
$${\hat W_{ij}} = \Box_{{\rm{ret}}}^{- 1}\left[ {- 4\pi G\left({{\sigma _{ij}} - {\delta _{ij}}{{{\sigma _{kk}}} \over {d - 2}}} \right) - {{d - 1} \over {2(d - 2)}}{\partial _i}V{\partial _j}V} \right]\,.$$
(177b)

Here \(\square_{{\rm{ret}}}^{- 1}\) means the retarded integral in d + 1 space-time dimensions, which admits, though, no simple expression generalizing Eq. (31) in physical (t, x) space.Footnote 46

As reviewed in Section 6.1, the generic functions F(x) we have to deal with in 3 dimensions, are smooth on ℝ3 except at y1 and y2, around which they admit singular Laurent-type expansions in powers and inverse powers of r1 ≡ ∣xy1∣ and r2 ≡ ∣xy2∣, given by Eq. (178). In d spatial dimensions, there is an analogue of the function F, which results from the post-Newtonian iteration process performed in d dimensions as we just outlined. Let us call this function F(d)(x), where x ∈ ℝd. When r1 → 0 the function F(d) admits a singular expansion which is more involved than in 3 dimensions, as it reads

$${F^{(d)}}(x) = \sum\limits_{\matrix{{{{p_0} \le p \le {\cal N}}} \\ {{{q_0} \le q \le {q_1}}}}} {r_1^{p + q\varepsilon }\mathop f\limits_1 } {}_{p,q}^{(\varepsilon )}({n_1}) + o(r_1^{\cal N}).$$
(178)

The coefficients \({f_1}\,_{p,q}^{(\varepsilon)}({{\rm{n}}_1})\) depend on ε = d − 3, and the powers of r1 involve the relative integers p and q whose values are limited by some p0, q0 and q1 as indicated. Here we will be interested in functions F(d)(x) which have no poles as ε → 0 (this will always be the case at 3PN order). Therefore, we can deduce from the fact that F(d)(x) is continuous at d = 3 the constraint

$$\sum\limits_{q = {q_0}}^{{q_1}} {\mathop {{\rm{ }}f}\limits_1 1mu_{p,q}^{(\varepsilon = 0)}} ({{\bf{n}}_1}) = \mathop {{\rm{ }}f}\limits_1 {\;_p}({{\bf{n}}_1})1mu.$$
(179)

For the problem at hand, we essentially have to deal with the regularization of Poisson integrals, or iterated Poisson integrals (and their gradients needed in the equations of motion), of the generic function F(d). The Poisson integral of F(d), in d dimensions, is given by the Green’s function for the Laplace operator,

$${P^{(d)}}({\bf{x}\prime}) = {\Delta ^{- 1}}\left[ {{F^{(d)}}({\bf{x}})} \right] \equiv - {{\tilde k} \over {4\pi}}\int {{{{{\rm{d}}^d}{\bf{x}}} \over {|{\bf{x}} - {\bf{x}}\prime{|^{d - 2}}}}} {F^{(d)}}({\bf{x}})\,,$$
(180)

where \(\tilde k\) is a constant related to the usual Eulerian Γ-function byFootnote 47

$$\tilde k = {{\Gamma \left({{{d - 2} \over 2}} \right)} \over {\pi {{d - 2} \over 2}}}.$$
(181)

We need to evaluate the Poisson integral at the point x′ = y1 where it is singular; this is quite easy in dimensional regularization, because the nice properties of analytic continuation allow simply to get \([{P^{(d)}}({\rm{x}}\prime)]_{{\rm{X\prime =}}{y_1}}\) by replacing x′ by y1 inside the explicit integral (180). So we simply have

$${P^{(d)}}({y_1}) = - {{\tilde k} \over {4\pi}}\int {{{{{\rm{d}}^d}{\bf{x}}} \over {r_1^{d - 2}}}} {F^{(d)}}({\bf{x}})\,.$$
(182)

It is not possible at present to compute the equations of motion in the general d-dimensional case, but only in the limit where ε → 0 [163, 61]. The main technical step of our strategy consists of computing, in the limit ε → 0, the difference between the d-dimensional Poisson potential (182), and its Hadamard 3-dimensional counterpart given by (P)1, where the Hadamard partie finie is defined by Eq. (160). But we must be precise when defining the Hadamard partie finie of a Poisson integral. Indeed, the definition (160) stricto sensu is applicable when the expansion of the function F, for r1 → 0, does not involve logarithms of r1; see Eq. (160). However, the Poisson integral P(x′) of F(x) will typically involve such logarithms at the 3PN order, namely some ln r1 where r1≡ ∣x′ − y1∣ formally tends to zero (hence ln r1 is formally infinite). The proper way to define the Hadamard partie finie in this case is to include the ln r1 into its definition; we arrive at [70]

$${(P)_1} = - {1 \over {4\pi}}{\rm{P}}{{\rm{f}}_{r_1^{\prime},{s_2}}}\int {{{{{\rm{d}}^3}{\bf{x}}} \over {{r_1}}}} F({\bf{x}}) - {(r_1^2\,F)_1}\,.$$
(183)

The first term follows from Hadamard’s partie finie integral (162); the second one is given by Eq. (160). Notice that in this result the constant s1 entering the partie finie integral (162) has been “replaced” by r1, which plays the role of a new regularization constant (together with r2 for the other particle), and which ultimately parametrizes the final Hadamard regularized 3PN equations of motion. It was shown that r1 and r2 are unphysical, in the sense that they can be removed by a coordinate transformation [69, 71]. On the other hand, the constant s2 remaining in the result (183) is the source for the appearance of the physical ambiguity parameter λ. Denoting the difference between the dimensional and Hadamard regularizations by means of the script letter \({\mathcal D}\), we pose (for what concerns the point 1)

$${\mathcal D}{P_1} \equiv {P^{(d)}}({y_1}) - {(P)_1}\,.$$
(184)

That is, \({\mathcal D}{P_1}\) is what we shall have to add to the Hadamard-regularization result in order to get the d-dimensional result. However, we shall only compute the first two terms of the Laurent expansion of \({\mathcal D}{P_1}\) when ε → 0, say \({\mathcal D}{P_1} = {a_{- 1\,}}{\varepsilon ^{- 1}} + {a_0} + {\mathcal O}(\varepsilon)\). This is the information we need to clear up the ambiguity parameter. We insist that the difference \({\mathcal D}{P_1}\) comes exclusively from the contribution of terms developing some poles ∝ 1/ε in the d-dimensional calculation.

Next we outline the way we obtain, starting from the computation of the “difference”, the 3PN equations of motion in dimensional regularization, and show how the ambiguity parameter λ is determined. By contrast to r1 and r2 which are pure gauge, is a genuine physical ambiguity, introduced in Refs. [70, 71] as the single unknown numerical constant parametrizing the ratio between s2 and r2 [where s2 is the constant left in Eq. (183)] as

$$\ln \left({{{r_2^\prime} \over {{s_2}}}} \right) = {{159} \over {308}} + \lambda {{{m_1} + {m_2}} \over {{m_2}}}\qquad ({\rm{and}}\, 1 \leftrightarrow 2)\,,$$
(185)

where m1 and m2 are the two masses. The terms corresponding to the λ-ambiguity in the acceleration a1 = dv1/dt of particle 1 read simply

$$\Delta {a_1}[\lambda ] = - {{44\lambda} \over 3}\,{{G_{\rm{N}}^4\,{m_1}\,m_2^2\,({m_1} + {m_2})} \over {r_{12}^5\,{c^6}}}\,{n_{12}}\,,$$
(186)

where the relative distance between particles is denoted y1y2r12 n12 (with n12 being the unit vector pointing from particle 2 to particle 1). We start from the end result of Ref. [71] for the 3PN harmonic coordinates acceleration a1 in Hadamard’s regularization, abbreviated as HR. Since the result was obtained by means of the specific extended variant of Hadamard’s regularization (in short EHR, see Section 6.1) we write it as

$$a_1^{({\rm{HR}})} = a_1^{({\rm{EHR}})} + \Delta {a_1}[\lambda ]\,,$$
(187)

where \(a_1^{({\rm{EHR}})}\) is a fully determined functional of the masses m1 and m2, the relative distance r12 n12, the coordinate velocities v1 and v2, and also the gauge constants r1 and r2. The only ambiguous term is the second one and is given by Eq. (186).

Our strategy is to extract from both the dimensional and Hadamard regularizations their common core part, obtained by applying the so-called “pure-Hadamard-Schwartz” (pHS) regularization. Following the definition in Ref. [61], the pHS regularization is a specific, minimal Hadamard-type regularization of integrals, based on the partie finie integral (162), together with a minimal treatment of “contact” terms, in which the definition (162) is applied separately to each of the elementary potentials V, Vi, etc. (and gradients) that enter the post-Newtonian metric. Furthermore, the regularization of a product of these potentials is assumed to be distributive, i.e., (FG)1 = (F)1(G)1 in the case where F and G are given by such elementary potentials; this is thus in contrast with Eq. (161). The pHS regularization also assumes the use of standard Schwartz distributional derivatives [381]. The interest of the pHS regularization is that the dimensional regularization is equal to it plus the “difference”; see Eq. (190).

To obtain the pHS-regularized acceleration we need to substract from the EHR result a series of contributions, which are specific consequences of the use of EHR [70, 72]. For instance, one of these contributions corresponds to the fact that in the EHR the distributional derivative is given by Eqs. (167)(168) which differs from the Schwartz distributional derivative in the pHS regularization. Hence we define

$$a_1^{({\rm{pHS}})} = a_1^{({\rm{EHR}})} - \sum \delta {a_1}\,,$$
(188)

where the δa1’s denote the extra terms following from the EHR prescriptions. The pHS-regularized acceleration (188) constitutes essentially the result of the first stage of the calculation of a1, as reported in Ref. [193].

The next step consists of evaluating the Laurent expansion, in powers of ε = d − 3, of the difference between the dimensional regularization and the pHS (3-dimensional) computation. As we reviewed above, this difference makes a contribution only when a term generates a pole ∼ 1/ε, in which case the dimensional regularization adds an extra contribution, made of the pole and the finite part associated with the pole [we consistently neglect all terms \({\mathcal O}(\varepsilon)\)]. One must then be especially wary of combinations of terms whose pole parts finally cancel but whose dimensionally regularized finite parts generally do not, and must be evaluated with care. We denote the above defined difference by

$${\mathcal D}{a_1} = \sum {\mathcal D} {P_1}\,.$$
(189)

It is made of the sum of all the individual differences of Poisson or Poisson-like integrals as computed in Eq. (184). The total difference (189) depends on the Hadamard regularization scales r1 and s2(or equivalently on λ and r1, r2), and on the parameters associated with dimensional regularization, namely ε and the characteristic length scale 0 introduced in Eq. (174). Finally, the result is the explicit computation of the ε-expansion of the dimensional regularization (DR) acceleration as

$$a_1^{({\rm{DR}})} = a_1^{({\rm{pHS}})} + {\mathcal D}{a_1}\,.$$
(190)

With this result we can prove two theorems [61].

Theorem 8. The pole part ∝ 1/ε of the DR acceleration (190) can be re-absorbed (i.e. renormalized) into some shifts of the two “bare” world-lines: y1y1 + ξ1 and y2y2 + ξ2, with ξ1,2 ∝ 1/ε say, so that the result, expressed in terms of the “dressed” quantities, is finite when ε → 0.

The situation in harmonic coordinates is to be contrasted with the calculation in ADM-type coordinates within the Hamiltonian formalism, where it was shown that all pole parts directly cancel out in the total 3PN Hamiltonian: No renormalization of the world-lines is needed [163]. The central result is then:

Theorem 9. The renormalized (finite) DR acceleration is physically equivalent to the Hadamard-regularized (HR) acceleration (end result of Ref. [71]), in the sense that

$$a_1^{({\rm{HR}})} = {\lim\limits_{\varepsilon \to 0}} \,\left[ {a_1^{({\rm{DR}})} + {\delta _\xi}\,{a_1}} \right]\,,$$
(191)

where δξa1 denotes the effect of the shifts on the acceleration, if and only if the HR ambiguity parameter λ entering the harmonic-coordinates equations of motion takes the unique value (172).

