Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries, Luc Blanchet, published in LivingReviews in Relativity on 2002-04-30

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_{i j}

generated by some isolated source, at the location of some detector in the wave zone of the source, to the stress-energy tensor

T^{α β}

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 ery precise mathematical way to the first principles of the theory. Therefore, a general wave-generation formalism must somehow manage the non-linearity of the field equations by imposing some suitable approximation series in one or several small physical parameters. Of ourse the ultimate aim of approximation methods is to extract from the theory some firm predictions for the outcome of experiments such as VIRGO and LIGO. Some important approximations that we shall use in this article are the 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. [143, 53, 54, 144, 150, 8, 13] .

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 most “downstream” approximation that we shall use in this article is the post-Newtonian one; therefore this is the approximation that dictates the allowed physical properties of our matter source. We assume mainly that the source is at once slowly moving and weakly stressed, and we abbreviate this by saying that the source is post-Newtonian. For post-Newtonian sources, the parameter defined from the components of the matter stress-energy tensor

T^{α β}

and the source's Newtonian potential

U

\begin{matrix} ε = max {| \frac{T^{0 i}}{T^{00}} |, {| \frac{T^{i j}}{T^{00}} |}^{1 / 2}, {| \frac{U}{c^{2}} |}^{1 / 2}}, \end{matrix}

(1)

is much less than one. This parameter represents essentially a slow motion estimate

ε \sim v / c

, where

v

denotes a typical internal velocity. By a slight abuse of notation, following Chandrasekhar et al. [40, 42, 41] , we shall henceforth write

ε \equiv 1 / c

, even though

ε

is dimensionless whereas

c

has the dimension of a velocity. The small post-Newtonian remainders will be denoted

O (1 / c^{n})

. Thus,

1 / c ≪ 1

in the case of post-Newtonian sources.

We have

| U / c^{2} |^{1 / 2} ≪ 1 / c

for sources with negligible self-gravity, and whose dynamics are therefore driven by non-gravitational forces. However, we shall generally assume that the source is self-gravitating; in that case we see that it is necessarily weakly (but not negligibly) self-gravitating, i.e.

| U / c^{2} |^{1 / 2} = O (1 / c)

. Note that the adjective “slow-motion” is a bit clumsy because we shall in fact consider very relativistic sources such as inspiralling compact binaries, for which

1 / c

can be as large as

30 %

in the last rotations, and whose description necessitates the control of high post-Newtonian approximations.

The lowest-order wave generation formalism, in the Newtonian limit

1 / c \to 0

, is the famous quadrupole formalism of Einstein [68] and Landau and Lifchitz [97] . This formalism can also be referred to as Newtonian because the evolution of the quadrupole moment of the source is computed using Newton's laws of gravity.

It expresses the gravitational field

h_{i j}^{T T}

in a transverse and traceless (TT) coordinate system, covering the far zone of the source^$2$ , as

\begin{matrix} h_{i j}^{T T} = \frac{2 G}{c^{4} R} P_{i j a b} (N) {\frac{d^{2} Q_{a b}}{d T^{2}} (T - R / c) + O (\frac{1}{c})} + O (\frac{1}{R^{2}}), \end{matrix}

(2)

where

R = | X |

is the distance to the source,

N = X / R

is the unit direction from the source to the observer, and

P_{i j a b} = P_{i a} P_{j b} - \frac{1}{2} δ_{i j} P_{i j} P_{a b}

is the TT projection operator, with

P_{i j} = δ_{i j} - N_{i} N_{j}

being the projector onto the plane orthogonal to

N

. The source's quadrupole moment takes the familiar Newtonian form

\begin{matrix} Q_{i j} (t) = \int_{s o u r c e} d^{3} x ρ (x, t) (x_{i} x_{j} - \frac{1}{3} δ_{i j} x^{2}), \end{matrix}

(3)

where

ρ

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

\begin{matrix} ℒ = \frac{G}{5 c^{5}} {\frac{d^{3} Q_{a b}}{d T^{3}} \frac{d^{3} Q_{a b}}{d T^{3}} + O (\frac{1}{c^{2}})} . \end{matrix}

(4)

Our notation

ℒ

stands for the total gravitational “luminosity” of the source. The cardinal virtues of the Einstein–Landau–Lifchitz quadrupole formalism are its generality – the only restrictions are that the source be Newtonian and bounded – its simplicity, as it necessitates only the computation of the time derivatives of the Newtonian quadrupole moment (using the Newtonian laws of motion), and, most importantly, its agreement with the observation of the dynamics of the Hulse-Taylor binary pulsar PSR 1913+16 [140, 141, 139] .

Indeed the prediction of the quadrupole formalism for the waves emitted by the binary pulsar system comes from applying Eq. ( 4 ) to a system of two point masses moving on an eccentric orbit (the classic reference is Peters and Mathews [117] ; see also Refs. [71, 148] ). Then, relying on the energy equation

\begin{matrix} \frac{d E}{d t} = - ℒ, \end{matrix}

(5)

where

E

is the Newtonian binary's center-of-mass energy, we deduce from Kepler 's third law the expression of the “observable”, that is, the change in the orbital period

P

of the pulsar, or

\dot{P}

, as a function of

P

itself. From the binary pulsar test, we can say that the post-Newtonian corrections to the quadrupole formalism, which we shall compute in this article, have already received, in the case of compact binaries, strong observational support (in addition to having, as we shall demonstrate, a sound theoretical basis).

The multipole expansion is one of the most useful tools of physics, but its use in general relativity is difficult because of the 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, 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 [111, 110] , and obtained the corresponding formulas for linear momentum [111, 110, 1, 124] and angular momentum [116, 46] . The general structure of the infinite multipole series in the linearized theory was investigated by several works [128, 126, 119, 142] , 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 [142] . The explicit expressions of the multipole moments (for instance in STF guise) as integrals over the source, valid in the linearized theory but irrespective of a slow motion hypothesis, are completely known [101, 39, 38, 57] .

In the full non-linear theory, the (radiative) multipole moments can be read off the coefficient of

1 / R

in the expansion of the metric when

R \to + \infty

, with a null coordinate

T - R / c = c o n s t .

. 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. [32] and Sachs [127] . The precise way under which such radiative space-times fall off asymptotically has been formulated geometrically by Penrose [114, 115] in the concept of an asymptotically simple space-time (see also Ref. [76] ). 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 \to + \infty

, 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 [14, 3] , following pioneering work by Bonnor and collaborators [33, 34, 35, 81] and Thorne [142] .

In this approach the basic multipole moments are the source moments, rather than the radiative ones. In a first stage, the moments are left unspecified, as being some arbitrary functions of time, supposed to describe an actual physical source. They are iterated by means of a 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 [147, 47] ). 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 [6, 11] . 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 [37, 36] .

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 [14] . 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 [3] that the asymptotic behaviour of the solution at future null infinity is in agreement with the findings of the Bondi–Sachs–Penrose [32, 127, 114, 115, 76] approach to gravitational radiation.

However, checking that the asymptotic structure of the radiative field is correct is not sufficient by itself, because the ultimate aim is to relate the far field to the properties of the source, and we are now obliged to ask: What are the multipole moments corresponding to a given stress-energy tensor

T^{α β}

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. [6] , and was subsequently proved to be in fact valid up to any post-Newtonian order in Ref. [11] . The source moments are given by some integrals extending over the post-Newtonian expansion of the total (pseudo) stress-energy tensor

τ^{α β}

, which is made of a matter part described by

T^{α β}

and a crucial non-linear gravitational source term

Λ^{α β}

. These moments carry in front a particular operation of taking the finite part (

ℱ P

as we call it below), which makes them mathematically well-defined despite the fact that the gravitational part

Λ^{α β}

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. [16] for the mass-type moments (strangely enough, the mass moments admit a compact-support expression at 1PN order), and in Ref. [58] for the current-type ones.

The wave-generation formalism resulting from matching the exterior multipolar and post-Minkowskian field [14, 3] to the post-Newtonian source [6, 11] 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. [17, 12, 10] . 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 [15, 5, 9] . Now, recall that the leading radiation reaction force, which is quadrupolar, occurs already at the 2.5PN order in the source's equations of motion. Therefore the 1.5PN “relative” order in the radiation reaction corresponds in fact to the 4PN order in the equations of motion, beyond the Newtonian acceleration. It has been shown that the gravitational wave tails enter the radiation reaction at precisely the 1.5PN relative order, which means 4PN “absolute” order [15] .

A different wave-generation formalism has been devised by Will and Wiseman [152] (see also Refs. [151, 112] ), after earlier attempts by Epstein and Wagoner [70] and Thorne [142] . This formalism has exactly the same scope as ours, i.e. it applies to any isolated post-Newtonian sources, but it differs in the definition of the source multipole moments and in many technical details when properly implemented [152] . In both formalisms, the moments are generated by the post-Newtonian expansion of the pseudo-tensor

τ^{α β}

, but in the Will–Wiseman formalism they are defined by some compact-support integrals terminating at some finite radius

ℛ

enclosing the source (e.g. the radius of the near zone). By contrast, in our case [6, 11] , the moments are given by some integrals covering the whole space and regularized by means of the finite part

ℱ 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 aaa 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.

^$1$ In this article Greek indices take the values $0, 1, 2, 3$ and Latin $1, 2, 3$ . Our signature is +2. $G$ and $c$ are Newton's constant and the speed of light.

^$2$ The TT coordinate system can be extended to the near zone of the source as well; see for instance Ref. [95] .

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 \times 10^{- 6}

– a value calculated from the Peters [116] law, which is itself based on the quadrupole formula ( 2 ).

The main point about modelling the inspiralling compact binary is that a model made of two structureless point particles, characterized solely by two mass parameters

m_{1}

and

m_{2}

(and possibly two spins), is sufficient.

Indeed, most of the non-gravitational effects usually plaguing the dynamics of binary star systems, such as the effects of a magnetic field, of an interstellar medium, and so on, are dominated by gravitational effects.

However, the real justification for a model of point particles is that the effects due to the finite size of the compact bodies are small. Consider for instance the influence of the Newtonian quadrupole moments

Q_{1}

and

Q_{2}

induced by tidal interaction between two neutron stars. Let

a_{1}

and

a_{2}

be the radius of the stars, and

L

the distance between the two centers of mass. We have, for tidal moments,

\begin{matrix} Q_{1} = k_{1} m_{2} \frac{a_{1}^{5}}{L^{3}}, Q_{2} = k_{2} m_{1} \frac{a_{2}^{5}}{L^{3}}, \end{matrix}

(6)

where

k_{1}

and

k_{2}

are the star's dimensionless (second) Love numbers [103] , which depend on their internal structure, and are, typically, of the order unity. On the other hand, for compact objects, we can introduce their “compactness”, defined by the dimensionless ratios

\begin{matrix} K_{1} = \frac{G m_{1}}{a_{1} c^{2}}, K_{2} = \frac{G m_{2}}{a_{2} c^{2}}, \end{matrix}

(7)

which equal

\sim 0.2

for neutron stars (depending on their equation of state). The quadrupoles

Q_{1}

and

Q_{2}

will affect both sides of Eq. ( 5 ), i.e. the Newtonian binding energy

E

of the two bodies, and the emitted total gravitational flux

ℒ

as computed using the Newtonian quadrupole formula ( 4 ). It is known that for inspiralling compact binaries the neutron stars are not co-rotating because the tidal synchronization time is much larger than the time left till the coalescence. As shown by Kochanek [92] the best models for the fluid motion inside the two neutron stars are the so-called Roche–Riemann ellipsoids, which have tidally locked figures (the quadrupole moments face each other at any instant during the inspiral), but for which the fluid motion has zero circulation in the inertial frame. In the Newtonian approximation we find that within such a model (in the case of two identical neutron stars) the orbital phase, deduced from Eq. ( 5 ), reads

\begin{matrix} φ^{f i n i t e s i z e} - φ_{0} = - \frac{1}{8 x^{5 / 2}} {1 + c o n s t . k {(\frac{x}{K})}^{5}}, \end{matrix}

(8)

where

x = (G m ω / c^{3})^{2 / 3}

is a standard dimensionless post-Newtonian parameter

\sim 1 / c^{2}

(

ω

is the orbital frequency), and where

k

is the Love number and

K

is the compactness of the neutron star. The first term in the right-hand side of ( 8 ) corresponds to the gravitational-wave damping of two point masses; the second term is the finite-size effect, which appears as a relative correction, proportional to

(x / K)^{5}

, to the latter radiation damping effect. Because the finite-size effect is purely Newtonian, its relative correction

\sim (x / K)^{5}

should not depend on

c

; and indeed the factors

1 / c^{2}

cancel out in the ratio

x / K

. However, the compactness

K

of compact objects is by Eq. ( 7 ) of the order unity (or, say, one half ), therefore the

1 / c^{2}

it contains should not be taken into account numerically in this case, and so the real order of magnitude of the relative contribution of the finite-size effect in Eq. ( 8 ) is given by

x^{5}

alone. This means that for compact objects the finite-size effect should be comparable, numerically, to a post-Newtonian correction of order 5PN or

1 / c^{10}

(see Ref. [52] for the proof in the context of relativistic equations of motion). This is a much higher post-Newtonian order than the one at which we shall investigate the gravitational effects on the phasing formula. Using

k^{'} \equiv c o n s t . k \sim 1

and

K \sim 0.2

for neutron stars (and the bandwidth of a VIRGO detector between 10 Hz and 1000 Hz), we find that the cumulative phase error due to the finite-size effect amounts to less that one orbital rotation over a total of

\sim 16, 000

produced by the gravitational-wave damping of point masses. The conclusion is that the finite-size effect can in general be neglected in comparison with purely gravitational-wave damping effects. But note that for non-compact or moderately compact objects (such as white dwarfs for instance) the Newtonian tidal interaction dominates over the radiation damping.

The 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 0.3

c

in the last rotations (as compared to

\sim 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 [48, 49, 72, 50, 135, 121, 122, 96, 59] , 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 (

\sim 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

ℒ

In summary, the theoretical problem posed by inspiralling compact binaries is two-fold: On the one hand

E

, and on the other hand

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

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 [28, 29] .

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 [98] . Subsequently, Einstein, Infeld and Hoffmann [69] 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 [118] and Fock [73] (see also Ref. [109] ).

The 2PN approximation was tackled by Otha et al. [105, 107, 106] , 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 [56, 55, 67, 51, 52] , building on a non-linear iteration of the metric of two particles initiated in Ref. [2] . The corresponding result for the ADM-Hamiltonian of two particles at the 2PN order was given in Ref. [63] (see also Refs. [130, 131] ). Kopeikin [93] derived the 2.5PN equations of motion for two extended compact objects. The 2.5PN-accurate harmonic-coordinate equations as well as the complete gravitational field (namely the metric

g_{α β}

) generated by two point masses were computed by Blanchet, Faye and Ponsot [25] , following a method based on previous work on wave generation [6] .

Up to the 2PN level the equations of motion are conservative. Only at the 2.5PN order appears the first 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 [56, 55, 67, 51, 52] , have been used for the study of the radiation damping of the binary pulsar – its orbital

\dot{P}

[52] . It is important to realize that the 2.5PN equations of motion have been proved to hold in the case of binary systems of strongly self-gravitating bodies [52] . This is via an “effacing” principle (in the terminology of Damour [52] ) for the internal structure of the bodies. As a result, the equations depend only on the “Schwarzschild” masses,

m_{1}

and

m_{2}

, of the compact objects. Notably their compactness parameters

K_{1}

and

K_{2}

, defined by Eq. ( 7 ), do not enter the equations of motion, as has been explicitly verified up to the 2.5PN order by Kopeikin [93] and Grishchuk and Kopeikin [79] , who made a “physical” computation, à la Fock, taking into account the internal structure of two self-gravitating extended bodies. The 2.5PN equations of motion have also been established by Itoh, Futamase and Asada [83, 84] , who use a variant of the surface-integral approach of Einstein, Infeld and Hoffmann [69] , that is valid for compact bodies, independently of the strength of the internal gravity.

The present state of the art is the 3PN approximation^$3$ . To this order the equations have been worked out independently by two groups, by means of different methods, and with equivalent results. On the one hand, Jaranowski and Schäfer [87, 88, 89] , and Damour, Jaranowski and Schäfer [60, 62, 61] , following the line of research of Refs. [105, 107, 106, 63] , employ the ADM-Hamiltonian formalism of general relativity; on the other hand, Blanchet and Faye [21, 22, 20, 23] , and de Andrade, Blanchet and Faye [66] , founding their approach on the post-Newtonian iteration initiated in Ref. [25] , compute directly the equations of motion (instead of a Hamiltonian) in harmonic coordinates. The end results have been shown [62, 66] to be physically equivalent in the sense that there exists a unique “contact” transformation of the dynamical variables that changes the harmonic-coordinates Lagrangian obtained in Ref. [66] into a new Lagrangian, whose Legendre transform coincides exactly with the Hamiltonian given in Ref. [60] . The 3PN equations of motion, however, depend on one unspecified numerical coefficient,

ω_{s t a t i c}

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 in Ref. [61] .

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 bbb of this article, we shall present the most complete results for the 3PN equations of motion, and for the associated Lagrangian and Hamiltonian formulations (from which we deduce the center-of-mass energy

E

The second sub-problem, that of the computation of the energy flux

ℒ

, has been carried out by application of the wave-generation formalism described previously. Following earliest computations at the 1PN level [149, 30] , 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 [18, 77] , and, independently, by Will and Wiseman [152] , using their own formalism (see Refs. [19, 27] for joint reports of these calculations). The preceding approximation, 1.5PN, which represents in fact the dominant contribution of tails in the wave zone, had been obtained in Refs. [153, 31] by application of the formula for tail integrals given in Ref. [17] . 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 [7, 10] . However, unlike the 1.5PN, 2.5PN and 3.5PN orders that are entirely composed of tail terms, the 3PN approximation also involves, besides the tails of tails, many non-tail contributions coming from the relativistic corrections in the (source) multipole moments of the binary. These have been “almost” completed by Blanchet, Iyer and Joguet [26, 24] , in the sense that the result still involves one unknown numerical coefficient, due to the use of the Hadamard regularization, which is a combination of the parameter

λ

in the equations of motion, and a new parameter

θ

coming from the computation of the 3PN quadrupole moment.

In Part bbb 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 equation ( 5 ), supposed to be valid at this order.

The post-Newtonian flux

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 \to 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 [120] at the 1.5PN order (following the pioneering work of Galt'sov et al [75] ), and by Tagoshi and Nakamura [135] , using a numerical code, up to the 4PN order. This technique has culminated with the beautiful analytical methods of Sasaki, Tagoshi and Tanaka [129, 137, 138] (see also Ref. [102] ), who solved the problem up to the extremely high 5.5PN order.

^$3$ Let us mention that the 3.5PN terms in the equations of motion are also known, both for point-particle binaries [85, 86, 113] and extended fluid bodies [5, 9] ; they correspond to 1PN “relative” corrections in the radiation reaction force. Known also is the contribution of wave tails in the equations of motion, which arises at the 4PN order and represents a 1.5PN modification of the gravitational radiation damping [15] .

