Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries
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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.
Keywords
Gravitational radiation Post-Newtonian approximations Multipolar expansion Inspiralling compact binary1 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.
- 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.
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.
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.
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.1
1.1 Analytic approximations and wave generation formalism
The basic problem we face is to relate the asymptotic gravitational-wave form h ij 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 T αβ , using approximation methods in general relativity.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.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.
We have ∣U/c2∣1/2 ≪ 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., \(|U/{c^2}{|^{1/2}} = {\mathcal O}{\rm{(1/}}c)\).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 T − R/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.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 T αβ 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 τ αβ , which is made of a matter part described by T αβ and a crucial non-linear gravitational source term Λ αβ . 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 Λ αβ has a spatially infinite support, which would have made the bound of the integral at spatial infinity singular (of course the finite part is not added a posteriori to restore the 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 τ αβ , 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.
1.2 The quadrupole moment formalism
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]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.
1.3 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).
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].
1.4 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].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 Ṗ [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.
- 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.
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.
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.
Foffa & Sturani [203] derive the 3PN Lagrangian in harmonic coordinates within the effective field theory approach pioneered by Goldberger & Rothstein [223].
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.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]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.
1.5 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 (ℓ, 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, ν ≡ μ/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]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.
2 Part A: Post-Newtonian Sources
3 Non-linear Iteration of the Vacuum Field Equations
3.1 Einstein’s field equations
- 1.The matter stress-energy tensor is of spatially compact support, i.e., can be enclosed into some time-like world tube, say r ⩽ a, 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.
The matter distribution inside the source is smooth: T αβ ∈ C∞(ℝ3).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.
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.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)
3.2 Linearized vacuum equations
The conservation of the lowest-order moments gives the constancy of the total mass of the source, M ≡ I = const, center-of-mass position, X i ≡ I i /I = const, total linear momentum \({{\rm{P}}_i} \equiv {\rm{I}}_i^{(1)} = 0\),18 and total angular momentum, J i = const. It is always possible to achieve X i = 0 by translating the origin of our coordinates to the center of mass. The total mass M is the ADM mass of the Hamiltonian formulation of general relativity. Note that the quantities M, X i , P i and J i include the contributions due to the waves emitted by the source. They describe the initial state of the source, before the emission of gravitational radiation.
The multipole functions I L (u) and J L (u), which thoroughly encode the physical properties of the source at the linearized level (because the other moments W L , …, Z L , parametrize a gauge transformation), will be referred to as the 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 T αβ of the source: Theorem 1 follows merely from the algebraic and differential properties of the vacuum field equations outside the source.
3.3 The multipolar post-Minkowskian solution
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.
3.4 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].
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, M L and S L , instead of the six types, I L , J L , …, Z L ? The reason is that we shall determine in Theorem 6 below the explicit closed-form expressions of the six source moments I L , J L , …, Z L , but that, by contrast, it seems to be impossible to obtain some similar closed-form expressions for the canonical moments M L and S L . The only thing we can do is to write down the explicit non-linear algorithm that computes M L , S L starting from I L , J L , …, Z L . In consequence, it is better to view the moments I L , J L , …, Z L as more “fundamental” than M L and S L , 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 I L , J L , …, Z L as the multipole moments of the source. This being said, the moments M L and S L 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 M L and S L differ from the corresponding I L and J L 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.
3.5 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.
4 Asymptotic Gravitational Waveform
4.1 The radiative multipole moments
4.2 Gravitational-wave tails and tails-of-tails
We learned from Theorem 4 the general method which permits the computation of the radiative multipole moments U L , V L in terms of the source moments I L , J L , …, Z L , or in terms of the intermediate canonical moments M L , S L discussed in Section 2.4. We shall now show that the relation between U L , V L and M L , S L (say) includes tail effects starting at the relative 1.5PN order.
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 \(- {\mathcal T}\).
4.3 Radiative versus source moments
- 1.The instantaneous (i.e., non-hereditary) piece \({\rm{U}}_{ij}^{{\rm{inst}}}\) up to 3.5PN order readsThe Newtonian term in this expression contains the Newtonian quadrupole moment Q ij and recovers the standard quadrupole formalism [see Eq. (67)];$$\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)
- 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: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);$${\rm{U}}_{ij}^{{\rm{tail}}} = {{2G{\rm{M}}} \over {{c^3}}}\int\nolimits_0^{+ \infty} {{\rm{d}}\tau \left[ {\ln \left({{{c\tau} \over {2{r_0}}}} \right) + {{11} \over {12}}} \right]{\rm{M}}_{ij}^{(4)}(U - \tau)\,.}$$(90)
- 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.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)
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.
5 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 T αβ , 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.
5.1 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 r ≪ λ (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.
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.
5.2 General expression of the multipole expansion
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 I ij 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.
5.3 Equivalence with the Will-Wiseman formalism
5.4 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 I L , J L ,…, Z L parametrizing the linearized metric (35)–(37) at the basis of that construction.30
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 U L and V L 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.
6 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.
- 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.
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: r ≪ λ. 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].
6.1 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., T αβ ∈ C∞(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.
6.2 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).
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.
6.3 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 g αβ , 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, V i , \({\hat W_{ij}}\), …, 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.
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.
6.4 Radiation reaction potentials to 4PN order
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].
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].
7 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].
8 Regularization of the Field of Point Particles
- 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.
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).
8.1 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].
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.
8.2 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.
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.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.
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.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 I ij , 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].
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.
8.3 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].
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, V i , 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).
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: y1 → y1 + ξ1 and y2 → y2 + 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:
8.4 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].
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, σ a , and σ ab , 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.
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.
9 Newtonian-like Equations of Motion
9.1 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).
- 1.
Stay manifestly invariant — at least in harmonic coordinates — when we perform a global post-Newtonian-expanded Lorentz transformation;
- 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.
Be conservative, i.e., to admit a Lagrangian or Hamiltonian formulation, when the gravitational radiation reaction is turned off.
9.2 Lagrangian and Hamiltonian formulations
9.3 Equations of motion in the center-of-mass frame
9.4 Equations of motion and energy for quasi-circular orbits
9.5 The 2.5PN metric in the near zone
10 Conservative Dynamics of Compact Binaries
10.1 Concept of innermost circular orbit
Having in hand the conserved energy E(x) for circular orbits given by Eq. (232), or even more accurate by (233), we define the innermost circular orbit (ICO) as the minimum, when it exists, of the energy function E(x) — see e.g., Ref. [51]. Notice that the ICO is not defined as a point of dynamical general-relativistic unstability. Hence, we prefer to call this point the ICO rather than, strictly speaking, an innermost stable circular orbit or ISCO. A study of the dynamical stability of circular binary orbits in the post-Newtonian approximation is reported in Section 8.2.