The precise shifts ξ1 and ξ2 needed in Theorem 9 involve not only a pole contribution ∝ 1/ε, but also a finite contribution when ε → 0. Their explicit expressions read:Footnote 48

$${\xi _1} = {{11} \over 3}{{G_{\rm{N}}^2\,m_1^2} \over {{c^6}}}\left[ {{1 \over \varepsilon} - 2\ln \left({{{{{r\prime}_1}{{\bar q}^{1/2}}} \over {{\ell _0}}}} \right) - {{327} \over {1540}}} \right]a_1^{\rm{N}}\qquad {\rm{(together with}}1 \leftrightarrow 2)\,,$$
(192)

where GN is Newton’s constant, 0 is the characteristic length scale of dimensional regularization, cf. Eq. (174), where \(a_1^{\rm{N}}\) is the Newtonian acceleration of the particle 1 in d dimensions, and \(\bar q \equiv 4\pi {e^{\gamma {\rm{E}}}}\) depends on Euler’s constant γE ≃ 0.577.

Dimensional regularization of the radiation field

We now address the similar problem concerning the binary’s radiation field — to 3PN order beyond Einstein’s quadrupole formalism (2)(3). As reviewed in Section 6.2, three ambiguity parameters: ξ, κ and ζ, have been shown to appear in the 3PN expression of the quadrupole moment [81, 80].

To apply dimensional regularization, we must use as in Section 6.3 the d-dimensional post-Newtonian iteration leading to potentials such as those in Eqs. (177); and we have to generalize to d dimensions some key results of the wave generation formalism of Part A. Essentially, we need the d-dimensional analogues of the multipole moments of an isolated source IL and JL in Eqs. (123). Here we report the result we find in the case of the mass-type moment:

$$\begin{array}{*{20}c} {{\rm{I}}_L^{(d)}(t) = {{d - 1} \over {2(d - 2)}}\,{\mathcal F}{\mathcal P}\int {{{\rm{d}}^d}} {\bf{x}}\left\{{{{\hat x}_L}\,{\Sigma _{[\ell ]}}({\bf{x}},t) - {{4(d + 2\ell - 2)} \over {{c^2}(d + \ell - 2)(d + 2\ell)}}\,{{\hat x}_{aL}}\,\underset {[\ell + 1]} \Sigma {}_a^{(1)}({\bf{x}},t)} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\left. {+ {{2(d + 2\ell - 2)} \over {{c^4}(d + \ell - 1)(d + \ell - 2)(d + 2\ell + 2)}}{{\hat x}_{abL}}\,\underset {[\ell + 2]} \Sigma {}_{ab}^{(2)}({\bf{x}},t)} \right\},} \\ \end{array}$$
(193)

in which we denote, generalizing Eqs. (124),

$$\Sigma = {2 \over {d - 1}}{{(d - 2){{\bar \tau}^{00}} + {{\bar \tau}^{ii}}} \over {{c^2}}}\,,$$
(194a)
$${\Sigma _i} = {{{{\bar \tau}^{0i}}} \over c}\,,$$
(194b)
$${\Sigma _{ij}} = {\bar \tau ^{ij}}\,,$$
(194c)

and where for any source densities the underscript [] means the infinite series

$$\underset {[\ell ]} \Sigma ({\bf{x}},t) = \sum\limits_{k = 0}^{+ \infty} {{1 \over {{2^{2k}}k!}}} {{\Gamma \left({{d \over 2} + \ell} \right)} \over {\Gamma \left({{d \over 2} + \ell + k} \right)}}{\left({{r \over c}{\partial \over {\partial t}}} \right)^{2k}}\,\,\Sigma ({\bf{x}},t)\,.$$
(195)

The latter definition represents the d-dimensional version of the post-Newtonian expansion series (126). At Newtonian order, the expression (193) reduces to the standard result \({\rm{I}}_L^{(d)} = \int {{{\rm{d}}^d}{\rm{x}}\,\rho {{\hat x}_L}} + {\mathcal O}(1/{c^2})\) with ρ = T00/c2 denoting the usual Newtonian density.

The ambiguity parameters ξ, κ and come from the Hadamard regularization of the mass quadrupole moment Iij at the 3PN order. The terms corresponding to these ambiguities were found to be

$$\Delta {{\rm{I}}_{ij}}[\xi, \kappa, \zeta ] = {{44} \over 3}{{G_{\rm{N}}^2\,m_1^3} \over {{c^6}}}\left[ {\left({\xi + \kappa {{{m_1} + {m_2}} \over {{m_1}}}} \right)y_1^{\langle i}a_1^{j\rangle} + \,\zeta \,v_1^{\langle i}v_1^{j\rangle}} \right] + 1 \leftrightarrow 2\,,$$
(196)

where y1, v1 and a1 denote the first particle’s position, velocity and acceleration (and the brackets 〈〉 surrounding indices refer to the STF projection). Like in Section 6.3, we express both the Hadamard and dimensional results in terms of the more basic pHS regularization. The first step of the calculation [80] is therefore to relate the Hadamard-regularized quadrupole moment \({\rm{I}}_{ij}^{({\rm{HR}})}\), for general orbits, to its pHS part:

$${\rm{I}}_{ij}^{({\rm{HR}})} = {\rm{I}}_{ij}^{({\rm{pHS}})} + \Delta {{\rm{I}}_{ij}}\left[ {\xi + {1 \over {22}},\kappa, \zeta + {9 \over {110}}} \right]\,.$$
(197)

In the right-hand side we find both the pHS part, and the effect of adding the ambiguities, with some numerical shifts of the ambiguity parameters (ξξ + 1/22, → ζ + 9/110) due to the difference between the specific Hadamard-type regularization scheme used in Ref. [81] and the pHS one. The pHS part is free of ambiguities but depends on the gauge constants r1 and r2 introduced in the harmonic-coordinates equations of motion [69, 71].

We next use the d-dimensional moment (193) to compute the difference between the dimensional regularization (DR) result and the pHS one [62, 63]. As in the work on equations of motion, we find that the ambiguities arise solely from the terms in the integration regions near the particles, that give rise to poles ℝ 1/ε, corresponding to logarithmic ultra-violet (UV) divergences in 3 dimensions. The infra-red (IR) region at infinity, i.e., ∣x∣ → +∞, does not contribute to the difference between DR and pHS. The compact-support terms in the integrand of Eq. (193), proportional to the matter source densities σ, σa, and σab, are also found not to contribute to the difference. We are therefore left with evaluating the difference linked with the computation of the non-compact terms in the expansion of the integrand of (193) near the singularities that produce poles in d dimensions.

Let F(d)(x) be the non-compact part of the integrand of the quadrupole moment (193) (with indices L = ij), where F(d) includes the appropriate multipolar factors such as \({\hat x_{ij}}\), so that

$${\rm{I}}_{ij}^{(d)} = \int {{{\rm{d}}^d}} {\bf{x}}\,{F^{(d)}}({\bf{x}})\,.$$
(198)

We do not indicate that we are considering here only the non-compact part of the moments. Near the singularities the function F(d)(x) admits a singular expansion of the type (178). In practice, the various coefficients \({}_1f_{p,q}^{(\varepsilon)}\) are computed by specializing the general expressions of the non-linear retarded potentials V, Va, \({\hat W_{ab}}\), etc. (valid for general extended sources) to point particles in d dimensions. On the other hand, the analogue of Eq. (198) in 3 dimensions is

$${{\rm{I}}_{ij}} = {\rm{Pf}}\int {{{\rm{d}}^3}} {\bf{x}}\,F({\bf{x}})\,,$$
(199)

where Pf refers to the Hadamard partie finie defined in Eq. (162). The difference \({\mathcal D}{{\rm{I}}_{ij}}\) between the DR evaluation of the d-dimensional integral (198) and its corresponding three-dimensional evaluation (199), reads then

$${\mathcal D}{{\rm{I}}_{ij}} = {\rm{I}}_{ij}^{(d)} - {{\rm{I}}_{ij}}\,.$$
(200)

Such difference depends only on the UV behaviour of the integrands, and can therefore be computed “locally”, i.e., in the vicinity of the particles, when r1 → 0 and r2 → 0. We find that Eq. (200) depends on two constant scales s1 and s2 coming from Hadamard’s partie finie (162), and on the constants belonging to dimensional regularization, i.e., ε = d − 3 and 0 defined by Eq. (174). The dimensional regularization of the 3PN quadrupole moment is then obtained as the sum of the pHS part, and of the difference computed according to Eq. (200), namely

$${\rm{I}}_{ij}^{({\rm{DR}})} = {\rm{I}}_{ij}^{({\rm{pHS}})} + {\mathcal D}{{\rm{I}}_{ij}}\,.$$
(201)

An important fact, hidden in our too-compact notation (201), is that the sum of the two terms in the right-hand side of Eq. (201) does not depend on the Hadamard regularization scales s1 and s2. Therefore it is possible without changing the sum to re-express these two terms (separately) by means of the constants r1 and r′2 instead of s1 and s2, where r1, r2 are the two fiducial scales entering the Hadamard-regularization result (197). This replacement being made the pHS term in Eq. (201) is exactly the same as the one in Eq. (197). At this stage all elements are in place to prove the following theorem [62, 63].

Theorem 10. The DR quadrupole moment (201) is physically equivalent to the Hadamard-regularized one (end result of Refs. [81, 80]), in the sense that

$${\rm{I}}_{ij}^{({\rm{HR}})} = {\lim\limits_{\varepsilon \to 0}} \left[ {{\rm{I}}_{ij}^{({\rm{DR}})} + {\delta _\xi}{{\rm{I}}_{ij}}} \right]\,,$$
(202)

where δξIij denotes the effect of the same shifts as determined in Theorems 8 and 9, if and only if the HR ambiguity parameters ξ, κ and ζ take the unique values reported in Eqs. (173). Moreover, the poles 1/ε separately present in the two terms in the brackets of Eq. (202) cancel out, so that the physical (“dressed”) DR quadrupole moment is finite and given by the limit when ε → 0 as shown in Eq. (202).

This theorem finally provides an unambiguous determination of the 3PN radiation field by dimensional regularization. Furthermore, as reviewed in Section 6.2, several checks of this calculation could be done, which provide independent confirmations for the ambiguity parameters. Such checks also show the powerfulness of dimensional regularization and its validity for describing the classical general-relativistic dynamics of compact bodies.

Newtonian-like Equations of Motion

The 3PN acceleration and energy for particles

We present the acceleration of one of the particles, say the particle 1, at the 3PN order, as well as the 3PN energy of the binary, which is conserved in the absence of radiation reaction. To get this result we used essentially a “direct” post-Newtonian method (issued from Ref. [76]), which consists of reducing the 3PN metric of an extended regular source, worked out in Eqs. (144), to the case where the matter tensor is made of delta functions, and then curing the self-field divergences by means of the Hadamard regularization technique. The equations of motion are simply the 3PN geodesic equations explicitly provided in Eqs. (150)(152); the metric therein is the regularized metric generated by the system of particles itself. Hadamard’s regularization permits to compute all the terms but one, and the Hadamard ambiguity parameter is obtained from dimensional regularization; see Section 6.3. We also add the 3.5PN terms in harmonic coordinates which are known from Refs. [258, 259, 260, 336, 278, 322, 254]. These correspond to radiation reaction effects at relative 1PN order (see Section 5.4 for discussion on radiation reaction up to 1.5PN order).

Though the successive post-Newtonian approximations are really a consequence of general relativity, the final equations of motion must be interpreted in a Newtonian-like fashion. That is, once a convenient general-relativistic (Cartesian) coordinate system is chosen, we should express the results in terms of the coordinate positions, velocities, and accelerations of the bodies, and view the trajectories of the particles as taking place in the absolute Euclidean space of Newton. But because the equations of motion are actually relativistic, they must:

  1. 1.

    Stay manifestly invariant — at least in harmonic coordinates — when we perform a global post-Newtonian-expanded Lorentz transformation;

  2. 2.

    Possess the correct “perturbative” limit, given by the geodesics of the (post-Newtonian-expanded) Schwarzschild metric, when one of the masses tends to zero;

  3. 3.

    Be conservative, i.e., to admit a Lagrangian or Hamiltonian formulation, when the gravitational radiation reaction is turned off.