2 Einstein's Field Equations

The field equations of general relativity form a system of ten second-order partial differential equations obeyed by the space-time metric

g_{α β}

\begin{matrix} G^{α β} [g, \partial g, \partial^{2} g] = \frac{8 π G}{c^{4}} T^{α β} [g], \end{matrix}

(9)

where the Einstein curvature tensor

G^{α β} \equiv R^{α β} - \frac{1}{2} R g^{α β}

is generated, through the gravitational coupling

κ = 8 π G / c^{4}

, by the matter stress-energy tensor

T^{α β}

. Among these ten equations, four govern, via the contracted Bianchi identity, the evolution of the matter system,

\begin{matrix} \nabla_{μ} G^{α μ} \equiv 0 ⟹ \nabla_{μ} T^{α μ} = 0 . \end{matrix}

(10)

The space-time geometry is constrained by the six remaining equations, which place six independent constraints on the ten components of the metric

g_{α β}

, leaving four of them to be fixed by a choice of a coordinate system.

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

\begin{matrix} h^{α β} = \sqrt{- g} g^{α β} - η^{α β}, \end{matrix}

(11)

where

g^{α β}

denotes the contravariant metric (satisfying

g^{α μ} g_{μ β} = δ_{β}^{α}

), where

g

is the determinant of the covariant metric,

g = d e t (g_{α β})

, and where

η^{α β}

represents an auxiliary Minkowskian metric. The harmonic-coordinate condition, which accounts exactly for the four equations ( 10 ) corresponding to the conservation of the matter tensor, reads

\begin{matrix} \partial_{μ} h^{α μ} = 0 . \end{matrix}

(12)

The equations ( 11 , 12 ) introduce into the definition of our coordinate system a preferred Minkowskian structure, with Minkowski metric

η_{α β}

. Of course, this is not contrary to the spirit of general relativity, where there is only one physical metric

g_{α β}

without any flat prior geometry, because the coordinates are not governed by geometry (so to speak), but rather are chosen by researchers when studying physical phenomena and doing experiments. Actually, the coordinate condition ( 12 ) is especially useful when we view the gravitational waves as perturbations of space-time propagating on the fixed Minkowskian manifold with the background metric

η_{α β}

. This view is perfectly legitimate and represents a fructuous and rigorous way to think of the problem when using approximation methods. Indeed, the metric

η_{α β}

, originally introduced in the coordinate condition ( 12 ), does exist at any finite order of approximation (neglecting higher-order terms), and plays in a sense the role of some “prior” flat geometry.

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

\begin{matrix} □ h^{α β} = \frac{16 π G}{c^{4}} τ^{α β}, \end{matrix}

(13)

where

□ \equiv □_{η} = η^{μ ν} \partial_{μ} \partial_{ν}

. The source term,

τ^{α β}

, can rightly be interpreted as the stress-energy pseudo-tensor (actually,

τ^{α β}

is a Lorentz tensor) of the matter fields, described by

T^{α β}

, and the gravitational field, given by the gravitational source term

Λ^{α β}

, i.e.

\begin{matrix} τ^{α β} = | g | T^{α β} + \frac{c^{4}}{16 π G} Λ^{α β} . \end{matrix}

(14)

The exact expression of

Λ^{α β}

, including all non-linearities, reads

\begin{matrix} Λ^{α β} = & - & h^{μ ν} \partial_{μ ν}^{2} h^{α β} + \partial_{μ} h^{α ν} \partial_{ν} h^{β μ} + \frac{1}{2} g^{α β} g_{μ ν} \partial_{λ} h^{μ τ} \partial_{τ} h^{ν λ} \end{matrix}

\begin{matrix} - & g^{α μ} g_{ν τ} \partial_{λ} h^{β τ} \partial_{μ} h^{ν λ} - g^{β μ} g_{ν τ} \partial_{λ} h^{α τ} \partial_{μ} h^{ν λ} + g_{μ ν} g^{λ τ} \partial_{λ} h^{α μ} \partial_{τ} h^{β ν} \end{matrix}

\begin{matrix} + & \frac{1}{8} (2 g^{α μ} g^{β ν} - g^{α β} g^{μ ν}) (2 g_{λ τ} g_{ε π} - g_{τ ε} g_{λ π}) \partial_{μ} h^{λ π} \partial_{ν} h^{τ ε} . \end{matrix}

(15)

As is clear from this expression,

Λ^{α β}

is made of terms at least quadratic in the gravitational-field strength

h

and its first and second space-time derivatives. In the following, for the highest post-Newtonian order that we consider (3PN), we need the quadratic, cubic and quartic pieces of

Λ^{α β}

. With obvious notation, we can write them as

\begin{matrix} Λ^{α β} = N^{α β} [h, h] + M^{α β} [h, h, h] + L^{α β} [h, h, h, h] + O (h^{5}) . \end{matrix}

(16)

These various terms can be straightforwardly computed from Eq. ( 15 ); see Eqs. (3.8) in Ref. [22] for explicit expressions.

As said above, the condition ( 12 ) is equivalent to the matter equations of motion, in the sense of the conservation of the total pseudo-tensor

τ^{α β}

\begin{matrix} \partial_{μ} τ^{α μ} = 0 ⟺ \nabla_{μ} T^{α μ} = 0 . \end{matrix}

(17)

In this article, we look for the solutions of the field equations ( 13 , 14 , 15 , 17 ) under the following four hypotheses:

1. The matter stress-energy tensor $T^{α β}$ is of spatially compact support, i.e. can be enclosed into some time-like world tube, say $r \leq a$ , where $r = | x |$ is the harmonic-coordinate radial distance.
Outside the domain of the source, when $r > a$ , the gravitational source term, according to Eq. ( 17 ), is divergence-free,
$\begin{matrix} \partial_{μ} Λ^{α μ} = 0 (w h e n r > a) . \end{matrix}$ (18)
2. The matter distribution inside the source is smooth^$4$ : $T^{α β} \in C^{\infty} (R^{3})$ . 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 Eq. ( 1 ).
For such a source we assume the legitimacy of the method of matched asymptotic expansions for identifying the inner post-Newtonian field and the outer multipolar decomposition in the source's exterior near zone.

4. The gravitational field has been independent of time (stationary) in some remote past, i.e. before some finite instant

- T

in the past, in the sense that

\begin{matrix} \frac{\partial}{\partial t} [h^{α β} (x, t)] = 0 when t \leq - T . \end{matrix}

(19)

The latter condition is a means to impose, by brute force, the famous no-incoming radiation condition, ensuring that the matter source is isolated from the rest of the Universe and does not receive any radiation from infinity. Ideally, the no-incoming radiation condition should be imposed at past null infinity. We shall later argue (see Section 6 ) that our condition of stationarity in the past ( 19 ), although much weaker than the real no-incoming radiation condition, does not entail any physical restriction on the general validity of the formulas we derive.

Subject to the condition ( 19 ), the Einstein differential field equations ( 13 ) can be written equivalently into the form of the integro-differential equations

\begin{matrix} h^{α β} = \frac{16 π G}{c^{4}} □_{r e t}^{- 1} τ^{α β}, \end{matrix}

(20)

containing the usual retarded inverse d'Alembertian operator, given by

\begin{matrix} (□_{r e t}^{- 1} f) (x, t) \equiv - \frac{1}{4 π} \int \int \int \frac{d^{3} x^{'}}{| x - x^{'} |} f (x^{'}, t - | x - x^{'} | / c), \end{matrix}

(21)

extending over the whole three-dimensional space

R^{3}

3 Linearized Vacuum Equations

In what follows we solve the field equations ( 12 , 13 ), in the vacuum region outside the compact-support source, in the form of a formal non-linearity or post-Minkowskian expansion, considering the field variable

h^{α β}

as a non-linear metric perturbation of Minkowski space-time. At the linearized level (or first-post-Minkowskian approximation), we write:

\begin{matrix} h_{e x t}^{α β} = G h_{1}^{α β} + O (G^{2}), \end{matrix}

(22)

where the subscript ”ext” reminds us that the solution is valid only in the exterior of the source, and where we have introduced Newton's constant

G

as a book-keeping parameter, enabling one to label very conveniently the successive post-Minkowskian approximations. Since

h^{α β}

is a dimensionless variable, with our convention the linear coefficient

h_{1}^{α β}

in Eq. ( 22 ) has the dimension of the inverse of

G

– a mass squared in a system of units where

ħ = c = 1

. In vacuum, the harmonic-coordinate metric coefficient

h_{1}^{α β}

satisfies

\begin{matrix} □ h_{1}^{α β} = 0, \end{matrix}

(23)

\begin{matrix} \partial_{μ} h_{1}^{α μ} = 0 . \end{matrix}

(24)

We want to solve those equations by means of an infinite multipolar series valid outside a time-like world tube containing the source. Indeed the multipole expansion is the correct method for describing the physics of the source as seen from its exterior (

r > a

). On the other hand, the post-Minkowskian series is physically valid in the weak-field region, which surely includes the exterior of any source, starting at a sufficiently large distance.

For post-Newtonian sources the exterior weak-field region, where both multipole and post-Minkowskian expansions are valid, simply coincides with the exterior

r > a

. It is therefore quite natural, and even, one would say inescapable when considering general sources, to combine the post-Minkowskian approximation with the multipole decomposition. This is the original idea of the “double-expansion” series of Bonnor [33] , which combines the

G

-expansion (or

m

-expansion in his notation) with the

a

-expansion (equivalent to the multipole expansion, since the

l

th order multipole moment scales like

a^{l}

with the source radius).

The multipolar-post-Minkowskian method will be implemented systematically, using STF-harmonics to describe the multipole expansion [142] , and looking for a definite algorithm for the approximation scheme [14] .

The solution of the system of equations ( 23 , 24 ) takes the form of a series of retarded multipolar waves^$5$

\begin{matrix} h_{1}^{α β} = \sum_{l = 0}^{+ \infty} \partial_{L} (\frac{K_{L}^{α β} (t - r / c)}{r}), \end{matrix}

(25)

where

r = | x |

, and where the functions

K_{L}^{α β} \equiv K_{i_{1} . . . i_{l}}^{α β}

are smooth functions of the retarded time

u \equiv t - r / c

[

K_{L} (u) \in C^{\infty} (R)

], which become constant in the past, when

t \leq - T

. It is evident, since a monopolar wave satisfies

□ (K_{L} (u) / r) = 0

and the d'Alembertian commutes with the multi-derivative

\partial_{L}

, that Eq. ( 25 ) represents the most general solution of the wave equation ( 23 ) (see Section 2 in Ref. [14] for a proof based on the Euler–Poisson–Darboux equation). The gauge condition ( 24 ), however, is not fulfilled in general, and to satisfy it we must algebraically decompose the set of functions

K_{L}^{00}

K_{L}^{0 i}

K_{L}^{i j}

into ten tensors which are STF with respect to all their indices, including the spatial indices

i

i j

. Imposing the condition ( 24 ) reduces the number of independent tensors to six, and we find that the solution takes an especially simple “canonical” form, parametrized by only two moments, plus some arbitrary linearized gauge transformation [142, 14] .

Theorem 1

The most general solution of the linearized field equations ( 23 , 24 ), outside some time-like world tube enclosing the source (

r > a

), and stationary in the past (see Eq. ( 19 )), reads

\begin{matrix} h_{1}^{α β} = k_{1}^{α β} + \partial^{α} φ_{1}^{β} + \partial^{β} φ_{1}^{α} - η^{α β} \partial_{μ} φ_{1}^{μ} . \end{matrix}

(26)

The first term depends on two STF-tensorial multipole moments,

I_{L} (u)

and

J_{L} (u)

, which are arbitrary functions of time except for the laws of conservation of the monopole:

I = c o n s t .

, and dipoles:

I_{i} = c o n s t .

J_{i} = c o n s t .

. It is given by

\begin{matrix} k_{1}^{00} & = & - \frac{4}{c^{2}} \sum_{l \geq 0} \frac{(-)^{l}}{l!} \partial_{L} (\frac{1}{r} I_{L} (u)), \end{matrix}

(27)

\begin{matrix} k_{1}^{0 i} & = & \frac{4}{c^{3}} \sum_{l \geq 1} \frac{(-)^{l}}{l!} {\partial_{L - 1} (\frac{1}{r} I_{i L - 1}^{(1)} (u)) \end{matrix}

\begin{matrix} + \frac{l}{l + 1} ɛ_{i a b} \partial_{a L - 1} (\frac{1}{r} J_{b L - 1} (u))}, \end{matrix}

(28)

\begin{matrix} k_{1}^{i j} & = & - \frac{4}{c^{4}} \sum_{l \geq 2} \frac{(-)^{l}}{l!} {\partial_{L - 2} (\frac{1}{r} I_{i j L - 2}^{(2)} (u)) \end{matrix}

\begin{matrix} + \frac{2 l}{l + 1} \partial_{a L - 2} (\frac{1}{r} ɛ_{a b (i} J_{j) b L - 2}^{(1)} (u))} . \end{matrix}

(29)

The other terms represent a linearized gauge transformation, with gauge vector

φ_{1}^{α}

of the type ( 25 ), and parametrized for four other multipole moments, say

W_{L} (u)

X_{L} (u)

Y_{L} (u)

and

Z_{L} (u)

The conservation of the lowest-order moments gives the constancy of the total mass of the source,

M \equiv I = c o n s t .

, center-of-mass position^$6$ ,

X_{i} \equiv I_{i} / I = c o n s t .

, total linear momentum

P_{i} \equiv I_{i}^{(1)} = 0

, and total angular momentum,

S_{i} \equiv J_{i} = c o n s t .

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 Arnowitt-Deser-Misner (ADM) mass of the Hamiltonian formulation of general relativity. Note that the quantities

M

X_{i}

P_{i}

and

S_{i}

include the contributions due to the waves emitted by the source. They describe the “initial” state of the source, before the emission of gravitational radiation.

The multipole functions

I_{L} (u)

and

J_{L} (u)

, which thoroughly encode the physical properties of the source at the linearized level (because the other moments

W_{L}

. . .

Z_{L}

parametrize a gauge transformation), will be referred to as the 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^{α β}

of the source: the above theorem follows merely from the algebraic and differential properties of the vacuum equations outside the source.

For completeness, let us give the components of the gauge-vector

φ_{1}^{α}

entering Eq. ( 26 ):

\begin{matrix} φ_{1}^{0} & = & \frac{4}{c^{3}} \sum_{l \geq 0} \frac{(-)^{l}}{l!} \partial_{L} (\frac{1}{r} W_{L} (u)), \end{matrix}

(30)

\begin{matrix} φ_{1}^{i} & = & - \frac{4}{c^{4}} \sum_{l \geq 0} \frac{(-)^{l}}{l!} \partial_{i L} (\frac{1}{r} X_{L} (u)) \end{matrix}

\begin{matrix} - \frac{4}{c^{4}} \sum_{l \geq 1} \frac{(-)^{l}}{l!} {\partial_{L - 1} (\frac{1}{r} Y_{i L - 1} (u)) + \frac{l}{l + 1} ɛ_{i a b} \partial_{a L - 1} (\frac{1}{r} Z_{b L - 1} (u))} . \end{matrix}

\begin{matrix}  \end{matrix}

(31)

Because the theory is covariant with respect to non-linear diffeomorphisms and not merely with respect to linear gauge transformations, the moments

W_{L}

. . .

Z_{L}

do play a physical role starting at the non-linear level, in the following sense. If one takes these moments equal to zero and continues the calculations one ends up with a metric depending on

I_{L}

and

J_{L}

only, but that metric will not describe the same physical source as the one constructed from the six moments

I_{L}

. . .

Z_{L}

. In other words, the two non-linear metrics associated with the sets of multipole moments {

I_{L}

J_{L}

, 0,

. . .

, 0} and {

I_{L}

J_{L}

W_{L}

. . .

Z_{L}

} are not isometric.

We point out in Section 4.2 below that the full set of moments {

I_{L}

J_{L}

W_{L}

. . .

Z_{L}

} is in fact physically equivalent to some reduced set {

M_{L}

S_{L}

, 0,

. . .

, 0}, but with some moments

M_{L}

S_{L}

that differ from

I_{L}

J_{L}

by non-linear corrections (see Eq. ( 100 )). All the multipole moments

I_{L}

J_{L}

W_{L}

X_{L}

Y_{L}

Z_{L}

will be computed in Section 5 .

^$5$ 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 $a L - 1 = a i_{1} . . . i_{l - 1}$ . Always understood in expressions such as Eq. ( 25 ) are $l$ summations over the $l$ indices $i_{1}, . . ., i_{l}$ ranging from 1 to 3. The derivative operator $\partial_{L}$ is a short-hand for $\partial_{i_{1}} . . . \partial_{i_{l}}$ . 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 $i_{p}, i_{q} \in L$ , we have $K_{. . . i_{p} . . . i_{q} . . .} = K_{. . . i_{q} . . . i_{p} . . .}$ and that $δ_{i_{p} i_{q}} K_{. . . i_{p} . . . i_{q} . . .} = 0$ (see Ref. [142] and Appendices A and B in Ref. [14] for reviews about the STF formalism). The STF projection is denoted with a hat, so $K_{L} \equiv {\hat{K}}_{L}$ , or sometimes with carets around the indices, $K_{L} \equiv K_{〈 L 〉}$ . In particular, ${\hat{n}}_{L} = n_{〈 L 〉}$ is the STF projection of the product of unit vectors $n_{L} = n_{i_{1}} . . . n_{i_{l}}$ ; an expansion into STF tensors ${\hat{n}}_{L} = {\hat{n}}_{L} (θ, φ)$ is equivalent to the usual expansion in spherical harmonics $Y_{l m} = Y_{l m} (θ, φ)$ . Similarly, we denote $x_{L} = x_{i_{1}} . . . x_{i_{l}} = r^{l} n_{L}$ and ${\hat{x}}_{L} = x_{〈 L 〉}$ . Superscripts like $(p)$ indicate $p$ successive time-derivations.

^$6$ The constancy of the center of mass $X_{i}$ – rather than a linear variation with time – results from our assumption of stationarity before the date $- T$ . Hence, $P_{i} = 0$ .

4 Non-linear Iteration of the Field Equations

By Theorem 1 we know the most general solution of the linearized equations in the exterior of the source. We then tackle the problem of the post-Minkowskian iteration of that solution. We consider the full post-Minkowskian series

\begin{matrix} h_{e x t}^{α β} = \sum_{n = 1}^{+ \infty} G^{n} h_{n}^{α β}, \end{matrix}

(32)

where the first term is composed of the result given by Eqs. ( 26 , 29 , 31 ). In this article, we shall always understand the infinite sums such as the one in Eq. ( 32 ) in the sense of formal power series, i.e. as an ordered collection of coefficients, e.g.

{(h_{n}^{α β})}_{n \in N}

. We do not attempt to control the mathematical nature of the series and refer to the mathematical-physics literature for discussion (in the present context, see notably Refs. [45, 64] ).

4.1 The post-Minkowskian solution

We insert the ansatz ( 32 ) into the vacuum Einstein field equations ( 12 , 13 ), i.e. with

τ^{α β} = c^{4} / (16 π G) Λ^{α β}

, and we equate term by term the factors of the successive powers of our book-keeping parameter

G

. We get an infinite set of equations for each of the

h_{n}^{α β}

's:

\forall n \geq 2

\begin{matrix} □ h_{n}^{α β} = Λ_{n}^{α β} [h_{1}, h_{2}, . . ., h_{n - 1}], \end{matrix}

(33)

\begin{matrix} \partial_{μ} h_{n}^{α μ} = 0 . \end{matrix}

(34)

The right-hand side of the wave equation ( 33 ) is obtained from inserting the previous iterations, up to the order

n - 1

, into the gravitational source term. In more details, the series of equations ( 33 ) reads

\begin{matrix} □ h_{2}^{α β} = N^{α β} [h_{1}, h_{1}], \end{matrix}

(35)

\begin{matrix} □ h_{3}^{α β} = M^{α β} [h_{1}, h_{1}, h_{1}] + N^{α β} [h_{1}, h_{2}] + N^{α β} [h_{2}, h_{1}], \end{matrix}

(36)

\begin{matrix} □ h_{4}^{α β} & = & L^{α β} [h_{1}, h_{1}, h_{1}, h_{1}] \end{matrix}

\begin{matrix} + M^{α β} [h_{1}, h_{1}, h_{2}] + M^{α β} [h_{1}, h_{2}, h_{1}] + M^{α β} [h_{2}, h_{1}, h_{1}] \end{matrix}

\begin{matrix} + N^{α β} [h_{2}, h_{2}] + N^{α β} [h_{1}, h_{3}] + N^{α β} [h_{3}, h_{1}] \end{matrix}

\begin{matrix} . . . \end{matrix}

(37)

The quadratic, cubic and quartic pieces of

Λ^{α β}

are defined by Eq. ( 16 ).

