Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries
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Abstract
The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian 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 we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many nonlinear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail effects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary’s orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.
1 Introduction
The theory of gravitational radiation from isolated sources, in the context of general relativity, is a fascinating science that can be explored by means of what was referred to in the French XVIIIth century as l’analyse sublime: the analytical (i.e. mathematical) method, and more specifically the resolution of partial differential equations. Indeed, the field equations of general relativity, when use is made of the harmonic-coordinate conditions, take the form of a quasi-linear hyperbolic differential system of equations, involving the famous wave operator or d’Alembertian (denoted □), invented by d’Alembert in his Traité de dynamique of 1743.
Nowadays, the importance of the field lies in the exciting possibility of comparing the theory with contemporary astrophysical observations, made by a new generation of detectors — large-scale optical interferometers LIGO, VIRGO, GEO and TAMA — that should routinely observe the gravitational waves produced by massive and rapidly evolving systems such as inspiralling compact binaries. To prepare these experiments, the required theoretical work consists of carrying out a sufficiently general solution of the Einstein field equations, valid for a large class of matter systems, and describing the physical processes of the emission and propagation of the waves from the source to the distant detector, as well as their back-reaction onto the source.
1.1 Gravitational-wave generation formalisms
The basic problem we face is to relate the asymptotic gravitational-wave form h_{ ij } generated by some isolated source, at the location of some detector in the wave zone of the source, to the stress-energy tensor T^{ αβ } of the matter fields^{1}. For general sources it is hopeless to solve the problem via a rigorous deduction within the exact theory of general relativity, and we have to resort to approximation methods, keeping in mind that, sadly, such methods are often not related in a very precise mathematical way to the first principles of the theory. Therefore, a general 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. Of ourse the ultimate aim of approximation methods is to extract from the theory some firm predictions for the outcome of experiments such as VIRGO and LIGO. Some important approximations that we shall use in this article are the 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), and the far-zone expansion (1/R-expansion in the distance). In particular, the post-Newtonian expansion has provided us in the past with our best insights into the problems of motion and radiation in general relativity. The most successful wave-generation formalisms make a gourmet cocktail of all these approximation methods. For reviews on analytic approximations and applications to the motion and the gravitational wave-generation see Refs. [211, 83, 84, 212, 218, 17, 22].
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. The main problem with this approximation is its domain of validity, which is limited to the near zone of the source — the region surrounding the source that is of small extent with respect to the wavelength of waves. A serious consequence is the a priori inability of the 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 first to perform the post-Minkowskian expansion. Notice that the post-Minkowskian method is also upstream (in the previous sense) with respect to the multipole expansion, when considered outside the source, and with respect to the far-zone expansion, when considered far from the source.
The 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 [120, 129, 201], 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 formula (4) to include the current-quadrupole and mass-octupole moments [171, 170], and obtained the corresponding formulas for linear momentum [171, 170, 10, 186] and angular momentum [177, 75]. The general structure of the infinite multipole series in the linearized theory was investigated by several works [191, 192, 181, 210], 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 [210]. The explicit expressions of the multipole moments (for instance in STF guise) as integrals over the source, valid in the linearized theory but irrespective of a slow motion hypothesis, are completely known [159, 65, 64, 89].
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. [53] and Sachs [193]. The precise way under which such radiative space-times fall off asymptotically has been formulated geometrically by Penrose [175, 176] in the concept of an asymptotically simple space-time (see also Ref. [121]). 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 behaviour of the source. In particular the multipole moments that one considers in this approach are those measured at infinity — we call them the radiative multipole moments. These moments are distinct, because of 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 and Damour [26, 12], following pioneering work by Bonnor and collaborators [54, 55, 56, 130] and Thorne [210]. 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 (whereas, the direct attack of the post-Minkowskian expansion, valid at once inside and outside the source, faces some calculational difficulties [215, 76]). In this “multipolar-post-Minkowskian” 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 [15, 20]. 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 [63, 62].
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 [26]. 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 find [12] that the asymptotic behaviour of the solution at future null infinity is in agreement with the findings of the Bondi-Sachs-Penrose [53, 193, 175, 176, 121] approach to gravitational radiation. However, checking that the asymptotic structure of the radiative field is correct is not sufficient by itself, because the ultimate aim is to relate the far field to the properties of the source, and we are now obliged to ask: What are the multipole moments corresponding to a given stress-energy tensor T^{ αβ } describing the source? Only in the case of post-Newtonian sources has it been possible to answer this question. The general expression of the moments was obtained at the level of the second post-Newtonian (2PN) order in Ref. [15], and was subsequently proved to be in fact valid up to any post-Newtonian order in Ref. [20]. 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. [28] for the mass-type moments (strangely enough, the mass moments admit a compact-support expression at 1PN order), and in Ref. [90] for the current-type ones.
The wave-generation formalism resulting from matching the exterior multipolar and post-Minkowskian field [26, 12] to the post-Newtonian source [15, 20] 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 3PN order in Refs. [29, 21, 19]. 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 [27, 14, 18]. Now, recall that the leading radiation reaction force, which is quadrupolar, occurs already at the 2.5PN order in the source’s equations of motion. Therefore the 1.5PN “relative” order in the radiation reaction corresponds in fact to the 4PN order in the equations of motion, beyond the Newtonian acceleration. It has been shown that the gravitational wave tails enter the radiation reaction at precisely the 1.5PN relative order, which means 4PN “absolute” order [27]. A systematic post-Newtonian iteration scheme for the near-zone field, formally taking into account all radiation reaction effects, has been recently proposed, consistent with the present formalism [185, 41].
A different wave-generation formalism has been devised by Will and Wiseman [220] (see also Refs. [219, 173, 174]), after earlier attempts by Epstein and Wagoner [107] and Thorne [210]. This formalism has exactly the same scope as ours, 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 [220]. 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 \({\mathcal R}\) enclosing the source, e.g., the radius of the near zone). By contrast, in our case [15, 20], the moments are given by some integrals covering the whole space and regularized by means of the finite part \({\mathcal F}{\mathcal P}\). We shall prove the complete equivalence, at the most general level, between the two formalisms. What is interesting about both formalisms is that the source multipole moments, which involve a whole series of relativistic corrections, are coupled together, in the true non-linear solution, in a very complicated way. These multipole couplings give rise to the many tail and related non-linear effects, which form an integral part of the radiative moments at infinity and thereby of the observed signal.
Part A of this article is devoted to a presentation of the post-Newtonian wave generation formalism. We try to state the main results in a form that is simple enough to be understood without the full details, but at the same time we outline some of the proofs when they present some interest on their own. To emphasize the importance of some key results, we present them in the form of mathematical theorems.
1.2 Problem posed by compact binary systems
Inspiralling compact binaries, containing neutron stars and/or black holes, are promising sources of gravitational waves detectable by the detectors LIGO, VIRGO, GEO and TAMA. The two compact objects steadily lose their orbital binding energy by emission of gravitational radiation; as a result, the orbital separation between them decreases, and the orbital frequency increases. Thus, the frequency of the 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. For instance, the eccentricity of the orbit of the Hulse-Taylor binary pulsar is presently e_{0} = 0.617. At the time when the gravitational waves emitted by the binary system will become visible by the detectors, i.e. when the signal frequency reaches about 10 Hz (in a few hundred million years from now), the eccentricity will be e = 5.3 × 10^{−6} — a value calculated from the Peters [177] law, which is itself based on the quadrupole formula (2).