We denote by r12 = ∣y1(t) − y2(t)∣ the harmonic-coordinate distance between the two particles, with \({y_1} = (y_1^i)\) and \({y_2} = (y_2^i)\), by n12 = (y1y2)/r12 the corresponding unit direction, and by v1 = dy1/dt and a1 = dv1/dt the coordinate velocity and acceleration of the particle 1 (and idem for 2). Sometimes we pose v12 = v1v2 for the relative velocity. The usual Euclidean scalar product of vectors is denoted with parentheses, e.g., (n12v1) = n12 · v1 and (v1v2) = v1 · v2. The equations of the body 2 are obtained by exchanging all the particle labels 1 ↔ 2 (remembering that n12 and v12 change sign in this operation):

$$\begin{array}{*{20}c} {{a_1} = - {{G{m_2}} \over {r_{12}^2}}{n_{12}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\quad + {1 \over {{c^2}}}\left\{{\left[ {{{5{G^2}{m_1}{m_2}} \over {r_{12}^3}} + {{4{G^2}m_2^2} \over {r_{12}^3}} + {{G{m_2}} \over {r_{12}^2}}\left({{3 \over 2}{{({n_{12}}{v_2})}^2} - v_1^2 + 4({v_1}{v_2}) - 2v_2^2} \right)} \right]{n_{12}}\quad \quad \quad \quad} \right.} \\ {\left. {+ {{G{m_2}} \over {r_{12}^2}}\left({4({n_{12}}{v_1}) - 3({n_{12}}{v_2})} \right){v_{12}}} \right\}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\quad \quad + {1 \over {{c^4}}}\left\{{\left[ {- {{57{G^3}m_1^2{m_2}} \over {4r_{12}^4}} - {{69{G^3}{m_1}m_2^2} \over {2r_{12}^4}} - {{9{G^3}m_2^3} \over {r_{12}^4}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \right.} \right.} \\ {\quad \quad \quad \quad + {{G{m_2}} \over {r_{12}^2}}\left({- {{15} \over 8}{{({n_{12}}{v_2})}^4} + {3 \over 2}{{({n_{12}}{v_2})}^2}v_1^2 - 6{{({n_{12}}{v_2})}^2}({v_1}{v_2}) - 2{{({v_1}{v_2})}^2} + {9 \over 2}{{({n_{12}}{v_2})}^2}v_2^2} \right.} \\ {\left. {+ 4({v_1}{v_2})v_2^2 - 2v_2^4} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\quad \quad \quad \quad + {{{G^2}{m_1}{m_2}} \over {r_{12}^3}}\left({{{39} \over 2}{{({n_{12}}{v_1})}^2} - 39({n_{12}}{v_1})({n_{12}}{v_2}) + {{17} \over 2}{{({n_{12}}{v_2})}^2} - {{15} \over 4}v_1^2 - {5 \over 2}({v_1}{v_2}) + {5 \over 4}v_2^2} \right)} \\ {\left. {\quad \quad \quad \quad + {{{G^2}m_2^2} \over {r_{12}^3}}\left({2{{({n_{12}}{v_1})}^2} - 4({n_{12}}{v_1})({n_{12}}{v_2}) - 6{{({n_{12}}{v_2})}^2} - 8({v_1}{v_2}) + 4v_2^2} \right)} \right]{n_{12}}\quad \quad \quad \quad \quad} \\ {\quad \quad \quad \quad + \left[ {{{{G^2}m_2^2} \over {r_{12}^3}}\left({- 2({n_{12}}{v_1}) - 2({n_{12}}{v_2})} \right) + {{{G^2}{m_1}{m_2}} \over {r_{12}^3}}\left({\,\, - {{63} \over 4}({n_{12}}{v_1}) + {{55} \over 4}({n_{12}}{v_2})} \right)} \right.\quad \quad \quad \quad} \\ {\quad \quad \quad \quad \quad + {{G{m_2}} \over {r_{12}^2}}\left({- 6({n_{12}}{v_1}){{({n_{12}}{v_2})}^2} + {9 \over 2}{{({n_{12}}{v_2})}^3} + ({n_{12}}{v_2})v_1^2 - 4({n_{12}}{v_1})({v_1}{v_2})} \right.\quad \quad \quad \quad \quad \quad \quad} \\ {\left. {\left. {\left. {+ \, 4({n_{12}}{v_2})({v_1}{v_2}) + 4({n_{12}}{v_1})v_2^2 - 5({n_{12}}{v_2})v_2^2} \right)} \right]{v_{12}}} \right\}\quad \quad \quad} \\ {\quad + {1 \over {{c^5}}}\left\{{\left[ {{{208{G^3}{m_1}m_2^2} \over {15r_{12}^4}}({n_{12}}{v_{12}}) - {{24{G^3}m_1^2{m_2}} \over {5r_{12}^4}}({n_{12}}{v_{12}}) + {{12{G^2}{m_1}{m_2}} \over {5r_{12}^3}}({n_{12}}{v_{12}})v_{12}^2} \right]} \right.{n_{12}}} \\ {\>\left. {+ \left[ {{{8{G^3}m_1^2{m_2}} \over {5r_{12}^4}} - {{32\,{G^3}{m_1}m_2^2} \over {5r_{12}^4}} - {{4{G^2}{m_1}{m_2}} \over {5r_{12}^3}}v_{12}^2} \right]{v_{12}}} \right\}\quad \quad \quad \quad \quad \quad} \\ {\quad + {1 \over {{c^6}}}\left\{{\left[ {{{G{m_2}} \over {r_{12}^2}}\left({{{35} \over {16}}{{({n_{12}}{v_2})}^6} - {{15} \over 8}{{({n_{12}}{v_2})}^4}v_1^2 + {{15} \over 2}{{({n_{12}}{v_2})}^4}({v_1}{v_2}) + 3{{({n_{12}}{v_2})}^2}{{({v_1}{v_2})}^2}} \right.} \right.} \right.} \\ {- {{15} \over 2}{{({n_{12}}{v_2})}^4}v_2^2 + {3 \over 2}{{({n_{12}}{v_2})}^2}v_1^2v_2^2 - 12{{({n_{12}}{v_2})}^2}({v_1}{v_2})v_2^2 - 2{{({v_1}{v_2})}^2}v_2^2} \\ {\left. {+ {{15} \over 2}{{({n_{12}}{v_2})}^2}v_2^4 + 4({v_1}{v_2})v_2^4 - 2v_2^6} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\quad + {{{G^2}{m_1}{m_2}} \over {r_{12}^3}}\left({- {{171} \over 8}{{({n_{12}}{v_1})}^4} + {{171} \over 2}{{({n_{12}}{v_1})}^3}({n_{12}}{v_2}) - {{723} \over 4}{{({n_{12}}{v_1})}^2}{{({n_{12}}{v_2})}^2}} \right.} \\ {\quad \quad \quad + {{383} \over 2}({n_{12}}{v_1}){{({n_{12}}{v_2})}^3} - {{455} \over 8}{{({n_{12}}{v_2})}^4} + {{229} \over 4}{{({n_{12}}{v_1})}^2}v_1^2} \\ {- {{205} \over 2}({n_{12}}{v_1})({n_{12}}{v_2})v_1^2 + {{191} \over 4}{{({n_{12}}{v_2})}^2}v_1^2 - {{91} \over 8}v_1^4 - {{229} \over 2}{{({n_{12}}{v_1})}^2}({v_1}{v_2})} \\ {+ 244({n_{12}}{v_1})({n_{12}}{v_2})({v_1}{v_2}) - {{225} \over 2}{{({n_{12}}{v_2})}^2}({v_1}{v_2}) + {{91} \over 2}v_1^2({v_1}{v_2})\quad \quad \quad} \\ {- {{177} \over 4}{{({v_1}{v_2})}^2} + {{229} \over 4}{{({n_{12}}{v_1})}^2}v_2^2 - {{283} \over 2}({n_{12}}{v_1})({n_{12}}{v_2})v_2^2\quad \quad \quad \quad \quad} \\ {\left. {+ {{259} \over 4}{{({n_{12}}{v_2})}^2}v_2^2 - {{91} \over 4}v_1^2v_2^2 + 43({v_1}{v_2})v_2^2 - {{81} \over 8}v_2^4} \right)\quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{G^2}m_2^2} \over {r_{12}^3}}\left({- 6{{({n_{12}}{v_1})}^2}{{({n_{12}}{v_2})}^2} + 12({n_{12}}{v_1}){{({n_{12}}{v_2})}^3} + 6{{({n_{12}}{v_2})}^4}} \right.\quad \quad \quad \quad} \\ {+ \, 4({n_{12}}{v_1})({n_{12}}{v_2})({v_1}{v_2}) + 12{{({n_{12}}{v_2})}^2}({v_1}{v_2}) + 4{{({v_1}{v_2})}^2}} \\ {\left. {- \, 4({n_{12}}{v_1})({n_{12}}{v_2})v_2^2 - 12{{({n_{12}}{v_2})}^2}v_2^2 - 8({v_1}{v_2})v_2^2 + 4v_2^4} \right)} \\ {+ {{{G^3}m_2^3} \over {r_{12}^4}}\left({- {{({n_{12}}{v_1})}^2} + 2({n_{12}}{v_1})({n_{12}}{v_2}) + {{43} \over 2}{{({n_{12}}{v_2})}^2} + 18({v_1}{v_2}) - 9v_2^2} \right)} \\ {\quad \quad \quad \quad + {{{G^3}{m_1}m_2^2} \over {r_{12}^4}}\left({{{415} \over 8}{{({n_{12}}{v_1})}^2} - {{375} \over 4}({n_{12}}{v_1})({n_{12}}{v_2}) + {{1113} \over 8}{{({n_{12}}{v_2})}^2} - {{615} \over {64}}{{({n_{12}}{v_{12}})}^2}{\pi ^2}} \right.} \\ {\left. {+ 18v_1^2 + {{123} \over {64}}{\pi ^2}v_{12}^2 + 33({v_1}{v_2}) - {{33} \over 2}v_2^2} \right)\quad \quad \quad} \\ {\quad \quad \quad \quad \quad + {{{G^3}m_1^2{m_2}} \over {r_{12}^4}}\left({- {{45887} \over {168}}{{({n_{12}}{v_1})}^2} + {{24025} \over {42}}({n_{12}}{v_1})({n_{12}}{v_2}) - {{10469} \over {42}}{{({n_{12}}{v_2})}^2} + {{48197} \over {840}}v_1^2} \right.} \\ {\left. {\quad \quad \quad \quad \quad \quad \quad \quad - {{36227} \over {420}}({v_1}{v_2}) + {{36227} \over {840}}v_2^2 + 110{{({n_{12}}{v_{12}})}^2}\ln \left({{{{r_{12}}} \over {{{r\prime}_1}}}} \right) - 22v_{12}^2\ln \left({{{{r_{12}}} \over {{{r\prime}_1}}}} \right)} \right)} \\ {\quad + {{16{G^4}m_2^4} \over {r_{12}^5}} + {{{G^4}m_1^2m_2^2} \over {r_{12}^5}}\left({175 - {{41} \over {16}}{\pi ^2}} \right) + {{{G^4}m_1^3{m_2}} \over {r_{12}^5}}\left({\,\, - {{3187} \over {1260}} + {{44} \over 3}\ln \left({{{{r_{12}}} \over {{{r\prime}_1}}}} \right)} \right)} \\ {\left. {+ {{{G^4}{m_1}m_2^3} \over {r_{12}^5}}\left({{{110741} \over {630}} - {{41} \over {16}}{\pi ^2} - {{44} \over 3}\ln \left({{{{r_{12}}} \over {{{r\prime}_2}}}} \right)} \right)} \right]{n_{12}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ \left[ {{{G{m_2}} \over {r_{12}^2}}\left({{{15} \over 2}({n_{12}}{v_1}){{({n_{12}}{v_2})}^4} - {{45} \over 8}{{({n_{12}}{v_2})}^5} - {3 \over 2}{{({n_{12}}{v_2})}^3}v_1^2 + 6({n_{12}}{v_1}){{({n_{12}}{v_2})}^2}({v_1}{v_2})} \right.} \right.} \\ {\quad \quad \quad - \, 6{{({n_{12}}{v_2})}^3}({v_1}{v_2}) - 2({n_{12}}{v_2}){{({v_1}{v_2})}^2} - 12({n_{12}}{v_1}){{({n_{12}}{v_2})}^2}v_2^2 + 12{{({n_{12}}{v_2})}^3}v_2^2} \\ {\quad \quad \quad + ({n_{12}}{v_2})v_1^2v_2^2 - 4({n_{12}}{v_1})({v_1}{v_2})v_2^2 + 8({n_{12}}{v_2})({v_1}{v_2})v_2^2 + 4({n_{12}}{v_1})v_2^4\quad \quad \quad} \\ {\left. {- \, 7({n_{12}}{v_2})v_2^4} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\, + {{{G^2}m_2^2} \over {r_{12}^3}}\left({- 2{{({n_{12}}{v_1})}^2}({n_{12}}{v_2}) + 8({n_{12}}{v_1}){{({n_{12}}{v_2})}^2} + 2{{({n_{12}}{v_2})}^3} + 2({n_{12}}{v_1})({v_1}{v_2})} \right.\quad \quad \quad} \\ {\left. {+ 4({n_{12}}{v_2})({v_1}{v_2}) - 2({n_{12}}{v_1})v_2^2 - 4({n_{12}}{v_2})v_2^2} \right)\quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{G^2}{m_1}{m_2}} \over {r_{12}^3}}\left({- {{243} \over 4}{{({n_{12}}{v_1})}^3} + {{565} \over 4}{{({n_{12}}{v_1})}^2}({n_{12}}{v_2}) - {{269} \over 4}({n_{12}}{v_1}){{({n_{12}}{v_2})}^2}} \right.\quad \quad \quad \quad} \\ {\quad \quad \quad - {{95} \over {12}}{{({n_{12}}{v_2})}^3} + {{207} \over 8}({n_{12}}{v_1})v_1^2 - {{137} \over 8}({n_{12}}{v_2})v_1^2 - 36({n_{12}}{v_1})({v_1}{v_2})} \\ {\left. {+ {{27} \over 4}({n_{12}}{v_2})({v_1}{v_2}) + {{81} \over 8}({n_{12}}{v_1})v_2^2 + {{83} \over 8}({n_{12}}{v_2})v_2^2} \right)\quad \quad} \\ {+ {{{G^3}m_2^3} \over {r_{12}^4}}\left({4({n_{12}}{v_1}) + 5({n_{12}}{v_2})} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\, + {{{G^3}{m_1}m_2^2} \over {r_{12}^4}}\left({\,\, - {{307} \over 8}({n_{12}}{v_1}) + {{479} \over 8}({n_{12}}{v_2}) + {{123} \over {32}}({n_{12}}{v_{12}}){\pi ^2}} \right)\quad \quad \quad \quad \quad \quad \quad \quad} \\ {\left. {\left. {+ {{{G^3}m_1^2{m_2}} \over {r_{12}^4}}\left({{{31397} \over {420}}({n_{12}}{v_1}) - {{36227} \over {420}}({n_{12}}{v_2}) - 44({n_{12}}{v_{12}})\ln \left({{{{r_{12}}} \over {{{r\prime}_1}}}} \right)} \right)} \right]{v_{12}}} \right\}\quad \quad \quad} \\ {+ {1 \over {{c^7}}}\left\{{\left[ {{{{G^4}m_1^3{m_2}} \over {r_{12}^5}}\left({{{3992} \over {105}}({n_{12}}{v_1}) - {{4328} \over {105}}({n_{12}}{v_2})} \right)} \right.} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{G^4}m_1^2m_2^2} \over {r_{12}^6}}\left({- {{13576} \over {105}}({n_{12}}{v_1}) + {{2872} \over {21}}({n_{12}}{v_2})} \right) - {{3172} \over {21}}{{{G^4}{m_1}m_2^3} \over {r_{12}^6}}({n_{12}}{v_{12}})\quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{G^3}m_1^2{m_2}} \over {r_{12}^4}}\left({48{{({n_{12}}{v_1})}^3} - {{696} \over 5}{{({n_{12}}{v_1})}^2}({n_{12}}{v_2}) + {{744} \over 5}({n_{12}}{v_1}){{({n_{12}}{v_2})}^2} - {{288} \over 5}{{({n_{12}}{v_2})}^3}} \right.\quad \quad} \\ {- {{4888} \over {105}}({n_{12}}{v_1})v_1^2 + {{5056} \over {105}}({n_{12}}{v_2})v_1^2 + {{2056} \over {21}}({n_{12}}{v_1})({v_1}{v_2})\quad \quad \quad} \\ {\left. {- {{2224} \over {21}}({n_{12}}{v_2})({v_1}{v_2}) - {{1028} \over {21}}({n_{12}}{v_1})v_2^2 + {{5812} \over {105}}({n_{12}}{v_2})v_2^2} \right)\quad \quad} \\ {\, + {{{G^3}{m_1}m_2^2} \over {r_{12}^4}}\left({- {{582} \over 5}{{({n_{12}}{v_1})}^3} + {{1746} \over 5}{{({n_{12}}{v_1})}^2}({n_{12}}{v_2}) - {{1954} \over 5}({n_{12}}{v_1}){{({n_{12}}{v_2})}^2}} \right.\quad \quad \quad \quad \quad \,\,} \\ {+ 158{{({n_{12}}{v_2})}^3} + {{3568} \over {105}}({n_{12}}{v_{12}})v_1^2 - {{2864} \over {35}}({n_{12}}{v_1})({v_1}{v_2})\quad \quad \quad \quad \quad} \\ {\left. {+ {{10048} \over {105}}({n_{12}}{v_2})({v_1}{v_2}) + {{1432} \over {35}}({n_{12}}{v_1})v_2^2 - {{5752} \over {105}}({n_{12}}{v_2})v_2^2} \right)\quad \quad \quad} \\ {+ {{{G^2}{m_1}{m_2}} \over {r_{12}^3}}\left({- 56{{({n_{12}}{v_{12}})}^5} + 60{{({n_{12}}{v_1})}^3}v_{12}^2 - 180{{({n_{12}}{v_1})}^2}({n_{12}}{v_2})v_{12}^2} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ 174({n_{12}}{v_1}){{({n_{12}}{v_2})}^2}v_{12}^2 - 54{{({n_{12}}{v_2})}^3}v_{12}^2 - {{246} \over {35}}({n_{12}}{v_{12}})v_1^4\quad \quad \quad \quad} \\ {+ {{1068} \over {35}}({n_{12}}{v_1})v_1^2({v_1}{v_2}) - {{984} \over {35}}({n_{12}}{v_2})v_1^2({v_1}{v_2}) - {{1068} \over {35}}({n_{12}}{v_1}){{({v_1}{v_2})}^2}} \\ {+ {{180} \over 7}({n_{12}}{v_2}){{({v_1}{v_2})}^2} - {{534} \over {35}}({n_{12}}{v_1})v_1^2v_2^2 + {{90} \over 7}({n_{12}}{v_2})v_1^2v_2^2\quad \quad \quad \quad} \\ {+ {{984} \over {35}}({n_{12}}{v_1})({v_1}{v_2})v_2^2 - {{732} \over {35}}({n_{12}}{v_2})({v_1}{v_2})v_2^2 - {{204} \over {35}}({n_{12}}{v_1})v_2^4\quad \quad} \\ {\left. {\left. {+ {{24} \over 7}({n_{12}}{v_2})v_2^4} \right)} \right]{n_{12}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ \left[ {- {{184} \over {21}}{{{G^4}m_1^3{m_2}} \over {r_{12}^5}} + {{6224} \over {105}}{{{G^4}m_1^2m_2^2} \over {r_{12}^6}} + {{6388} \over {105}}{{{G^4}{m_1}m_2^3} \over {r_{12}^6}}} \right.\quad \quad \quad \quad \,\,\,} \\ {\quad \quad \quad \quad \quad \quad \quad + {{{G^3}m_1^2{m_2}} \over {r_{12}^4}}\left({{{52} \over {15}}{{({n_{12}}{v_1})}^2} - {{56} \over {15}}({n_{12}}{v_1})({n_{12}}{v_2}) - {{44} \over {15}}{{({n_{12}}{v_2})}^2} - {{132} \over {35}}v_1^2 + {{152} \over {35}}({v_1}{v_2})} \right.} \\ {\left. {- {{48} \over {35}}v_2^2} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\quad \quad \quad \quad + {{{G^3}{m_1}m_2^2} \over {r_{12}^4}}\left({{{454} \over {15}}{{({n_{12}}{v_1})}^2} - {{372} \over 5}({n_{12}}{v_1})({n_{12}}{v_2}) + {{854} \over {15}}{{({n_{12}}{v_2})}^2} - {{152} \over {21}}v_1^2} \right.} \\ {\left. {+ {{2864} \over {105}}({v_1}{v_2}) - {{1768} \over {105}}v_2^2} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{G^2}{m_1}{m_2}} \over {r_{12}^3}}\left({60{{({n_{12}}{v_{12}})}^4} - {{348} \over 5}{{({n_{12}}{v_1})}^2}v_{12}^2 + {{684} \over 5}({n_{12}}{v_1})({n_{12}}{v_2})v_{12}^2} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {- 66{{({n_{12}}{v_2})}^2}v_{12}^2 + {{334} \over {35}}v_1^4 - {{1336} \over {35}}v_1^2({v_1}{v_2}) + {{1308} \over {35}}{{({v_1}{v_2})}^2} + {{654} \over {35}}v_1^2v_2^2} \\ {\left. {\left. {\left. {- {{1252} \over {35}}({v_1}{v_2})v_2^2 + {{292} \over {35}}v_2^4} \right)} \right]{v_{12}}} \right\} + {\mathcal O}\left({{1 \over {{c^8}}}} \right)\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(203)