The previous definition of the ICO is motivated by the comparison with some results of numerical relativity. Indeed we shall confront the prediction of the standard (Taylor-based) post-Newtonian approximation with numerical computations of the energy of binary black holes under the assumptions of conformal flatness for the spatial metric and of exactly circular orbits [228, 232, 133, 121]. The latter restriction is implemented by requiring the existence of an “helical” Killing vector (HKV), which is time-like inside the light cylinder associated with the circular motion, and space-like outside. The HKV will be defined in Eq. (273) below. In the numerical approaches of Refs. [228, 232, 133, 121] there are no gravitational waves, the field is periodic in time, and the gravitational potentials tend to zero at spatial infinity within a restricted model equivalent to solving five out of the ten Einstein field equations (the so-called Isenberg-Wilson-Mathews approximation; see Ref. [228] for a discussion). Considering an evolutionary sequence of equilibrium configurations the circular-orbit energy E(Ω) and the ICO of binary black holes are obtained numerically (see also Refs. [92, 229, 301] for related calculations of binary neutron stars and strange quark stars).
To take into account the spin effects our first task is to replace all the masses entering the energy function (232) by their equivalent expressions in terms of ωa and the irreducible masses μa, and then to replace ωa in terms of Ω according to the corotation prescription (247).55 It is clear that the leading contribution is that of the spin kinetic energy given in Eq. (246b), and it comes from the replacement of the rest mass-energy m = m1 + m2. From Eq. (246b) this effect is of order Ω2 in the case of corotating binaries, which means by comparison with Eq. (232) that it is equivalent to an “orbital” effect at the 2PN order (i.e., ℝ x2). Higher-order corrections in Eq. (246b), which behave at least like Ω4, will correspond to the orbital 5PN order at least and are negligible for the present purpose. In addition there will be a subdominant contribution, of the order of Ω8/3 equivalent to 3PN order, which comes from the replacement of the masses into the Newtonian part, proportional to x ℝ Ω2/3, of the energy E; see Eq. (232). With the 3PN accuracy we do not need to replace the masses that enter into the post-Newtonian corrections in E, so in these terms the masses can be considered to be the irreducible ones.
The binding energy EICO versus ΩICO in the equal-mass case (ν = 1/4). Left panel: Comparison with the numerical relativity result of Gourgoulhon, Grandclément et al. [228, 232] valid in the corotating case (marked by a star). Points indicated by nPN are computed from the minimum of Eq. (232), and correspond to irrotational binaries. Points denoted by nPNcorot come from the minimum of the sum of Eqs. (232) and (250), and describe corotational binaries. Note the very good convergence of the standard (Taylor-expanded) PN series. Right panel: Numerical relativity results of Cook, Pfeiffer et al. [133, 121] for quasi-equilibrium (QE) configurations and various boundary conditions for the lapse function, in the non-spinning (NS), leading-order non spinning (LN) and corotating (CO) cases. The point from [228, 232] (HKV-GGB) is also reported as in the left panel, together with IVP, the initial value approach with effective potential [132, 342], as well as standard PN predictions from the left panel and non-standard (EOB) ones. The agreement between the QE computation and the standard non-resummed 3PN point is excellent especially in the irrotational NS case.
However, we recall that the numerical works [228, 232, 133, 121] assume that the spatial metric is conformally flat, which is incompatible with the post-Newtonian approximation starting from the 2PN order (see [196] for a discussion). Nevertheless, the agreement found in Figure 1 constitutes an appreciable improvement of the previous situation, because the first estimations of the ICO in post-Newtonian theory [274] and numerical relativity [132, 342, 29] disagreed with each other, and do not match with the present 3PN results.
10.2 Dynamical stability of circular orbits
In this section, following Ref. [79], we shall investigate the problem of the stability, against dynamical perturbations, of circular orbits at the 3PN order. We propose to use two different methods, one based on a linear perturbation at the level of the center-of-mass equations of motion (219)–(220) in (standard) harmonic coordinates, the other one consisting of perturbing the Hamiltonian equations in ADM coordinates for the center-of-mass Hamiltonian (223). We shall find a criterion for the stability of circular orbits and shall present it in an invariant way — the same in different coordinate systems. We shall check that our two methods agree on the result.
The stability criterion (269) has been compared in great details to various other stability criteria by Favata [191] and shown to perform very well, and has also been generalized to spinning black hole binaries in Ref. [190]. Note that this criterion is based on the physical requirement that a stable perturbation should have a real frequency. It gives an innermost stable circular orbit, when it exists, which differs from the innermost circular orbit or ICO defined in Section 8.1; see Ref. [378] for a discussion on the difference between an ISCO and the ICO in the PN context. Note also that the criterion (269) is based on systematic post-Newtonian expansions, without resorting for instance to Padé approximants. Nevertheless, it performs better than other criteria based on various resummation techniques, as discussed in Ref. [191].
10.3 The first law of binary point-particle mechanics
In this section we shall review a very interesting relation for binary systems of point particles modelling black hole binaries and moving on circular orbits, known as the “first law of point-particle mechanics”. This law was obtained using post-Newtonian methods in Ref. [289], but is actually a particular case of a more general law, valid for systems of black holes and extended fluid balls, derived by Friedman, Uryū & Shibata [208].
Before tackling the problem it is necessary to make more precise the notion of circular orbits. These are obtained from the conservative part of the dynamics, neglecting the dissipative radiation-reaction force responsible for the gravitational-wave inspiral. In post-Newtonian theory this means neglecting the radiation-reaction force at 2.5PN and 3.5PN orders, i.e., considering only the conservative dynamics at the even-parity 1PN, 2PN and 3PN orders. We have seen in Sections 5.2 and 5.4 that this clean separation between conservative even-parity and dissipative odd-parity terms breaks at 4PN order, because of a contribution originating from gravitational-wave tails in the radiation-reaction force. We expect that at any higher order 4PN, 4.5PN, 5PN, etc. there will be a mixture of conservative and dissipative effects; here we assume that at any higher order we can neglect the radiation-reaction dissipation effects.
This law was proved in a very general way in Ref. [208] for systems of black holes and extended bodies under some arbitrary Killing symmetry. The particular form given in Eq. (280) is a specialization to the case of point particle binaries with helical Killing symmetry. It has been proved directly in this form in Ref. [289] up to high post-Newtonian order, namely 3PN order plus the logarithmic contributions occurring at 4PN and 5PN orders.
The first law of binary point-particle mechanics (280) is of course reminiscent of the celebrated first law of black hole mechanics \(\delta {\rm{M -}}{\omega _{\rm{H}}}\delta {\rm{J =}}{\kappa \over {8\pi}}\delta A\), which holds for any non-singular, asymptotically flat perturbation of a stationary and axisymmetric black hole of mass M, intrinsic angular momentum (or spin) J ≡ Ma, surface area A, uniform surface gravity κ, and angular frequency ω H on the horizon [26, 417]; see Ref. [289] for a discussion.