The inspiralling compact binaries are ideally suited for application of a high-order post-Newtonian wave generation formalism. The main reason is that these systems are very relativistic, with orbital velocities as high as 0.5c in the last rotations (as compared to ∼ 10^{−3}c for the binary pulsar), and it is not surprising that the quadrupole-moment formalism (2, 3, 4, 5) constitutes a poor description of the emitted gravitational waves, since many post-Newtonian corrections play a substantial role. This expectation has been confirmed in recent years by several measurement-analyses [77, 78, 111, 79, 203, 183, 184, 152, 92], which have demonstrated that the post-Newtonian precision needed to implement successively the optimal filtering technique in the LIGO/VIRGO detectors corresponds grossly, in the case of neutron-star binaries, to the 3PN approximation, or 1/c^{6} beyond the quadrupole moment approximation. Such a high precision is necessary because of the large number of orbital rotations that will be monitored in the detector’s frequency bandwidth (∼ 16, 000 in the case of neutron stars), giving the possibility of measuring very accurately the orbital phase of the binary. Thus, the 3PN order is required mostly to compute the time evolution of the orbital phase, which depends, via the energy equation (5), on the center-of-mass binding energy E and the total gravitational-wave energy flux \({\mathcal L}\).
In summary, the theoretical problem posed by inspiralling compact binaries is two-fold: On the one hand E, and on the other hand \({\mathcal L}\), are to be deduced from general relativity with the 3PN precision or better. To obtain E we must control the 3PN equations of motion of the binary in the case of general, not necessarily circular, orbits. As for \({\mathcal L}\) it necessitates the application of a 3PN wave generation formalism (actually, things are more complicated because the equations of motion are also needed during the computation of the flux). It is quite interesting that such a high order approximation as the 3PN one should be needed in preparation for LIGO and VIRGO data analysis. As we shall see, the signal from compact binaries contains at the 3PN order the signature of several non-linear effects which are specific to general relativity. Therefore, we have here the possibility of probing, experimentally, some aspects of the non-linear structure of Einstein’s theory [47, 48].
1.3 Post-Newtonian equations of motion and radiation
By equations of motion we mean the explicit expression of the accelerations of the bodies in terms of the positions and velocities. In Newtonian gravity, writing the equations of motion for a system of N particles is trivial; in general relativity, even writing the equations in the case N = 2 is difficult. The first relativistic term, at the 1PN order, was derived by Lorentz and Droste [156]. Subsequently, Einstein, Infeld and Hoffmann [106] obtained the 1PN corrections by means of their famous “surface-integral” method, in which the equations of motion are deduced from the vacuum field equations, and which are therefore applicable to any compact objects (be they neutron stars, black holes, or, perhaps, naked singularities). The 1PN-accurate equations were also obtained, for the motion of the centers of mass of extended bodies, by Petrova [179] and Fock [112] (see also Ref. [169]).
The 2PN approximation was tackled by Ohta et al. [165, 167, 166], who considered the post-Newtonian iteration of the Hamiltonian of N point-particles. We refer here to the Hamiltonian as the Fokker-type Hamiltonian, which is obtained from the matter-plus-field Arnowitt-Deser-Misner (ADM) Hamiltonian by eliminating the field degrees of freedom. The result for the 2PN and even 2.5PN equations of binary motion in harmonic coordinates was obtained by Damour and Deruelle [86, 85, 104, 80, 81], building on a non-linear iteration of the metric of two particles initiated in Ref. [11]. The corresponding result for the ADM-Hamiltonian of two particles at the 2PN order was given in Ref. [98] (see also Refs. [195, 196]). Kopeikin [149] derived the 2.5PN equations of motion for two extended compact objects. The 2.5PN-accurate harmonic-coordinate equations as well as the complete gravitational field (namely the metric g_{ αβ }) generated by two point masses were computed in Ref. [42], following a method based on previous work on wave generation [15].
Up to the 2PN level the equations of motion are conservative. Only at the 2.5PN order appears the first non-conservative effect, associated with the gravitational radiation reaction. The (harmonic-coordinate) equations of motion up to that level, as derived by Damour and Deruelle [86, 85, 104, 80, 81], have been used for the study of the radiation damping of the binary pulsar — its orbital Ṗ [81, 82, 102]. It is important to realize that the 2.5PN equations of motion have been proved to hold in the case of binary systems of strongly self-gravitating bodies [81]. This is via an “effacing” principle (in the terminology of Damour [81]) for the internal structure of the bodies. As a result, the equations depend only on the “Schwarzschild” masses, m_{1} and m_{2}, of the compact objects. Notably their compactness parameters K_{1} and K_{2}, defined by Equation (7), do not enter the equations of motion, as has been explicitly verified up to the 2.5PN order by Kopeikin et al. [149, 127], who made a “physical” computation, à la Fock, taking into account the internal structure of two self-gravitating extended bodies. The 2.5PN equations of motion have also been established by Itoh, Futamase and Asada [134, 135], who use a variant of the surface-integral approach of Einstein, Infeld and Hoffmann [106], that is valid for compact bodies, independently of the strength of the internal gravity.
The present state of the art is the 3PN approximation^{4}. To this order the equations have been worked out independently by two groups, by means of different methods, and with equivalent results. On the one hand, Jaranowski and Schafer [139, 140, 141], and Damour, Jaranowski, and Schafer [95, 97, 96], following the line of research of Refs. [165, 167, 166, 98], employ the ADM-Hamiltonian formalism of general relativity; on the other hand, Blanchet and Faye [37, 38, 36, 39], and de Andrade, Blanchet, and Faye [103], founding their approach on the post-Newtonian iteration initiated in Ref. [42], compute directly the equations of motion (instead of a Hamiltonian) in harmonic coordinates. The end results have been shown [97, 103] to be physically equivalent in the sense that there exists a unique “contact” transformation of the dynamical variables that changes the harmonic-coordinates Lagrangian obtained in Ref. [103] into a new Lagrangian, whose Legendre transform coincides exactly with the Hamiltonian given in Ref. [95]. The 3PN equations of motion, however, depend on one unspecified numerical coefficient, ω_{static} in the ADM-Hamiltonian formalism and λ in the harmonic-coordinates approach, which is due to some incompleteness of the Hadamard self-field regularization method. This coefficient has been fixed by means of a dimensional regularization, both within the ADM-Hamiltonian formalism [96], and the harmonic-coordinates equations of motion [30]. The works [96, 30] have demonstrated the power of dimensional regularization and its perfect adequateness for the problem of the interaction between point masses in general relativity. Furthermore, an important work by Itoh and Futamase [133, 132] (using the same surface-integral method as in Refs. [134, 135]) succeeded in obtaining the complete 3PN equations of motion in harmonic coordinates directly, i.e. without ambiguity and containing the correct value for the parameter λ.
So far the status of the post-Newtonian equations of motion is quite satisfying. There is mutual agreement between all the results obtained by means of different approaches and techniques, whenever it is possible to compare them: point particles described by Dirac delta-functions, extended post-Newtonian fluids, surface-integrals methods, mixed post-Minkowskian and post-Newtonian expansions, direct post-Newtonian iteration and matching, harmonic coordinates versus ADM-type coordinates, and different processes or variants of the regularization of the self field of point particles. In Part B of this article, we shall present the complete results for the 3PN equations of motion, and for the associated Lagrangian and Hamiltonian formulations (from which we deduce the center-of-mass energy E).
The second sub-problem, that of the computation of the energy flux \({\mathcal L}\), has been carried out by application of the wave-generation formalism described previously. Following earliest computations at the 1PN level [217, 49], at a time when the post-Newtonian corrections in L had a purely academic interest, the energy flux of inspiralling compact binaries was completed to the 2PN order by Blanchet, Damour and Iyer [33, 122], and, independently, by Will and Wiseman [220], using their own formalism (see Refs. [35, 46] for joint reports of these calculations). The preceding approximation, 1.5PN, which represents in fact the dominant contribution of tails in the wave zone, had been obtained in Refs. [221, 50] by application of the formula for tail integrals given in Ref. [29]. 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 by Blanchet [16, 19]. However, unlike the 1.5PN, 2.5PN, and 3.5PN orders that are entirely composed of tail terms, the 3PN approximation also involves, besides the tails of tails, many non-tail contributions coming from the relativistic corrections in the (source) multipole moments of the binary. These have been “almost” completed in Refs. [45, 40, 44], in the sense that the result still involves one unknown numerical coefficient, due to the use of the Hadamard regularization, which is a combination of the parameter λ in the equations of motion, and a new parameter θ coming from the computation of the 3PN quadrupole moment. The latter parameter is itself a linear combination of three unknown parameters, θ = ξ + 2κ + ζ. We shall review the computation of the three parameters ξ, κ, and ζ by means of dimensional regularization [31, 32]. In Part B of this article, we shall present the most up-to-date results for the 3.5PN energy flux and orbital phase, deduced from the energy balance equation (5), supposed to be valid at this order.