Let us now proceed by induction. Some

n

being given, we assume that we succeeded in constructing, from the linearized coefficient

h_{1}

, the sequence of 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 Eq. ( 33 ),

Λ_{n}^{α β}

, 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 Eq. ( 33 ) is composed of the product of many multipole expansions, which are singular at the origin of the spatial coordinates

r = 0

, and which make the retarded integral divergent at that point. This does not mean that there are no solutions to the wave equation, but simply that the retarded integral does not constitute the appropriate solution in that context.

What we need is a solution which takes the same structure as the source term

Λ_{n}^{α β}

, i.e. is expanded into multipole contributions, with a singularity at

r = 0

, and satisfies the d'Alembertian equation as soon as

r > 0

. Such a particular solution can be obtained, following the suggestion in Ref. [14] , by means of a mathematical trick in which one first “regularizes” the source term

Λ_{n}^{α β}

by multiplying it by the factor

r^{B}

, where

B \in C

. Let us assume, for definiteness, that

Λ_{n}^{α β}

is composed of multipolar pieces with maximal multipolarity

l_{m a x}

. This means that we start the iteration from the linearized metric ( 26 , 29 , 31 ) in which the multipolar sums are actually finite^$7$ . The divergences when

r \to 0

of the source term are typically power-like, say

1 / r^{k}

(there are also powers of the logarithm of

r

), and with the previous assumption there will exist a maximal order of divergency, say

k_{m a x}

. Thus, when the real part of

B

is large enough, i.e.

ℜ (B) > k_{m a x} - 3

, the “regularized” source term

r^{B} Λ_{n}^{α β}

is regular enough when

r \to 0

so that one can perfectly apply the retarded integral operator. This defines the

B

-dependent retarded integral

\begin{matrix} I^{α β} (B) \equiv □_{r e t}^{- 1} [{\tilde{r}}^{B} Λ_{n}^{α β}], \end{matrix}

(38)

where the symbol

□_{r e t}^{- 1}

stands for the retarded integral ( 21 ). It is convenient to introduce inside the regularizing factor some arbitrary constant length scale

r_{0}

in order to make it dimensionless. Everywhere in this article we pose

\begin{matrix} \tilde{r} \equiv \frac{r}{r_{0}} . \end{matrix}

(39)

The fate of the constant

r_{0}

in a detailed calculation will be interesting to follow, as we shall see, because it provides some check that the calculation is going well. Now the point for our purpose is that the function

I^{α β} (B)

on the complex plane, which was originally defined only when

ℜ (B) > k_{m a x} - 3

, admits a unique analytic continuation to all values of

B \in C

except at some integer values. Furthermore, the analytic continuation of

I^{α β} (B)

can be expanded, when

B \to 0

(namely the limit of interest to us) into a Laurent expansion involving in general some multiple poles. The key idea, as we shall prove, is that the finite part, or the coefficient of the zeroth power of

B

in that expansion, represents the particular solution we are looking for. We write the Laurent expansion of

I^{α β} (B)

, when

B \to 0

, in the form

\begin{matrix} I^{α β} (B) = \sum_{p = p_{0}}^{+ \infty} ι_{p}^{α β} B^{p}, \end{matrix}

(40)

where

p \in Z

, and the various coefficients

ι_{p}^{α β}

are functions of the field point

(x, t)

. When

p_{0} \leq - 1

there are poles;

- p_{0}

, which depends on

n

, refers to the maximal order of the poles. By applying the box operator onto both sides of ( 40 ), and equating the different powers of

B

, we arrive at

\begin{matrix} p_{0} \leq p \leq - 1 ⟹ □ ι_{p}^{α β} & = & 0, \end{matrix}

(41)

\begin{matrix} p \geq 0 ⟹ □ ι_{p}^{α β} & = & \frac{(ln r)^{p}}{p!} Λ_{n}^{α β} . \end{matrix}

(42)

As we see, the case

p = 0

shows that the finite-part coefficient in Eq. ( 40 ), namely

ι_{0}^{α β}

, is a particular solution of the requested equation:

□ ι_{0}^{α β} = Λ_{n}^{α β}

. Furthermore, we can prove that this term, by its very construction, owns the same structure made of a multipolar expansion singular at

r = 0

Let us forget about the intermediate name

ι_{0}^{α β}

, and denote, from now on, the latter solution by

u_{n}^{α β} \equiv ι_{0}^{α β}

, or, in more explicit terms,

\begin{matrix} u_{n}^{α β} = ℱ P_{B = 0} □_{r e t}^{- 1} [{\tilde{r}}^{B} Λ_{n}^{α β}], \end{matrix}

(43)

where the finite-part symbol

ℱ P_{B = 0}

means the previously detailed operations of considering the analytic continuation, taking the Laurent expansion, and picking up the finite-part coefficient when

B \to 0

. The story is not complete, however, because

u_{n}^{α β}

does not fulfill the constraint of harmonic coordinates ( 34 ); its divergence, say

w_{n}^{α} = \partial_{μ} u_{n}^{α μ}

, is different from zero in general. From the fact that the source term is divergence-free in vacuum,

\partial_{μ} Λ_{n}^{α μ} = 0

(see Eq. ( 18 )), we find instead

\begin{matrix} w_{n}^{α} = ℱ P_{B = 0} □_{r e t}^{- 1} [B {\tilde{r}}^{B} \frac{n_{i}}{r} Λ_{n}^{α i}] . \end{matrix}

(44)

The factor

B

comes from the differentiation of the regularization factor

{\tilde{r}}^{B}

. So,

w_{n}^{α}

is zero only in the special case where the Laurent expansion of the retarded integral in Eq. ( 44 ) does not develop any simple pole when

B \to 0

. Fortunately, when it does, the structure of the pole is quite easy to control. We find that it necessarily consists of a solution of the source-free d'Alembertian equation, and, what is more (from its stationarity in the past), the solution is a retarded one. Hence, taking into account the index structure of

w_{n}^{α}

, there must exist four STF-tensorial functions of the retarded time

u = t - r / c

, say

N_{L} (u)

P_{L} (u)

Q_{L} (u)

and

R_{L} (u)

, such that

\begin{matrix} w_{n}^{0} & = & \sum_{l = 0}^{+ \infty} \partial_{L} [r^{- 1} N_{L} (u)], \end{matrix}

(45)

\begin{matrix} w_{n}^{i} & = & \sum_{l = 0}^{+ \infty} \partial_{i L} [r^{- 1} P_{L} (u)] \end{matrix}

\begin{matrix} + \sum_{l = 1}^{+ \infty} {\partial_{L - 1} [r^{- 1} Q_{i L - 1} (u)] + ɛ_{i a b} \partial_{a L - 1} [r^{- 1} R_{b L - 1} (u)]} . \end{matrix}

(46)

From that expression we are able to find a new object, say

v_{n}^{α β}

, which takes the same structure as

w_{n}^{α}

(a retarded solution of the source-free wave equation) and, furthermore, whose divergence is exactly the opposite of the divergence of

u_{n}^{α β}

, i.e.

\partial_{μ} v_{n}^{α μ} = - w_{n}^{α}

. Such a

v_{n}^{α β}

is not unique, but we shall see that it is simply necessary to make a choice for

v_{n}^{α β}

(the simplest one) in order to obtain the general solution. The formulas that we adopt are

\begin{matrix} v_{n}^{00} & = & - r^{- 1} N^{(- 1)} + \partial_{a} [r^{- 1} (- N_{a}^{(- 1)} + C_{a}^{(- 2)} - 3 P_{a})], \end{matrix}

(47)

\begin{matrix} v_{n}^{0 i} & = & r^{- 1} (- Q_{i}^{(- 1)} + 3 P_{i}^{(1)}) - ɛ_{i a b} \partial_{a} [r^{- 1} R_{b}^{(- 1)}] \end{matrix}

\begin{matrix} - \sum_{l = 2}^{+ \infty} \partial_{L - 1} [r^{- 1} N_{i L - 1}], \end{matrix}

(48)

\begin{matrix} v_{n}^{i j} & = & - δ_{i j} r^{- 1} P + \sum_{l = 2}^{+ \infty} {2 δ_{i j} \partial_{L - 1} [r^{- 1} P_{L - 1}] - 6 \partial_{L - 2 (i} [r^{- 1} P_{j) L - 2}] \end{matrix}

\begin{matrix} + \partial_{L - 2} [r^{- 1} (N_{i j L - 2}^{(1)} + 3 P_{i j L - 2}^{(2)} - Q_{i j L - 2})] \end{matrix}

\begin{matrix} - 2 \partial_{a L - 2} [r^{- 1} ɛ_{a b (i} R_{j) b L - 2}]} . \end{matrix}

(49)

Notice the presence of anti-derivatives, denoted e.g. by

N^{(- 1)} (u) = \int_{- \infty}^{u} d v N (v)

; there is no problem with the limit

v \to - \infty

since all the corresponding functions are zero when

t \leq - T

. The choice made in Eqs. ( 49 ) is dictated by the fact that the

00

component involves only some monopolar and dipolar terms, and that the spatial trace

i i

is monopolar:

v_{n}^{i i} = - 3 r^{- 1} P

. Finally, if we pose

\begin{matrix} h_{n}^{α β} = u_{n}^{α β} + v_{n}^{α β}, \end{matrix}

(50)

we see that we solve at once the d'Alembertian equation ( 33 ) and the coordinate condition ( 34 ). That is, we have succeeded in finding a solution of the field equations at the

n

th post-Minkowskian order.

By induction the same method applies to any order

n

, and, therefore, we have constructed a complete post-Minkowskian series ( 32 ) based on the linearized approximation

h_{1}^{α β}

given by ( 26 , 29 , 31 ). The previous procedure constitutes an algorithm, which could be implemented by an algebraic computer programme.

^$7$ 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 , 29 , 31 ), there is a maximal multipolarity $l_{m a x} (n)$ at any post-Minkowskian order $n$ , which grows linearly with $n$ .

4.2 Generality of the solution

We have a solution, but is that a general solution? The answer, yes, is provided by the following result [14] :

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^{α β} = \sum_{n = 1}^{+ \infty} G^{n} h_{n}^{α β} [I_{L}, J_{L}, . . ., 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 Eqs. ( 29 ), and on four supplementary functions

W_{L} (u)

. . .

Z_{L} (u)

parametrizing the gauge vector ( 31 ).

The proof is quite easy. With Eq. ( 50 ) we obtained a particular solution of the system of equations ( 33 , 34 ).

To it we should add the most general solution of the corresponding homogeneous system of equations, which is obtained by setting

Λ_{n}^{α β} = 0

into Eqs. ( 33 , 34 ). But this homogeneous system of equations is nothing but the linearized vacuum field equations ( 23 , 24 ), for which we know the most general solution

h_{1}^{α β}

given by ( 26 , 29 , 31 ). Thus, we must add to our “particular” solution

h_{n}^{α β}

a general homogeneous solution that is necessarily of the type

h_{1}^{α β} [δ I_{L}, . . ., δ Z_{L}]

, where

δ I_{L}

. . .

δ Z_{L}

denote some “corrections” to the multipole moments at the

n

th post-Minkowskian order. It is then clear, since precisely the linearized metric is a linear functional of all these moments, that the previous corrections to the moments can be absorbed into a re-definition of the original ones

I_{L}, . . ., Z_{L}

by posing

\begin{matrix} I_{L}^{n e w} & = & I_{L} + G^{n - 1} δ I_{L}, \end{matrix}

(51)

\begin{matrix} . . . \end{matrix}

\begin{matrix} Z_{L}^{n e w} & = & Z_{L} + G^{n - 1} δ Z_{L} . \end{matrix}

(52)

After re-arranging the metric in terms of these new moments, taking into account the fact that the precision of the metric is limited to the

n

th post-Minkowskian order, and dropping the superscript “new”, we find exactly the same solution as the one we had before (indeed, the moments are arbitrary functions of time) – hence the proof.

The six sets of multipole moments

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

δ x^{α} = φ_{1}^{α}

, where

φ_{1}^{α}

is the gauge vector given by Eqs. ( 31 ). So, at the linearized level, we have only the two types of moments

M_{L}

and

S_{L}

, parametrizing

k_{1}^{α β}

by the same formulas as in Eqs. ( 29 ). 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 ( 43 , 44 , 46 , 49 , 50 ) as we did above, but starting from the gauge-transformed linear metric

k_{1}^{α β}

instead of

h_{1}^{α β}

. The result of the iteration is therefore some

k^{α β} = \sum_{n = 1}^{+ \infty} G^{n} k_{n}^{α β} [M_{L}, 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^{α β} [M_{L}, S_{L}]

is isometric to the original one

h^{α β} [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}^{α β} [M_{L}, S_{L}]

instead of the more complicated

k_{1}^{α β} [I_{L}, J_{L}] + \partial^{α} φ_{1}^{β} + \partial^{β} φ_{1}^{α} - η^{α β} \partial_{μ} φ_{1}^{μ}

, 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. [7] and Eq. ( 100 ) below. Indeed, this is to be expected because the physical difference between both types of moments stems only from non-linearities.

4.3 Near-zone and far-zone structures

In our presentation of the post-Minkowskian algorithm ( 43 , 44 , 46 , 49 , 50 ) 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 \to 0

, of each one of the post-Minkowskian coefficients

h_{n}^{α β}

owns a certain structure. This hypothesis is established as a theorem once the mathematical induction succeeds.

Theorem 3

The general structure of the expansion of the post-Minkowskian exterior metric in the near-zone (when

r \to 0

) is of the type:

\forall N \in N

\begin{matrix} h_{n} (x, t) = \sum {\hat{n}}_{L} r^{m} (ln r)^{p} F_{L, m, p, n} (t) + o (r^{N}), \end{matrix}

(53)

where

m \in Z

, with

m_{0} \leq m \leq N

(and

m_{0}

becoming more and more negative as

n

grows),

p \in N

with

p \leq n - 1

. The functions

F_{L, m, p, n}

are multilinear functionals of the source multipole moments

I_{L}, . . ., Z_{L}

For the proof see Ref. [14] ^$8$ .

As we see, the near-zone expansion involves, besides the simple powers of

r

, some powers of the logarithm of

r

, with a maximal power of

n - 1

. As a corollary of that theorem, we find (by restoring all the powers of

c

in Eq. ( 53 ) and using the fact that each

r

goes into the combination

r / c

), that the general structure of the post-Newtonian expansion (

c \to + \infty

) is necessarily of the type

\begin{matrix} h_{n} (c) ≃ \sum_{p, q \in N} \frac{(ln c)^{p}}{c^{q}}, \end{matrix}

(54)

where

p \leq n - 1

(and

q \geq 2

). The post-Newtonian expansion proceeds not only with the normal powers of

1 / c

but also with powers of the logarithm of

c

[14] .

Paralleling the structure of the near-zone expansion, we have a similar result concerning the structure of the far-zone expansion at Minkowskian future null infinity, i.e. when

r \to + \infty

with

u = t - r / c = c o n s t .

\forall N \in N

\begin{matrix} h_{n} (x, t) = \sum \frac{{\hat{n}}_{L} (ln r)^{p}}{r^{k}} G_{L, k, p, n} (u) + o (\frac{1}{r^{N}}), \end{matrix}

(55)

where

k, p \in N

, with

1 \leq k \leq N

, and where, likewise in the near-zone expansion ( 53 ), some powers of logarithms, such that

p \leq n - 1

, appear. The appearance of logarithms in the far-zone expansion of the harmonic-coordinates metric has been known since the work of Fock [74] . One knows also that this is a coordinate effect, because the study of the “asymptotic” structure of space-time at future null infinity by Bondi et al. [32] , Sachs [127] and Penrose [114, 115] , has revealed the existence of other coordinate systems that avoid the appearance of any logarithms: the so-called radiative coordinates, in which the far-zone expansion of the metric proceeds with simple powers of the inverse radial distance. Hence, the logarithms are simply an artifact of the use of harmonic coordinates [82, 99] . The following theorem, proved in Ref. [3] , shows that our general construction of the metric in the exterior of the source, when developed at future null infinity, is consistent with the Bondi–Sachs–Penrose [32, 127, 114, 115] approach to gravitational radiation.

Theorem 4

The most general multipolar-post-Minkowskian solution, stationary in the past (see Eq. ( 19 )), admits some radiative coordinates

(T, X)

, for which the expansion at future null infinity,

R \to + \infty

with

U \equiv T - R / c = c o n s t .

, takes the form

\begin{matrix} H_{n} (X, T) = \sum \frac{{\hat{N}}_{L}}{R^{k}} K_{L, k, n} (U) + O (\frac{1}{R^{N}}) . \end{matrix}

(56)

The functions

K_{L, k, n}

are computable functionals of the source multipole moments. In radiative coordinates the retarded time

U = T - R / c

is a null coordinate in the asymptotic limit. The metric

H_{e x t}^{α β} = \sum_{n \geq 1} G^{n} H_{n}^{α β}

is asymptotically simple in the sense of Penrose [114, 115] , perturbatively to any post-Minkowskian order.

Proof: We introduce a linearized “radiative” metric by performing a gauge transformation of the harmonic-coordinates metric defined by Eqs. ( 26 , 29 , 31 ), namely

\begin{matrix} H_{1}^{α β} = h_{1}^{α β} + \partial^{α} ξ_{1}^{β} + \partial^{β} ξ_{1}^{α} - η^{α β} \partial_{μ} ξ_{1}^{μ}, \end{matrix}

(57)

where the gauge vector

ξ_{1}^{α}

\begin{matrix} ξ_{1}^{α} = 2 M η^{0 α} ln (\frac{r}{r_{0}}) . \end{matrix}

(58)

This gauge transformation is non-harmonic:

\begin{matrix} \partial_{μ} H_{1}^{α μ} = □ ξ_{1}^{α} = \frac{2 M}{r^{2}} η^{0 α} . \end{matrix}

(59)

Its effect is to “correct” for the well-known logarithmic deviation of the harmonic coordinates' retarded time with respect to the true space-time characteristic or light cones. After the change of gauge, the coordinate

u = t - r / c

coincides with a null coordinate at the linearized level^$9$ . This is the requirement to be satisfied by a linearized metric so that it can constitute the linearized approximation to a full (post-Minkowskian) radiative field [99] .

One can easily show that, at the dominant order when

r \to + \infty

\begin{matrix} k_{μ} k_{ν} H_{1}^{μ ν} = O (\frac{1}{r^{2}}), \end{matrix}

(60)

where

k^{α} = (1, n)

is the outgoing Minkowskian null vector. Given any

n \geq 2

, let us recursively assume that we have obtained all the previous radiative post-Minkowskian coefficients

H_{m}^{α β}

, i.e.