The 2.5PN and 3.5PN terms are associated with gravitational radiation reaction.Footnote 49 The 3PN harmonic-coordinates equations of motion depend on two arbitrary length scales r1 and r2 associated with the logarithms present at the 3PN order. It has been proved in Ref. [71] that r1 and r′2 are merely linked with the choice of coordinates — we can refer to r1 and r2 as “gauge constants”. In our approach [69, 71], the harmonic coordinate system is not uniquely fixed by the coordinate condition αhαμ = 0. In fact there are infinitely many “locally-defined” harmonic coordinate systems. For general smooth matter sources, as in the general formalism of Part A, we expect the existence and uniqueness of a global harmonic coordinate system. But here we have some point-particles, with delta-function singularities, and in this case we do not have the notion of a global coordinate system. We can always change the harmonic coordinates by means of the gauge vector ηα = δxα, satisfying Δηα = 0 except at the location of the two particles (we assume that the transformation is at the 3PN level, so we can consider simply a flat-space Laplace equation). More precisely, we can show that the logarithms appearing in Eq. (203), together with the constants r1and r2 therein, can be removed by the coordinate transformation associated with the 3PN gauge vector (with r1 = ∣xy1(t)∣ and r2 = ∣xy2(t)∣; and α = ηαμμ):

$${\eta ^\alpha} = - {{22} \over 3}{{{G^2}{m_1}{m_2}} \over {{c^6}}}\,{\partial ^\alpha}\left[ {{{G{m_1}} \over {{r_2}}}\ln \left({{{{r_{12}}} \over {{{r\prime}_1}}}} \right) + {{G{m_2}} \over {{r_1}}}\ln \left({{{{r_{12}}} \over {{{r\prime}_2}}}} \right)} \right]\,.$$
(204)

Therefore, the arbitrariness in the choice of the constants r1 and r2 is innocuous on the physical point of view, because the physical results must be gauge invariant. Indeed we shall verify that r1and r2 cancel out in our final results.