The post-Newtonian flux \({\mathcal L}\), which comes from a “standard” post-Newtonian calculation, is in complete agreement (up to the 3.5PN order) with the result given by the very different technique of linear black-hole perturbations, valid in the “test-mass” limit where the mass of one of the bodies tends to zero (limit ν → 0, where ν = μ/m). Linear black-hole perturbations, triggered by the geodesic motion of a small mass around the black hole, have been applied to this problem by Poisson [182] at the 1.5PN order (following the pioneering work of Galt’sov et al. [116]), and by Tagoshi and Nakamura [203], using a numerical code, up to the 4PN order. This technique has culminated with the beautiful analytical methods of Sasaki, Tagoshi and Tanaka [194, 205, 206] (see also Ref. [160]), who solved the problem up to the extremely high 5.5PN order.
2 Part A: Post-Newtonian Sources
3 Einstein’s Field Equations
- 1.The matter stress-energy tensor T^{ αβ } 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 Equation (17), is divergence-free,$${\partial _\mu}{\Lambda ^{\alpha \mu}} = 0\quad \quad ({\rm{when}}\,\,r\,{\rm{>}}\,a).$$(18)
- 2.
The matter distribution inside the source is smooth^{6}: T^{ αβ } ∈ C^{∞}(ℝ^{3}). We have in mind a smooth hydrodynamical “fluid” system, without any singularities nor shocks (a priori), that is described by some Eulerian equations including high relativistic corrections. In particular, we exclude from the start any black holes (however we shall return to this question when we find a model for describing compact objects).
- 3.
The source is post-Newtonian in the sense of the existence of the small parameter defined by Equation (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, in the sense that$${\partial \over {\partial t}}[{h^{\alpha \beta}}({\bf{x}},t)] = 0\quad \quad {\rm{when}}\,t \leq - \mathcal{T}.$$(19)
4 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^{8}, X_{ i } ≡ I_{ i }/I = const, total linear momentum \({{\rm{P}}_i} \equiv {\rm{I}}_i^{(1)} = 0\), and total angular momentum, S_{ i } ≡ J_{ i } = const. It is always possible to achieve X_{ i } = 0 by translating the origin of our coordinates to the center of mass. The total mass M is the ADM mass of the Hamiltonian formulation of general relativity. Note that the quantities M, X_{ i }, P_{ i } and Si 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: the above theorem follows merely from the algebraic and differential properties of the vacuum equations outside the source.
5 Non-linear Iteration of the Field Equations
5.1 The post-Minkowskian solution
Let us now proceed by induction. Some n being given, we assume that we succeeded in constructing, from the linearized coefficient h_{1}, the sequence of post-Minkowskian coefficients h_{2}, h_{3}, …, h_{n−1}, and from this we want to infer the next coefficient h_{ n }. The right-hand side of Equation (30), \(\Lambda _n^{\alpha \beta}\), is known by induction hypothesis. Thus the problem is that of solving a wave equation whose source is given. The point is that this wave equation, instead of being valid everywhere in ℝ^{3}, is correct only outside the matter (r > a), and it makes no sense to solve it by means of the usual retarded integral. Technically speaking, the right-hand side of Equation (30) is composed of the product of many multipole expansions, which are singular at the origin of the spatial coordinates r = 0, and which make the retarded integral divergent at that point. This does not mean that there are no solutions to the wave equation, but simply that the retarded integral does not constitute the appropriate solution in that context.
5.2 Generality of the solution
We have a solution, but is that a general solution? The answer, yes, is provided by the following result [26]:
Theorem 2 The most general solution of the harmonic-coordinates Einstein field equations in the vacuum region outside an isolated source, admitting some post-Minkowskian and multipolar expansions, is given by the previous construction as \({h^{\alpha \beta}} = \sum\nolimits_{n = 1}^{+ \infty} {{G^n}h_n^{\alpha \beta}[{{\rm{I}}_L},{{\rm{J}}_L}, \ldots, {{\rm{Z}}_L}]}\). It depends on two sets of arbitrary STF-tensorial functions of time I_{ L } (u) and J_{ L }(u) (satisfying the conservation laws) defined by Equations (27) , and on four supplementary functions W_{ L }(u), …, Z_{ L }(u) parametrizing the gauge vector (28) .
The six sets of multipole moments I_{ L }(u), …, Z_{ L }(u) contain the physical information about any isolated source as seen in its exterior. However, as we now discuss, it is always possible to find two, and only two, sets of multipole moments, M_{ L }(u) and S_{ L }(u), for parametrizing the most general isolated source as well. The route for constructing such a general solution is to get rid of the moments W_{ L }, X_{ L }, Y_{ L }, Z_{ L } at the linearized level by performing the linearized gauge transformation \(\delta {x^\alpha} = \varphi _1^\alpha\), where \(\varphi _1^\alpha\) is the gauge vector given by Equations (28). So, at the linearized level, we have only the two types of moments M_{ L } and S_{ L }, parametrizing \(k_1^{\alpha \beta}\) by the same formulas as in Equations (27). We must be careful to denote these moments with some names different from I_{ L } and J_{ L } because they will ultimately correspond to a different physical source. Then we apply exactly the same post-Minkowskian algorithm, following the formulas (39, 40, 41, 42, 43) as we did above, but starting from the gauge-transformed linear metric \(k_1^{\alpha \beta}\) instead of \(h_1^{\alpha \beta}\). The result of the iteration is therefore some \({k^{\alpha \beta}} = \sum\nolimits_{n = 1}^{+ \infty} {{G^n}k_n^{\alpha \beta}[{{\rm{M}}_L},{{\rm{S}}_L}]}\). Obviously this post-Minkowskian algorithm yields some simpler calculations as we have only two multipole moments to iterate. The point is that one can show that the resulting metric k^{ αβ }[M_{ L }, S_{ L }] is isometric to the original one h^{ αβ } [I_{ L }, J_{ L }, …, Z_{ L }] if and only if M_{ L } and S_{ L } are related to the moments I_{ L }, J_{ L }, …, Z_{ L } by some (quite involved) non-linear equations. Therefore, the most general solution of the field equations, modulo a coordinate transformation, can be obtained by starting from the linearized metric \(k_1^{\alpha \beta}[{{\rm{M}}_L},{{\rm{S}}_L}]\) instead of the more complicated \(k_1^{\alpha \beta}[{{\rm{I}}_L},{{\rm{J}}_L}] + {\partial ^\alpha}\varphi _1^\beta + {\partial ^\beta}\varphi _1^\alpha - {\eta ^{\alpha \beta}}{\partial _\mu}\varphi _1^\mu\), and continuing the post-Minkowskian calculation.
So why not consider from the start that the best description of the isolated source is provided by only the two types of multipole moments, M_{ L } and S_{ L }, instead of the six, 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 moments I_{ L }, J_{ L }, …, Z_{ L }, but that, by contrast, it seems to be impossible to obtain some similar closed-form expressions for 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 often 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 Ref. [16] and Equation (96) below. Indeed, this is to be expected because the physical difference between both types of moments stems only from non-linearities.
5.3 Near-zone and far-zone structures
In our presentation of the post-Minkowskian algorithm (39, 40, 41, 42, 43) we have 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 (and 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 \(h_n^{\alpha \beta}\) has a certain structure. This hypothesis is established as a theorem once the mathematical induction succeeds.