Notice that the relation (282) has been derived for point particles and arbitrary aligned spins. We would like now to derive the analogous relation for binary black holes. The key difference is that black holes are extended finite-size objects while point particles have by definition no spatial extension. For point particle binaries the spins can have arbitrary magnitude and still be compatible with the HKV. In this case the law (282) would describe also (super-extremal) naked singularities. For black hole binaries the HKV constraints the rotational state of each black hole and the binary system must be corotating.
10.4 Post-Newtonian approximation versus gravitational self-force
The high-accuracy predictions from general relativity we have drawn up to now are well suited to describe the inspiralling phase of compact binaries, when the post-Newtonian parameter (1) is small independently of the mass ratio q ≡ m1/m2 between the compact bodies. In this section we investigate how well does the post-Newtonian expansion compare with another very important approximation scheme in general relativity: The gravitational self-force approach, based on black-hole perturbation theory, which gives an accurate description of extreme mass ratio binaries having q ≪ 1 or equivalently ν ≪ 1, even in the strong field regime. The gravitational self-force analysis [317, 360, 178, 231] (see [348, 177, 23] for reviews) is thus expected to provide templates for extreme mass ratio inspirals (EMRI) anticipated to be present in the bandwidth of space-based detectors.
Consider a system of two (non-spinning) compact objects with q = m1/m2 ≪ 1; we shall call the smaller mass m1 the “particle”, and the larger mass m2 the “black hole”. The orbit of the particle around the black hole is supposed to be exactly circular as we neglect the radiation-reaction effects. With circular orbits and no dissipation, we are considering the conservative part of the dynamics, and the geometry admits the HKV field (273). Note that in self-force theory there is a clean split between the dissipative and conservative parts of the dynamics (see e.g., [22]). This split is particularly transparent for an exact circular orbit, since the radial component (along r) is the only non-vanishing component of the conservative self-force, while the dissipative part of the self-force are the components along t and φ.
Different analytical approximation schemes and numerical techniques to study black hole binaries, depending on the mass ratio q = m1/m2 and the post-Newtonian parameter ε2 ∼ v2/c2 ∼ Gm/(c2r12). Post-Newtonian theory and perturbative self-force analysis can be compared in the post-Newtonian regime (ε ≪ 1 thus r12 ≫ Gm/c2) of an extreme mass ratio (m1 ≪ m2) binary.
For the PN-SF comparison, we require two physical quantities which are precisely defined in the context of each of the approximation schemes. The orbital frequency Ω of the circular orbit as measured by a distant observer is one such quantity and has been introduced in Eq. (273); the second quantity is the redshift observable \(u_1^T\) (or equivalently \({z_1} = 1/u_1^T\)) associated with the smaller mass m1 ≪ m2 and defined by Eqs. (274) or (275). The truly coordinate and perturbative-gauge independent properties of Ω and the redshift observable \(u_1^T\) play a crucial role in this comparison. In the perturbative self-force approach we use Schwarzschild coordinates for the background, and we refer to “gauge invariance” as a property which holds within the restricted class of gauges for which (273) is a helical Killing vector. In all other respects, the gauge choice is arbitrary. In the post-Newtonian approach we work with harmonic coordinates and compute the explicit expression (276) of the redshift observable.
In Eq. (288) we denote by u4(ν), v4(ν) and u5(ν), v5(ν) some unknown 4PN and 5PN coefficients, which are however polynomials of the symmetric mass ratio ν. They can be entirely determined from the related coefficients e4(ν), j4(ν) and e5(ν), j5(ν) in the expressions of the energy and angular momentum in Eqs. (233) and (234). To this aim it suffices to apply the differential first law (280) up to 5PN order; see Ref. [289] for more details.
| 3PN coefficient | SF value |
|---|---|
| \({\alpha _3} \equiv - {{121} \over 3} + {{41} \over {32}}{\pi^2} = - 27.6879026 \ldots\) | −27.6879034 ± 0.0000004 |
| PN coefficient | SF value |
|---|---|
| α 4 | −114.34747(5) |
| α 5 | −245.53(1) |
| α 6 | −695(2) |
| β 6 | +339.3(5) |
| α 7 | −5837(16) |
Gladly we discover that the more recent analytical value of the 4PN coefficient, Eq. (293), matches the numerical value which was earlier measured in Ref. [67] (see Table 2). This highlights the predictive power of perturbative self-force calculations in determining numerically new post-Newtonian coefficients [176, 68, 67]. This ability is obviously due to the fact (illustrated in Figure 2) that perturbation theory is legitimate in the strong field regime of the coalescence of black hole binary systems, which is inaccessible to the post-Newtonian method. Of course, the limitation of the self-force approach is the small mass-ratio limit; in this respect it is taken over by the post-Newtonian approximation.
To conclude, the consistency of this “cross-cultural” comparison between the analytical post-Newtonian and the perturbative self-force approaches confirms the soundness of both approximations in describing the dynamics of compact binaries. Furthermore this interplay between PN and SF efforts (which is now rapidly growing [383]) is important for the synthesis of template waveforms of EMRIs to be analysed by space-based gravitational wave detectors, and has also an impact on efforts of numerical relativity in the case of comparable masses.
11 Gravitational Waves from Compact Binaries
Because the orbit of inspiralling binaries is circular, the energy balance equation is sufficient, and there is no need to invoke the angular momentum balance equation for computing the evolution of the orbital period Ṗ and eccentricity ė, see Eqs. (9)–(13) in the case of the binary pulsar. Furthermore the time average over one orbital period as in Eqs. (9) is here irrelevant, and the energy and angular momentum fluxes are related by \({\mathcal F}{\rm{=}}\Omega {\mathcal G}\). This all sounds good, but it is important to remind that we shall use the balance equation (295) at the very high 3.5PN order, and that at such order one is missing a complete proof of it (following from first principles in general relativity). Nevertheless, in addition to its physically obvious character, Eq. (295) has been verified by radiation-reaction calculations, in the cases of point-particle binaries [258, 259] and extended post-Newtonian fluids [43, 47], at the 1PN order and even at 1.5PN, the latter order being especially important because of the first appearance of wave tails; see Section 5.4. One should also quote here Refs. [260, 336, 278, 322, 254] for the 3.5PN terms in the binary’s equations of motion, fully consistent with the balance equations.
Obtaining the energy flux \({\mathcal F}\) can be divided into two equally important steps: Computing the source multipole moments I L and J L of the compact binary system with due account of a self-field regularization; and controlling the tails and related non-linear effects occurring in the relation between the binary’s source moments and the radiative ones U L and V L observed at future null infinity (cf. the general formalism of Part A).