When retaining the “even” relativistic corrections at the 1PN, 2PN and 3PN orders, and neglecting the “odd” radiation reaction terms at the 2.5PN and 3.5PN orders, we find that the equations of motion admit a conserved energy (and a Lagrangian, as we shall see); that energy can be straightforwardly obtained by guess-work starting from Eq. (203), with the result

$$\begin{array}{*{20}c} {E = {{{m_1}v_1^2} \over 2} - {{G{m_1}{m_2}} \over {2{r_{12}}}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {1 \over {{c^2}}}\left\{{{{{G^2}m_1^2{m_2}} \over {2r_{12}^2}} + {{3{m_1}v_1^4} \over 8} + {{G{m_1}{m_2}} \over {{r_{12}}}}\left({- {1 \over 4}({n_{12}}{v_1})({n_{12}}{v_2}) + {3 \over 2}v_1^2 - {7 \over 4}({v_1}{v_2})} \right)} \right\}} \\ {+ {1 \over {{c^4}}}\left\{{- {{{G^3}m_1^3{m_2}} \over {2r_{12}^3}} - {{19{G^3}m_1^2m_2^2} \over {8r_{12}^3}} + {{5{m_1}v_1^6} \over {16}}} \right.\,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,} \\ {\quad \quad \, + {{G{m_1}{m_2}} \over {{r_{12}}}}\left({{3 \over 8}{{({n_{12}}{v_1})}^3}({n_{12}}{v_2}) + {3 \over {16}}{{({n_{12}}{v_1})}^2}{{({n_{12}}{v_2})}^2} - {9 \over 8}({n_{12}}{v_1})({n_{12}}{v_2})v_1^2} \right.} \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad - {{13} \over 8}{{({n_{12}}{v_2})}^2}v_1^2 + {{21} \over 8}v_1^4 + {{13} \over 8}{{({n_{12}}{v_1})}^2}({v_1}{v_2}) + {3 \over 4}({n_{12}}{v_1})({n_{12}}{v_2})({v_1}{v_2})} \\ {\left. {- {{55} \over 8}v_1^2({v_1}{v_2}) + {{17} \over 8}{{({v_1}{v_2})}^2} + {{31} \over {16}}v_1^2v_2^2} \right)\quad \,\quad} \\ {\left. {\quad + {{{G^2}m_1^2{m_2}} \over {r_{12}^2}}\left({{{29} \over 4}{{({n_{12}}{v_1})}^2} - {{13} \over 4}({n_{12}}{v_1})({n_{12}}{v_2}) + {1 \over 2}{{({n_{12}}{v_2})}^2} - {3 \over 2}v_1^2 + {7 \over 4}v_2^2} \right)} \right\}} \\ {+ {1 \over {{c^6}}}\left\{{{{35{m_1}v_1^8} \over {128}}} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\quad + {{G{m_1}{m_2}} \over {{r_{12}}}}\left({- {5 \over {16}}{{({n_{12}}{v_1})}^5}({n_{12}}{v_2}) - {5 \over {16}}{{({n_{12}}{v_1})}^4}{{({n_{12}}{v_2})}^2} - {5 \over {32}}{{({n_{12}}{v_1})}^3}{{({n_{12}}{v_2})}^3}} \right.} \\ {\quad \quad \quad \quad \quad \quad \quad + {{19} \over {16}}{{({n_{12}}{v_1})}^3}({n_{12}}{v_2})v_1^2 + {{15} \over {16}}{{({n_{12}}{v_1})}^2}{{({n_{12}}{v_2})}^2}v_1^2 + {3 \over 4}({n_{12}}{v_1}){{({n_{12}}{v_2})}^3}v_1^2} \\ {\quad + {{19} \over {16}}{{({n_{12}}{v_2})}^4}v_1^2 - {{21} \over {16}}({n_{12}}{v_1})({n_{12}}{v_2})v_1^4 - 2{{({n_{12}}{v_2})}^2}v_1^4} \\ {\,\, + {{55} \over {16}}v_1^6 - {{19} \over {16}}{{({n_{12}}{v_1})}^4}({v_1}{v_2}) - {{({n_{12}}{v_1})}^3}({n_{12}}{v_2})({v_1}{v_2})} \\ {- {{15} \over {32}}{{({n_{12}}{v_1})}^2}{{({n_{12}}{v_2})}^2}({v_1}{v_2}) + {{45} \over {16}}{{({n_{12}}{v_1})}^2}v_1^2({v_1}{v_2})} \\ {\quad \quad \quad \quad \quad + {5 \over 4}({n_{12}}{v_1})({n_{12}}{v_2})v_1^2({v_1}{v_2}) + {{11} \over 4}{{({n_{12}}{v_2})}^2}v_1^2({v_1}{v_2}) - {{139} \over {16}}v_1^4({v_1}{v_2})} \\ {\quad \quad \quad \quad \quad \quad \quad - {3 \over 4}{{({n_{12}}{v_1})}^2}{{({v_1}{v_2})}^2} + {5 \over {16}}({n_{12}}{v_1})({n_{12}}{v_2}){{({v_1}{v_2})}^2} + {{41} \over 8}v_1^2{{({v_1}{v_2})}^2} + {1 \over {16}}{{({v_1}{v_2})}^3}} \\ {\left. {\quad \quad \quad \quad \quad \quad - {{45} \over {16}}{{({n_{12}}{v_1})}^2}v_1^2v_2^2 - {{23} \over {32}}({n_{12}}{v_1})({n_{12}}{v_2})v_1^2v_2^2 + {{79} \over {16}}v_1^4v_2^2 - {{161} \over {32}}v_1^2({v_1}{v_2})v_2^2} \right)} \\ {+ {{{G^2}m_1^2{m_2}} \over {r_{12}^2}}\left({- {{49} \over 8}{{({n_{12}}{v_1})}^4} + {{75} \over 8}{{({n_{12}}{v_1})}^3}({n_{12}}{v_2}) - {{187} \over 8}{{({n_{12}}{v_1})}^2}{{({n_{12}}{v_2})}^2}} \right.\quad} \\ {\quad \quad \quad \quad + {{247} \over {24}}({n_{12}}{v_1}){{({n_{12}}{v_2})}^3} + {{49} \over 8}{{({n_{12}}{v_1})}^2}v_1^2 + {{81} \over 8}({n_{12}}{v_1})({n_{12}}{v_2})v_1^2} \\ {\quad \quad \quad \quad \quad \quad \quad - {{21} \over 4}{{({n_{12}}{v_2})}^2}v_1^2 + {{11} \over 2}v_1^4 - {{15} \over 2}{{({n_{12}}{v_1})}^2}({v_1}{v_2}) - {3 \over 2}({n_{12}}{v_1})({n_{12}}{v_2})({v_1}{v_2})} \\ {\quad \quad \quad \quad \quad + {{21} \over 4}{{({n_{12}}{v_2})}^2}({v_1}{v_2}) - 27v_1^2({v_1}{v_2}) + {{55} \over 2}{{({v_1}{v_2})}^2} + {{49} \over 4}{{({n_{12}}{v_1})}^2}v_2^2} \\ {\left. {\quad \quad \quad \quad \quad \quad \quad \quad - {{27} \over 2}({n_{12}}{v_1})({n_{12}}{v_2})v_2^2 + {3 \over 4}{{({n_{12}}{v_2})}^2}v_2^2 + {{55} \over 4}v_1^2v_2^2 - 28({v_1}{v_2})v_2^2 + {{135} \over {16}}v_2^4} \right)} \\ {+ {{3{G^4}m_1^4{m_2}} \over {8r_{12}^4}} + {{{G^4}m_1^3m_2^2} \over {r_{12}^4}}\left({{{9707} \over {420}} - {{22} \over 3}\ln \left({{{{r_{12}}} \over {{{r\prime}_1}}}} \right)\,} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,\,} \\ {+ {{{G^3}m_1^2m_2^2} \over {r_{12}^3}}\left({{{547} \over {12}}{{({n_{12}}{v_1})}^2} - {{3115} \over {48}}({n_{12}}{v_1})({n_{12}}{v_2}) - {{123} \over {64}}({n_{12}}{v_1})({n_{12}}{v_{12}}){\pi ^2} - {{575} \over {18}}v_1^2} \right.} \\ {\left. {+ {{41} \over {64}}{\pi ^2}({v_1}{v_{12}}) + {{4429} \over {144}}({v_1}{v_2})} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{G^3}m_1^3{m_2}} \over {r_{12}^3}}\left({- {{44627} \over {840}}{{({n_{12}}{v_1})}^2} + {{32027} \over {840}}({n_{12}}{v_1})({n_{12}}{v_2}) + {3 \over 2}{{({n_{12}}{v_2})}^2} + {{24187} \over {2520}}v_1^2} \right.} \\ {\left. {\left. {\quad \quad \quad \quad \quad \quad \quad - {{27967} \over {2520}}({v_1}{v_2}) + {5 \over 4}v_2^2 + 22({n_{12}}{v_1})({n_{12}}{v_{12}})\ln \,\left({\,{{{r_{12}}} \over {{{r\prime}_1}}}\,} \right)\, - {{22} \over 3}({v_1}{v_{12}})\ln \,\left({\,{{{r_{12}}} \over {{{r\prime}_1}}}\,} \right)} \right)} \right\}} \\ {+ 1 \leftrightarrow 2 + {\mathcal O}\left({{1 \over {{c^7}}}} \right)\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(205)

To the terms given above, we must add the same terms but corresponding to the relabelling 1 ↔ 2. Actually, this energy is not conserved because of the radiation reaction. Thus its time derivative, as computed by means of the 3PN equations of motion themselves (i.e., by order-reducing all the accelerations), is purely equal to the 2.5PN effect,

$$\begin{array}{*{20}c} {{{{\rm{d}}E} \over {{\rm{d}}t}} = {4 \over 5}{{{G^2}m_1^2{m_2}} \over {{c^5}r_{12}^3}}\,\left[ {({v_1}{v_{12}})\left({\,\, - v_{12}^2 + 2{{G{m_1}} \over {{r_{12}}}}\, - 8{{G{m_2}} \over {{r_{12}}}}\,} \right) + ({n_{12}}{v_1})({n_{12}}{v_{12}})\left({\, 3v_{12}^2 - 6{{G{m_1}} \over {{r_{12}}}}\, + {{52} \over 3}{{G{m_2}} \over {{r_{12}}}}\,} \right)\,} \right]} \\ {+ 1 \leftrightarrow 2 + {\mathcal O}\left({{1 \over {{c^7}}}} \right)\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(206)

The resulting energy balance equation can be better expressed by transfering to the left-hand side certain 2.5PN terms so that we recognize in the right-hand side the familiar form of a total energy flux. Posing

$${E^{{\rm{2}}{\rm{.5PN}}}} = E + {{4{G^2}m_1^2{m_2}} \over {5{c^5}r_{12}^2}}({n_{12}}{v_1})\left[ {v_{12}^2 - {{2G({m_1} - {m_2})} \over {{r_{12}}}}} \right] + 1 \leftrightarrow 2\,,$$
(207)

we find agreement with the standard Einstein quadrupole formula (4):

$${{{\rm{d}}{E^{{\rm{2}}{\rm{.5PN}}}}} \over {{\rm{d}}t}} = - {G \over {5{c^5}}}{{{{\rm{d}}^3}{{\rm{Q}}_{ij}}} \over {{\rm{d}}{t^3}}}{{{{\rm{d}}^3}{{\rm{Q}}_{ij}}} \over {{\rm{d}}{t^3}}} + {\mathcal O}\left({{1 \over {{c^7}}}} \right)\,,$$
(208)

where the Newtonian trace-free quadrupole moment reads \({{\rm{Q}}_{ij}} = {m_1}(y_1^iy_1^i - {1 \over 3}{\delta ^{ij}}y_1^2) + 1 \leftrightarrow 2\). We refer to [258, 259] for the discussion of the energy balance equation up to the next 3.5PN order. See also Eq. (158) for the energy balance equation at relative 1.5PN order for general fluid systems.