5.4 The radiative multipole moments
6 Exterior Field of 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, as well as all their perturbations. By Theorem 4 we learned how to construct the radiative moments at infinity. We now want to understand how a specific choice of stress-energy tensor T^{ αβ } (i.e. a choice of some physical model describing the source) selects a particular physical exterior solution among our general class.
6.1 The matching equation
We shall provide the answer in the case of a post-Newtonian source for which the post-Newtonian parameter 1/c defined by Equation (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 and the post-Newtonian expansions are valid. This region is nothing but the exterior near zone, such that r > a (exterior) and r ≪ λ (near zone). It always exists around post-Newtonian sources.
6.2 General expression of the multipole expansion
The latter proof makes it clear how crucial the analytic-continuation finite part \({\mathcal F}{\mathcal P}\) is, which we recall is the same as in our iteration of the exterior post-Minkowskian field (see Equation (39)). Without a finite part, the multipole moment (75) would be strongly divergent, because the pseudo-tensor \({\bar \tau ^{a\beta}}\) has a non-compact support owing to the contribution of the gravitational field, and the multipolar factor x_{ L } behaves like r^{ l } when r → +∞. In applications (Part B of this article) we must carefully follow the rules for handling the \({\mathcal F}{\mathcal P}\) operator.
The two terms in the right-hand side of Equation (67) depend separately on the length scale r_{0} that we have introduced into the definition of the finite part, through the analytic-continuation factor \({\tilde r^B} = {(r/{r_0})^B}\) (see Equation (36)). However, the sum of these two terms, i.e. the exterior multipolar field \({\mathcal M}(h)\) itself, is independent of r_{0}. To see this, the simplest way is to differentiate formally \({\mathcal M}(h)\) with respect to r_{0}. The independence of the field upon r_{0} is quite useful in applications, since in general many intermediate calculations do depend on r_{0}, and only in the final stage does the cancellation of the r_{0}’s occur. For instance, we shall see that the source quadrupole moment depends on r_{0} starting from the 3PN level [45], but that this r_{0} is compensated by another r_{0} coming from the non-linear “tails of tails” at the 3PN order.
6.3 Equivalence with the Will-Wiseman formalism
6.4 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 Sections 3 and 4. Namely, we must find the expressions of the six STF source multipole moments I_{ L }, J_{ L }, …, Z_{ L } parametrizing the linearized metric (26, 27, 28) at the basis of that construction^{12}.
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 τ^{ αβ } (this necessitates solving the field equations inside the matter, see Section 5.5) before inserting them into the source moments (85, 86, 82, 83, 91, 87, 88, 89, 90). The formula (91) is used to express all the terms up to that post-Newtonian order by means of more tractable integrals extending over ℝ^{3}. Given a specific model for the matter source we then have to find a way to compute all these spatial integrals (we do it in Section 10 in the case of point-mass binaries). Next, we must substitute the source multipole moments into the linearized metric (26, 27, 28), 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 shall work out these multipole interactions for general sources in the next section up to the 3PN order. 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 reflects simply 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.
6.5 Post-Newtonian field in the near zone
Theorem 6 solves in principle the question of the generation of gravitational waves by extended post-Newtonian sources. However, note that this result has to be completed by the definition of an explicit algorithm for the post-Newtonian iteration, analogous to the post-Minkowskian algorithm we defined in Section 4, so that the source multipole moments, which contain the full post-Newtonian expansion of the pseudo-tensor τ^{ αβ }, can be completely specified. Such a systematic post-Newtonian iteration scheme, valid (formally) to any post-Newtonian order, has been implemented [185, 41] using matched asymptotic expansions. The solution of this problem 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^{14}.
Before proceeding, let us recall that the “standard” post-Newtonian approximation, as it was used until, say, the early 1980’s (see for instance Refs. [2, 142, 143, 172]), is plagued with some apparently inherent difficulties, which crop up at some high post-Newtonian order. The first problem 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 over all space), and that as a result the standard Poisson integral diverges at the bound of the integral at spatial infinity, i.e. r = ∣x∣ → +∞, with t = const.
The second problem is related with the a priori limitation of the approximation to the near zone, which is the region surrounding the source of small extent with respect to the wavelength of the emitted radiation: r ≪ λ. 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. In particular, 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 not possible, a priori, to implement within the post-Newtonian iteration 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 into account, a priori, into the scheme. In a sense the post-Newtonian approximation is not “self-supporting”, because it necessitates some information taken from outside its own domain of validity.
Here we present, following Refs. [185, 41], 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 (65). This solution is free of the divergences of Poisson-type integrals we mentionned above, and it incorporates the effects of gravitational radiation reaction appropriate to an isolated system.
Importantly, we find that the post-Newtonian expansion \({\bar h^{\alpha \beta}}\) given by Theorem 7 is a functional not only of the related expansion of the pseudo-tensor, \({\bar \tau ^{\alpha \beta}}\), but also, by Equation (95), of its multipole expansion \({\mathcal M}({\tau ^{\alpha \beta}})\), which is valid in the exterior of the source, and in particular in the asymptotic regions far from the source. This can be understood by the fact that the post-Newtonian solution (93) depends on the boundary conditions imposed at infinity, that describe a matter system isolated from the rest of the universe.
Equation (93) is interesting for providing a practical recipe for performing the post-Newtonian iteration ad infinitum. Moreover, it gives some insights on the structure of radiation reaction terms. Recall that the anti-symmetric waves, regular in the source, are associated with radiation reaction effects. More precisely, it has been shown [185] that the specific anti-symmetric wave given by the second term of Equation (93) is linked with some non-linear contribution due to gravitational wave tails in the radiation reaction force. Such a contribution constitutes a generalization of the tail-transported radiation reaction term at the 4PN order, i.e. 1.5PN order relative to the dominant radiation reaction order, as determined in Ref. [27]. This term is in fact required by energy conservation and the presence of tails in the wave zone (see, e.g., Equation (97) below). Hence, the second term of Equation (93) is dominantly of order 4PN and can be neglected in computations of the radiation reaction up to 3.5PN order (as in Ref. [164]). The usual radiation reaction terms, up to 3.5PN order, which are linear in the source multipole moments (for instance the usual radiation reaction term at 2.5PN order), are contained in the first term of Equation (93), and are given by the terms with odd powers of 1/c in the post-Newtonian expansion (94). It can be shown [41] that such terms take also the form of some anti-symmetric multipolar wave, which turn out to be parametrized by the same moments as in the exterior field, namely the moments which are the STF analogues of Equations (68).
7 Non-linear Multipole Interactions
The tail integrals in Equations (97, 98) involve all the instants from −∞ in the past up to the current time U. However, strictly speaking, the integrals must not extend up to minus infinity in the past, because we have assumed from the start that the metric is stationary before the date \(- {\mathcal T}\); see Equation (19). The range of integration of the tails is therefore limited a priori to the time interval \([ - {\mathcal T},U]\). But now, once we have derived the tail integrals, thanks in part to the 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 bound system at our current epoch. In this situation we can check, using a simple Newtonian model for the behaviour of the quadrupole moment M_{ ij }(U − τ) when τ → +∞, that the tail integrals, when assumed to extend over the whole time interval [−∞, U], remain perfectly well-defined (i.e. convergent) at the integration bound τ = +∞. We regard this fact as a solid a posteriori justification (though not a proof) of our a priori too restrictive assumption of stationarity in the past. This assumption does not seem to yield any physical restriction on the applicability of the final formulas.
8 The Third Post-Newtonian Metric
It is important to remark that the above 3PN metric represents 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-type potentials (117, 118, 119).
9 Part B: Compact Binary Systems
The problem of the motion and gravitational radiation of compact objects in post-Newtonian approximations of general relativity is of crucial importance, for at least three reasons. First, the motion of N objects at the 1PN level (1/c^{2}), according to the Einstein-Infeld-Hoffmann equations [106], is routinely taken into account to describe the Solar System dynamics (see Ref. [163]). Second, the gravitational radiation-reaction force, which appears in the equations of motion at the 2.5PN order, has been experimentally verified, by the observation of the secular acceleration of the orbital motion of the binary pulsar PSR 1913+16 [208, 209, 207].