11.1 The binary’s multipole moments
The general expressions of the source multipole moments given by Theorem 6, Eqs. (123), are worked out explicitly for general fluid systems at the 3.5PN order. For this computation one uses the formula (126), and we insert the 3.5PN metric coefficients (in harmonic coordinates) expressed in Eqs. (144) by means of the retarded-type elementary potentials (146)–(148). Then we specialize each of the (quite numerous) terms to the case of point-particle binaries by inserting, for the matter stress-energy tensor T αβ , the standard expression made out of Dirac delta-functions. In Section 11 we shall consider spinning point particle binaries, and in that case the stress-energy tensor is given by Eq. (378). The infinite self-field of point-particles is removed by means of the Hadamard regularization; and, as we discussed in Section 6.4, dimensional regularization is used to fix the values of a few ambiguity parameters. This computation has been performed in Ref. [86] at the 1PN order, and in [64] at the 2PN order; we report below the most accurate 3PN results obtained in Refs. [81, 80, 62, 63] for the flux and [11, 74, 197] for the waveform.
11.2 Gravitational wave energy flux
We shall see that the tails play a crucial role in the predicted signal of compact binaries. It is quite remarkable that so small an effect as a “tail of tail” should be relevant to the data analysis of the current generation of gravitational wave detectors. By contrast, the non-linear memory effects, given by the integrals inside the 2.5PN and 3.5PN terms in Eq. (92), do not contribute to the gravitational-wave energy flux before the 4PN order in the case of circular-orbit binaries. Indeed the memory integrals are actually “instantaneous” in the flux, and a simple general argument based on dimensional analysis shows that instantaneous terms cannot contribute to the energy flux for circular orbits.64 Therefore the memory effect has rather poor observational consequences for future detections of inspiralling compact binaries.
The effects due to the spins of the two black holes arise at the 1.5PN order for the spin-orbit (SO) coupling, and at the 2PN order for the spin-spin (SS) coupling, for maximally rotating black holes. Spin effects will be discussed in Section 11. On the other hand, the terms due to the radiating energy flowing into the black-hole horizons and absorbed rather than escaping to infinity, have to be added to the standard post-Newtonian calculation based on point particles as presented here; such terms arise at the 4PN order for Schwarzschild black holes [349] and at 2.5PN order for Kerr black holes [392].
11.3 Orbital phase evolution
Post-Newtonian contributions to the accumulated number of gravitational-wave cycles \({{\mathcal N}_{{\rm{cycle}}}}\) for compact binaries detectable in the bandwidth of LIGO-VIRGO detectors. The entry frequency is fseismic = 10 Hz and the terminal frequency is \({f_{{\rm{ISCO}}}} = {{{c^3}} \over {{6^3}{/^2}\pi G\;m}}\). The main origin of the approximation (instantaneous vs. tail) is indicated. See also Table 4 in Section 11 below for the contributions of spin-orbit effects.
| PN order | 1.4 M⊙ + 1.4 M⊙ | 10 M⊙ + 1.4 M⊙ | 10 M⊙ + 10 M⊙ | |
|---|---|---|---|---|
| N | (inst) | 15952.6 | 3558.9 | 598.8 |
| 1PN | (inst) | 439.5 | 212.4 | 59.1 |
| 1.5PN | (leading tail) | −210.3 | −180.9 | −51.2 |
| 2PN | (inst) | 9.9 | 9.8 | 4.0 |
| 2.5PN | (1PN tail) | −11.7 | −20.0 | −7.1 |
| 3PN | (inst + tail-of-tail) | 2.6 | 2.3 | 2.2 |
| 3.5PN | (2PN tail) | −0.9 | −1.8 | −0.8 |
11.4 Polarization waveforms for data analysis
The theoretical templates of the compact binary inspiral follow from insertion of the previous solutions for the 3.5PN-accurate orbital frequency and phase into the binary’s two polarization waveforms h+ and h× defined with respect to a choice of two polarization vectors P = (P i ) and Q = (Q i ) orthogonal to the direction N of the observer; see Eqs. (69).
Our convention for the two polarization vectors is that they form with N a right-handed triad, and that P and Q lie along the major and minor axis, respectively, of the projection onto the plane of the sky of the circular orbit. This means that P is oriented toward the orbit’s ascending node — namely the point \({\mathcal N}\) at which the orbit intersects the plane of the sky and the bodies are moving toward the observer located in the direction N. The ascending node is also chosen for the origin of the orbital phase ϕ. We denote by i the inclination angle between the direction of the detector N as seen from the binary’s center-of-mass, and the normal to the orbital plane (we always suppose that the normal is right-handed with respect to the sense of motion, so that 0 ⩽ i ⩽ π). We use the shorthands c i ≡ cos i and s i ≡ sin i for the cosine and sine of the inclination angle.
To conclude, the use of theoretical templates based on the preceding 3PN/3.5PN waveforms, and having their frequency evolution built in via the 3.5PN phase evolution (318) [recall also the “tail-distorted” phase variable (321)], should yield some accurate detection and measurement of the binary signals, whose inspiral phase takes place in the detector’s bandwidth [105, 106, 159, 156, 3, 18, 111]. Interestingly, it should also permit some new tests of general relativity, because we have the possibility of checking that the observed signals do obey each of the terms of the phasing formula (318) — particularly interesting are those terms associated with non-linear tails — exactly as they are predicted by Einstein’s theory [84, 85, 15, 14]. Indeed, we don’t know of any other physical systems for which it would be possible to perform such tests.
11.5 Spherical harmonic modes for numerical relativity
The spin-weighted spherical harmonic modes of the polarization waveforms have been defined in Eq. (71). They can be evaluated either from applying the angular integration formula (72), or alternatively from using the relations (73)–(74) giving the individual modes directly in terms of separate contributions of the radiative moments U L and V L . The latter route is actually more interesting [272] if some of the radiative moments are known to higher PN order than others. In this case the comparison with the numerical calculation for these particular modes can be made with higher post-Newtonian accuracy.
12 Eccentric Compact Binaries
Inspiralling compact binaries are usually modelled as moving in quasi-circular orbits since gravitational radiation reaction circularizes the orbit towards the late stages of inspiral [340, 339], as we discussed in Section 1.2. Nevertheless, there is an increased interest in inspiralling binaries moving in quasi-eccentric orbits. Astrophysical scenarios currently exist which lead to binaries with non-zero eccentricity in the gravitational-wave detector bandwidth, both terrestrial and space-based. For instance, inner binaries of hierarchical triplets undergoing Kozai oscillations [283, 300] could not only merge due to gravitational radiation reaction but a fraction of them should have non negligible eccentricities when they enter the sensitivity band of advanced ground based interferometers [419]. On the other hand the population of stellar mass binaries in globular clusters is expected to have a thermal distribution of eccentricities [32]. In a study of the growth of intermediate black holes [235] in globular clusters it was found that the binaries have eccentricities between 0.1 and 0.2 in the eLISA bandwidth. Though, supermassive black hole binaries are powerful gravitational wave sources for eLISA, it is not known if they would be in quasi-circular or quasi-eccentric orbits. If a Kozai mechanism is at work, these supermassive black hole binaries could be in highly eccentric orbits and merge within the Hubble time [40]. Sources of the kind discussed above provide the prime motivation for investigating higher post-Newtonian order modelling for quasi-eccentric binaries.