Lagrangian and Hamiltonian formulations

The conservative part of the equations of motion in harmonic coordinates (203) is derivable from a generalized Lagrangian, depending not only on the positions and velocities of the bodies, but also on their accelerations: a1 = dv1/dt and a2 = dv2/dt. As shown in Ref. [147], the accelerations in the harmonic-coordinates Lagrangian occur already from the 2PN order. This fact is in accordance with a general result [308] that N-body equations of motion cannot be derived from an ordinary Lagrangian beyond the 1PN level, provided that the gauge conditions preserve the manifest Lorentz invariance. Note that we can always arrange for the dependence of the Lagrangian upon the accelerations to be linear, at the price of adding some so-called “multi-zero” terms to the Lagrangian, which do not modify the equations of motion (see, e.g., Ref. [169]). At the 3PN level, we find that the Lagrangian also depends on accelerations. It is notable that these accelerations are sufficient — there is no need to include derivatives of accelerations. Note also that the Lagrangian is not unique because we can always add to it a total time derivative dF/dt, where F is any function depending on the positions and velocities, without changing the dynamics. We find [174]

$$\begin{array}{*{20}c} {{L^{{\rm{harm}}}} = {{G{m_1}{m_2}} \over {2{r_{12}}}} + {{{m_1}v_1^2} \over 2}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {1 \over {{c^2}}}\left\{{- {{{G^2}m_1^2{m_2}} \over {2r_{12}^2}} + {{{m_1}v_1^4} \over 8} + {{G{m_1}{m_2}} \over {{r_{12}}}}\left({- {1 \over 4}({n_{12}}{v_1})({n_{12}}{v_2}) + {3 \over 2}v_1^2 - {7 \over 4}({v_1}{v_2})} \right)} \right\}} \\ {+ {1 \over {{c^4}}}\left\{{{{{G^3}m_1^3{m_2}} \over {2r_{12}^3}} + {{19{G^3}m_1^2m_2^2} \over {8r_{12}^3}}} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,\,\,} \\ {\quad \quad \quad \quad \quad + {{{G^2}m_1^2{m_2}} \over {r_{12}^2}}\left({{7 \over 2}{{({n_{12}}{v_1})}^2} - {7 \over 2}({n_{12}}{v_1})({n_{12}}{v_2}) + {1 \over 2}{{({n_{12}}{v_2})}^2} + {1 \over 4}v_1^2 - {7 \over 4}({v_1}{v_2}) + {7 \over 4}v_2^2} \right)} \\ {\quad \quad \quad \quad + {{G{m_1}{m_2}} \over {{r_{12}}}}\left({{3 \over {16}}{{({n_{12}}{v_1})}^2}{{({n_{12}}{v_2})}^2} - {7 \over 8}{{({n_{12}}{v_2})}^2}v_1^2 + {7 \over 8}v_1^4 + {3 \over 4}({n_{12}}{v_1})({n_{12}}{v_2})({v_1}{v_2})} \right.} \\ {\left. {- 2v_1^2({v_1}{v_2}) + {1 \over 8}{{({v_1}{v_2})}^2} + {{15} \over {16}}v_1^2v_2^2} \right) + {{{m_1}v_1^6} \over {16}}} \\ {\left. {+ G{m_1}{m_2}\left({- {7 \over 4}({a_1}{v_2})({n_{12}}{v_2}) - {1 \over 8}({n_{12}}{a_1}){{({n_{12}}{v_2})}^2} + {7 \over 8}({n_{12}}{a_1})v_2^2} \right)} \right\}} \\ {\quad \quad \quad \quad \quad + {1 \over {{c^6}}}\left\{{{{{G^2}m_1^2{m_2}} \over {r_{12}^2}}\left({{{13} \over {18}}{{({n_{12}}{v_1})}^4} + {{83} \over {18}}{{({n_{12}}{v_1})}^3}({n_{12}}{v_2}) - {{35} \over 6}{{({n_{12}}{v_1})}^2}{{({n_{12}}{v_2})}^2} - {{245} \over {24}}{{({n_{12}}{v_1})}^2}v_1^2} \right.} \right.} \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + {{179} \over {12}}({n_{12}}{v_1})({n_{12}}{v_2})v_1^2 - {{235} \over {24}}{{({n_{12}}{v_2})}^2}v_1^2 + {{373} \over {48}}v_1^4 + {{529} \over {24}}{{({n_{12}}{v_1})}^2}({v_1}{v_2})} \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - {{97} \over 6}({n_{12}}{v_1})({n_{12}}{v_2})({v_1}{v_2}) - {{719} \over {24}}v_1^2({v_1}{v_2}) + {{463} \over {24}}{{({v_1}{v_2})}^2} - {7 \over {24}}{{({n_{12}}{v_1})}^2}v_2^2} \\ {\left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - {1 \over 2}({n_{12}}{v_1})({n_{12}}{v_2})v_2^2 + {1 \over 4}{{({n_{12}}{v_2})}^2}v_2^2 + {{463} \over {48}}v_1^2v_2^2 - {{19} \over 2}({v_1}{v_2})v_2^2 + {{45} \over {16}}v_2^4} \right)} \\ {+ {{5{m_1}v_1^8} \over {128}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\quad \quad \quad \quad \quad \quad + G{m_1}{m_2}\left({{3 \over 8}({a_1}{v_2})({n_{12}}{v_1}){{({n_{12}}{v_2})}^2} + {5 \over {12}}({a_1}{v_2}){{({n_{12}}{v_2})}^3} + {1 \over 8}({n_{12}}{a_1})({n_{12}}{v_1}){{({n_{12}}{v_2})}^3}} \right.} \\ {\quad \quad \quad \quad \quad \quad \quad + {1 \over {16}}({n_{12}}{a_1}){{({n_{12}}{v_2})}^4} + {{11} \over 4}({a_1}{v_1})({n_{12}}{v_2})v_1^2 - ({a_1}{v_2})({n_{12}}{v_2})v_1^2} \\ {\quad \quad - 2({a_1}{v_1})({n_{12}}{v_2})({v_1}{v_2}) + {1 \over 4}({a_1}{v_2})({n_{12}}{v_2})({v_1}{v_2})} \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + {3 \over 8}({n_{12}}{a_1}){{({n_{12}}{v_2})}^2}({v_1}{v_2}) - {5 \over 8}({n_{12}}{a_1}){{({n_{12}}{v_1})}^2}v_2^2 + {{15} \over 8}({a_1}{v_1})({n_{12}}{v_2})v_2^2} \\ {\quad \quad \quad - {{15} \over 8}({a_1}{v_2})({n_{12}}{v_2})v_2^2 - {1 \over 2}({n_{12}}{a_1})({n_{12}}{v_1})({n_{12}}{v_2})v_2^2} \\ {\left. {- {5 \over {16}}({n_{12}}{a_1}){{({n_{12}}{v_2})}^2}v_2^2} \right)\quad \quad \quad \quad \quad \quad} \\ {\quad + {{{G^2}m_1^2{m_2}} \over {{r_{12}}}}\left({- {{235} \over {24}}({a_2}{v_1})({n_{12}}{v_1}) - {{29} \over {24}}({n_{12}}{a_2}){{({n_{12}}{v_1})}^2} - {{235} \over {24}}({a_1}{v_2})({n_{12}}{v_2})} \right.} \\ {\quad \quad - {{17} \over 6}({n_{12}}{a_1}){{({n_{12}}{v_2})}^2} + {{185} \over {16}}({n_{12}}{a_1})v_1^2 - {{235} \over {48}}({n_{12}}{a_2})v_1^2} \\ {\left. {- {{185} \over 8}({n_{12}}{a_1})({v_1}{v_2}) + {{20} \over 3}({n_{12}}{a_1})v_2^2} \right)\quad \quad \quad \quad} \\ {+ {{G{m_1}{m_2}} \over {{r_{12}}}}\left({- {5 \over {32}}{{({n_{12}}{v_1})}^3}{{({n_{12}}{v_2})}^3} + {1 \over 8}({n_{12}}{v_1}){{({n_{12}}{v_2})}^3}v_1^2 + {5 \over 8}{{({n_{12}}{v_2})}^4}v_1^2} \right.} \\ {- {{11} \over {16}}({n_{12}}{v_1})({n_{12}}{v_2})v_1^4 + {1 \over 4}{{({n_{12}}{v_2})}^2}v_1^4 + {{11} \over {16}}v_1^6\quad} \\ {\quad \quad - {{15} \over {32}}{{({n_{12}}{v_1})}^2}{{({n_{12}}{v_2})}^2}({v_1}{v_2}) + ({n_{12}}{v_1})({n_{12}}{v_2})v_1^2({v_1}{v_2})} \\ {\quad \quad \quad \quad \quad + {3 \over 8}{{({n_{12}}{v_2})}^2}v_1^2({v_1}{v_2}) - {{13} \over {16}}v_1^4({v_1}{v_2}) + {5 \over {16}}({n_{12}}{v_1})({n_{12}}{v_2}){{({v_1}{v_2})}^2}} \\ {\quad \quad \quad \quad \quad \quad + {1 \over {16}}{{({v_1}{v_2})}^3} - {5 \over 8}{{({n_{12}}{v_1})}^2}v_1^2v_2^2 - {{23} \over {32}}({n_{12}}{v_1})({n_{12}}{v_2})v_1^2v_2^2 + {1 \over {16}}v_1^4v_2^2} \\ {\left. {- {1 \over {32}}v_1^2({v_1}{v_2})v_2^2} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {- {{3{G^4}m_1^4{m_2}} \over {8r_{12}^4}} + {{{G^4}m_1^3m_2^2} \over {r_{12}^4}}\left({- {{9707} \over {420}} + {{22} \over 3}\ln \left({{{{r_{12}}} \over {{{r\prime}_1}}}} \right)} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{G^3}m_1^2m_2^2} \over {r_{12}^3}}\left({{{383} \over {24}}{{({n_{12}}{v_1})}^2} - {{889} \over {48}}({n_{12}}{v_1})({n_{12}}{v_2}) - {{123} \over {64}}({n_{12}}{v_1})({n_{12}}{v_{12}}){\pi ^2} - {{305} \over {72}}v_1^2} \right.} \\ {\left. {+ {{41} \over {64}}{\pi ^2}({v_1}{v_{12}}) + {{439} \over {144}}({v_1}{v_2})} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{G^3}m_1^3{m_2}} \over {r_{12}^3}}\left({- {{8243} \over {210}}{{({n_{12}}{v_1})}^2} + {{15541} \over {420}}({n_{12}}{v_1})({n_{12}}{v_2}) + {3 \over 2}{{({n_{12}}{v_2})}^2} + {{15611} \over {1260}}v_1^2} \right.\quad} \\ \end{array}$$
(209)

Witness the accelerations occurring at the 2PN and 3PN orders; see also the gauge-dependent logarithms of r12/r1 and r12/r2. We refer to [174] for the explicit expressions of the ten conserved quantities corresponding to the integrals of energy [also given in Eq. (205)], linear and angular momenta, and center-of-mass position. Notice that while it is strictly forbidden to replace the accelerations by the equations of motion in the Lagrangian, this can and should be done in the final expressions of the conserved integrals derived from that Lagrangian.

Now we want to exhibit a transformation of the particles’ dynamical variables — or contact transformation, as it is called in the jargon — which transforms the 3PN harmonic-coordinates Lagrangian (209) into a new Lagrangian, valid in some ADM or ADM-like coordinate system, and such that the associated Hamiltonian coincides with the 3PN Hamiltonian that has been obtained by Jaranowski & Schäfer [261, 262]. In ADM coordinates the Lagrangian will be ordinary, depending only on the positions and velocities of the bodies. Let this contact transformation be Y1(t) = y1(t) + δy1(t) and 1 ↔ 2, where Y1 and y1 denote the trajectories in ADM and harmonic coordinates, respectively. For this transformation to be able to remove all the accelerations in the initial Lagrangian Lharm up to the 3PN order, we determine [174] it to be necessarily of the form

$$\delta {y_1} = {1 \over {{m_1}}}\left[ {{{\partial {L^{{\rm{harm}}}}} \over {\partial {a_1}}} + {{\partial F} \over {\partial {v_1}}} + {1 \over {{c^6}}}{X_1}} \right] + {\mathcal O}\left({{1 \over {{c^8}}}} \right)\qquad ({\rm{and}}\,idem\, 1 \leftrightarrow 2)\,,$$
(210)

where F is a freely adjustable function of the positions and velocities, made of 2PN and 3PN terms, and where X1 represents a special correction term, that is purely of order 3PN. The point is that once the function F is specified there is a unique determination of the correction term X1 for the contact transformation to work (see Ref. [174] for the details). Thus, the freedom we have is entirely encoded into the function F, and the work then consists in showing that there exists a unique choice of F for which our Lagrangian Lharm is physically equivalent, via the contact transformation (210), to the ADM Hamiltonian of Refs. [261, 262]. An interesting point is that not only the transformation must remove all the accelerations in Lharm, but it should also cancel out all the logarithms ln(r12/r1) and ln(r12/r2), because there are no logarithms in ADM coordinates. The result we find, which can be checked to be in full agreement with the expression of the gauge vector in Eq. (204), is that F involves the logarithmic terms

$$F = {{22} \over 3}{{{G^3}{m_1}{m_2}} \over {{c^6}r_{12}^2}}\left[ {m_1^2({n_{12}}{v_1})\ln \left({{{{r_{12}}} \over {{{r\prime}_1}}}} \right) - m_2^2({n_{12}}{v_2})\ln \left({{{{r_{12}}} \over {{{r\prime}_2}}}} \right)} \right] + \cdots \,,$$
(211)

together with many other non-logarithmic terms (indicated by dots) that are entirely specified by the isometry of the harmonic and ADM descriptions of the motion. For this particular choice of F the ADM Lagrangian reads

$${L^{{\rm{ADM}}}} = {L^{{\rm{harm}}}} + {{\delta {L^{{\rm{harm}}}}} \over {\delta y_1^i}}\delta y_1^i + {{\delta {L^{{\rm{harm}}}}} \over {\delta y_2^i}}\delta y_2^i + {{{\rm{d}}F} \over {{\rm{d}}t}} + {\mathcal O}\left({{1 \over {{c^8}}}} \right)\,.$$
(212)