Last but not least, the forthcoming detection and analysis of gravitational waves emitted by inspiralling compact binaries — two neutron stars or black holes driven into coalescence by emission of gravitational radiation — will necessitate the prior knowledge of the equations of motion and radiation field up to high post-Newtonian order. As discussed in the introduction in Section 1 (see around Equations (6, 7, 8)), the appropriate theoretical description of inspiralling compact binaries is by two structureless point-particles, characterized solely by their masses m_{1} and m_{2} (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. Several analyses [77, 78, 111, 79, 203, 183, 184, 152, 92, 93, 59, 58, 91, 1, 6] have shown that, in order to get sufficiently accurate theoretical templates, one must include post-Newtonian effects up to the 3PN level at least.
To date, the templates have been completed through 3.5PN order for the phase evolution [35, 40, 31], and 2.5PN order for the amplitude corrections [46, 4]. Spin effects are known for the dominant relativistic spin-orbit coupling term at 1.5PN order and the spin-spin coupling term at 2PN order [146, 3, 144, 119, 118, 117, 70], and also for the next-to-leading spin-orbit coupling at 2.5PN order [168, 204, 110, 25].
10 Regularization of the Field of Point Particles
- 1.
Hadamard’s 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, 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 permits therefore 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).
10.1 Hadamard self-field regularization
In most practical computations we employ the Hadamard regularization [128, 199] (see Ref. [200] 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 investigations detailed in Refs. [36, 39].
The distributional derivative (129, 130, 131) does not satisfy the Leibniz rule for the derivation of a product, in accordance with a general result of Schwartz [198]. Rather, the investigation [36] suggests that, in order to construct a consistent theory (using the “ordinary” product for pseudo-functions), the Leibniz rule should be weakened, and replaced by the rule of integration by part, Equation (128), 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.
The Hadamard regularization (F)_{1} is defined by Equation (122) in a preferred spatial hyper-surface 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 [42]. To deal with the problem at 3PN order, a Lorentz-invariant variant of the regularization, denoted [F]_{1}, was introduced in Ref. [39]. 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/c^{2} at least. See Ref. [39] for the formulas defining this regularization in the form of some infinite power series in 1/c^{2}. The regularization [F]_{1} plays a crucial role in obtaining the equations of motion at the 3PN order in Refs. [37, 38]. In particular, the use of the Lorentz-invariant regularization [F]_{1} permits to obtain the value of the ambiguity parameter ω_{kinetic} in Equation (132) below.
10.2 Hadamard regularization ambiguities
The “standard” Hadamard regularization yields some ambiguous results for the computation of certain integrals at the 3PN order, as Jaranowski and Schäfer [139, 140, 141] first noticed 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, Equation (122), and the partie finie of a divergent integral, Equation (124) (i.e. without using a theory of pseudo-functions and generalized distributional derivatives as proposed in Refs. [36, 39]). It was shown in Refs. [139, 140, 141] that there are two and only two types of ambiguous terms in the 3PN Hamiltonian, which were then parametrized by two unknown numerical coefficients ω_{static} and ω_{kinetic}.
Damour, Jaranowski, and Schäfer [95] recovered the value of ω_{kinetic} given in Equation (132) by directly proving that this 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, they had 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 should be manifestly Lorentz-invariant, as was indeed found to be the case in Refs. [37, 38].
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^{24}. Technically speaking, the presence of the ambiguity parameter λ is associated with the non-distributivity of Hadamard’s regularization, in the sense of Equation (123). Mathematically speaking, λ is probably related to the fact that it is impossible to construct a distributional derivative operator, such as Equations (129, 130, 131), satisfying the Leibniz rule for the derivation of the product [198]. 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 (129, 130, 131) violates the Leibniz rule they become inequivalent for point particles. Finally, physically speaking, let us argue that λ 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, we expect 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, Equation (135), and will be valid for any compact objects, for instance black holes.
Let us comment here 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 by Kopeikin et al. [149, 127] who derived the equations of motion of two extended fluid balls, and obtained equations of motion depending only on the two masses m_{1} and m_{2} of the compact bodies^{26}. At the 3PN order we expect that the extended-body program should give the value of the regularization parameter λ (maybe after some gauge transformation to remove the terms depending on the internal structure). Ideally, its value should be confirmed by independent and more physical methods (like those of Refs. [214, 150, 101]).
An important work, in several respects more physical than the formal use of regularizations, is the one of Itoh and Futamase [133, 132], following previous investigations in Refs. [134, 135]. These authors derived the 3PN equations of motion in harmonic coordinates by means of a particular variant of the famous “surface-integral” method introduced long ago by Einstein, Infeld, and Hoffmann [106]. The aim is to describe extended relativistic compact binary systems in the strong-field point particle limit defined in Ref. [115]. This approach is very interesting because it is based on the physical notion of extended compact bodies in general relativity, and is free of the problems of ambiguities due to the Hadamard self-field regularization. The end result of Refs. [133, 132] is in agreement with the 3PN harmonic coordinates equations of motion [37, 38] and, moreover, is unambiguous, as it does determine the ambiguity parameter λ to exactly the value (135).
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, and Joguet [45], in their computation of 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 additional to and 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, involves only the linear combination of θ and λ given by \(\hat \theta = \theta - 7\lambda/3\), as shown in [40].
10.3 Dimensional regularization of the equations of motion
As reviewed in Section 8.2, 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 which appeared in the 3PN equations of motion (recall that λ is equivalent to the static ambiguity parameter ω_{static}, see Equation (133)).
Dimensional regularization was invented as a means to preserve the gauge symmetry of perturbative quantum field theories [202, 51, 57, 73]. 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 ω_{kinetic} (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.
Our strategy is to express both the dimensional and Hadamard regularizations in terms of their common “core” part, obtained by applying the so-called “pure-Hadamard-Schwartz” (pHS) regularization. Following the definition of Ref. [30], the pHS regularization is a specific, minimal Hadamard-type regularization of integrals, based on the partie finie integral (124), together with a minimal treatment of “contact” terms, in which the definition (124) is applied separately to each of the elementary potentials V, V_{ i }, … (and gradients) that enter the post-Newtonian metric in the form given in Section 7. 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 in contrast with Equation (123)). The pHS regularization also assumes the use of standard Schwartz distributional derivatives [199]. The interest of the pHS regularization is that the dimensional regularization is equal to it plus the “difference”; see Equation (155).
Theorem 8 The pole part ∝ 1/ε of the DR acceleration (155) can be re-absorbed (i.e. renormalized) into some shifts of the two “bare” world-lines: y_{1} → y_{1} + ξ_{1} and y_{2} → y_{2} + ξ_{2}, with, say, ξ_{1,2} ∝ 1/ε, 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 [96]. A central result is then as follows:
10.4 Dimensional regularization of the radiation field
We now address the similar problem concerning the binary’s radiation field (3PN beyond the Einstein quadrupole formalism), for which three ambiguity parameters, ξ, κ, ζ, have been shown to appear [45, 44] (see Section 8.2).
We next use the d-dimensional moment (158) to compute the difference between the dimensional regularization (DR) result and the pHS one [31, 32]. 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 (i.e. r_{1} = ∣x − y_{1}∣ → 0 or r_{2} = ∣x − y_{2}∣ → 0) that give rise to poles ∝ 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 DR — pHS. The compact-support terms in the integrand of Equation (158), proportional to the matter source densities σ, σ_{ α }, 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 in (158) 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 8.2, several checks of this calculation could be done, which provide, together with comparisons with alternative methods [96, 30, 133, 132], independent confirmations for the four ambiguity parameters λ, ξ, κ, and ζ, and confirm the consistency of dimensional regularization and its validity for describing the general-relativistic dynamics of compact bodies.