12.1 Doubly periodic structure of the motion of eccentric binaries
In Section 7.3 we have given the equations of motion of non-spinning compact binary systems in the frame of the center-of-mass for general orbits at the 3PN and even 3.5PN order. We shall now investigate (in this section and the next one) the explicit solution to those equations. In particular, let us discuss the general “doubly-periodic” structure of the post-Newtonian solution, closely following Refs. [142, 143, 149].
The 3PN equations of motion admit, when neglecting the radiation reaction terms at 2.5PN order, ten first integrals of the motion corresponding to the conservation of energy, angular momentum, linear momentum, and center of mass position. When restricted to the frame of the center of mass, the equations admit four first integrals associated with the energy E and the angular momentum vector J, given in harmonic coordinates at 3PN order by Eqs. (4.8)–(4.9) of Ref. [79].
12.2 Quasi-Keplerian representation of the motion
The quasi-Keplerian (QK) representation of the motion of compact binaries is an elegant formulation of the solution of the 1PN equations of motion parametrized by the eccentric anomaly u (entering a specific generalization of Kepler’s equation) and depending on various orbital elements, such as three types of eccentricities. It was introduced by Damour & Deruelle [149, 150] to study the problem of binary pulsar timing data including relativistic corrections at the 1PN order, where the relativistic periastron precession complicates the simpler Keplerian solution.
The above QK representation of the compact binary motion at 1PN order has been generalized at the 2PN order in Refs. [170, 379, 420], and at the 3PN order by Memmesheimer, Gopakumar & Schäfer [312]. The construction of a generalized QK representation at 3PN order exploits the fact that the radial equation given by Eq. (331a) is a polynomial in 1/r (of seventh degree at 3PN order). However, this is true only in coordinate systems avoiding the appearance of terms with the logarithm ln r; the presence of logarithms in the standard harmonic (SH) coordinates at the 3PN order will obstruct the construction of the QK parametrization. Therefore Ref. [312] obtained it in the ADM coordinate system and also in the modified harmonic (MH) coordinates, obtained by applying the gauge transformation given in Eq. (204) on the SH coordinates. The equations of motion in the center-of-mass frame in MH coordinates have been given in Eqs. (222); see also Ref. [9] for details about the transformation between SH and MH coordinates.
12.3 Averaged energy and angular momentum fluxes
The gravitational wave energy and angular momentum fluxes from a system of two point masses in elliptic motion was first computed by Peters & Mathews [340, 339] at Newtonian level. The 1PN and 1.5PN corrections to the fluxes were provided in Refs. [416, 86, 267, 87, 366] and used to study the associated secular evolution of orbital elements under gravitational radiation reaction using the QK representation of the binary’s orbit at 1PN order [149]. These results were extended to 2PN order in Refs. [224, 225] for the instantaneous terms (leaving aside the tails) using the generalized QK representation [170, 379, 420]; the energy flux and waveform were in agreement with those of Ref. [424] obtained using a different method. Arun et al. [10, 9, 12] have fully generalized the results at 3PN order, including all tails and related hereditary contributions, by computing the averaged energy and angular momentum fluxes for quasi-elliptical orbits using the QK representation at 3PN order [312], and deriving the secular evolution of the orbital elements under 3PN gravitational radiation reaction.74
The secular evolution of orbital elements under gravitational radiation reaction is in principle only the starting point for constructing templates for eccentric binary orbits. To go beyond the secular evolution one needs to include in the evolution of the orbital elements, besides the averaged contributions in the fluxes, the terms rapidly oscillating at the orbital period. An analytic approach, based on an improved method of variation of constants, has been discussed in Ref. [153] for dealing with this issue at the leading 2.5PN radiation reaction order.
Variation of the enhancement factor φ(e) with the eccentricity e. This function agrees with the numerical calculation of Ref. [87] modulo a trivial rescaling with the Peters-Mathews function (356a). The inset graph is a zoom of the function at a smaller scale. The dots represent the numerical computation and the solid line is a fit to the numerical points. In the circular orbit limit we have φ(0) = 1.
The latter analytical result is very important for checking that the arbitrary constant x0 disappears from the final result. Indeed we immediately verify from comparing the last term in Eq. (356d) with Eq. (359c) that x0 cancels out from the sum of the instantaneous and hereditary contributions in the 3PN energy flux. This fact was already observed for the circular orbit case in Ref. [81]; see also the discussions around Eqs. (93)–(94) and at the end of Section 4.2.
Finally we can check that the correct circular orbit limit, which is given by Eq. (314), is recovered from the sum \(\left\langle {{{\mathcal F}_{{\rm{inst}}}}} \right\rangle + \left\langle {{{\mathcal F}_{{\rm{hered}}}}} \right\rangle\). The next correction of order \(e_t^2\) when e t → 0 can be deduced from Eqs. (360)–(361) in analytic form; having the flux in analytic form may be useful for studying the gravitational waves from binary black hole systems with moderately high eccentricities, such as those formed in globular clusters [235].
As we are interested in the phasing of binaries moving in quasi-eccentric orbits in the adiabatic approximation, we require the orbital averages not only of the energy flux \({\mathcal F}\) but also of the angular momentum flux \({{\mathcal G}_i}\). Since the quasi-Keplerian orbit is planar, we only need to average the magnitude \({\mathcal G}\) of the angular momentum flux. The complete computation thus becomes a generalisation of the previous computation of the averaged energy flux requiring similar steps (see Ref. [12]): The angular momentum flux is split into instantaneous \({{\mathcal G}_{{\rm{inst}}}}\) and hereditary \({{\mathcal G}_{{\rm{hered}}}}\) contributions; the instantaneous part is averaged using the QK representation in either MH or ADM coordinates; the hereditary part is evaluated separately and defined by means of several types of enhancement functions of the time eccentricity e t ; finally these are obtained numerically as well as analytically to next-to-leading order \(e_t^2\). At this stage we dispose of both the averaged energy and angular momentum fluxes \(\left\langle {\mathcal F} \right\rangle\) and \(\left\langle {\mathcal G} \right\rangle\).