\forall m \leq n - 1

, and that all of them satisfy

\begin{matrix} k_{μ} k_{ν} H_{m}^{μ ν} = O (\frac{1}{r^{2}}) . \end{matrix}

(61)

From this induction hypothesis one can prove that the

n

th post-Minkowskian source term

Λ_{n}^{α β} = Λ_{n}^{α β} (H_{1}, . . ., H_{n - 1})

is such that

\begin{matrix} Λ_{n}^{α β} = \frac{k^{α} k^{β}}{r^{2}} σ_{n} (u, n) + O (\frac{1}{r^{3}}) . \end{matrix}

(62)

To the leading order this term takes the classic form of the stress-energy tensor for a swarm of massless particles, with

σ_{n}

being related to the power in the waves. One can show that all the problems with the appearance of logarithms come from the retarded integral of the terms in Eq. ( 62 ) that behave like

1 / r^{2}

See indeed the integration formula ( 121 ), which behaves like

ln r / r

at infinity. But now, thanks to the particular index structure of the term ( 62 ), we can correct for the effect by adjusting the gauge at the

n

th post-Minkowskian order. We pose, as a gauge vector,

\begin{matrix} ξ_{n}^{α} = ℱ P □_{r e t}^{- 1} [\frac{k^{α}}{2 r^{2}} \int_{- \infty}^{u} d v σ_{n} (v, n)], \end{matrix}

(63)

where

ℱ P

refers to the same finite part operation as in Eq. ( 43 ). This vector is such that the logarithms that will appear in the corresponding gauge terms cancel out the logarithms coming from the retarded integral of the source term ( 62 ); see Ref. [3] for the details. Hence, to the

n

th post-Minkowskian order, we define the radiative metric as

\begin{matrix} H_{n}^{α β} = U_{n}^{α β} + V_{n}^{α β} + \partial^{α} ξ_{n}^{β} + \partial^{β} ξ_{n}^{α} - η^{α β} \partial_{μ} ξ_{n}^{μ} . \end{matrix}

(64)

Here

U_{n}^{α β}

and

V_{n}^{α β}

denote the quantities that are the analogues of

u_{n}^{α β}

and

v_{n}^{α β}

, which were introduced into the harmonic-coordinates algorithm: See Eqs. ( 43 , 44 , 46 , 49 ). In particular, these quantities are constructed in such a way that the sum

U_{n}^{α β} + V_{n}^{α β}

is divergence-free, so we see that the radiative metric does not obey the harmonic-gauge condition:

\begin{matrix} \partial_{μ} H_{n}^{α μ} = □ ξ_{n}^{α} = \frac{k^{α}}{2 r^{2}} \int_{- \infty}^{u} d v σ_{n} (v, n) . \end{matrix}

(65)

The far-zone expansion of the latter metric is of the type ( 56 ), i.e. is free of any logarithms, and the retarded time in these coordinates tends asymptotically toward a null coordinate at infinity. The property of asymptotic simplicity, in the mathematical form given by Geroch and Horowitz [76] , is proved by introducing the conformal factor

Ω = 1 / r

in radiative coordinates (see Ref. [3] ). Finally, it can be checked that the metric so constructed, which is a functional of the source multipole moments

I_{L}

. . .

Z_{L}

(from the definition of the algorithm), is as general as the general harmonic-coordinate metric of Theorem 2 , since it merely differs from it by a coordinate transformation

(t, x) ⟶ (T, X)

, where

(t, x)

are the harmonic coordinates and

(T, X)

the radiative ones, together with a re-definition of the multipole moments.

^$8$ The $o$ and $O$ Landau symbols for remainders have their standard meaning.

^$9$ 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 $(T, R)$ .

4.4 The radiative multipole moments

The leading-order term

1 / R

of the metric in radiative coordinates, neglecting

O (1 / R^{2})

, yields the operational definition of two sets of STF radiative multipole moments, mass-type

U_{L} (U)

and current-type

V_{L} (U)

. By definition, we have

\begin{matrix} H_{i j}^{T T} (U, N) & = & \frac{4 G}{c^{2} R} P_{i j a b} (N) \sum_{l = 2}^{+ \infty} \frac{1}{c^{l} l!} {N_{L - 2} U_{a b L - 2} (U) \end{matrix}

\begin{matrix} - \frac{2 l}{c (l + 1)} N_{c L - 2} ɛ_{c d (a} V_{b) d L - 2} (U)} + O (\frac{1}{R^{2}}) . \end{matrix}

(66)

This multipole decomposition represents the generalization, up to any post-Newtonian order (witness the factors of

1 / c

in front of each of the multipolar pieces) of the quadrupole-moment formalism reviewed in Eq. ( 2 ). The corresponding total gravitational flux reads

\begin{matrix} ℒ (U) & = & \sum_{l = 2}^{+ \infty} \frac{G}{c^{2 l + 1}} {\frac{(l + 1) (l + 2)}{(l - 1) l l! (2 l + 1)!!} U_{L}^{(1)} (U) U_{L}^{(1)} (U) \end{matrix}

\begin{matrix} + \frac{4 l (l + 2)}{c^{2} (l - 1) (l + 1)! (2 l + 1)!!} V_{L}^{(1)} (U) V_{L}^{(1)} (U)} . \end{matrix}

(67)

Notice that the meaning of such formulas is rather empty, because we do not know yet how the radiative moments are given in terms of the actual source parameters. Only at the Newtonian level do we know this relation, which from the comparison with the quadrupole formalism of Eqs. ( 2 , 3 , 4 ) reduces to

\begin{matrix} U_{i j} (U) = Q_{i j}^{(2)} (U) + O (\frac{1}{c^{2}}), \end{matrix}

(68)

where

Q_{i j}

is the Newtonian quadrupole given by Eq. ( 3 ). Fortunately, we are not in such bad shape because we have learned from Theorem 4 the general method that permits us to compute the radiative multipole moments

U_{L}

V_{L}

in terms of the source moments

I_{L}

J_{L}

. . .

Z_{L}

. Therefore, what is missing is the explicit dependence of the source multipole moments as functions of the actual parameters of some isolated source.

We come to grips with this question in the next section.

5 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^{α β}

(i.e. a choice of some physical model describing the source) selects a particular physical exterior solution among our general class.

5.1 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 Eq. ( 1 ) is small. The fundamental fact that permits the connection of the exterior field to the inner field of the source is the existence of a “matching” region, in which both the multipole 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.

Let us denote by

ℳ (h)

the multipole expansion of

h

(for simplicity, we suppress the space-time indices).

ℳ (h)

we really mean the multipolar-post-Minkowskian exterior metric that we have constructed in Sections 3 and 4 :

\begin{matrix} ℳ (h) \equiv h_{e x t} = \sum_{n = 1}^{+ \infty} G^{n} h_{n} [I_{L}, . . ., Z_{L}] . \end{matrix}

(69)

Of course,

h

agrees with its own multipole expansion in the exterior of the source,

\begin{matrix} r > a ⟹ ℳ (h) = h . \end{matrix}

(70)

By contrast, inside the source,

h

and

ℳ (h)

disagree with each other because

h

is a fully-fledged solution of the field equations with matter source, while

ℳ (h)

is a vacuum solution becoming singular at

r = 0

. Now let us denote by

\bar{h}

the post-Newtonian expansion of

h

. We have already anticipated the general structure of this expansion as given in Eq. ( 54 ). In the matching region, where both the multipolar and post-Newtonian expansions are valid, we write the numerical equality

\begin{matrix} a < r ≪ λ ⟹ ℳ (h) = \bar{h} . \end{matrix}

(71)

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

We now transform Eq. ( 71 ) into a matching equation, by replacing in the left-hand side

ℳ (h)

by its near-zone re-expansion

\bar{ℳ (h)}

, and in the right-hand side

\bar{h}

by its multipole expansion

ℳ (\bar{h})

. The structure of the near-zone expansion (

r \to 0

) of the exterior multipolar field has been found in Eq. ( 53 ). We denote the corresponding infinite series

\bar{ℳ (h)}

with the same overbar as for the post-Newtonian expansion because it is really an expansion when

r / c \to 0

, equivalent to an expansion when

c \to \infty

. Concerning the multipole expansion of the post-Newtonian metric,

ℳ (\bar{h})

, we simply postulate its existence. Therefore, the matching equation is the statement that

\begin{matrix} \bar{ℳ (h)} = ℳ (\bar{h}), \end{matrix}

(72)

by which we really mean an infinite set of functional identities, valid

\forall (x, t) \in R_{*}^{3} \times R

, between the coefficients of the series in both sides of the equation. Note that such a meaning is somewhat different from that of a numerical equality like Eq. ( 71 ), which is valid only when

x

belongs to some limited spatial domain.

The matching equation ( 72 ) tells us that the formal near-zone expansion of the multipole decomposition is identical, term by term, to the multipole expansion of the post-Newtonian solution. However, the former expansion is nothing but the formal far-zone expansion, when

r \to \infty

, of each of the post-Newtonian coefficients. Most importantly, it is possible to write down, within the present formalism, the general structure of these identical expansions as a consequence of Theorem 3 , Eq. ( 53 ):

\begin{matrix} \bar{ℳ (h)} = \sum {\hat{n}}_{L} r^{m} (ln r)^{p} F_{L, m, p} (t) = ℳ (\bar{h}), \end{matrix}

(73)

where the functions

F_{L, m, p} = \sum_{n \geq 1} G^{n} F_{L, m, p, n}

. The latter expansion can be interpreted either as the singular re-expansion of the multipole decomposition when

r \to 0

(first equality in Eq. ( 73 )), or the singular re-expansion of the post-Newtonian series when

r \to + \infty

(second equality). We recognize the beauty of singular perturbation theory, where two asymptotic expansions, taken formally outside their respective domains of validity, are matched together. Of course, the method works because there exists, physically, an overlapping region in which the two approximation series are expected to be numerically close to the exact solution.

5.2 General expression of the multipole expansion

Theorem 5

Under the hypothesis of matching, Eq. ( 72 ), the multipole expansion of the solution of the Einstein field equation outside a post-Newtonian source reads as

\begin{matrix} ℳ (h^{α β}) = ℱ P_{B = 0} □_{r e t}^{- 1} [{\tilde{r}}^{B} ℳ (Λ^{α β})] - \frac{4 G}{c^{4}} \sum_{l = 0}^{+ \infty} \frac{(-)^{l}}{l!} \partial_{L} {\frac{1}{r} ℋ_{L}^{α β} (t - r / c)}, \end{matrix}

(74)

where the “multipole moments” are given by

\begin{matrix} ℋ_{L}^{α β} (u) = ℱ P_{B = 0} \int d^{3} x | \tilde{x} |^{B} x_{L} {\bar{τ}}^{α β} (x, u) . \end{matrix}

(75)

Here,

{\bar{τ}}^{α β}

denotes the post-Newtonian expansion of the stress-energy pseudo-tensor defined by Eq. ( 14 ).

Proof [6, 11] : First notice where the physical restriction of considering a post-Newtonian source enters this theorem: the multipole moments ( 75 ) depend on the post-Newtonian expansion

{\bar{τ}}^{α β}

, rather than on

τ^{α β}

itself. Consider

Δ^{α β}

, namely the difference between

h^{α β}

, which is a solution of the field equations everywhere inside and outside the source, and the first term in Eq. ( 74 ), namely the finite part of the retarded integral of the multipole expansion

ℳ (Λ^{α β})

\begin{matrix} Δ^{α β} \equiv h^{α β} - ℱ P □_{r e t}^{- 1} [ℳ (Λ^{α β})] . \end{matrix}

(76)

From now on we shall generally abbreviate the symbols concerning the finite-part operation at

B = 0

by a mere

ℱ P

. According to Eq. ( 20 ),

h^{α β}

is given by the retarded integral of the pseudo-tensor

τ^{α β}

. So,

\begin{matrix} Δ^{α β} = \frac{16 π G}{c^{4}} □_{r e t}^{- 1} τ^{α β} - ℱ P □_{r e t}^{- 1} [ℳ (Λ^{α β})] . \end{matrix}

(77)

In the second term the finite part plays a crucial role because the multipole expansion

ℳ (Λ^{α β})

is singular at

r = 0

. By contrast, the first term in Eq. ( 77 ), as it stands, is well-defined because we are considering only some smooth field distributions:

τ^{α β} \in C^{\infty} (R^{4})

. There is no need to include a finite part

ℱ P

in the first term, but a contrario there is no harm to add one in front of it, because for convergent integrals the finite part simply gives back the value of the integral. The advantage of adding “artificially” the

ℱ P

in the first term is that we can re-write Eq. ( 77 ) into the much more interesting form

\begin{matrix} Δ^{α β} = \frac{16 π G}{c^{4}} ℱ P □_{r e t}^{- 1} [τ^{α β} - ℳ (τ^{α β})], \end{matrix}

(78)

in which we have also used the fact that

ℳ (Λ^{α β}) = 16 π G / c^{4} \cdot ℳ (τ^{α β})

because

T^{α β}

has a compact support. The interesting point about Eq. ( 78 ) is that

Δ^{α β}

appears now to be the (finite part of a) retarded integral of a source with spatially compact support. This follows from the fact that the pseudo-tensor agrees numerically with its own multipole expansion when

r > a

(same equation as ( 70 )). Therefore,

ℳ (Δ^{α β})

can be obtained from the known formula for the multipole expansion of the retarded solution of a wave equation with compact-support source. This formula, given in Appendix B of Ref. [16] , yields the second term in Eq. ( 74 ),

\begin{matrix} ℳ (Δ^{α β}) = - \frac{4 G}{c^{4}} \sum_{l = 0}^{+ \infty} \frac{(-)^{l}}{l!} \partial_{L} {\frac{1}{r} ℋ_{L}^{α β} (u)}, \end{matrix}

(79)

but in which the moments do not yet match the result ( 75 ); instead,

\begin{matrix} ℋ_{L}^{α β} = ℱ P \int d^{3} x x_{L} [τ^{α β} - ℳ (τ^{α β})] . \end{matrix}

(80)

The reason is that we have not yet applied the assumption of a post-Newtonian source. Such sources are entirely covered by their own near zone (i.e.

a ≪ λ

), and, in addition, the integral ( 80 ) has a compact support limited to the domain of the source. In consequence, we can replace the integrand in Eq. ( 80 ) by its post-Newtonian expansion, valid over all the near zone, i.e.

\begin{matrix} ℋ_{L}^{α β} = ℱ P \int d^{3} x x_{L} [{\bar{τ}}^{α β} - \bar{ℳ (τ^{α β})}] . \end{matrix}

(81)

Strangely enough, we do not get the expected result because of the presence of the second term in Eq. ( 81 ).

Actually, this term is a bit curious, because the object

\bar{ℳ (τ^{α β})}

it contains is only known in the form of the formal series whose structure is given by the first equality in Eq. ( 73 ) (indeed

τ

and

h

have the same type of structure). Happily (because we would not know what to do with this term in applications), we are now going to prove that the second term in Eq. ( 81 ) is in fact identically zero. The proof is based on the properties of the analytic continuation as applied to the formal structure ( 73 ) of

\bar{ℳ (τ^{α β})}

. Each term of this series yields a contribution to Eq. ( 81 ) that takes the form, after performing the angular integration, of the integral

ℱ P_{B = 0} \int_{0}^{+ \infty} d r r^{B + b} (ln r)^{p}

, and multiplied by some function of time. We want to prove that the radial integral

\int_{0}^{+ \infty} d r r^{B + b} (ln r)^{p}

is zero by analytic continuation (

\forall B \in C

). First we can get rid of the logarithms by considering some repeated differentiations with respect to

B

; thus we need only to consider the simpler integral

\int_{0}^{+ \infty} d r r^{B + b}

. We split the integral into a “near-zone” integral

\int_{0}^{ℛ} d r r^{B + b}

and a “far-zone” one

\int_{ℛ}^{+ \infty} d r r^{B + b}

, where

ℛ

is some constant radius. When

ℜ (B)

is a large enough positive number, the value of the near-zone integral is

ℛ^{B + b + 1} / (B + b + 1)

, while when

ℜ (B)

is a large negative number, the far-zone integral reads the opposite,

- ℛ^{B + b + 1} / (B + b + 1)

. Both obtained values represent the unique analytic continuations of the near-zone and far-zone integrals for any

B \in C

except

- b - 1

. The complete integral

\int_{0}^{+ \infty} d r r^{B + b}

is equal to the sum of these analytic continuations, and is therefore identically zero (

\forall B \in C

, including the value

- b - 1

). At last we have completed the proof of Theorem 5 :

\begin{matrix} ℋ_{L}^{α β} = ℱ P \int d^{3} x x_{L} {\bar{τ}}^{α β} . \end{matrix}

(82)

The latter proof makes it clear how crucial the analytic-continuation finite part

ℱ P

is, which we recall is the same as in our iteration of the exterior post-Minkowskian field (see Eq. ( 43 )). Without a finite part, the multipole moment ( 82 ) would be strongly divergent, because the pseudo-tensor

{\bar{τ}}^{α β}

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 \to + \infty

. In applications (Part bbb of this article) we must carefully follow the rules for handling the

ℱ P

operator.

The two terms in the right-hand side of Eq. ( 74 ) 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 Eq. ( 39 )). However, the sum of these two terms, i.e. the exterior multipolar field

ℳ (h)

itself, is independent of

r_{0}

. To see this, the simplest way is to differentiate formally

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 [26] , but that this

r_{0}

is compensated by another

r_{0}

coming from the non-linear “tails of tails” at the 3PN order.

5.3 Equivalence with the Will–Wiseman formalism

Recently, Will and Wiseman [152] (see also Refs. [151, 112] ), extending previous work of Epstein and Wagoner [70] and Thorne [142] , have obtained a different-looking multipole decomposition, with different definitions for the multipole moments of a post-Newtonian source. They find, instead of our multipole decomposition given by Eq. ( 74 ),

\begin{matrix} ℳ (h^{α β}) = □_{r e t}^{- 1} [ℳ (Λ^{α β})]_{|_{ℛ}} - \frac{4 G}{c^{4}} \sum_{l = 0}^{+ \infty} \frac{(-)^{l}}{l!} \partial_{L} {\frac{1}{r} W_{L}^{α β} (t - r / c)} . \end{matrix}

(83)

There is no

ℱ P

operation in the first term, but instead the retarded integral is truncated, as indicated by the subscript

ℛ

, to extend only in the “far zone”: i.e.

| x^{'} | > ℛ

in the notation of Eq. ( 21 ), where

ℛ

is a constant radius enclosing the source (

ℛ > a

). The near-zone part of the retarded integral is thereby removed, and there is no problem with the singularity of the multipole expansion

ℳ (Λ^{α β})

at the origin.