11 Newtonian-like Equations of Motion
11.1 The 3PN acceleration and energy
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. [42]), which consists of reducing the 3PN metric of an extended regular source, worked out in Equations (115), 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 geodesic equations associated with the regularized metric (see Ref. [39] for a proof). The Hadamard ambiguity parameter λ is computed from dimensional regularization in Section 8.3. We also add the 3.5PN terms which are known from Refs. [136, 137, 138, 174, 148, 164].
- (i)
stay manifestly invariant — at least in harmonic coordinates — when we perform a global post-Newtonian-expanded Lorentz transformation,
- (ii)
possess the correct “perturbative” limit, given by the geodesics of the (post-Newtonian-expanded) Schwarzschild metric, when one of the masses tends to zero, and
- (iii)
be conservative, i.e. to admit a Lagrangian or Hamiltonian formulation, when the gravitational radiation reaction is turned off.
11.2 Lagrangian and Hamiltonian formulations
11.3 Equations of motion in the center-of-mass frame
11.4 Equations of motion and energy for circular orbits
11.5 The innermost circular orbit (ICO)
Having in hand the circular-orbit energy, we define the innermost circular orbit (ICO) as the minimum, when it exists, of the energy function E(x). Notice that we do not define the ICO 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 of general relativity can be found in Ref. [43].
The previous definition of the ICO is motivated by our comparison with the results of numerical relativity. Indeed we shall confront the prediction of the standard (Taylor-based) post-Newtonian approach with a recent result of numerical relativity by Gourgoulhon, Grandclément, and Bonazzola [123, 126]. These authors computed numerically the energy of binary black holes under the assumptions of conformai flatness for the spatial metric and of exactly circular orbits. The latter restriction is implemented by requiring the existence of an “helical” Killing vector, which is timelike inside the light cylinder associated with the circular motion, and space-like outside. In the numerical approach [123, 126] 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. [114] for a discussion). Considering an evolutionary sequence of equilibrium configurations Refs. [123, 126] obtained numerically the circular-orbit energy E(ω) and looked for the ICO of binary black holes (see also Refs. [52, 124, 154] for related calculations of binary neutron and strange quark stars).
To take into account the spin effects our first task is to replace all the masses entering the energy function (194) by their equivalent expressions in terms of ω and the two irreducible masses. It is clear that the leading contribution is that of the spin kinetic energy given by Equation (199), and it comes from the replacement of the rest mass-energy m c^{2} (where m = M_{1} + M_{2}). From Equation (199) this effect is of order ω^{2} in the case of corotating binaries, which means by comparison with Equation (194) that it is equivalent to an “orbital” effect at the 2PN order (i.e. ∝ x^{2}). Higher-order corrections in Equation (199), which behave at least like ω^{4}, 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 ω^{8/3} equivalent to 3PN order, which comes from the replacement of the masses into the “Newtonian” part, proportional to x ∝ ω^{2/3}, of the energy E (see Equation (194)). 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.
In conclusion, we find that the location of the ICO as computed by numerical relativity, under the helical-symmetry and conformal-flatness approximations, is in good agreement with the post-Newtonian prediction. See also Ref. [88] for the results calculated within the effective-one-body approach method [60, 61] at the 3PN order, which are close to the ones reported in Figure 1. This agreement constitutes an appreciable improvement of the previous situation, because the earlier estimates of the ICO in post-Newtonian theory [145] and numerical relativity [180, 9] strongly disagreed with each other, and do not match with the present 3PN results. The numerical calculation of quasi-equilibrium configurations has been since then redone and refined by a number of groups, for both corotational and irrotational binaries (see in particular Ref. [74]). These works confirm the previous findings.
11.6 Accuracy of the post-Newtonian approximation
In this section we want to assess the validity of the post-Newtonian approximation, and, more precisely, to address, and to some extent to answer, the following questions: How accurate is the post-Newtonian expansion for describing the dynamics of binary black hole systems? Is the ICO of binary black holes, defined by the minimum of the energy function E(ω), accurately determined at the highest currently known post-Newtonian order? The latter question is pertinent because the ICO represents a point in the late stage of evolution of the binary which is very relativistic (orbital velocities of the order of 50% of the speed of light). How well does the 3PN approximation as compared with the prediction provided by numerical relativity (see Section 9.5)? What is the validity of the various post-Newtonian resummation techniques [92, 93, 60, 61] which aim at “boosting” the convergence of the standard post-Newtonian approximation?
The previous questions are interesting but difficult to settle down rigorously. Indeed the very essence of an approximation is to cope with our ignorance of the higher-order terms in some expansion, but the higher-order terms are precisely the ones which would be needed for a satisfying answer to these problems. So we shall be able to give only some educated guesses and/or plausible answers, that we cannot justify rigorously, but which seem very likely from the standard point of view on the post-Newtonian theory, in particular that the successive orders of approximation get smaller and smaller as they should (in average), with maybe only few accidents occuring at high orders where a particular approximation would be abnormally large with respect to the lower-order ones. Admittedly, in addition, our faith in the estimation we shall give regarding the accuracy of the 3PN order for instance, comes from the historical perspective, thanks to the many successes achieved in the past by the post-Newtonian approximation when confronting the theory and observations. It is indeed beyond question, from our past experience, that the post-Newtonian method does work.
Establishing the post-Newtonian expansion rigorously has been the subject of numerous mathematical oriented works, see, e.g., [187, 188, 189]. In the present section we shall simply look (much more modestly) at what can be said by inspection of the explicit post-Newtonian coefficients which have been computed so far. Basically, the point we would like to emphasize^{35} is that the post-Newtonian approximation, in standard form (without using the resummation techniques advocated in Refs. [92, 60, 61]), is able to located the ICO of two black holes, in the case of comparable masses (m_{1} m_{2}), with a very good accuracy. At first sight this statement is rather surprising, because the dynamics of two black holes at the point of the ICO is so relativistic. Indeed one sometimes hears about the “bad convergence”, or the “fundamental breakdown”, of the post-Newtonian series in the regime of the ICO. However our estimates do show that the 3PN approximation is good in this regime, for comparable masses, and we have already confirmed this by the remarkable agreement with the numerical calculations, as detailed in Section 9.5.
Let us now discuss a few order-of-magnitude estimates. At the location of the ICO we have found (see Figure 1 in Section 9.5) that the frequency-related parameter x defined by Equation (192) is approximately of the order of x ∼ (0.1)^{2/3} ∼ 20% for equal masses. Therefore, we might a priori expect that the contribution of the 1PN approximation to the energy at the ICO should be of that order. For the present discussion we take the pessimistic view that the order of magnitude of an approximation represents also the order of magnitude of the higher-order terms which are neglected. We see that the 1PN approximation should yield a rather poor estimate of the “exact” result, but this is quite normal at this very relativistic point where the orbital velocity is v/c ∼ x^{1/2} ∼ 50%. By the same argument we infer that the 2PN approximation should do much better, with fractional errors of the order of x^{2} ∼ 5%, while 3PN will be even better, with the accuracy x^{3} ∼ 1%.
Now the previous estimate makes sense only if the numerical values of the post-Newtonian coefficients in Equations (204) stay roughly of the order of one. If this is not the case, and if the coefficients increase dangerously with the post-Newtonian order n, one sees that the post-Newtonian approximation might in fact be very bad. It has often been emphasized in the litterature (see, e.g., Refs. [77, 183, 92]) that in the test-mass limit ν → 0 the post-Newtonian series converges slowly, so the post-Newtonian approximation is not very good in the regime of the ICO. Indeed we have seen that when ν = 0 the radius of convergence of the series is 1/3 (not so far from \(x_{{\rm{ICO}}}^{{\rm{Sch}}} = 1/6\)), and that accordingly the post-Newtonian coefficients increase by a factor ∼ 3 at each order. So it is perfectly correct to say that in the case of test particles in the Schwarzschild background the post-Newtonian approximation is to be carried out to a high order in order to locate the turning point of the ICO.