13 Spinning Compact Binaries
The post-Newtonian templates have been developed so far for compact binary systems which can be described with great precision by point masses without spins. Here by spin, we mean the intrinsic (classical) angular momentum S of the individual compact body. However, including the effects of spins is essential, as the astrophysical evidence indicates that stellar-mass black holes [2, 390, 311, 227, 323] and supermassive black holes [188, 101, 102] (see Ref. [364] for a review) can be generically close to maximally spinning. The presence of spins crucially affects the dynamics of the binary, in particular leading to orbital plane precession if they are not aligned with the orbital angular momentum (see for instance [138, 8]), and thereby to strong modulations in the observed signal frequency and phase.
- 1.
Dynamics. The goal is to obtain the equations of motion and related conserved integrals of the motion, the equations of precession of the spins, and the post-Newtonian metric in the near zone. For this step we need a formulation of the dynamics of particles with spins (either Lagrangian or Hamiltonian);
- 2.
Radiation. The mass and current radiative multipole moments, including tails and all hereditary effects, are to be computed. One then deduces the gravitational waveform and the fluxes, from which we compute the secular evolution of the orbital phase. This step requires plugging the previous dynamics into the general wave generation formalism of Part A.
As usual we shall make a distinction between spin-orbit (SO) effects, which are linear in the spins, and spin-spin (SS) ones, which are quadratic. In this article we shall especially review the SO effects as they play the most important role in gravitational wave detection and parameter estimation. As we shall see a good deal is known on spin effects (both SO and SS), but still it will be important in the future to further improve our knowledge of the waveform and gravitational-wave phasing, by computing still higher post-Newtonian SO and SS terms, and to include at least the dominant spin-spin-spin (SSS) effect [305]. For the computations of SSS and even SSSS effects see Refs. [246, 245, 296, 305, 413].
The SO effects have been known at the leading level since the seminal works of Tulczyjew [411, 412], Barker & O’Connell [27, 28] and Kidder et al. [275, 271]. With our post-Newtonian counting such leading level corresponds to the 1.5PN order. The SO terms have been computed to the next-to-leading level which corresponds to 2.5PN order in Refs. [394, 194, 165, 292, 352, 241] for the equations of motion or dynamics, and in Refs. [53, 54] for the gravitational radiation field. Note that Refs. [394, 194, 165, 241] employ traditional post-Newtonian methods (both harmonic-coordinates and Hamiltonian), but that Refs. [292, 352] are based on the effective field theory (EFT) approach. The next-to-next-to-leading SO level corresponding to 3.5PN order has been obtained in Refs. [242, 244] using the Hamiltonian method for the equations of motion, in Ref. [297] using the EFT, and in Refs. [307, 90] using the harmonic-coordinates method. Here we shall focus on the harmonic-coordinates approach [307, 90, 89, 306] which is in fact best formulated using a Lagrangian, see Section 11.1. With this approach the next-to-next-to-leading SO level was derived not only for the equations of motion including precession, but also for the radiation field (energy flux and orbital phasing) [89, 306]. An analytic solution for the SO precession effects will be presented in Section 11.2. Note that concerning the radiation field the highest known SO level actually contains specific tail-induced contributions at 3PN [54] and 4PN [306] orders, see Section 11.3.
The SS effects are known at the leading level corresponding to 2PN order from Barker & O’Connell [27, 28] in the equations of motion (see [271, 351, 110] for subsequent derivations), and from Refs. [275, 271] in the radiation field. Next-to-leading SS contributions are at 3PN order and have been obtained with Hamiltonian [387, 389, 388, 247, 241], EFT [354, 356, 355, 293, 299] and harmonic-coordinates [88] techniques (with [88] obtaining also the next-to-leading SS terms in the gravitational-wave flux). With SS effects in a compact binary system one must make a distinction between the spin squared terms, involving the coupling between the two same spins S1 or S2, and the interaction terms, involving the coupling between the two different spins S1 and S2. The spin-squared terms \(S_1^2\) and \(S_2^2\) arise due to the effects on the dynamics of the quadrupole moments of the compact bodies that are induced by their spins [347]. They have been computed through 2PN order in the fluxes and orbital phase in Refs. [217, 218, 314]. The interaction terms S1 × S2can be computed using a simple pole-dipole formalism like the one we shall review in Section 11.1. The interaction terms S1 × S2 between different spins have been derived to next-to-next-to-leading 4PN order for the equations of motion in Refs. [294, 298] (EFT) and [243] (Hamiltonian). In this article we shall generally neglect the SS effects and refer for these to the literature quoted above.
13.1 Lagrangian formalism for spinning point particles
Some necessary material for constructing a Lagrangian for a spinning point particle in curved spacetime is presented here. The formalism is issued from early works [239, 19] and has also been developed in the context of the EFT approach [351]. Variants and alternatives (most importantly the associated Hamiltonian formalism) can be found in Refs. [389, 386, 25]. The formalism yields for the equations of motion of spinning particles and the equations of precession of the spins the classic results known in general relativity [411, 412, 310, 331, 135, 409, 179].
- 1.
The action is a covariant scalar, i.e., behaves as a scalar with respect to general space-time diffeomorphisms;
- 2.
It is a global Lorentz scalar, i.e., stays invariant under an arbitrary change of the tetrad vectors: \({e_A}^\alpha (\tau) \to {\Lambda ^B}_A{e_B}^\alpha (\tau)\) where \({\Lambda ^B}_A\) is a constant Lorentz matrix;
- 3.
It is reparametrization-invariant, i.e., its form is independent of the parameter τ used to follow the particle’s worldline.
In applications (e.g., the construction of gravitational wave templates for the compact binary inspiral) it is very useful to introduce new spin variables that are designed to have a conserved three-dimensional Euclidean norm (numerically equal to s). Using conserved-norm spin vector variables is indeed the most natural choice when considering the dynamics of compact binaries reduced to the frame of the center of mass or to circular orbits [90]. Indeed the evolution equations of such spin variables reduces, by construction, to ordinary precession equations, and these variables are secularly constant (see Ref. [423]).
13.2 Equations of motion and precession for spin-orbit effects
Geometric definitions for the precessional motion of spinning compact binaries [54, 306]. We show (i) the source frame defined by the fixed orthonormal basis {x, y, z}; (ii) the instantaneous orbital plane which is described by the orthonormal basis {x ℓ , y ℓ , ℓ}; (iii) the moving triad {n, λ, ℓ} and the associated three Euler angles α, ι and Φ; (v) the direction of the total angular momentum J which coincides with the z-direction. Dashed lines show projections into the x-y plane.
We now investigate an analytical solution for the dynamics of compact spinning binaries on quasi-circular orbits, including the effects of spin precession [54, 306]. This solution will be valid whenever the radiation reaction effects can be neglected, and is restricted to the linear SO level.