Inserting into this equation all our explicit expressions we find

$$\begin{array}{*{20}c} {{L^{{\rm{ADM}}}} = {{G{m_1}{m_2}} \over {2{R_{12}}}} + {1 \over 2}{m_1}V_1^2\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {1 \over {{c^2}}}\left\{{- {{{G^2}m_1^2{m_2}} \over {2R_{12}^2}} + {1 \over 8}{m_1}V_1^4 + {{G{m_1}{m_2}} \over {{R_{12}}}}\left({- {1 \over 4}({N_{12}}{V_1})({N_{12}}{V_2}) + {3 \over 2}V_1^2 - {7 \over 4}({V_1}{V_2})} \right)} \right\}} \\ {+ {1 \over {{c^4}}}\left\{{{{{G^3}m_1^3{m_2}} \over {4R_{12}^3}} + {{5{G^3}m_1^2m_2^2} \over {8R_{12}^3}} + {{{m_1}V_1^6} \over {16}}} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{G^2}m_1^2{m_2}} \over {R_{12}^2}}\left({{{15} \over 8}{{({N_{12}}{V_1})}^2} + {{11} \over 8}V_1^2 - {{15} \over 4}({V_1}{V_2}) + 2V_2^2} \right)\quad \quad \quad \quad \quad \quad} \\ {\quad \quad \quad \quad + {{G{m_1}{m_2}} \over {{R_{12}}}}\left({{3 \over {16}}{{({N_{12}}{V_1})}^2}{{({N_{12}}{V_2})}^2} - {1 \over 4}({N_{12}}{V_1})({N_{12}}{V_2})V_1^2 - {5 \over 8}{{({N_{12}}{V_2})}^2}V_1^2 + {7 \over 8}V_1^4} \right.} \\ {\left. {\left. {\quad \quad \quad \quad \quad \quad \quad + {3 \over 4}({N_{12}}{V_1})({N_{12}}{V_2})({V_1}{V_2}) - {7 \over 4}V_1^2({V_1}{V_2}) + {1 \over 8}{{({V_1}{V_2})}^2} + {{11} \over {16}}V_1^2V_2^2} \right)} \right\}} \\ {+ {1 \over {{c^6}}}\left\{{{{5{m_1}V_1^8} \over {128}} - {{{G^4}m_1^4{m_2}} \over {8R_{12}^4}} + {{{G^4}m_1^3m_2^2} \over {R_{12}^4}}\left({- {{227} \over {24}} + {{21} \over {32}}{\pi ^2}} \right)} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{G{m_1}{m_2}} \over {{R_{12}}}}\left({- {5 \over {32}}{{({N_{12}}{V_1})}^3}{{({N_{12}}{V_2})}^3} + {3 \over {16}}{{({N_{12}}{V_1})}^2}{{({N_{12}}{V_2})}^2}V_1^2} \right.\quad \quad \quad \quad} \\ {\quad \quad \quad \quad \quad \quad + {9 \over {16}}({N_{12}}{V_1}){{({N_{12}}{V_2})}^3}V_1^2 - {3 \over {16}}({N_{12}}{V_1})({N_{12}}{V_2})V_1^4 - {5 \over {16}}{{({N_{12}}{V_2})}^2}V_1^4} \\ {\quad \quad \quad \quad \quad \quad + {{11} \over {16}}V_1^6 - {{15} \over {32}}{{({N_{12}}{V_1})}^2}{{({N_{12}}{V_2})}^2}({V_1}{V_2}) + {3 \over 4}({N_{12}}{V_1})({N_{12}}{V_2})V_1^2({V_1}{V_2})} \\ {\quad \quad \quad \quad \quad \quad - {1 \over {16}}{{({N_{12}}{V_2})}^2}V_1^2({V_1}{V_2}) - {{21} \over {16}}V_1^4({V_1}{V_2}) + {5 \over {16}}({N_{12}}{V_1})({N_{12}}{V_2}){{({V_1}{V_2})}^2}} \\ {+ {1 \over 8}V_1^2{{({V_1}{V_2})}^2} + {1 \over {16}}{{({V_1}{V_2})}^3} - {5 \over {16}}{{({N_{12}}{V_1})}^2}V_1^2V_2^2\quad \quad} \\ {\left. {\quad - {9 \over {32}}({N_{12}}{V_1})({N_{12}}{V_2})V_1^2V_2^2 + {7 \over 8}V_1^4V_2^2 - {{15} \over {32}}V_1^2({V_1}{V_2})V_2^2} \right)} \\ {+ {{{G^2}m_1^2{m_2}} \over {R_{12}^2}}\left({- {5 \over {12}}{{({N_{12}}{V_1})}^4} - {{13} \over 8}{{({N_{12}}{V_1})}^3}({N_{12}}{V_2}) - {{23} \over {24}}{{({N_{12}}{V_1})}^2}{{({N_{12}}{V_2})}^2}} \right.\quad \quad} \\ {\quad \quad \quad \quad + {{13} \over {16}}{{({N_{12}}{V_1})}^2}V_1^2 + {1 \over 4}({N_{12}}{V_1})({N_{12}}{V_2})V_1^2 + {5 \over 6}{{({N_{12}}{V_2})}^2}V_1^2 + {{21} \over {16}}V_1^4} \\ {\quad \quad - {1 \over 2}{{({N_{12}}{V_1})}^2}({V_1}{V_2}) + {1 \over 3}({N_{12}}{V_1})({N_{12}}{V_2})({V_1}{V_2}) - {{97} \over {16}}V_1^2({V_1}{V_2})} \\ {\quad \quad \quad + {{341} \over {48}}{{({V_1}{V_2})}^2} + {{29} \over {24}}{{({N_{12}}{V_1})}^2}V_2^2 - ({N_{12}}{V_1})({N_{12}}{V_2})V_2^2 + {{43} \over {12}}V_1^2V_2^2} \\ {- \left. {{{71} \over 8}({V_1}{V_2})V_2^2 + {{47} \over {16}}V_2^4} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{G^3}m_1^2m_2^2} \over {R_{12}^3}}\left({{{73} \over {16}}{{({N_{12}}{V_1})}^2} - 11({N_{12}}{V_1})({N_{12}}{V_2}) + {3 \over {64}}{\pi ^2}({N_{12}}{V_1})({N_{12}}{V_{12}})} \right.\quad \quad \quad} \\ {\left. {- {{265} \over {48}}V_1^2 - {1 \over {64}}{\pi ^2}({V_1}{V_{12}}) + {{59} \over 8}({V_1}{V_2})} \right)\quad \quad \quad \quad \quad} \\ {\left. {+ {{{G^3}m_1^3{m_2}} \over {R_{12}^3}}\left({- 5{{({N_{12}}{V_1})}^2} - {1 \over 8}({N_{12}}{V_1})({N_{12}}{V_2}) + {{173} \over {48}}V_1^2 - {{27} \over 8}({V_1}{V_2}) + {{13} \over 8}V_2^2} \right)} \right\}} \\ {+ 1 \leftrightarrow 2 + {\mathcal O}\left({{1 \over {{c^7}}}} \right)\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(213)

The notation is the same as in Eq. (209), except that we use upper-case letters to denote the ADM-coordinates positions and velocities; thus, for instance N12 = (Y1Y2)/R12 and (N12V1) = N12 ·V1. The Hamiltonian is simply deduced from the latter Lagrangian by applying the usual Legendre transformation. Posing P1 = ∂LADM/V1 and 1 ↔ 2, we get [261, 262, 263, 162, 174]