Newtonian | a_{1}(ν) | a_{2}(ν) | a_{3}(ν) | ||
---|---|---|---|---|---|
ν=0 | 1 | −0.75 | −3.37 | −10.55 | |
\(\nu = {1 \over 4}\), | ω*_{static} ≃ −9.34 | 1 | −0.77 | −2.78 | −8.75 |
\(\nu = {1 \over 4}\), | ω_{static} = 0 (GR) | 1 | −0.77 | −2.78 | −0.97 |
It is impossible of course to be thoroughly confident about the validity of the previous statement because we know only the coefficients up to 3PN order. Any tentative conclusion based on 3PN can be “falsified” when we obtain the next 4PN order. Nevertheless, we feel that the mere fact that \({a_3}({1 \over 4}) = - 0.97\) in Table 1 is sufficient to motivate our conclusion that the gravitational field generated by two bodies is more complicated than the Schwarzschild space-time. This appraisal should look cogent to relativists and is in accordance with the author’s respectfulness of the complexity of the Einstein field equations.
We want next to comment on a possible implication of our conclusion as regards the so-called post-Newtonian resummation techniques, i.e. Padé approximants [92, 93, 94], which aim at “boosting” the convergence of the post-Newtonian series in the pre-coalescence stage, and the effective-one-body (EOB) method [60, 61, 94], which attempts at describing the late stage of the coalescence of two black holes. These techniques are based on the idea that the gravitational two-body interaction is a “deformation” — with \(\nu \leq {1 \over 4}\) being the deformation parameter — of the Schwarzschild space-time. The Padé approximants are valuable tools for giving accurate representations of functions having some singularities. In the problem at hands they would be justified if the “exact” expression of the energy (whose 3PN expansion is given by Equations (203, 204)) would admit a singularity at some reasonable value of x (e.g., ≤ 0.5). In the Schwarzschild case, for which Equation (210) holds, the Padé series converges rapidly [92]: The Padé constructed only from the 2PN approximation of the energy — keeping only \(a_1^{{\rm{Sch}}}\) and \(a_2^{{\rm{Sch}}}\) — already coincide with the exact result given by Equation (208). On the other hand, the EOB method maps the post-Newtonian two-body dynamics (at the 2PN or 3PN orders) on the geodesic motion on some effective metric which happens to be a ν-deformation of the Schwarzschild space-time. In the EOB method the effective metric looks like Schwarzschild by definition, and we might of course expect the two-body interaction to own the main Schwarzschild-like features.
Finally we come to the good news that, if really the post-Newtonian coefficients when \(\nu = {1 \over 4}\) stay of the order of one (or minus one) as it seems to, this means that the standard post-Newtonian approach, based on the standard Taylor approximants, is probably very accurate. The post-Newtonian series is likely to “converge well”, with a “convergence radius” of the order of one^{36}. Hence the order-of-magnitude estimate we proposed at the beginning of this section is probably correct. In particular the 3PN order should be close to the “exact” solution for comparable masses even in the regime of the ICO.
12 Gravitational Waves from Compact Binaries
Obtaining \({\mathcal L}\) can be divided into two equally important steps: (1) the computation of the source multipole moments I_{ L } and J_{ L } of the compact binary and (2) the control and determination of the tails and related non-linear effects occuring in the relation between the binary’s source moments and the radiative ones U_{ L } and V_{ L } (cf. the general formalism of Part A).
12.1 The binary’s multipole moments
The general expressions of the source multipole moments given by Theorem 6, Equations (85), are first to be worked out explicitly for general fluid systems at the 3PN order. For this computation one uses the formula (91), and we insert the 3PN metric coefficients (in harmonic coordinates) expressed in Equations (115) by means of the retarded-type elementary potentials (117, 118, 119). 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. The infinite self-field of point-particles is removed by means of the Hadamard regularization; and dimensional regularization is used to compute the few ambiguity parameters (see Section 8). This computation has been performed in [49] at the 1PN order, and in [33] at the 2PN order; we report below the most accurate 3PN results obtained in Refs. [45, 44, 31, 32].
12.2 Contribution of wave tails
12.3 Orbital phase evolution
Contributions of post-Newtonian orders to the accumulated number of gravitational-wave cycles \({\mathcal N}\) (defined by Equation ( 236 )) in the bandwidth of VIRGO and LIGO detectors. Neutron stars have mass 1.4 M_{⊙}, and black holes 10 M_{⊙}. The entry frequency is f_{seismic} = 10 Hz, and the terminal frequency is f_{ISCO} = c^{3}/(6^{3/2}πGm).
2 × 1.4 M_{⊙} | 10 M_{⊙} + 1.4 M_{⊙} | 2 × 10 M_{⊙} | |
---|---|---|---|
Newtonian order | 16031 | 3576 | 602 |
1PN | 441 | 213 | 59 |
1.5PN (dominant tail) | −211 | −181 | −51 |
2PN | 9.9 | 9.8 | 4.1 |
2.5PN | −11.7 | −20.0 | −7.1 |
3PN | 2.6 | 2.3 | 2.2 |
3.5PN | −0.9 | −1.8 | −0.8 |
12.4 The two polarization waveforms
To conclude, the use of theoretical templates based on the preceding 2.5PN wave forms, and having their frequency evolution built in via the 3.5PN phase evolution (234, 235), should yield some accurate detection and measurement of the binary signals. 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 formulas (234, 235), e.g., those associated with the specific non-linear tails, exactly as they are predicted by Einstein’s theory [47, 48, 5]. Indeed, we don’t know of any other physical systems for which it would be possible to perform such tests.
Footnotes
- 1.
In this article Greek indices take the values 0, 1, 2, 3 and Latin 1, 2, 3. Our signature is +2. G and c are Newton’s constant and the speed of light.
- 2.
The TT coordinate system can be extended to the near zone of the source as well; see for instance Ref. [151].
- 3.
See Ref. [81] for the proof of such an “effacement” principle in the context of relativistic equations of motion.
- 4.
Let us mention that the 3.5PN terms in the equations of motion are also known, both for point-particle binaries [136, 137, 138, 174, 148, 164] and extended fluid bodies [14, 18]; they correspond to 1PN “relative” corrections in the radiation reaction force. Known also is the contribution of wave tails in the equations of motion, which arises at the 4PN order and represents a 1.5PN modification of the gravitational radiation damping [27].
- 5.
See also Equation (140) for the expression in d + 1 space-time dimensions.
- 6.
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 ≤ +∞).
- 7.
Our notation is the following: L = i_{1}i_{2} … i_{ l } denotes a multi-index, made of l (spatial) indices. Similarly we write for instance P = j_{1} … j_{ p } (in practice, we generally do not need to consider the carrier letter i or j), or aL − 1 = ai_{1} … i_{l−1} Always understood in expressions such as Equation (25) are l summations over the l indices i_{1}, …, i_{ l } ranging from 1 to 3. The derivative operator ∂_{ L } is a short-hand for \({\partial _{{i_1}}} \ldots {\partial _{{i_l}}}\). The function K_{ L } 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 \({K_{\ldots {i_p} \ldots {i_q} \ldots}} = {K_{\ldots {i_q} \ldots {i_p} \ldots}}\) and that \({\delta _{{i_p}{i_q}}}{K_{\ldots {i_p} \ldots {i_q} \ldots}} = 0\) (see Ref. [210] and Appendices A and B in Ref. [26] for reviews about the STF formalism). The STF projection is denoted with a hat, so \({K_L} \equiv {\hat K_L}\), or sometimes with carets around the indices, K_{ L } = K_{〈L〉}. In particular, \({\hat n_L} = {n_{\langle L\rangle}}\) is the STF projection of the product of unit vectors \({n_L} = {n_{{i_1}}} \ldots {n_{ij}}\); 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). Similarly, we denote \({x_L} = {x_{{i_1}}} \ldots {x_{{i_l}}} = {r^l}{n_L}\) and \({\hat x_L} = {x_{\langle L\rangle}}\). Superscripts like (p) indicate p successive time-derivations.