The multipole moments

W_{L}

are then given, in contrast with our result ( 75 ), by an integral extending over the “near zone” only:

\begin{matrix} W_{L}^{α β} (u) = \int_{| x | < ℛ} d^{3} x x_{L} {\bar{τ}}^{α β} (x, u) . \end{matrix}

(84)

Since the integrand is compact-supported there is no problem with the bound at infinity and the integral is well-defined (no need of a

ℱ P

Let us show that the two different formalisms are equivalent. We compute the difference between our moment

ℋ_{L}

, defined by Eq. ( 75 ), and the Will–Wiseman moment

W_{L}

, given by Eq. ( 84 ). For the comparison we split

ℋ_{L}

into far-zone and near-zone integrals corresponding to the radius

ℛ

. Since the finite part

ℱ P

present in

ℋ_{L}

deals only with the bound at infinity, it can be removed from the near-zone integral, which is then seen to be exactly equal to

W_{L}

. So the difference between the two moments is simply given by the far-zone integral:

\begin{matrix} ℋ_{L}^{α β} (u) - W_{L}^{α β} (u) = ℱ P \int_{| x | > ℛ} d^{3} x x_{L} {\bar{τ}}^{α β} (x, u) . \end{matrix}

(85)

Next, we transform this expression. Successively we write

{\bar{τ}}^{α β} = ℳ ({\bar{τ}}^{α β})

because we are outside the source, and

ℳ ({\bar{τ}}^{α β}) = \bar{ℳ (τ^{α β})}

from the matching equation ( 72 ). At this stage, we recall from our reasoning right after Eq. ( 81 ) that the finite part of an integral over the whole space

R^{3}

of a quantity having the same structure as

\bar{ℳ (τ^{α β})}

is identically zero by analytic continuation. The main trick of the proof is made possible by this fact, as it allows us to transform the far-zone integration

| x | > ℛ

in Eq. ( 85 ) into a near-zone one

| x | < ℛ

, at the price of changing the overall sign in front of the integral. So,

\begin{matrix} ℋ_{L}^{α β} (u) - W_{L}^{α β} (u) = - ℱ P \int_{| x | < ℛ} d^{3} x x_{L} \bar{ℳ (τ^{α β})} (x, u) . \end{matrix}

(86)

Finally, it is straightforward to check that the right-hand side of this equation, when summed up over all multipolarities

l

, accounts exactly for the near-zone part that was removed from the retarded integral of

ℳ (Λ^{α β})

(first term in Eq. ( 83 )), so that the “complete” retarded integral as given by the first term in our own definition ( 74 ) is exactly reconstituted. In conclusion, the formalism of Ref. [152] is equivalent to the one of Refs. [6, 11] .

5.4 The source multipole moments

In principle the bridge between the exterior gravitational field generated by the post-Newtonian source and its inner field is provided by Theorem 5 ; however, we still have to make the connection with the explicit construction of the general multipolar and post-Minkowskian metric in 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 , 29 , 31 ) at the basis of that construction^$10$ . To do this we impose the harmonic-gauge conditions ( 12 ) onto the multipole decomposition as given by ( 74 ), and we decompose the multipole functions

ℋ_{L}^{α β} (u)

into STF irreducible pieces (refer to [11] for the details).

Theorem 6

The STF multipole moments

I_{L}

and

J_{L}

of a post-Newtonian source are given, formally up to any post-Newtonian order, by (

l \geq 2

)

\begin{matrix} I_{L} (u) & = & ℱ P \int d^{3} x \int_{- 1}^{1} d z {δ_{l} {\hat{x}}_{L} Σ - \frac{4 (2 l + 1)}{c^{2} (l + 1) (2 l + 3)} δ_{l + 1} {\hat{x}}_{i L} Σ_{i}^{(1)} \end{matrix}

\begin{matrix} + \frac{2 (2 l + 1)}{c^{4} (l + 1) (l + 2) (2 l + 5)} δ_{l + 2} {\hat{x}}_{i j L} Σ_{i j}^{(2)}} (x, u + z | x | / c), \end{matrix}

(87)

\begin{matrix} J_{L} (u) & = & ℱ P \int d^{3} x \int_{- 1}^{1} d z ɛ_{a b 〈 i_{l}} {δ_{l} {\hat{x}}_{L - 1 〉 a} Σ_{b} \end{matrix}

\begin{matrix} - \frac{2 l + 1}{c^{2} (l + 2) (2 l + 3)} δ_{l + 1} {\hat{x}}_{L - 1 〉 a c} Σ_{b c}^{(1)}} (x, u + z | x | / c) . \end{matrix}

(88)

These moments are the ones that are to be inserted into the linearized metric

h_{1}^{α β}

that represents the lowest approximation to the post-Minkowskian field

h_{e x t}^{α β} = \sum_{n \geq 1} G^{n} h_{n}^{α β}

defined in Section 4 .

In these formulas the notation is as follows: Some convenient source densities are defined from the post-Newtonian expansion of the pseudo-tensor

τ^{α β}

\begin{matrix} Σ & = & \frac{{\bar{τ}}^{00} + {\bar{τ}}^{i i}}{c^{2}}, \end{matrix}

(89)

\begin{matrix} Σ_{i} & = & \frac{{\bar{τ}}^{0 i}}{c}, \end{matrix}

(90)

\begin{matrix} Σ_{i j} & = & {\bar{τ}}^{i j} \end{matrix}

(91)

(where

{\bar{τ}}^{i i} \equiv δ_{i j} {\bar{τ}}^{i j}

). As indicated in Eqs. ( 88 ) these quantities are to be evaluated at the spatial point

x

and at time

u + z | x | / c

. Notice the presence of an extra integration variable

z

, ranging from

- 1

to 1. The

z

-integration involves the weighting function^$11$

\begin{matrix} δ_{l} (z) = \frac{(2 l + 1)!!}{2^{l + 1} l!} (1 - z^{2})^{l}, \end{matrix}

(92)

which is normalized in such a way that

\begin{matrix} \int_{- 1}^{1} d z δ_{l} (z) = 1 . \end{matrix}

(93)

For completeness, we give also the formulas for the four auxiliary source moments

W_{L}, . . ., Z_{L}

, which parametrize the gauge vector

φ_{1}^{α}

as defined in Eqs. ( 31 ):

\begin{matrix} W_{L} (u) & = & ℱ P \int d^{3} x \int_{- 1}^{1} d z {\frac{2 l + 1}{(l + 1) (2 l + 3)} δ_{l + 1} {\hat{x}}_{i L} Σ_{i} \end{matrix}

\begin{matrix} - \frac{2 l + 1}{2 c^{2} (l + 1) (l + 2) (2 l + 5)} δ_{l + 2} {\hat{x}}_{i j L} Σ_{i j}^{(1)}}, \end{matrix}

(94)

\begin{matrix} X_{L} (u) = ℱ P \int d^{3} x \int_{- 1}^{1} d z {\frac{2 l + 1}{2 (l + 1) (l + 2) (2 l + 5)} δ_{l + 2} {\hat{x}}_{i j L} Σ_{i j}}, \end{matrix}

(95)

\begin{matrix} Y_{L} (u) & = & ℱ P \int d^{3} x \int_{- 1}^{1} d z {- δ_{l} {\hat{x}}_{L} Σ_{i i} + \frac{3 (2 l + 1)}{(l + 1) (2 l + 3)} δ_{l + 1} {\hat{x}}_{i L} Σ_{i}^{(1)} \end{matrix}

\begin{matrix} - \frac{2 (2 l + 1)}{c^{2} (l + 1) (l + 2) (2 l + 5)} δ_{l + 2} {\hat{x}}_{i j L} Σ_{i j}^{(2)}}, \end{matrix}

(96)

\begin{matrix} Z_{L} (u) = ℱ P \int d^{3} x \int_{- 1}^{1} d z ɛ_{a b 〈 i_{l}} {- \frac{2 l + 1}{(l + 2) (2 l + 3)} δ_{l + 1} {\hat{x}}_{L - 1 〉 b c} Σ_{a c}} . \end{matrix}

(97)

As discussed in Section 4 , one can always find two intermediate “packages” of multipole moments,

M_{L}

and

S_{L}

, which are some non-linear functionals of the source moments ( 88 ) and ( 94 , 95 , 96 , 97 ), and such that the exterior field depends only on them, modulo a change of coordinates. See e.g. Eq. ( 100 ) below.

In fact, all these source moments make sense only in the form of a post-Newtonian expansion, so in practice we need to know how to expand all the

z

-integrals as series when

c \to + \infty

. Here is the appropriate formula:

\begin{matrix} \int_{- 1}^{1} d z δ_{l} (z) τ (x, u + z | x | / c) = \sum_{k = 0}^{+ \infty} \frac{(2 l + 1)!!}{2^{k} k! (2 l + 2 k + 1)!!} {(\frac{| x |}{c} \frac{\partial}{\partial u})}^{2 k} τ (x, u) . \end{matrix}

(98)

Since the right-hand side involves only even powers of

1 / c

, the same result holds equally well for the “advanced” variable

u + z | x | / c

or the “retarded” one

u - z | x | / c

. Of course, in the Newtonian limit, the moments

I_{L}

and

J_{L}

(and also

M_{L}

S_{L}

) reduce to the standard expressions. For instance, we have

\begin{matrix} I_{L} (u) = Q_{L} (u) + O (\frac{1}{c^{2}}), \end{matrix}

(99)

where

Q_{L}

is the Newtonian mass-type multipole moment (see Eq. ( 3 )). (The moments

W_{L}

. . .

Z_{L}

have also a Newtonian limit, but it is not particularly illuminating.) 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^{α β}

, can be completely specified. Such a systematic post-Newtonian iteration scheme, valid (formally) to any post-Newtonian order, has been recently implemented by Poujade and Blanchet [123] using matched asymptotic expansions (see Section 7 below for the metric developed explicitly up to the 3PN order).

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^$12$ .

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^{α β}

(this necessitates solving the field equations inside the matter) before inserting them into the source moments ( 88 , 91 , 92 , 93 , 98 , 94 , 95 , 96 , 97 ). The formula ( 98 ) 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 , 29 , 31 ), 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.

^$10$ 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.

^$11$ This function approaches the Dirac delta-function (hence its name) in the limit of large multipoles: ${lim}_{l \to + \infty} δ_{l} (z) = δ (z)$ . Indeed the source looks more and more like a point mass as we increase the multipolar order $l$ .

^$12$ 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. [43] .

6 Non-linear Multipole Interactions

We shall now show that the radiative mass-type quadrupole moment

U_{i j}

includes a quadratic tail at the relative 1.5PN order (or

1 / c^{3}

), corresponding to the interaction of the mass

M

of the source and its quadrupole moment

I_{i j}

. This is due to the back-scattering of quadrupolar waves off the Schwarzschild curvature generated by

M

. Next,

U_{i j}

includes a so-called non-linear memory integral at the 2.5PN order, due to the quadrupolar radiation of the stress-energy distribution of linear quadrupole waves themselves, i.e. of multipole interactions

I_{i j} \times I_{k l}

. Finally, we have also a cubic tail, or “tail of tail”, arising at the 3PN order, and associated with the multipole interaction

M^{2} \times I_{i j}

. The result for

U_{i j}

is better expressed in terms of the intermediate quadrupole moment

M_{i j}

already discussed in Section 4.2 . This moment reads [7]

\begin{matrix} M_{i j} = I_{i j} - \frac{4 G}{c^{5}} [W^{(2)} I_{i j} - W^{(1)} I_{i j}^{(1)}] + O (\frac{1}{c^{7}}), \end{matrix}

(100)

where

W

means

W_{L}

as given by Eq. ( 94 ) in the case

l = 0

(of course, in Eq. ( 100 ) we need only the Newtonian value of

W

). The difference between the two moments

M_{i j}

and

I_{i j}

is a small 2.5PN quantity.

Henceforth, we shall express many of the results in terms of the mass moments

M_{L}

and the corresponding current ones

S_{L}

. The complete formula for the radiative quadrupole, valid through the 3PN order, reads [12, 10]

\begin{matrix} U_{i j} (U) = M_{i j}^{(2)} (U) + \frac{2 G M}{c^{3}} \int_{0}^{+ \infty} d τ M_{i j}^{(4)} (U - τ) [ln (\frac{c τ}{2 r_{0}}) + \frac{11}{12}] \end{matrix}

\begin{matrix} + \frac{G}{c^{5}} {- \frac{2}{7} \int_{0}^{+ \infty} d τ M_{a 〈 i}^{(3)} (U - τ) M_{j 〉 a}^{(3)} (U - τ) \end{matrix}

\begin{matrix} - \frac{2}{7} M_{a 〈 i}^{(3)} M_{j 〉 a}^{(2)} - \frac{5}{7} M_{a 〈 i}^{(4)} M_{j 〉 a}^{(1)} + \frac{1}{7} M_{a 〈 i}^{(5)} M_{j 〉 a} + \frac{1}{3} ɛ_{a b 〈 i} M_{j 〉 a}^{(4)} S_{b}} \end{matrix}

\begin{matrix} + \frac{2 G^{2} M^{2}}{c^{6}} \int_{0}^{+ \infty} d τ M_{i j}^{(5)} (U - τ) [{ln}^{2} (\frac{c τ}{2 r_{0}}) + \frac{57}{70} ln (\frac{c τ}{2 r_{0}}) + \frac{124627}{44100}] \end{matrix}

\begin{matrix} + O (\frac{1}{c^{7}}) . \end{matrix}

(101)

The retarded time in radiative coordinates is denoted

U = T - R / c

. The constant

r_{0}

is the one that enters our definition of the finite-part operation

ℱ P

(see Eq. ( 39 )). The “Newtonian” term in Eq. ( 101 ) contains the Newtonian quadrupole moment

Q_{i j}

(see Eq. ( 99 )). The dominant radiation tail at the 1.5PN order was computed within the present formalism in Ref. [17] . The 2.5PN non-linear memory integral – the first term inside the coefficient of

G / c^{5}

– has been obtained using both post-Newtonian methods [4, 154, 145] and rigorous studies of the field at future null infinity [44] . The other multipole interactions at the 2.5PN order can be found in Ref. [12] . Finally the “tail of tail” integral appearing at the 3PN order has been derived in this formalism in Ref. [10] . Be careful to note that the latter post-Newtonian orders correspond to “relative” orders when counted in the local radiation-reaction force, present in the equations of motion: For instance, the 1.5PN tail integral in Eq. ( 101 ) is due to a 4PN radiative effect in the equations of motion [15] ; similarly, the 3PN tail-of-tail integral is (presumably) associated with some radiation-reaction terms occuring at the 5.5PN order.

Notice that all the radiative multipole moments, for any

l

, get some tail-induced contributions. They are computed at the 1.5PN level in Appendix C of Ref. [6] . We find

\begin{matrix} U_{L} (U) & = & M_{L}^{(l)} (U) \end{matrix}

\begin{matrix} + \frac{2 G M}{c^{3}} \int_{0}^{+ \infty} d τ M_{L}^{(l + 2)} (U - τ) [ln (\frac{c τ}{2 r_{0}}) + κ_{l}] + O (\frac{1}{c^{5}}), \end{matrix}

(102)

\begin{matrix} V_{L} (U) & = & S_{L}^{(l)} (U) \end{matrix}

\begin{matrix} + \frac{2 G M}{c^{3}} \int_{0}^{+ \infty} d τ S_{L}^{(l + 2)} (U - τ) [ln (\frac{c τ}{2 r_{0}}) + π_{l}] + O (\frac{1}{c^{5}}), \end{matrix}

(103)

where the constants

κ_{l}

and

π_{l}

are given by

\begin{matrix} κ_{l} & = & \frac{2 l^{2} + 5 l + 4}{l (l + 1) (l + 2)} + \sum_{k = 1}^{l - 2} \frac{1}{k}, \end{matrix}

(104)

\begin{matrix} π_{l} & = & \frac{l - 1}{l (l + 1)} + \sum_{k = 1}^{l - 1} \frac{1}{k} . \end{matrix}

(105)

Recall that the retarded time

U

in radiative coordinates is given by

\begin{matrix} U = t - \frac{r}{c} - \frac{2 G M}{c^{3}} ln (\frac{r}{r_{0}}) + O (G^{2}), \end{matrix}

(106)

where

(t, r)

are harmonic coordinates; recall the gauge vector

ξ_{1}^{α}

in Eq. ( 58 ). Inserting

U

as given by Eq. ( 106 ) into Eqs. ( 6 ) we obtain the radiative moments expressed in terms of source-rooted coordinates

(t, r)

, e.g.

\begin{matrix} U_{L} & = & M_{L}^{(l)} (t - r / c) \end{matrix}

\begin{matrix} + \frac{2 G M}{c^{3}} \int_{0}^{+ \infty} d τ M_{L}^{(l + 2)} (t - τ - r / c) [ln (\frac{c τ}{2 r}) + κ_{l}] + O (\frac{1}{c^{5}}) . \end{matrix}

(107)

This expression no longer depends on the constant

r_{0}

(i.e. the

r_{0}

gets replaced by

r

)^$13$ . If we now change the harmonic coordinates

(t, r)

to some new ones, such as, for instance, some “Schwarzschild-like” coordinates

(t^{'}, r^{'})

such that

t^{'} = t

and

r^{'} = r + G M / c^{2}

, we get

\begin{matrix} U_{L} & = & M_{L}^{(l)} (t^{'} - r^{'} / c) \end{matrix}

\begin{matrix} + \frac{2 G M}{c^{3}} \int_{0}^{+ \infty} d τ M_{L}^{(l + 2)} (t^{'} - τ - r^{'} / c) [ln (\frac{c τ}{2 r^{'}}) + κ_{l}^{'}] + O (\frac{1}{c^{5}}), \end{matrix}

(108)

where

κ_{l}^{'} = κ_{l} + 1 / 2

. Therefore the constant

κ_{l}

(and

π_{l}

as well) depends on the choice of source-rooted coordinates

(t, r)

: E.g. we have

κ_{2} = 11 / 12

in harmonic coordinates (see Eq. ( 101 )), but

κ_{2}^{'} = 17 / 12

in Schwarzschild coordinates [31] .

The tail integrals in Eqs. ( 101 , 6 ) involve all the instants from

- \infty

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

- T

; see Eq. ( 19 ). The range of integration of the tails is therefore limited a priori to the time interval [

- T

U

]. But now, once we have derived the tail integrals, thanks 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_{i j} (U - τ)

when

τ \to + \infty

, that the tail integrals, when assumed to extend over the whole time interval [

- \infty

U

], remain perfectly well-defined (i.e. convergent) at the integration bound

τ = + \infty

. 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.