13.3 Spin-orbit effects in the gravitational wave flux and orbital phase
Like in Section 9 our main task is to control up to high post-Newtonian order the mass and current radiative multipole moments U L and V L which parametrize the asymptotic waveform and gravitational fluxes far away from the source, cf. Eqs. (66)–(68). The radiative multipole moments are in turn related to the source multipole moments I L and J L through complicated relationships involving tails and related effects; see e.g., Eqs. (76).86
However, in writing the latter equation it is important to justify that the spin quantities S ℓ and Σ ℓ are secularly constant, i.e., do not evolve on a gravitational radiation reaction time scale so we can neglect their variations when taking the time derivative of Eq. (415). Fortunately, this is the case of the conserved-norm spin variables, as proved in Ref. [423] up to relative 1PN order, i.e., considering radiation reaction effects up to 3.5PN order. Furthermore this can also be shown from the following structural general argument valid at linear order in spins [54, 89]. In the center-of-mass frame, the only vectors at our disposal, except for the spins, are n and v. Recalling that the spin vectors are pseudovectors regarding parity transformation, we see that the only way SO contributions can enter scalars such as the energy E or the flux \({\mathcal F}\) is through the mixed products (n, v, Sa), i.e., through the components \(S_\ell ^{\rm{a}}\). Now, the same reasoning applies to the precession vectors Σa in Eqs. (388): They must be pseudovectors, and, at linear order in spin, they must only depend on n and v; so that they must be proportional to ℓ, as can be explicitly seen for instance in Eq. (394). Now, the time derivative of the components along ℓ of the spins are given by \({\rm{d}}S_\ell ^{\rm{a}}/{\rm{d}}t = {{\rm{S}}_{\rm{a}}} \cdot ({\rm{d}}\ell {\rm{/d}}t + \ell + {\Omega _{\rm{a}}})\). The second term vanishes because Σa ℝ ℓ, and since \({\rm{d}}\ell/{\rm{d}}t = {\mathcal O}(S)\), we obtain that \(S_\ell ^{\rm{a}}\) is constant at linear order in the spins. We have already met an instance of this important fact in Eq. (409c). This argument is valid at any post-Newtonian order and for general orbits, but is limited to spin-orbit terms; furthermore it does not specify any time scale for the variation, so it applies to short time scales such as the orbital and precessional periods, as well as to the long gravitational radiation reaction time scale (see also Ref. [218] and references therein for related discussions).
Spin-orbit contributions to the number of gravitational-wave cycles \({{\mathcal N}_{{\rm{cycle}}}}\) [defined by Eq. (319)] for binaries detectable by ground-based detectors LIGO-VIRGO. The entry frequency is fseismic = 10 Hz and the terminal frequency is \({f_{{\rm{ISCO}}}} = {{{c^3}} \over {{6^3}{/^2}\pi G\;m}}\). For each compact object the magnitude χa and the orientation κa of the spin are defined by \({S_a} = G\;m_a^2{\chi _a}\;{\hat S_a}\) and κa = Ŝa · ℓ; remind Eq. (366). The spin-spin (SS) terms are neglected.
| PN order | 1.4 M⊙ + 1.4 M⊙ | 10 M⊙ + 1.4 M⊙ | 10 M⊙ + 10 M⊙ | |
|---|---|---|---|---|
| 1.5PN | (leading SO) | 65.6κ1χ1 + 65.6κ2χ2 | 114.0κ1χ1 +11.7κ2χ2 | 16.0κ1χ1 + 16.0κ2χ2 |
| 2.5PN | (1PN SO) | 9.3κ1χ1 + 9.3κ2χ2 | 33.8κ1χ1 + 2.9κ2χ2 | 5.7κ1χ1 + 5.7κ2χ2 |
| 3pn | (leading SO-tail) | −3.2κ1χ1 − 3.2κ2χ2 | −13.2κ1χ1 − 1.3κ2χ2 | −2.6κ1χ1 − 2.6κ2χ2 |
| 3.5PN | (2PN SO) | 1.9κ1χ1 + 1.9κ2χ2 | 11.1κ1χ1 + 0.8κ2χ2 | 1.7κ1χ1 + 1.7κ2χ2 |
| 4PN | (1PN SO-tail) | −1.5κ1χ1 − 1.5κ2χ2 | −8.Oκ1χ1 − 0.7κ2χ2 | −1.5κ1χ1 − 1.5κ2χ2 |
Footnotes
- 1.
A few errata have been published in this intricate field; all formulas take into account the latest changes.
- 2.
In this article Greek indices αβ … μν … take space-time values 0, 1, 2, 3 and Latin indices ab…ij… spatial values 1, 2, 3. Cartesian coordinates are assumed throughout and boldface notation is often used for ordinary Euclidean vectors. In Section 11 upper Latin letters AB… will refer to tetrad indices 0, 1, 2, 3 with ab… the corresponding spatial values 1, 2, 3. Our signature is +2; hence the Minkowski metric reads η αβ = diag(−1, +1, +1, +1) = η AB . As usual G and c are Newton’s constant and the speed of light.
- 3.
- 4.
Note that for very eccentric binaries (with say e → 1−) the Newtonian potential U can be numerically much larger than the estimate \({\mathcal O}{\rm{(1/}}{c^2}) \sim {v^2}/{c^2}\) at the apastron of the orbit.
- 5.
- 6.
The TT coordinate system can be extended to the near zone of the source as well; see for instance Ref. [282].
- 7.
Namely (Gm)1 = Ω2a3, where m = m1 + m2 is the total mass and Ω = 2π/P is the orbital frequency. This law is also appropriately called the 1-2-3 law [319].
- 8.
This work entitled: “The last three minutes: Issues in gravitational-wave measurements of coalescing compact binaries” is sometimes coined the “3mn Caltech paper”.
- 9.
All the works reviewed in this section concern general relativity. However, let us mention here that the equations of motion of compact binaries in scalar-tensor theories are known up to 2.5PN order [318].
- 10.
The effective action should be equivalent, in the tree-level approximation, to the Fokker action [207], for which the field degrees of freedom (i.e., the metric), that are solutions of the field equations derived from the original matter + field action with gauge-fixing term, have been inserted back into the action, thus defining the Fokker action for the sole matter fields.
- 11.
This reference has an eloquent title: “Feynman graph derivation of the Einstein quadrupole formula”.
- 12.
In absence of a better terminology, we refer to the leading-order contribution to the recoil as “Newtonian”, although it really corresponds to a 3.5PN subdominant radiation-reaction effect in the binary’s equations of motion.