- 8.
The constancy of the center of mass Xi — rather than a linear variation with time — results from our assumption of stationarity before the date \(- {\mathcal T}\). Hence, Pi = 0.
- 9.
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 (26, 27, 28), there is a maximal multipolarity l_{max}(n) at any post-Minkowskian order n, which grows linearly with n.
- 10.
The o and \({\mathcal O}\) Landau symbols for remainders have their standard meaning.
- 11.
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).
- 12.
Recall that in actual applications we need mostly the mass-type moment I_{ l } and current-type one J_{ l }, because the other moments parametrize a linearized gauge transformation.
- 13.
This function approaches the Dirac delta-function (hence its name) in the limit of large multipoles: lim_{l→+∞} δl(z) = δ(z). Indeed the source looks more and more like a point mass as we increase the multipolar order l.
- 14.
An alternative approach to the problem of radiation reaction, besides the matching procedure, is to work only within a post-Minkowskian iteration scheme (which does not expand the retardations): see, e.g., Ref. [69].
- 15.
Notice that the normalization \(\int\nolimits_1^{+ \infty} {dz\;\gamma \iota (z)} = 1\) holds as a consequence of the corresponding normalization (83) for δ_{ l }(z), together with the fact that \(\int\nolimits_{- \infty}^{+ \infty} {dz\;\gamma \iota (z)} = 0\) by analytic continuation in the variable l ∈ ℂ.
- 16.
At the 3PN order (taking into account the tails of tails), we find that r_{0} does not completely cancel out after the replacement of U by the right-hand side of Equation (100). The reason is that the moment M_{ L } also depends on ro at the 3PN order. Considering also the latter dependence we can check that the 3PN radiative moment U_{ L } is actually free of the unphysical constant r_{0}.
- 17.
- 18.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.$${Q_l}(x) = {1 \over 2}\int\nolimits_{- 1}^1 {{{dz\,{P_l}(z)} \over {x - z}} = {1 \over 3}{P_l}(x)\ln \left({{{x + 1} \over {x - 1}}} \right) - \sum\limits_{j = 1}^l {{1 \over j}{P_{l - j}}(x){P_{j - 1}}(x)}}.$$
- 19.Equation (112) 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. [19] for the proof. This relation constitutes a generalization of the Neumann formula (see footnote after Equation (109)).$${1 \over 2}\int\nolimits_{- 1}^1 {{{dz\,{P_l}(z)} \over {\sqrt {{{(xy - z)}^2} - ({x^2} - 1)({y^2} - 1)}}} = {Q_l}(x){P_l}(y)}$$
- 20.
Actually, such a metric is valid up to 3.5PN order.
- 21.
It has been possible to “integrate directly” all the quartic contributions in the 3PN metric. See the terms composed of V^{4} and VX in the first of Equations (115).
- 22.
The function F(x) depends also on time t, through for instance its dependence on the velocities v_{1}(t) and v_{2}(t), but the (coordinate) t time is purely “spectator” in the regularization process, and thus will not be indicated.
- 23.
It was shown in Ref. [38] that using one or the other of these derivatives results in some equations of motion that differ by a mere coordinate transformation. This result indicates that the distributional derivatives introduced in Ref. [36] constitute merely some technical tools which are devoid of physical meaning.
- 24.
Note also that the harmonic-coordinates 3PN equations of motion as they have been obtained in Refs. [37, 38] depend, in addition to λ, on two arbitrary constants r′_{1} and r′_{2}, parametrizing some logarithmic terms. These constants are closely related to the constants s_{1} and s_{2} in the partie-finie integral (124); see Ref. [38] for the precise definition. However, r′_{1} and r′_{2}, are not “physical” in the sense that they can be removed by a coordinate transformation.
- 25.
One may wonder why the value of λ is a complicated rational fraction while ω_{static} is so simple. This is because ω_{static} was introduced precisely to measure the amount of ambiguities of certain integrals, while, by contrast, λ was introduced as an unknown constant entering the relation between the arbitrary scales r′_{1}, r′_{2} on the one hand, and s_{1}, s_{2} on the other hand, which has a priori nothing to do with ambiguities of integrals.
- 26.
See some comments on this work in Ref. [84], pp. 168–169.
- 27.
The result for ξ happens to be amazingly related to the one for λ by a cyclic permutation of digits; compare 3ξ = −9871/3080 with λ = −1987/3080.
- 28.
The work [34] provided also some new expressions for the multipole moments of an isolated post-Newtonian source, alternative to those given by Theorem 6, in the form of surface integrals extending on the outer part of the source’s near zone.
- 29.We have \({\lim\nolimits_{d \rightarrow 3}}\tilde k = 1\). Notice that \(\tilde k\) is closely linked to the volume Ω_{d−1} of the sphere with d − 1 dimensions (i.e. embedded into Euclidean d-dimensional space):$$\tilde k{\Omega _{d - 1}} = {{4\pi} \over {d - 2}}.$$
- 30.
When working at the level of the equations of motion (not considering the metric outside the world-lines), the effect of shifts can be seen as being induced by a coordinate transformation of the bulk metric as in Ref. [38].
- 31.Notice also the dependence upon π^{2}. Technically, the π^{2} terms arise from non-linear interactions involving some integrals such as$${1 \over \pi}\int {{{{d^3}{\bf{x}}} \over {r_1^2r_2^2}} = {{{\pi ^2}} \over {{r_{12}}}}.}$$
- 32.Note that in the result published in Ref. [95] the following terms are missing:This misprint has been corrected in an Erratum [95].$${{{{G^2}} \over {{c^6}r_{12}^2}}\left({- {{55} \over {12}}{m_1} - {{193} \over {48}}{m_2}} \right){{{{({N_{12}}{P_2})}^2}P_1^2} \over {{m_1}{m_2}}} + 1 \leftrightarrow 2.}$$
- 33.
Actually, in the present computation we do not need the radiation-reaction 2.5PN term in these relations; we give it only for completeness.
- 34.
In this section we pose G = 1 = c, and the two individual black hole masses are denoted M_{1} and M_{2}.
- 35.
- 36.
Actually, the post-Newtonian series could be only asymptotic (hence divergent), but nevertheless it should give excellent results provided that the series is truncated near some optimal order of approximation. In this discussion we assume that the 3PN order is not too far from that optimum.
- 37.
When computing the gravitational-wave flux in Ref. [45] we preferred to call the Hadamard-regularization constants u_{1} and u_{2}, in order to distinguish them from the constants s_{1} and s_{2} that were used in our previous computation of the equations of motion in Ref. [38]. Indeed these regularization constants need not neccessarily be the same when employed in different contexts.
- 38.
For circular orbits there is no difference at this order between I_{ l }, J_{ l } and M_{ l }, S_{ l }.
- 39.
- 40.
Generalizing the flux formula (231) to point masses moving on quasi elliptic orbits dates back to the work of Peters and Mathews [178] at Newtonian order. The result was obtained in [217, 49] at 1PN order, and then further extended by Gopakumar and Iyer [122] up to 2PN order using an explicit quasi-Keplerian representation of the motion [99, 197]. No complete result at 3PN order is yet available.
- 41.
Notice the “strange” post-Newtonian order of this time variable: \(\Theta = {\mathcal O}({c^{+ 8}})\).
- 42.
Notes
Acknowledgments
It is a great pleasure to thank Silvano Bonazzola, Alessandra Buonanno, Thibault Damour, Jürgen Ehlers, Gilles Esposito-Farèse, Guillaume Faye, Eric Gourgoulhon, Bala Iyer, Sergei Kopeikin, Misao Sasaki, Gerhard Schäfer, Bernd Schmidt, Kip Thorne, and Clifford Will for interesting discussions and/or collaborations.
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