To obtain the result ( 101 ), we must implement in details the post-Minkows-kian algorithm presented in Section 4.1 . Let us outline here this computation, limiting ourselves to the interaction between one or two masses

M \equiv M_{A D M} \equiv I

and the time-varying quadrupole moment

M_{a b} (u)

(that is related to the source quadrupole

I_{a b} (u)

by Eq. ( 100 )). For these moments the linearized metric ( 26 , 29 , 31 ) reads

\begin{matrix} h_{1}^{α β} = h_{(M)}^{α β} + h_{(M_{ab})}^{α β}, \end{matrix}

(109)

where the monopole part is nothing but the linearized piece of the Schwarzschild metric in harmonic coordinates,

\begin{matrix} h_{(M)}^{00} & = & - 4 r^{- 1} M, \end{matrix}

(110)

\begin{matrix} h_{(M)}^{0 i} & = & 0, \end{matrix}

(111)

\begin{matrix} h_{(M)}^{i j} & = & 0, \end{matrix}

(112)

and the quadrupole part is

\begin{matrix} h_{(M_{ab})}^{00} & = & - 2 \partial_{a b} [r^{- 1} M_{a b} (u)], \end{matrix}

(113)

\begin{matrix} h_{(M_{ab})}^{0 i} & = & 2 \partial_{a} [r^{- 1} M_{a i}^{(1)} (u)], \end{matrix}

(114)

\begin{matrix} h_{(M_{ab})}^{i j} & = & - 2 r^{- 1} M_{i j}^{(2)} (u) . \end{matrix}

(115)

(We pose

c = 1

until the end of this section.) Consider next the quadratically non-linear metric

h_{2}^{α β}

generated by these moments. Evidently it involves a term proportional to

M^{2}

, the mixed term corresponding to the interaction

M \times M_{a b}

, and the self-interaction term of

M_{a b}

. Say,

\begin{matrix} h_{2}^{α β} = h_{(M^{2})}^{α β} + h_{(M M_{ab})}^{α β} + h_{(M_{ab} M_{cd})}^{α β} . \end{matrix}

(116)

The first term represents the quadratic piece of the Schwarzschild metric,

\begin{matrix} h_{(M^{2})}^{00} & = & - 7 r^{- 2} M^{2}, \end{matrix}

(117)

\begin{matrix} h_{(M^{2})}^{0 i} & = & 0, \end{matrix}

(118)

\begin{matrix} h_{(M^{2})}^{i j} & = & - n_{i j} r^{- 2} M^{2} . \end{matrix}

(119)

The second term in Eq. ( 116 ) represents the dominant non-static multipole interaction, that is between the mass and the quadrupole moment, and that we now compute^$14$ . We apply the equations ( 43 , 44 , 46 , 49 , 50 ) in Section 4 . First we obtain the source for this term, viz.

\begin{matrix} Λ_{(M M_{ab})}^{α β} = N^{α β} [h_{(M)}, h_{(M_{ab})}] + N^{α β} [h_{(M_{ab})}, h_{(M)}], \end{matrix}

(120)

where

N^{α β} (h, h)

denotes the quadratic-order part of the gravitational source, as defined by Eq. ( 16 ). To integrate this term we need some explicit formulas for the retarded integral of an extended (non-compact-support) source having some definite multipolarity

l

. A thorough account of the technical formulas necessary for handling the quadratic and cubic interactions is given in the appendices of Refs. [12] and [10] . For the present computation the crucial formula corresponds to a source term behaving like

1 / r^{2}

\begin{matrix} □_{r e t}^{- 1} [\frac{{\hat{n}}_{L}}{r^{2}} F (t - r)] = - {\hat{n}}_{L} \int_{1}^{+ \infty} d x Q_{l} (x) F (t - r x), \end{matrix}

(121)

where

Q_{l}

is the Legendre function of the second kind^$15$ . With the help of this and other formulas we obtain the object

u_{2}^{α β}

given by Eq. ( 43 ). Next we compute the divergence

w_{2}^{α} = \partial_{μ} u_{2}^{α μ}

, and obtain the supplementary term

v_{2}^{α β}

by applying Eqs. ( 49 ). Actually, we find for this particular interaction

w_{2}^{α} = 0

and thus also

v_{2}^{α β} = 0

. Following Eq. ( 50 ), the result is the sum of

u_{2}^{α β}

and

v_{2}^{α β}

, and we get

\begin{matrix} M^{- 1} h_{(M M_{ab})}^{00} & = & n_{a b} r^{- 4} [- 21 M_{a b} - 21 r M_{a b}^{(1)} + 7 r^{2} M_{a b}^{(2)} + 10 r^{3} M_{a b}^{(3)}] \end{matrix}

\begin{matrix} + 8 n_{a b} \int_{1}^{+ \infty} d x Q_{2} (x) M_{a b}^{(4)} (t - r x), \end{matrix}

(122)

\begin{matrix} M^{- 1} h_{(M M_{ab})}^{0 i} & = & n_{i a b} r^{- 3} [- M_{a b}^{(1)} - r M_{a b}^{(2)} - \frac{1}{3} r^{2} M_{a b}^{(3)}] \end{matrix}

\begin{matrix} + n_{a} r^{- 3} [- 5 M_{a i}^{(1)} - 5 r M_{a i}^{(2)} + \frac{19}{3} r^{2} M_{a i}^{(3)}] \end{matrix}

\begin{matrix} + 8 n_{a} \int_{1}^{+ \infty} d x Q_{1} (x) M_{a i}^{(4)} (t - r x), \end{matrix}

(123)

\begin{matrix} M^{- 1} h_{(M M_{ab})}^{i j} & = & n_{i j a b} r^{- 4} [- \frac{15}{2} M_{a b} - \frac{15}{2} r M_{a b}^{(1)} - 3 r^{2} M_{a b}^{(2)} - \frac{1}{2} r^{3} M_{a b}^{(3)}] \end{matrix}

\begin{matrix} + δ_{i j} n_{a b} r^{- 4} [- \frac{1}{2} M_{a b} - \frac{1}{2} r M_{a b}^{(1)} - 2 r^{2} M_{a b}^{(2)} - \frac{11}{6} r^{3} M_{a b}^{(3)}] \end{matrix}

\begin{matrix} + n_{a (i} r^{- 4} [6 M_{j) a} + 6 r M_{j) a}^{(1)} + 6 r^{2} M_{j) a}^{(2)} + 4 r^{3} M_{j) a}^{(3)}] \end{matrix}

\begin{matrix} + r^{- 4} [- M_{i j} - r M_{i j}^{(1)} - 4 r^{2} M_{i j}^{(2)} - \frac{11}{3} r^{3} M_{i j}^{(3)}] \end{matrix}

\begin{matrix} + 8 \int_{1}^{+ \infty} d x Q_{0} (x) M_{i j}^{(4)} (t - r x) . \end{matrix}

(124)

The metric is composed of two types of terms: “instantaneous” ones depending on the values of the quadrupole moment at the retarded time

u = t - r

, and “non-local” or tail integrals, depending on all previous instants

t - r x \leq u

Let us investigate now the cubic interaction between two mass monopoles

M

with the quadrupole

M_{a b}

Obviously, the source term corresponding to this interaction reads

\begin{matrix} Λ_{(M^{2} M_{ab})}^{α β} & = & N^{α β} [h_{(M)}, h_{(M M_{ab})}] + N^{α β} [h_{(M M_{ab})}, h_{(M)}] \end{matrix}

\begin{matrix} + N^{α β} [h_{(M^{2})}, h_{(M_{ab})}] + N^{α β} [h_{(M_{ab})}, h_{(M^{2})}] \end{matrix}

\begin{matrix} + M^{α β} [h_{(M)}, h_{(M)}, h_{(M_{ab})}] + M^{α β} [h_{(M)}, h_{(M_{ab})}, h_{(M)}] \end{matrix}

\begin{matrix} + M^{α β} [h_{(M_{ab})}, h_{(M)}, h_{(M)}] \end{matrix}

(125)

(see Eq. ( 36 )). Notably, the

N

-terms in Eq. ( 125 ) involve the interaction between a linearized metric,

h_{(M)}

h_{(M_{ab})}

, and a quadratic one,

h_{(M^{2})}

h_{(M M_{ab})}

. So, included into these terms are the tails present in the quadratic metric

h_{(M M_{ab})}

computed previously with the result ( 124 ). These tails will produce in turn some “tails of tails” in the cubic metric

h_{(M^{2} M_{ab})}

. The rather involved computation will not be detailed here (see Ref. [10] ). Let us just mention the most difficult of the needed integration formulas^$16$ :

\begin{matrix} ℱ P □_{r e t}^{- 1} [\frac{{\hat{n}}_{L}}{r} \int_{1}^{+ \infty} d x Q_{m} (x) F (t - r x)] = {\hat{n}}_{L} \int_{1}^{+ \infty} d y F^{(- 1)} (t - r y) \end{matrix}

\begin{matrix} \times {Q_{l} (y) \int_{1}^{y} d x Q_{m} (x) \frac{d P_{l}}{d x} (x) + P_{l} (y) \int_{y}^{+ \infty} d x Q_{m} (x) \frac{d Q_{l}}{d x} (x)}, \end{matrix}

(126)

where

F^{(- 1)}

is the time anti-derivative of

F

. With this formula and others given in Ref. [10] we are able to obtain the closed algebraic form of the metric

h_{(M^{2} M_{ab})}^{α β}

, at the leading order in the distance to the source.

The net result is

\begin{matrix} M^{- 2} h_{(M^{2} M_{ab})}^{00} & = & \frac{n_{a b}}{r} \int_{0}^{+ \infty} d τ M_{a b}^{(5)} [- 4 {ln}^{2} (\frac{τ}{2 r}) - 4 ln (\frac{τ}{2 r}) \end{matrix}

\begin{matrix} + \frac{116}{21} ln (\frac{τ}{2 r_{0}}) - \frac{7136}{2205}] \end{matrix}

\begin{matrix} + O (\frac{1}{r^{2}}), \end{matrix}

(127)

\begin{matrix} M^{- 2} h_{(M^{2} M_{ab})}^{0 i} & = & \frac{{\hat{n}}_{i a b}}{r} \int_{0}^{+ \infty} d τ M_{a b}^{(5)} [- \frac{2}{3} ln (\frac{τ}{2 r}) - \frac{4}{105} ln (\frac{τ}{2 r_{0}}) - \frac{716}{1225}] \end{matrix}

\begin{matrix} + \frac{n_{a}}{r} \int_{0}^{+ \infty} d τ M_{a i}^{(5)} [- 4 {ln}^{2} (\frac{τ}{2 r}) - \frac{18}{5} ln (\frac{τ}{2 r}) \end{matrix}

\begin{matrix} + \frac{416}{75} ln (\frac{τ}{2 r_{0}}) - \frac{22724}{7875}] \end{matrix}

\begin{matrix} + O (\frac{1}{r^{2}}), \end{matrix}

(128)

\begin{matrix} M^{- 2} h_{(M^{2} M_{ab})}^{i j} & = & \frac{{\hat{n}}_{i j a b}}{r} \int_{0}^{+ \infty} d τ M_{a b}^{(5)} [- ln (\frac{τ}{2 r}) - \frac{191}{210}] \end{matrix}

\begin{matrix} + \frac{δ_{i j} n_{a b}}{r} \int_{0}^{+ \infty} d τ M_{a b}^{(5)} [- \frac{80}{21} ln (\frac{τ}{2 r}) - \frac{32}{21} ln (\frac{τ}{2 r_{0}}) - \frac{296}{35}] \end{matrix}

\begin{matrix} + \frac{{\hat{n}}_{a (i}}{r} \int_{0}^{+ \infty} d τ M_{j) a}^{(5)} [\frac{52}{7} ln (\frac{τ}{2 r}) + \frac{104}{35} ln (\frac{τ}{2 r_{0}}) + \frac{8812}{525}] \end{matrix}

\begin{matrix} + \frac{1}{r} \int_{0}^{+ \infty} d τ M_{i j}^{(5)} [- 4 {ln}^{2} (\frac{τ}{2 r}) - \frac{24}{5} ln (\frac{τ}{2 r}) \end{matrix}

\begin{matrix} + \frac{76}{15} ln (\frac{τ}{2 r_{0}}) - \frac{198}{35}] \end{matrix}

\begin{matrix} + O (\frac{1}{r^{2}}), \end{matrix}

(129)

where all the moments

M_{a b}

are evaluated at the instant

t - r - τ

(recall that

c = 1

). Notice that some of the logarithms in Eqs. ( 129 ) contain the ratio

τ / r

while others involve

τ / r_{0}

. The indicated remainders

O (1 / r^{2})

contain some logarithms of

r

; in fact they should be more accurately written as

o (r^{ε - 2})

for some

ε ≪ 1

The presence of logarithms of

r

in Eqs. ( 129 ) is an artifact of the harmonic coordinates

x^{α}

, and we need to gauge them away by introducing the radiative coordinates

X^{α}

at future null infinity (see Theorem 4 ).

As it turns out, it is sufficient for the present calculation to take into account the “linearized” logarithmic deviation of the light cones in harmonic coordinates so that

X^{α} = x^{α} + G ξ_{1}^{α} + O (G^{2})

, where

ξ_{1}^{α}

is the gauge vector defined by Eq. ( 58 ) (see also Eq. ( 106 )). With this coordinate change one removes all the logarithms of

r

in Eqs. ( 129 ). Hence, we obtain the radiative metric

\begin{matrix} M^{- 2} H_{(M^{2} M_{ab})}^{00} & = & \frac{N_{a b}}{R} \int_{0}^{+ \infty} d τ M_{a b}^{(5)} [- 4 {ln}^{2} (\frac{τ}{2 r_{0}}) + \frac{32}{21} ln (\frac{τ}{2 r_{0}}) - \frac{7136}{2205}] \end{matrix}

\begin{matrix} + O (\frac{1}{R^{2}}), \end{matrix}

(130)

\begin{matrix} M^{- 2} H_{(M^{2} M_{ab})}^{0 i} & = & \frac{{\hat{N}}_{i a b}}{R} \int_{0}^{+ \infty} d τ M_{a b}^{(5)} [- \frac{74}{105} ln (\frac{τ}{2 r_{0}}) - \frac{716}{1225}] \end{matrix}

\begin{matrix} + \frac{N_{a}}{R} \int_{0}^{+ \infty} d τ M_{a i}^{(5)} [- 4 {ln}^{2} (\frac{τ}{2 r_{0}}) + \frac{146}{75} ln (\frac{τ}{2 r_{0}}) - \frac{22724}{7875}] \end{matrix}

\begin{matrix} + O (\frac{1}{R^{2}}), \end{matrix}

(131)

\begin{matrix} M^{- 2} H_{(M^{2} M_{ab})}^{i j} & = & \frac{{\hat{N}}_{i j a b}}{R} \int_{0}^{+ \infty} d τ M_{a b}^{(5)} [- ln (\frac{τ}{2 r_{0}}) - \frac{191}{210}] \end{matrix}

\begin{matrix} + \frac{δ_{i j} N_{a b}}{R} \int_{0}^{+ \infty} d τ M_{a b}^{(5)} [- \frac{16}{3} ln (\frac{τ}{2 r_{0}}) - \frac{296}{35}] \end{matrix}

\begin{matrix} + \frac{{\hat{N}}_{a (i}}{R} \int_{0}^{+ \infty} d τ M_{j) a}^{(5)} [\frac{52}{5} ln (\frac{τ}{2 r_{0}}) + \frac{8812}{525}] \end{matrix}

\begin{matrix} + \frac{1}{R} \int_{0}^{+ \infty} d τ M_{i j}^{(5)} [- 4 {ln}^{2} (\frac{τ}{2 r_{0}}) + \frac{4}{15} ln (\frac{τ}{2 r_{0}}) - \frac{198}{35}] \end{matrix}

\begin{matrix} + O (\frac{1}{R^{2}}), \end{matrix}

(132)

where the moments are evaluated at time

U - τ \equiv T - R - τ

. It is trivial to compute the contribution of the radiative moments

U_{L} (U)

and

V_{L} (U)

corresponding to that metric. We find the “tail of tail” term reported in Eq. ( 101 ).

^$13$ 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 Eq. ( 106 ). The reason is that the moment $M_{L}$ also depends on $r_{0}$ 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 $r_{0}$ .

^$14$ The computation of the third term in Eq. ( 116 ), which corresponds to the interaction between two quadrupoles, $M_{a b} \times M_{c d}$ , can be found in Ref. [12] .

^$15$ The function $Q_{l}$ is given in terms of the Legendre polynomial $P_{l}$ by

Q_{l} (x) = \frac{1}{2} \int_{- 1}^{1} \frac{d z P_{l} (z)}{x - z} = \frac{1}{2} P_{l} (x) l n (\frac{x + 1}{x - 1}) - \sum_{j = 1}^{l} \frac{1}{j} P_{l - j} (x) P_{j - 1} (x) .

In the complex plane there is a branch cut from

- \infty

to 1. The first equality is known as the Neumann formula for the Legendre function.

^$16$ Eq. ( 126 ) has been obtained using a not so well known mathematical relation between the Legendre functions and polynomials:

\frac{1}{2} \int_{- 1}^{1} \frac{d z P_{l} (z)}{\sqrt{(x y - z)^{2} - (x^{2} - 1) (y^{2} - 1)}} = Q_{l} (x) P_{l} (y)

(where

1 \leq y < x

is assumed). See Appendix A in Ref. [10] for the proof. This relation constitutes a generalization of the Neumann formula (see footnote after Eq. ( 121 )).

7 The Third Post-Newtonian Metric

The detailed calculations that are called for in applications necessitate having ar one's disposal some explicit expressions of the metric coefficients

g_{α β}

, in harmonic coordinates, at the highest possible post-Newtonian order. The 3PN metric that we present below is expressed by means of some particular retarded-type potentials,

V

V_{i}

{\hat{W}}_{i j}

, etc., whose main advantages are to somewhat minimize the number of terms, so that even at the 3PN order the metric is still tractable, and to delineate the different problems associated with the computation of different categories of terms. Of course, these potentials have no physical significance by themselves. The basic idea in our post-Newtonian iteration is to use whenever possible a “direct” integration, with the help of some formulas like

□_{r e t}^{- 1} (\partial_{μ} V \partial^{μ} V + V □ V) = V^{2} / 2

. The 3PN harmonic-coordinates metric (issued from Ref. [22] ) reads

\begin{matrix} g_{00} & = & - 1 + \frac{2}{c^{2}} V - \frac{2}{c^{4}} V^{2} + \frac{8}{c^{6}} (\hat{X} + V_{i} V_{i} + \frac{V^{3}}{6}) \end{matrix}

\begin{matrix} + \frac{32}{c^{8}} (\hat{T} - \frac{1}{2} V \hat{X} + {\hat{R}}_{i} V_{i} - \frac{1}{2} V V_{i} V_{i} - \frac{1}{48} V^{4}) + O (\frac{1}{c^{10}}), \end{matrix}

(133)

\begin{matrix} g_{0 i} = - \frac{4}{c^{3}} V_{i} - \frac{8}{c^{5}} {\hat{R}}_{i} - \frac{16}{c^{7}} ({\hat{Y}}_{i} + \frac{1}{2} {\hat{W}}_{i j} V_{j} + \frac{1}{2} V^{2} V_{i}) + O (\frac{1}{c^{9}}), \end{matrix}

(134)

\begin{matrix} g_{i j} & = & δ_{i j} [1 + \frac{2}{c^{2}} V + \frac{2}{c^{4}} V^{2} + \frac{8}{c^{6}} (\hat{X} + V_{k} V_{k} + \frac{V^{3}}{6})] + \frac{4}{c^{4}} {\hat{W}}_{i j} \end{matrix}

\begin{matrix} + \frac{16}{c^{6}} ({\hat{Z}}_{i j} + \frac{1}{2} V {\hat{W}}_{i j} - V_{i} V_{j}) + O (\frac{1}{c^{8}}) . \end{matrix}

(135)

All the potentials are generated by the matter stress-energy tensor

T^{α β}

through the definitions (analogous to Eqs. ( 91 ))

\begin{matrix} σ & = & \frac{T^{00} + T^{i i}}{c^{2}}, \end{matrix}

(136)

\begin{matrix} σ_{i} & = & \frac{T^{0 i}}{c}, \end{matrix}

(137)

\begin{matrix} σ_{i j} & = & T^{i j} . \end{matrix}

(138)

V

and

V_{i}

represent some retarded versions of the Newtonian and gravitomagnetic potentials,

\begin{matrix} V & = & □_{r e t}^{- 1} [- 4 π G σ], \end{matrix}

(139)

\begin{matrix} V_{i} & = & □_{r e t}^{- 1} [- 4 π G σ_{i}] . \end{matrix}

(140)

From the 2PN order we have the potentials

\begin{matrix} \hat{X} & = & □_{r e t}^{- 1} [- 4 π G V σ_{i i} + {\hat{W}}_{i j} \partial_{i j} V + 2 V_{i} \partial_{t} \partial_{i} V + V \partial_{t}^{2} V \end{matrix}

\begin{matrix} + \frac{3}{2} (\partial_{t} V)^{2} - 2 \partial_{i} V_{j} \partial_{j} V_{i}], \end{matrix}