- 13.Considering the coordinates x α as a set of four scalars, a simple calculation shows thatwhere □ g ≡ g μν ∇ μ ∇ ν denotes the curved d’Alembertian operator. Hence the harmonic-coordinate condition tells that the coordinates x α themselves, considered as scalars, are harmonic, i.e., obey the vacuum (curved) d’Alembertian equation.$${\partial _\mu}{h^{\alpha \mu}} = \sqrt {- g} \,{\Box_g}{x^\alpha}\,,$$
- 14.
In d + 1 space-time dimensions, only one coefficient in this expression is modified; see Eq. (175) below.
- 15.
See Eqs. (3.8) in Ref. [71] for the cubic and quartic terms. We denote e.g., \(h_\mu ^\alpha = {\eta _{\mu v}}{h^{\alpha v}},\,\,h = {\eta _{\mu v}}{h^{\mu v}}\), and ∂ α = η αμ ∂ μ . A parenthesis around a pair of indices denotes the usual symmetrization: T(αβ) = ½(T αβ + T βα ).
- 16.
ℕ, ℤ, ℝ, and ℂ are the usual sets of non-negative integers, integers, real numbers, and complex numbers; C p (Ω) is the set of p-times continuously differentiable functions on the open domain Ω (p ⩽ +∞).
- 17.
Our notation is the following: L = i1i2 … i ℓ denotes a multi-index, made of ℓ (spatial) indices. Similarly, we write for instance K = j1 … j k (in practice, we generally do not need to write explicitly the “carrier” letter i or j), or aL − 1 = ai1 … iℓ−1. Always understood in expressions such as Eq. (34) are ℓ summations over the indices i1,…, i ℓ ranging from 1 to 3. The derivative operator ∂ L is a short-hand for \({\partial _{{i_1} \cdots}}{\partial _{{i_\ell}}}\). The function K L (for any space-time indices αβ) is symmetric and trace-free (STF) with respect to the ℓ indices composing L. This means that for any pair of indices i p , i q ∈ L, we have \({\rm{K}} \ldots {i_p} \ldots {i_q} \ldots = {\rm{K}} \ldots {i_q} \ldots {i_p} \ldots\) and that \({\delta _{{i_p}{i_q}}}{\rm{K}} \ldots {i_p} \ldots {i_q} = 0\) (see Ref. [403] and Appendices A and B in Ref. [57] for reviews about the STF formalism). The STF projection is denoted with a hat, so \({{\rm{K}}_L} \equiv {{\rm{\hat K}}_L}\), or sometimes with carets around the indices, K L ≡ K〈L〉. In particular, \({\hat n_L} = {n_{\left\langle L \right\rangle}}\) is the STF projection of the product of unit vectors \({n_L} = {n_{{i_1} \cdots}}{n_{{i_\ell}}}\), for instance \({\hat n_{ij}} = {n_{\left\langle {ij} \right\rangle}} = {n_{ij}} - {1 \over 3}{\delta _{ij}}\) and \({\hat n_{ijk}} = {n_{\left\langle {ijk} \right\rangle}} = {n_{ijk}} - {1 \over 5}({\delta _{ij}}{n_k} + {\delta _{ik}}{n_j} + {\delta _{jk}}{n_i})\); an expansion into STF tensors \({\hat n_L} = {\hat n_L}(\theta, \phi)\) is equivalent to the usual expansion in spherical harmonics Y lm = Y lm (θ, ϕ), see Eqs. (75) below. Similarly, we denote \({x_L} = {x_{{i_1} \cdots}}{x_{{i_\ell}}} = {r^l}{n_L}\) where r = ∣x∣, and \({\hat x_L} = {x_{\langle L\rangle}} = {\rm STF}[{x_L}]\). The Levi-Civita antisymmetric symbol is denoted ε ijk (with ε123 = 1). Parenthesis refer to symmetrization, T(ij) = ½(T ij + T ji ). Superscripts (q) indicate q successive time derivations.
- 18.
The constancy of the center of mass X i — rather than a linear variation with time — results from our assumption of stationarity before the date \(- {\mathcal T}\), see Eq. (29). Hence, P i = 0.
- 19.
This assumption is justified because we are ultimately interested in the radiation field at some given finite post-Newtonian precision like 3PN, and because only a finite number of multipole moments can contribute at any finite order of approximation. With a finite number of multipoles in the linearized metric (35)–(37), there is a maximal multipolarity ℓmax(n) at any post-Minkowskian order n, which grows linearly with n.
- 20.
We employ the Landau symbol o for remainders with its standard meaning. Thus, f(r) = o[g(r)] when r → 0 means that f(r)/g(r) → 0 when r → 0. Furthermore, we generally assume some differentiability properties such as d n f(r)/dr n = o[g(r)/r n ].
- 21.
In this proof the coordinates are considered as dummy variables denoted (t, r). At the end, when we obtain the radiative metric, we shall denote the associated radiative coordinates by (T, R).
- 22.
- 23.The function Q ℓ is given in terms of the Legendre polynomial P ℓ byIn the complex plane there is a branch cut from −∞ to 1. The first equality is known as the Neumann formula for the Legendre function.$${Q_\ell}(x) = {1 \over 2}\int\nolimits_{- 1}^1 {{{{\rm{d}}z\,{P_\ell}(z)} \over {x - z}}} = {1 \over 2}{P_\ell}(x)\,{\rm{ln}}\left({{{x + 1} \over {x - 1}}} \right) - \sum\limits_{j = 1}^\ell {{1 \over j}} {P_{\ell - j}}(x){P_{j - 1}}(x)\,.$$
- 24.
We pose c = 1 until the end of this section.
- 25.The equation (85) has been obtained using a not so well known mathematical relation between the Legendre functions and polynomials:where 1 ⩽ y < x is assumed. See Appendix A in Ref. [48] for the proof. This relation constitutes a generalization of the Neumann formula (see the footnote 23).$${1 \over 2}\int\nolimits_{- 1}^1 {{{{\rm{d}}z\,{P_\ell}(z)} \over {\sqrt {{{(xy - z)}^2} - ({x^2} - 1)({y^2} - 1)}}}} = {Q_\ell}(x){P_\ell}(y)\,,$$
- 26.
The neglected remainders are indicated by o(1/r) rather than \({\mathcal O}(1/{r^2})\) because they contain powers of the logarithm of r; in fact they could be more accurately written as o(rε−2) for some ε ≪ 1.
- 27.
The canonical moment M ij differs from the source moment I ij by small 2.5PN and 3.5PN terms; see Eq. (97).
- 28.
In all formulas below the STF projection 〈〉 applies only to the “free” indices denoted ijkl… carried by the moments themselves. Thus the dummy indices such as abc… are excluded from the STF projection.
- 29.
Recall that our abbreviated notation \({\mathcal F}{\mathcal P}\) includes the crucial regularization factor \({\tilde r^B}\).
- 30.
Recall that in actual applications we need mostly the mass-type moment I L and