$$\begin{array}{*{20}c} {{H^{{\rm{ADM}}}} = - {{G{m_1}{m_2}} \over {2{R_{12}}}} + {{P_1^2} \over {2{m_1}}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {1 \over {{c^2}}}\left\{{- {{P_1^4} \over {8m_1^3}} + {{{G^2}m_1^2{m_2}} \over {2R_{12}^2}} + {{G{m_1}{m_2}} \over {{R_{12}}}}\left({{1 \over 4}{{({N_{12}}{P_1})({N_{12}}{P_2})} \over {{m_1}{m_2}}} - {3 \over 2}{{P_1^2} \over {m_1^2}} + {7 \over 4}{{({P_1}{P_2})} \over {{m_1}{m_2}}}} \right)} \right\}} \\ {+ {1 \over {{c^4}}}\left\{{{{P_1^6} \over {16m_1^5}} - {{{G^3}m_1^3{m_2}} \over {4R_{12}^3}} - {{5{G^3}m_1^2m_2^2} \over {8R_{12}^3}}} \right.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{{G^2}m_1^2{m_2}} \over {R_{12}^2}}\left({- {3 \over 2}{{({N_{12}}{P_1})({N_{12}}{P_2})} \over {{m_1}{m_2}}} + {{19} \over 4}{{P_1^2} \over {m_1^2}} - {{27} \over 4}{{({P_1}{P_2})} \over {{m_1}{m_2}}} + {{5P_2^2} \over {2m_2^2}}} \right)\quad} \\ {+ {{G{m_1}{m_2}} \over {{R_{12}}}}\left({- {3 \over {16}}{{{{({N_{12}}{P_1})}^2}{{({N_{12}}{P_2})}^2}} \over {m_1^2m_2^2}} + {5 \over 8}{{{{({N_{12}}{P_2})}^2}P_1^2} \over {m_1^2m_2^2}}} \right.\quad \quad \quad \quad \quad} \\ {\left. {\left. {\quad \quad \quad \quad \quad \quad \quad + {5 \over 8}{{P_1^4} \over {m_1^4}} - {3 \over 4}{{({N_{12}}{P_1})({N_{12}}{P_2})({P_1}{P_2})} \over {m_1^2m_2^2}} - {1 \over 8}{{{{({P_1}{P_2})}^2}} \over {m_1^2m_2^2}} - {{11} \over {16}}{{P_1^2P_2^2} \over {m_1^2m_2^2}}} \right)} \right\}} \\ {+ {1 \over {{c^6}}}\left\{{- {{5P_1^8} \over {128m_1^7}} + {{{G^4}m_1^4{m_2}} \over {8R_{12}^4}} + {{{G^4}m_1^3m_2^2} \over {R_{12}^4}}\left({{{227} \over {24}} - {{21} \over {32}}{\pi ^2}} \right)\quad \quad \quad \quad \quad \quad \quad} \right.} \\ {\quad \quad \quad + {{{G^3}m_1^2m_2^2} \over {R_{12}^3}}\left({- {{43} \over {16}}{{{{({N_{12}}{P_1})}^2}} \over {m_1^2}} + {{119} \over {16}}{{({N_{12}}{P_1})({N_{12}}{P_2})} \over {{m_1}{m_2}}} - {3 \over {64}}{\pi ^2}{{{{({N_{12}}{P_1})}^2}} \over {m_1^2}}} \right.} \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad + {3 \over {64}}{\pi ^2}{{({N_{12}}{P_1})({N_{12}}{P_2})} \over {{m_1}{m_2}}} - {{473} \over {48}}{{P_1^2} \over {m_1^2}} + {1 \over {64}}{\pi ^2}{{P_1^2} \over {m_1^2}} + {{143} \over {16}}{{({P_1}{P_2})} \over {{m_1}{m_2}}}} \\ {\left. {- {1 \over {64}}{\pi ^2}{{({P_1}{P_2})} \over {{m_1}{m_2}}}} \right)\quad \quad \quad \quad \quad \quad \quad \quad} \\ {\quad \quad \quad \quad \quad \quad + {{{G^3}m_1^3{m_2}} \over {R_{12}^3}}\left({{5 \over 4}{{{{({N_{12}}{P_1})}^2}} \over {m_1^2}} + {{21} \over 8}{{({N_{12}}{P_1})({N_{12}}{P_2})} \over {{m_1}{m_2}}} - {{425} \over {48}}{{P_1^2} \over {m_1^2}} + {{77} \over 8}{{({P_1}{P_2})} \over {{m_1}{m_2}}} - {{25P_2^2} \over {8m_2^2}}} \right)} \\ {\quad \quad \quad + {{{G^2}m_1^2{m_2}} \over {R_{12}^2}}\left({{5 \over {12}}{{{{({N_{12}}{P_1})}^4}} \over {m_1^4}} - {3 \over 2}{{{{({N_{12}}{P_1})}^3}({N_{12}}{P_2})} \over {m_1^3{m_2}}} + {{10} \over 3}{{{{({N_{12}}{P_1})}^2}{{({N_{12}}{P_2})}^2}} \over {m_1^2m_2^2}}} \right.} \\ {\quad \quad \quad \quad \quad \quad \quad \quad + {{17} \over {16}}{{{{({N_{12}}{P_1})}^2}P_1^2} \over {m_1^4}} - {{15} \over 8}{{({N_{12}}{P_1})({N_{12}}{P_2})P_1^2} \over {m_1^3{m_2}}} - {{55} \over {12}}{{{{({N_{12}}{P_2})}^2}P_1^2} \over {m_1^2m_2^2}}} \\ {\quad \quad \quad \quad \quad \quad \quad + {{P_1^4} \over {16m_1^4}} - {{11} \over 8}{{{{({N_{12}}{P_1})}^2}({P_1}{P_2})} \over {m_1^3{m_2}}} + {{125} \over {12}}{{({N_{12}}{P_1})({N_{12}}{P_2})({P_1}{P_2})} \over {m_1^2m_2^2}}} \\ {\quad \quad \quad \quad \quad \quad \quad \quad - {{115} \over {16}}{{P_1^2({P_1}{P_2})} \over {m_1^3{m_2}}} + {{25} \over {48}}{{{{({P_1}{P_2})}^2}} \over {m_1^2m_2^2}} - {{193} \over {48}}{{{{({N_{12}}{P_1})}^2}P_2^2} \over {m_1^2m_2^2}} + {{371} \over {48}}{{P_1^2P_2^2} \over {m_1^2m_2^2}}} \\ {\left. {- {{27} \over {16}}{{P_2^4} \over {m_2^4}}} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {{G{m_1}{m_2}} \over {{R_{12}}}}\left({{5 \over {32}}{{{{({N_{12}}{P_1})}^3}{{({N_{12}}{P_2})}^3}} \over {m_1^3m_2^3}} + {3 \over {16}}{{{{({N_{12}}{P_1})}^2}{{({N_{12}}{P_2})}^2}P_1^2} \over {m_1^4m_2^2}}} \right.} \\ {\quad \quad \quad \quad \quad - {9 \over {16}}{{({N_{12}}{P_1}){{({N_{12}}{P_2})}^3}P_1^2} \over {m_1^3m_2^3}} - {5 \over {16}}{{{{({N_{12}}{P_2})}^2}P_1^4} \over {m_1^4m_2^2}} - {7 \over {16}}{{P_1^6} \over {m_1^6}}} \\ {\quad \quad \quad \quad \quad \quad \quad \quad + {{15} \over {32}}{{{{({N_{12}}{P_1})}^2}{{({N_{12}}{P_2})}^2}({P_1}{P_2})} \over {m_1^3m_2^3}} + {3 \over 4}{{({N_{12}}{P_1})({N_{12}}{P_2})P_1^2({P_1}{P_2})} \over {m_1^4m_2^2}}} \\ {\quad \quad \quad \quad \quad + {1 \over {16}}{{{{({N_{12}}{P_2})}^2}P_1^2({P_1}{P_2})} \over {m_1^3m_2^3}} - {5 \over {16}}{{({N_{12}}{P_1})({N_{12}}{P_2}){{({P_1}{P_2})}^2}} \over {m_1^3m_2^3}}} \\ {\quad \quad \quad + {1 \over 8}{{P_1^2{{({P_1}{P_2})}^2}} \over {m_1^4m_2^2}} - {1 \over {16}}{{{{({P_1}{P_2})}^3}} \over {m_1^3m_2^3}} - {5 \over {16}}{{{{({N_{12}}{P_1})}^2}P_1^2P_2^2} \over {m_1^4m_2^2}}} \\ {\left. {\left. {\quad \quad \quad \quad \quad \quad \quad + {7 \over {32}}{{({N_{12}}{P_1})({N_{12}}{P_2})P_1^2P_2^2} \over {m_1^3m_2^3}} + {1 \over 2}{{P_1^4P_2^2} \over {m_1^4m_2^2}} + {1 \over {32}}{{P_1^2({P_1}{P_2})P_2^2} \over {m_1^3m_2^3}}} \right)} \right\}} \\ {+ 1 \leftrightarrow 2 + {\mathcal O}\left({{1 \over {{c^7}}}} \right)\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(214)

Arguably, the results given by the ADM-Hamiltonian formalism (for the problem at hand) look simpler than their harmonic-coordinate counterparts. Indeed, the ADM Lagrangian is ordinary — no accelerations — and there are no logarithms nor associated gauge constants r1 and r2.Footnote 50 Of course, one is free to describe the binary motion in whatever coordinates one likes, and the two formalisms, harmonic (209) and ADM (213)(214), describe rigorously the same physics. On the other hand, the higher complexity of the harmonic-coordinates Lagrangian (209) enables one to perform more tests of the computations, notably by inquiring about the future of the constants r1and r2, that we know must disappear from physical quantities such as the center-of-mass energy and the total gravitational-wave flux.

Equations of motion in the center-of-mass frame

In this section we translate the origin of coordinates to the binary’s center-of-mass by imposing the vanishing of the binary’s mass dipole moment: Ii = 0 in the notation of Part A. Actually the dipole moment is computed as the center-of-mass conserved integral associated with the boost symmetry of the 3PN equations of motion [174, 79]. This condition results in the 3PN-accurate relationship between the individual positions in the center-of-mass frame y1 and y2, and the relative position xy1y2 and velocity vv1v2 = dx/dt (formerly denoted y12 and v12). We shall also use the orbital separation r ≡ ∣x∣, together with n = x/r and n · v. Mass parameters are: The total mass m = m1 + m2 (to be distinguished from the ADM mass denoted by M in Part A); the relative mass difference Δ = (m1m2)/m; the reduced mass μ = m1m2/m; and the very useful symmetric mass ratio

$$\nu \equiv {\mu \over m} \equiv {{{m_1}{m_2}} \over {{{({m_1} + {m_2})}^2}}}\,.$$
(215)

The usefulness of this ratio lies in its interesting range of variation: 0 < ν ⩽ 1/4, with ν = 1/4 in the case of equal masses, and ν → 0 in the test-mass limit for one of the bodies. Thus ν is numerically rather small and may be viewed as a small expansion parameter. We also pose X1 = m1/m and X2 = m2/m so that Δ = X1X2 and ν = X1X2.

For reference we give the 3PN-accurate expressions of the individual positions in the center-of-mass frame in terms of relative variables. They are in the form

$${y_1} = \left[ {{X_2} + \nu \,\Delta \,{\mathcal P}} \right]x + \nu \,\Delta \,{\mathcal Q}\,v + {\mathcal O}\left({{1 \over {{c^7}}}} \right)\,,$$
(216a)
$${y_2} = \left[ {- {X_1} + \nu \,\Delta \,{\mathcal P}} \right]x + \nu \,\Delta \,{\mathcal Q}\,v + {\mathcal O}\left({{1 \over {{c^7}}}} \right)\,,$$
(216b)

where all post-Newtonian corrections, beyond Newtonian order, are proportional to the mass ratio ν and the mass difference Δ. The two dimensionless coefficients \({\mathcal P}\) and \({\mathcal Q}\) read

$$\begin{array}{*{20}c} {{\mathcal P} = {1 \over {{c^2}}}\left\{{{{{v^2}} \over 2} - {{Gm} \over {2\,r}}} \right\}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {1 \over {{c^4}}}\left\{{{{3\,{v^4}} \over 8} - {{3\,\nu \,{v^4}} \over 2} + {{Gm} \over r}\,\left({- {{{{\dot r}^2}} \over 8} + {{3\,{{\dot r}^2}\,\nu} \over 4} + {{19\,{v^2}} \over 8} + {{3\,\nu \,{v^2}} \over 2}} \right) + {{{G^2}{m^2}} \over {{r^2}}}\left({{7 \over 4} - {\nu \over 2}} \right)} \right\}} \\ {+ {1 \over {{c^6}}}\left\{{{{5\,{v^6}} \over {16}} - {{11\,\nu \,{v^6}} \over 4} + 6\,{\nu ^2}\,{v^6}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \right.} \\ {+ {{Gm} \over r}\left({{{{{\dot r}^4}} \over {16}} - {{5\,{{\dot r}^4}\,\nu} \over 8} + {{21\,{{\dot r}^4}\,{\nu ^2}} \over {16}} - {{5\,{{\dot r}^2}\,{v^2}} \over {16}} + {{21\,{{\dot r}^2}\,\nu \,{v^2}} \over {16}}} \right.\quad \quad \quad \quad \quad \quad} \\ {\left. {- {{11\,{{\dot r}^2}\,{\nu ^2}\,{v^2}} \over 2} + {{53\,{v^4}} \over {16}} - 7\,\nu \,{v^4} - {{15\,{\nu ^2}\,{v^4}} \over 2}} \right)\quad \quad \quad \quad} \\ {+ {{{G^2}{m^2}} \over {{r^2}}}\left({- {{7\,{{\dot r}^2}} \over 3} + {{73\,{{\dot r}^2}\,\nu} \over 8} + 4\,{{\dot r}^2}\,{\nu ^2} + {{101\,{v^2}} \over {12}} - {{33\,\nu \,{v^2}} \over 8} + 3\,{\nu ^2}\,{v^2}} \right)\quad} \\ {\left. {+ {{{G^3}{m^3}} \over {{r^3}}}\left({- {{14351} \over {1260}} + {\nu \over 8} - {{{\nu ^2}} \over 2} + {{22} \over 3}\,\ln \left({{r \over {{{r\prime\prime}_0}}}} \right)} \right)} \right\}\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(217a)
$$\begin{array}{*{20}c} {{\mathcal Q} = {1 \over {{c^4}}}\left\{{- {{7\,Gm\,\dot r} \over 4}} \right\} + {1 \over {{c^5}}}\left\{{{{4\,Gm\,{v^2}} \over 5} - {{8\,{G^2}{m^2}} \over {5\,r}}} \right\}\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {1 \over {{c^6}}}\left\{{Gm\,\dot r\left({{{5\,{{\dot r}^2}} \over {12}} - {{19\,{{\dot r}^2}\,\nu} \over {24}} - {{15\,{v^2}} \over 8} + {{21\,\nu \,{v^2}} \over 4}} \right) + {{{G^2}{m^2}\,\dot r} \over r}\left({- {{235} \over {24}} - {{21\,\nu} \over 4}} \right)} \right\}\,.} \\ \end{array}$$
(217b)

Up to 2.5PN order there is agreement with the circular-orbit limit of Eqs. (6.4) in Ref. [45]. Notice the 2.5PN radiation-reaction term entering the coefficient \({\mathcal Q}\); such 2.5PN term is explicitly displayed for circular orbits in Eqs. (224) below. In Eqs. (217) the logarithms at the 3PN order appear only in the coefficient \({\mathcal P}\). They contain a particular combination \(r_0^{\prime\prime}\) of the two gauge-constants r1 and r2 defined by

$$\Delta \,\ln {r\prime\prime_0} = X_1^2\ln {r\prime_1} - X_2^2\ln {r\prime_2}\,,$$
(218)

and which happens to be different from a similar combination r0 we shall find in the equations of relative motion, see Eq. (221).

The 3PN and even 3.5PN center-of-mass equations of motion are obtained by replacing in the 3.5PN equations of motion (203) in a general frame, the positions and velocities by their center-of-mass expressions (216)(217), applying as usual the order-reduction of