(141)

\begin{matrix} {\hat{R}}_{i} & = & □_{r e t}^{- 1} [- 4 π G (V σ_{i} - V_{i} σ) - 2 \partial_{k} V \partial_{i} V_{k} - \frac{3}{2} \partial_{t} V \partial_{i} V], \end{matrix}

(142)

\begin{matrix} {\hat{W}}_{i j} & = & □_{r e t}^{- 1} [- 4 π G (σ_{i j} - δ_{i j} σ_{k k}) - \partial_{i} V \partial_{j} V] . \end{matrix}

(143)

Some parts of these potentials are directly generated by compact-support matter terms, while other parts are made of non-compact-support products of

V

-type potentials. There exists also a very important cubically non-linear term generated by the coupling between

{\hat{W}}_{i j}

and

V

, the second term in the

\hat{X}

-potential. At the 3PN level we have the most complicated of these potentials, namely

\begin{matrix} \hat{T} & = & □_{r e t}^{- 1} [- 4 π G (\frac{1}{4} σ_{i j} {\hat{W}}_{i j} + \frac{1}{2} V^{2} σ_{i i} + σ V_{i} V_{i}) \end{matrix}

\begin{matrix} + {\hat{Z}}_{i j} \partial_{i j} V + {\hat{R}}_{i} \partial_{t} \partial_{i} V - 2 \partial_{i} V_{j} \partial_{j} {\hat{R}}_{i} - \partial_{i} V_{j} \partial_{t} {\hat{W}}_{i j} \end{matrix}

\begin{matrix} + V V_{i} \partial_{t} \partial_{i} V + 2 V_{i} \partial_{j} V_{i} \partial_{j} V + \frac{3}{2} V_{i} \partial_{t} V \partial_{i} V + \frac{1}{2} V^{2} \partial_{t}^{2} V \end{matrix}

\begin{matrix} + \frac{3}{2} V (\partial_{t} V)^{2} - \frac{1}{2} (\partial_{t} V_{i})^{2}], \end{matrix}

(144)

\begin{matrix} {\hat{Y}}_{i} & = & □_{r e t}^{- 1} [- 4 π G (- σ {\hat{R}}_{i} - σ V V_{i} + \frac{1}{2} σ_{k} {\hat{W}}_{i k} + \frac{1}{2} σ_{i k} V_{k} + \frac{1}{2} σ_{k k} V_{i}) \end{matrix}

\begin{matrix} + {\hat{W}}_{k l} \partial_{k l} V_{i} - \partial_{t} {\hat{W}}_{i k} \partial_{k} V + \partial_{i} {\hat{W}}_{k l} \partial_{k} V_{l} - \partial_{k} {\hat{W}}_{i l} \partial_{l} V_{k} \end{matrix}

\begin{matrix} - 2 \partial_{k} V \partial_{i} {\hat{R}}_{k} - \frac{3}{2} V_{k} \partial_{i} V \partial_{k} V - \frac{3}{2} V \partial_{t} V \partial_{i} V \end{matrix}

\begin{matrix} - 2 V \partial_{k} V \partial_{k} V_{i} + V \partial_{t}^{2} V_{i} + 2 V_{k} \partial_{k} \partial_{t} V_{i}], \end{matrix}

(145)

\begin{matrix} {\hat{Z}}_{i j} & = & □_{r e t}^{- 1} [- 4 π G V (σ_{i j} - δ_{i j} σ_{k k}) - 2 \partial_{(i} V \partial_{t} V_{j)} + \partial_{i} V_{k} \partial_{j} V_{k} \end{matrix}

\begin{matrix} + \partial_{k} V_{i} \partial_{k} V_{j} - 2 \partial_{(i} V_{k} \partial_{k} V_{j)} - \frac{3}{4} δ_{i j} (\partial_{t} V)^{2} \end{matrix}

\begin{matrix} - δ_{i j} \partial_{k} V_{m} (\partial_{k} V_{m} - \partial_{m} V_{k})], \end{matrix}

(146)

which involve many types of compact-support contributions, as well as quadratic-order and cubic-order parts; but, surprisingly, there are no quartically non-linear terms^$17$ .

The above potentials are not independent. They are linked together by some differential identities issued from the harmonic gauge conditions, which are equivalent, via the Bianchi identities, to the equations of motion of the matter fields (see Eq. ( 17 )). These identities read

\begin{matrix} 0 & = & \partial_{t} {V + \frac{1}{c^{2}} [\frac{1}{2} {\hat{W}}_{k k} + 2 V^{2}] + \frac{4}{c^{4}} [\hat{X} + \frac{1}{2} {\hat{Z}}_{k k} + \frac{1}{2} V {\hat{W}}_{k k} + \frac{2}{3} V^{3}]} \end{matrix}

\begin{matrix} + \partial_{i} {V_{i} + \frac{2}{c^{2}} [{\hat{R}}_{i} + V V_{i}] + \frac{4}{c^{4}} [{\hat{Y}}_{i} - \frac{1}{2} {\hat{W}}_{i j} V_{j} + \frac{1}{2} {\hat{W}}_{k k} V_{i} + V {\hat{R}}_{i} + V^{2} V_{i}]} \end{matrix}

\begin{matrix} + O (\frac{1}{c^{6}}), \end{matrix}

(147)

\begin{matrix} 0 & = & \partial_{t} {V_{i} + \frac{2}{c^{2}} [{\hat{R}}_{i} + V V_{i}]} \end{matrix}

\begin{matrix} + \partial_{j} {{\hat{W}}_{i j} - \frac{1}{2} {\hat{W}}_{k k} δ_{i j} + \frac{4}{c^{2}} [{\hat{Z}}_{i j} - \frac{1}{2} {\hat{Z}}_{k k} δ_{i j}]} \end{matrix}

\begin{matrix} + O (\frac{1}{c^{4}}) . \end{matrix}

(148)

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 ( 140 , 143 , 146 ).

^$17$ It has been possible to “integrate directly” all the quartic contributions in the 3PN metric. See the terms composed of $V^{4}$ and $V \hat{X}$ in Eq. ( 133 ).

8 Hadamard Self-Field Regularization

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 [69] , is routinely taken into account to describe the Solar System dynamics (see Ref. [104] ). Second, the gravitational radiation-reaction force, which appears in the equations of motion at the 2.5PN order, has been verified by the observation of the secular acceleration of the orbital motion of the binary pulsar [140, 141, 139] . Third, the forthcoming detection and analysis of the gravitational waves emitted by inspiralling compact binaries will necessitate the prior knowledge of the equations of motion and radiation up to the high 3PN relative order [48, 49, 72, 50, 135, 121, 122, 96, 59] .

8.1 Definitions

A model of structureless point masses is expected to be sufficient to describe the inspiral phase of compact binaries (see the discussion around Eqs. ( 6 , 7 , 8 )). Thus we want to compute the metric (and its gradient needed in the equations of motion) at the 3PN order for a system of two point-like particles. A priori one is not allowed to use directly the metric expressions ( 133 , 134 , 135 , 138 , 140 , 143 , 146 ), as they have been derived under the assumption of a continuous (smooth) matter distribution. Applying them to a system of point particles, we find that most of the integrals become divergent at the location of the particles, i.e. when

x \to y_{1} (t)

y_{2} (t)

, where

y_{1} (t)

and

y_{2} (t)

denote the two trajectories. Consequently, we must supplement the calculation by a prescription for how to remove the “infinite part” of these integrals. We systematically employ the Hadamard regularization [80, 133] (see Ref. [134] for an entry to the mathematical literature). Let us present here an account of this regularization, as well as a theory of generalized functions (or pseudo-functions) associated with it, following the detailed investigations in Refs. [20, 23] .

Consider the class

ℱ

of functions

F (x)

which are smooth (

C^{\infty}

) on

R^{3}

except for the two points

y_{1}

and

y_{2}

, around which they admit a power-like singular expansion of the type

\begin{matrix} \forall n \in N, F (x) = \sum_{a_{0} \leq a \leq n} {r_{1}^{a}}_{1} f_{a} (n_{1}) + o (r_{1}^{n}), \end{matrix}

(149)

and similarly for the other point 2. Here

r_{1} = | x - y_{1} | \to 0

, and the coefficients

_{1} f_{a}

of the various powers of

r_{1}

depend on the unit direction

n_{1} = (x - y_{1}) / r_{1}

of approach to the singular point. The powers

a

r_{1}

are real, range in discrete steps [i.e.

a \in (a_{i})_{i \in N}

], and are bounded from below (

a_{0} \leq a

). The coefficients

_{1} f_{a}

(and

_{2} f_{a}

) for which

a < 0

can be referred to as the singular coefficients of

F

. If

F

and

G

belong to

ℱ

so does the ordinary product

F G

, as well as the ordinary gradient

\partial_{i} F

. We define the Hadamard “partie finie” of

F

at the location of the singular point 1 as

\begin{matrix} (F)_{1} = \int \frac{d Ω_{1}}{4 π}_{1} f_{0} (n_{1}), \end{matrix}

(150)

where

d Ω_{1} = d Ω (n_{1})

denotes the solid angle element centered on

y_{1}

and of direction

n_{1}

. The Hadamard partie finie is “non-distributive” in the sense that

(F G)_{1} \neq (F)_{1} (G)_{1}

in general. The second notion of Hadamard partie finie (

P f

) concerns that of the integral

\int d^{3} x F

, which is generically divergent at the location of the two singular points

y_{1}

and

y_{2}

(we assume no divergence at infinity). It is defined by

\begin{matrix} P f_{s_{1}, s_{2}} \int d^{3} x F & = & {lim}_{s \to 0} {\int_{S (s)} d^{3} x F \end{matrix}

\begin{matrix} + 4 π \sum_{a + 3 < 0} \frac{s^{a + 3}}{a + 3} {(\frac{F}{r_{1}^{a}})}_{1} + 4 π ln (\frac{s}{s_{1}}) {(r_{1}^{3} F)}_{1} + 1 \leftrightarrow 2} . \end{matrix}

(151)

The first term integrates over a domain

S (s)

defined as

R^{3}

to which the two spherical balls

r_{1} \leq s

and

r_{2} \leq s

of radius

s

and centered on the two singularities are excised:

S (s) \equiv R^{3} \ ℬ (y_{1}, s) \cup ℬ (y_{2}, s)

. The other terms, where the value of a function at 1 takes the meaning ( 150 ), are such that they cancel out the divergent part of the first term in the limit where

s \to 0

(the symbol

1 \leftrightarrow 2

means the same terms but corresponding to the other point 2). The Hadamard partie-finie integral depends on two strictly positive constants

s_{1}

and

s_{2}

, associated with the logarithms present in Eq. ( 151 ). See Ref. [20] for alternative expressions of the partie-finie integral.

To any

F \in ℱ

we associate the partie finie pseudo-function

P f F

, namely a linear form acting on

ℱ

, which is defined by the duality bracket

\begin{matrix} \forall G \in ℱ, 〈 P f F, G 〉 = P f \int d^{3} x F G . \end{matrix}

(152)

When restricted to the set

D

of smooth functions with compact support (we have

D \subset ℱ

), the pseudo-function

P f F

is a distribution in the sense of Schwartz [133] . The product of pseudo-functions coincides, by definition, with the ordinary pointwise product, namely

P f F . P f G = P f (F G)

. An interesting pseudo-function, constructed in Ref. [20] on the basis of the Riesz delta function [125] , is the delta-pseudo-function

P f δ_{1}

, which plays a role analogous to the Dirac measure in distribution theory,

δ_{1} (x) \equiv δ (x - y_{1})

. It is defined by

\begin{matrix} \forall F \in ℱ, 〈 P f δ_{1}, F 〉 = P f \int d^{3} x δ_{1} F = (F)_{1}, \end{matrix}

(153)

where

(F)_{1}

is the partie finie of

F

as given by Eq. ( 150 ). From the product of

P f δ_{1}

with any

P f F

we obtain the new pseudo-function

P f (F δ_{1})

, that is such that

\begin{matrix} \forall G \in ℱ, 〈 P f (F δ_{1}), G 〉 = (F G)_{1} . \end{matrix}

(154)

As a general rule, we are not allowed, in consequence of the “non-distributivity” of the Hadamard partie finie, to replace

F

within the pseudo-function

P f (F δ_{1})

by its regularized value:

P f (F δ_{1}) \neq (F)_{1} P f δ_{1}

. The object

P f (F δ_{1})

has no equivalent in distribution theory.

Next, we treat the spatial derivative of a pseudo-function of the type

P f F

, namely

\partial_{i} (P f F)

. Essentially, we require (in Ref. [20] ) that the so-called rule of integration by parts holds. By this we mean that we are allowed to freely operate by parts any duality bracket, with the all-integrated (“surface”) terms always zero, as in the case of non-singular functions. This requirement is motivated by our will that a computation involving singular functions be as much as possible the same as a computation valid for regular functions.

By definition,

\begin{matrix} \forall F, G \in ℱ, 〈 \partial_{i} (P f F), G 〉 = - 〈 \partial_{i} (P f G), F 〉 . \end{matrix}

(155)

Furthermore, we assume that when all the singular coefficients of

F

vanish, the derivative of

P f F

reduces to the ordinary derivative, i.e.

\partial_{i} (P f F) = P f (\partial_{i} F)

. Then it is trivial to check that the rule ( 155 ) contains as a particular case the standard definition of the distributional derivative [133] . Notably, we see that the integral of a gradient is always zero:

〈 \partial_{i} (P f F), 1 〉 = 0

. This should certainly be the case if we want to compute a quantity (e.g. a Hamiltonian density) which is defined only modulo a total divergence. We pose

\begin{matrix} \partial_{i} (P f F) = P f (\partial_{i} F) + D_{i} [F], \end{matrix}

(156)

where

P f (\partial_{i} F)

represents the “ordinary” derivative and

D_{i} [F]

the distributional term. The following solution of the basic relation ( 155 ) was obtained in Ref. [20] :

\begin{matrix} D_{i} [F] = 4 π P f (n_{1}^{i} [\frac{1}{2} {r_{1}}_{1} f_{- 1} + \sum_{k \geq 0} {\frac{1}{r_{1}^{k}}}_{1} f_{- 2 - k}] δ_{1}) + 1 \leftrightarrow 2, \end{matrix}

(157)

where we assume for simplicity that the powers

a

in the expansion ( 149 ) of

F

are relative integers. The distributional term ( 157 ) is of the form

P f (G δ_{1})

(plus

1 \leftrightarrow 2

). It is generated solely by the singular coefficients of

F

(the sum over

k

in Eq. ( 157 ) is always finite). The formula for the distributional term associated with the

l

th distributional derivative, i.e.

D_{L} [F] = \partial_{L} P f F - P f \partial_{L} F

, where

L = i_{1} i_{2} . . . i_{l}

, reads

\begin{matrix} D_{L} [F] = \sum_{k = 1}^{l} \partial_{i_{1} . . . i_{k - 1}} D_{i_{k}} [\partial_{i_{k + 1} . . . i_{l}} F] . \end{matrix}

(158)

We refer to Theorem 4 in Ref. [20] for the definition of another derivative operator, representing in fact the most general derivative satisfying the same properties as the one defined by Eq. ( 157 ) and, in addition, the commutation of successive derivatives (or Schwarz lemma)^$18$ .

The distributional derivative ( 156 , 157 , 158 ) does not satisfy the Leibniz rule for the derivation of a product, in accordance with a general theorem of Schwartz [132] . Rather, the investigation in Ref. [20] has suggested that, in order to construct a consistent theory (using the “ordinary” product for pseudo-functions), the Leibniz rule should in a sense be weakened, and replaced by the rule of integration by part ( 155 ), which is in fact nothing but an “integrated” version of the Leibniz rule.

The Hadamard regularization

(F)_{1}

is defined by Eq. ( 150 ) in a preferred spatial hypersurface

t = c o n s t .

of a coordinate system, and consequently is not a priori compatible with the requirement of global Lorentz invariance. Thus we expect that the equations of motion in harmonic coordinates (which, we recall, 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 in order to get some Lorentz-invariant equations of motion [25] .

To deal with the problem at 3PN a Lorentz-invariant regularization, denoted

[F]_{1}

, was introduced in Ref. [23] .

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. [23] for the formulas defining this regularization in the form of some infinite power series in the relativistic parameter

1 / c^{2}

. The regularization

[F]_{1}

plays a crucial role in obtaining the equations of motion at the 3PN order in Refs. [21, 22] .

^$18$ It was shown in Ref. [22] 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. [20] constitute merely some technical tools which are devoid of physical meaning.

8.2 Regularization ambiguities

The “standard” Hadamard regularization yields some ambiguous results for the computation of certain integrals at the 3PN order, as Jaranowski and Schäfer [87, 88, 89] 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, Eq. ( 150 ), and the partie finie of a divergent integral, Eq. ( 151 ) (i.e., without using a theory of pseudo-functions and generalized distributional derivatives as proposed in Refs. [20, 23] ). It was shown in Refs. [87, 88, 89] 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_{s t a t i c}

and

ω_{k i n e t i c}

Blanchet and Faye [20, 23] , motivated by the previous result, introduced their “improved” Hadamard regularization, the one we outlined in the previous Section 8.1 . This new regularization is mathematically well-defined and free of ambiguities; in particular it yields unique results for the computation of any of the integrals occuring in the 3PN equations of motion. Unfortunately, this regularization turned out to be in a sense incomplete, because it was found in Refs. [21, 22] that the 3PN equations of motion involve one and only one unknown numerical constant, called

λ

, which cannot be determined within the method. The comparison of this result with the work of Jaranowski and Schäfer [87, 88, 89] , on the basis of the computation of the invariant energy of binaries moving on circular orbits, showed [21] that

\begin{matrix} ω_{k i n e t i c} = \frac{41}{24}, \end{matrix}

(159)

\begin{matrix} ω_{s t a t i c} = - \frac{11}{3} λ - \frac{1987}{840} . \end{matrix}

(160)

Therefore, the ambiguity

ω_{k i n e t i c}

is fixed, while

λ

is equivalent to the other ambiguity

ω_{s t a t i c}

. Note that the harmonic-coordinates 3PN equations of motion as they have been obtained in Refs. [21, 22] depend also, in addition to

λ

, on two arbitrary constants

r_{1}^{'}

and

r_{2}^{'}

parametrizing some logarithmic terms^$19$ ; however, these constants are not “physical” in the sense that they can be removed by a coordinate transformation.

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. Mathematically speaking, the presence of

λ

is (probably) related to the fact that it is impossible to construct a distributional derivative operator, such as ( 156 , 157 , 158 ), satisfying the Leibniz rule for the derivation of the product [132] .

The Einstein field equations can be written into 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 they become inequivalent for point particles if the derivative operator violates the Leibniz rule. On the other hand, physically speaking,

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 finally expect

λ

to be independent of the internal structure of the compact bodies.

Damour, Jaranowski and Schäfer [60] recovered the value of

ω_{k i n e t i c}

given in Eq. ( 159 ) by proving that this value is the unique one for which the global Poincaré invariance of their formalism is verified. Since the coordinate conditions associated with the ADM approach do not manifestly respect the Poincaré symmetry, they had to prove that the 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 should be Lorentz-invariant, as was indeed found to be the case by Blanchet and Faye [21, 22] , thanks in particular to their use of a Lorentz-invariant regularization [23] (hence their determination of

ω_{k i n e t i c}

The other parameter

ω_{s t a t i c}

was computed by Damour, Jaranowski and Schäfer [61] by means of a dimensional regularization, instead of a Hadamard-type one, within the ADM-Hamiltonian formalism. Their result, which in principle fixes

λ

according to Eq. ( 160 ), is

\begin{matrix} ω_{s t a t i c} = 0 ⟺ λ = - \frac{1987}{3080} . \end{matrix}

(161)

As Damour et al. [61] argue, clearing up the ambiguity is made possible by the fact that the dimensional regularization, contrary to the Hadamard regularization, respects all the basic properties of the algebraic and differential calculus of ordinary functions. In this respect, the dimensional regularization is certainly better than the Hadamard one, which does not respect the “distributivity” of the product (recall that

(F G)_{1} \neq (F)_{1} (G)_{1}

) and unavoidably violates at some stage the Leibniz rule for the differentiation of a product.

Let us comment that the use of a self-field regularization in this problem, be it dimensional or based on the Hadamard 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 velocities, 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 has been achieved at the 2PN order by Kopeikin [93] and Grishchuk and Kopeikin [79] who derived the equations of motion of two extended fluid balls, and proved that for compact bodies the equations depend only on the two masses

m_{1}

and

m_{2}

. At the 3PN order we expect that the extended-body approach will give the value of the regularization parameter

λ

. In the following, we shall prefer to keep

λ

unspecified, until its value has been confirmed by independent and hopefully more physical methods (like in Refs. [146, 94, 65] ).

Blanchet, Iyer and Joguet [26] , in their computation of the 3PN radiation field of two point masses – the second half of the problem, besides the 3PN equations of motion – used the (standard) Hadamard regularization and found it necessary to introduce three additional regularization constants

ξ

κ

and

ζ

, which play a role analogous to the equation-of-motion

λ

. Such unknown constants come from the computation of the 3PN binary's quadrupole moment

I_{i j}

. Some good news is that the total gravitational-wave flux, in the case of circular orbits, depends in fact only on a single combination of the three latter constants,

\begin{matrix} θ = ξ + 2 κ + ζ . \end{matrix}

(162)

To summarize, the final result that we shall derive below for the binary's orbital phase will involve the two regularization constants:

λ

coming from the equations of motion and

θ

coming from the multipole moments.

But, interestingly, we shall find that there is only one unknown coefficient, in the form of a linear combination of

λ

and

θ

^$19$ The constants $r_{1}^{'}$ and $r_{2}^{'}$ are closely related to the constants $s_{1}$ and $s_{2}$ in the partie-finie integral ( 151 ). See Ref. [22] for the precise definition.

9 Newtonian-like Equations of Motion

9.1 The 3PN accelerations 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. [25] ), which consists of reducing the 3PN metric of an extended regular source, worked out in Eqs. ( 133 , 134 , 135 , 138 , 140 , 143 , 146 ), 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. [23] for a proof ).

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

(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.

We denote by

r_{12} = | y_{1} (t) - y_{2} (t) |

the harmonic-coordinate distance between the two particles, with

y_{1} = (y_{1}^{i})

and

y_{2} = (y_{2}^{i})

, by

n_{12}^{i} = (y_{1}^{i} - y_{2}^{i}) / r_{12}

the corresponding unit direction, and by

v_{1}^{i} = d y_{1}^{i} / d t

and

a_{1}^{i} = d v_{1}^{i} / d t

the coordinate velocity and acceleration of the particle 1 (and idem for 2). Sometimes we pose

v_{12}^{i} = v_{1}^{i} - v_{2}^{i}

for the relative velocity. The usual Euclidean scalar product of vectors is denoted with parentheses, e.g.

(n_{12} v_{1}) = n_{12} . v_{1}

and

(v_{1} v_{2}) = v_{1} . v_{2}

. The equations of the body 2 are obtained by exchanging all the particle labels

1 \leftrightarrow 2

(remembering that

n_{12}^{i}

and

v_{12}^{i}

change sign in this operation):

\begin{matrix} a_{1}^{i} = . . . \end{matrix}

(163)

The 3PN equations of motion depend on three arbitrary constants^$20$ : the dimensionless constant

λ

(e.g., a rational fraction), linked with an incompleteness of the Hadamard regularization as discussed in Section 8.2 ; and two arbitrary length scales

r_{1}^{'}

and

r_{2}^{'}

associated with the logarithms present at the 3PN order.

It has been proved in Ref. [22] that the two constants

r_{1}^{'}

and

r_{2}^{'}

are merely linked with the choice of coordinates – we can refer to

r_{1}^{'}

and

r_{2}^{'}

as “gauge constants”. In our approach [21, 22] , the harmonic coordinate system is not uniquely fixed by the coordinate condition

\partial_{μ} h^{α μ} = 0

. In fact there are infinitely many harmonic coordinate systems that are local. For general smooth sources, as in the general formalism of Part aaa , we expect the existence and uniqueness of a global harmonic coordinate system. But here we have some point-particles, with delta-function singularities, and in this case we don't have the notion of a global coordinate system. We can always change the harmonic coordinates by means of the gauge vector

η^{α} = δ x^{α}

, satisfying

Δ η^{α} = 0

except at the location of the two particles (we assume that the transformation is at the 3PN level, so we can consider simply a flat-space Laplace equation). More precisely, we can show that the logarithms appearing in Eq. ( 163 ), together with the constants

r_{1}^{'}

and

r_{2}^{'}

therein, can be removed by the coordinate transformation associated with the 3PN gauge vector (with

r_{1} = | x - y_{1} (t) |

and

r_{2} = | x - y_{2} (t) |

\begin{matrix} η^{α} = - \frac{22}{3} \frac{G^{2} m_{1} m_{2}}{c^{6}} \partial^{α} [\frac{G m_{1}}{r_{2}} ln (\frac{r_{12}}{r_{1}^{'}}) + \frac{G m_{2}}{r_{1}} ln (\frac{r_{12}}{r_{2}^{'}})] . \end{matrix}

(164)

Therefore, the “ambiguity” in the choice of the constants

r_{1}^{'}

and

r_{2}^{'}

is completely innocuous on the physical point of view, because the physical results must be gauge invariant. Indeed we shall verify that

r_{1}^{'}

and

r_{2}^{'}

cancel out in our final results.

When retaining the “even” relativistic corrections at the 1PN, 2PN and 3PN orders, and neglecting the “odd” radiation reaction term at the 2.5PN order, we find that the equations of motion admit a conserved energy (and a Lagrangian, as we shall see), and that energy can be straightforwardly obtained by guess-work starting from Eq. ( 163 ), with the result

\begin{matrix} E = . . . \end{matrix}

(165)

To the terms given above, we must add the terms corresponding to the relabelling

1 \leftrightarrow 2

. Actually, this energy is not conserved because of the radiation reaction. Thus its time derivative, as computed by means of the 3PN equations of motion themselves (i.e., order-reducing all the accelerations), is purely equal to the 2.5PN effect,

\begin{matrix} \frac{d E}{d t} & = & \frac{4}{5} \frac{G^{2} m_{1}^{2} m_{2}}{c^{5} r_{12}^{3}} [(v_{1} v_{12}) (- v_{12}^{2} + 2 \frac{G m_{1}}{r_{12}} - 8 \frac{G m_{2}}{r_{12}}) \end{matrix}

\begin{matrix} + (n_{12} v_{1}) (n_{12} v_{12}) (3 v_{12}^{2} - 6 \frac{G m_{1}}{r_{12}} + \frac{52}{3} \frac{G m_{2}}{r_{12}})] \end{matrix}

\begin{matrix} + 1 \leftrightarrow 2 + O (\frac{1}{c^{7}}) . \end{matrix}

(166)

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

\begin{matrix} \tilde{E} = E + \frac{4 G^{2} m_{1}^{2} m_{2}}{5 c^{5} r_{12}^{2}} (n_{12} v_{1}) [v_{12}^{2} - \frac{2 G (m_{1} - m_{2})}{r_{12}}] + 1 \leftrightarrow 2, \end{matrix}

(167)

we find agreement with the standard Einstein quadrupole formula ( 4 , 5 ):

\begin{matrix} \frac{d \tilde{E}}{d t} = - \frac{G}{5 c^{5}} \frac{d^{3} Q_{i j}}{d t^{3}} \frac{d^{3} Q_{i j}}{d t^{3}} + O (\frac{1}{c^{7}}), \end{matrix}

(168)

where the Newtonian trace-free quadrupole moment is

Q_{i j} = m_{1} (y_{1}^{i} y_{1}^{j} - \frac{1}{3} δ^{i j} y_{1}^{2}) + 1 \leftrightarrow 2

. As we can see, the 3PN equations of motion ( 163 ) are highly relativistic when describing the motion, but concerning the radiation they are in fact Newtonian, because they contain merely the “Newtonian” radiation reaction force at the 2.5PN order.

^$20$ Notice also the dependence upon $π^{2}$ . Technically, the $π^{2}$ terms arise from non-linear interactions involving some integrals such as

\frac{1}{π} \int \frac{d^{3} x}{r_{1}^{2} r_{2}^{2}} = \frac{π^{2}}{r_{12}} .

9.2 Lagrangian and Hamiltonian formulations

The conservative part of the equations of motion in harmonic coordinates ( 163 ) is derivable from a generalized Lagrangian, depending not only on the positions and velocities of the bodies, but also on their accelerations:

a_{1}^{i} = d v_{1}^{i} / d t

and

a_{2}^{i} = d v_{2}^{i} / d t

. As shown by Damour and Deruelle [55] , the accelerations in the harmonic-coordinates Lagrangian occur already from the 2PN order. This fact is in accordance with a general result of Martin and Sanz [100] that

N

-body equations of motion cannot be derived from an ordinary Lagrangian beyond the 1PN level, provided that the gauge conditions preserve the Lorentz invariance. Note that we can always arrange for the dependence of the Lagrangian upon the accelerations to be linear, at the price of adding some so-called “multi-zero” terms to the Lagrangian, which do not modify the equations of motion (see e.g. Ref. [63] ). At the 3PN level, we find that the Lagrangian also depends on accelerations. It is notable that these accelerations are sufficient – there is no need to include derivatives of accelerations. Note also that the Lagrangian is not unique because we can always add to it a total time derivative

d F / d t

, where

F

depends on the positions and velocities, without changing the dynamics. We find [66]

\begin{matrix} L^{h a r m} = . . . \end{matrix}

(169)

Witness the accelerations occuring at the 2PN and 3PN orders; see also the gauge-dependent logarithms of

r_{12} / r_{1}^{'}

and

r_{12} / r_{2}^{'}

, and the single term containing the regularization ambiguity

λ

. We refer to [66] for the explicit expressions of the ten conserved quantities corresponding to the integrals of energy (also given in Eq. ( 165 )), linear and angular momenta, and center-of-mass position. Notice that while it is strictly forbidden to replace the accelerations by the equations of motion in the Lagrangian, this can and should be done in the final expressions of the conserved integrals derived from that Lagrangian. Now we want to exhibit a transformation of the particles dynamical variables – or contact transformation, as it is called in the jargon – which transforms the 3PN harmonic-coordinates Lagrangian ( 169 ) into a new Lagrangian, valid in some ADM or ADM-like coordinate system, and such that the associated Hamiltonian coincides with the 3PN Hamiltonian that has been obtained by Damour, Jaranowski and Schäfer [60] . In ADM coordinates the Lagrangian will be “ordinary”, depending only on the positions and velocities of the bodies. Let this contact transformation be

Y_{1}^{i} (t) = y_{1}^{i} (t) + δ y_{1}^{i} (t)

and

1 \leftrightarrow 2

, where

Y_{1}^{i}

and

y_{1}^{i}

denote the trajectories in ADM and harmonic coordinates, respectively. For this transformation to be able to remove all the accelerations in the initial Lagrangian

L^{h a r m}

up to the 3PN order, we determine [66] it to be necessarily of the form

\begin{matrix} δ y_{1}^{i} = \frac{1}{m_{1}} [\frac{\partial L^{h a r m}}{\partial a_{1}^{i}} + \frac{\partial F}{\partial v_{1}^{i}} + \frac{1}{c^{6}} X_{1}^{i}] + O (\frac{1}{c^{8}}) \end{matrix}

(170)

(and idem

1 \leftrightarrow 2

), where

F

is a freely adjustable function of the positions and velocities, made of 2PN and 3PN terms, and where

X_{1}^{i}

represents a special correction term, that is purely of order 3PN. The point is that once the function

F

is specified there is a unique determination of the correction term

X_{1}^{i}

for the contact transformation to work (see Ref. [66] for the details). Thus, the freedom we have is entirely coded into the function

F

, and the work then consists in showing that there exists a unique choice of

F

for which our Lagrangian

L^{h a r m}

is physically equivalent, via the contact transformation ( 170 ), to the ADM Hamiltonian of Ref. [60] . An interesting point is that not only the transformation must remove all the accelerations in

L^{h a r m}

, but it should also cancel out all the logarithms

ln (r_{12} / r_{1}^{'})

and

ln (r_{12} / r_{2}^{'})

, because there are no logarithms in ADM coordinates. The result we find, which can be checked to be in full agreement with the expression of the gauge vector in Eq. ( 164 ), is that

F

involves the logarithmic terms

\begin{matrix} F = \frac{22}{3} \frac{G^{3} m_{1} m_{2}}{c^{6} r_{12}^{2}} [m_{1}^{2} (n_{12} v_{1}) ln (\frac{r_{12}}{r_{1}^{'}}) - m_{2}^{2} (n_{12} v_{2}) ln (\frac{r_{12}}{r_{2}^{'}})] + . . ., \end{matrix}

(171)

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

F

the ADM Lagrangian reads as

\begin{matrix} L^{A D M} = L^{h a r m} + \frac{δ L^{h a r m}}{δ y_{1}^{i}} δ y_{1}^{i} + \frac{δ L^{h a r m}}{δ y_{2}^{i}} δ y_{2}^{i} + \frac{d F}{d t} + O (\frac{1}{c^{8}}) . \end{matrix}

(172)

Inserting into this equation all our explicit expressions we find

\begin{matrix} L^{A D M} = . . . \end{matrix}

(173)

The notation is the same as in Eq. ( 169 ), except that we use upper-case letters to denote the ADM-coordinates positions and velocities; thus, for instance

N_{12} = (Y_{1} - Y_{2}) / R_{12}

and

(N_{12} V_{1}) = N_{12} . V_{1}

. The Hamiltonian is simply deduced from the latter Lagrangian by applying the usual Legendre transformation. Posing

P_{1}^{i} = \partial L^{A D M} / \partial V_{1}^{i}

and

1 \leftrightarrow 2

, we get [87, 88, 89, 60, 66] ^$21$

\begin{matrix} H^{A D M} = . . . \end{matrix}

(174)

Arguably, the results given by the ADM-Hamiltonian formalism (for the problem at hand) look simpler than their harmonic-coordinate counterparts. Indeed, the ADM Lagrangian is ordinary – no accelerations – and there are no logarithms nor associated gauge constants

r_{1}^{'}

and

r_{2}^{'}

. But of course, one is free to describe the binary motion in whatever coordinates one likes, and the two formalisms, harmonic ( 169 ) and ADM ( 173 , 174 ), describe rigorously the same physics. On the other hand, the higher complexity of the harmonic-coordinates Lagrangian ( 169 ) enables one to perform more tests of the computations, notably by inquiring about the future of the constants

r_{1}^{'}

and

r_{2}^{'}

, that we know must disappear from physical quantities such as the center-of-mass energy and the total gravitational-wave flux.

^$21$ Note that in the result published in Ref. [60] the following terms are missing:

\frac{G^{2}}{c^{6} r_{12}^{2}} (- \frac{55}{12} m_{1} - \frac{193}{48} m_{2}) \frac{(N_{12} P_{2})^{2} P_{1}^{2}}{m_{1} m_{2}} + 1 \leftrightarrow 2 .

This misprint has been corrected in an Erratum [60] .

9.3 Equations of motion for circular orbits

Most inspiralling compact binaries will have been circularized by the time they become visible by the detectors LIGO and VIRGO. In the case of orbits that are circular – apart from the gradual 2.5PN radiation-reaction inspiral – the quite complicated acceleration ( 163 ) simplifies drastically, since all the scalar products between

n_{12}

and the velocities are of small 2.5PN order: E.g.

(n_{12} v_{1}) = O (1 / c^{5})

, and the remainder can always be neglected here. Let us translate the origin of coordinates to the binary's center-of-mass by imposing that the binary's dipole

I_{i} = 0

(notation of Part aaa ). Up to the 2.5PN order, and in the case of circular orbits, this condition implies [7] ^$22$

\begin{matrix} m y_{1}^{i} & = & y_{12}^{i} [m_{2} + 3 γ^{2} ν δ m] - \frac{4}{5} \frac{G^{2} m^{2} ν δ m}{r_{12} c^{5}} v_{12}^{i} + O (\frac{1}{c^{6}}), \end{matrix}

(175)

\begin{matrix} m y_{2}^{i} & = & y_{12}^{i} [- m_{1} + 3 γ^{2} ν δ m] - \frac{4}{5} \frac{G^{2} m^{2} ν δ m}{r_{12} c^{5}} v_{12}^{i} + O (\frac{1}{c^{6}}) . \end{matrix}

(176)

Mass parameters are the total mass

m = m_{1} + m_{2}

(

m \equiv M

in the notation of Part aaa ), the mass difference

δ m = m_{1} - m_{2}

, the reduced mass

μ = m_{1} m_{2} / m

, and the very useful symmetric mass ratio

\begin{matrix} ν \equiv \frac{μ}{m} \equiv \frac{m_{1} m_{2}}{(m_{1} + m_{2})^{2}} . \end{matrix}

(177)

The usefulness of this ratio lies in its interesting range of variation:

0 < ν \leq 1 / 4

, with

ν = 1 / 4

in the case of equal masses, and

ν \to 0

in the “test-mass” limit for one of the bodies. To display conveniently the successive post-Newtonian corrections, we employ the post-Newtonian parameter

\begin{matrix} γ \equiv \frac{G m}{r_{12} c^{2}} = O (\frac{1}{c^{2}}) . \end{matrix}

(178)

Notice that there are no corrections of order 1PN in Eqs. ( 9.3 ) for circular orbits; the dominant term is of order 2PN, i.e. proportional to

γ^{2} = O (1 / c^{4})

The relative acceleration

a_{12}^{i} \equiv a_{1}^{i} - a_{2}^{i}

of two bodies moving on a circular orbit at the 3PN order is then given by

\begin{matrix} a_{12}^{i} = - ω^{2} y_{12}^{i} - \frac{32}{5} \frac{G^{3} m^{3} ν}{c^{5} r_{12}^{4}} v_{12}^{i} + O (\frac{1}{c^{7}}), \end{matrix}

(179)

where

y_{12}^{i} \equiv y_{1}^{i} - y_{2}^{i}

is the relative separation (in harmonic coordinates) and

ω

denotes the angular frequency of the circular motion. The second term in Eq. ( 179 ), opposite to the velocity

v_{12}^{i} \equiv v_{1}^{i} - v_{2}^{i}

, is the 2.5PN radiation reaction force, which comes from the reduction of the coefficient of

1 / c^{5}

in the expression ( 163 ). The main content of the 3PN equations ( 179 ) is the relation between the frequency

ω

and the orbital separation

r_{12}

, that we find to be given by the generalized version of Kepler's third law [21, 22] :

\begin{matrix} ω^{2} & = & \frac{G m}{r_{12}^{3}} {1 + (- 3 + ν) γ + (6 + \frac{41}{4} ν + ν^{2}) γ^{2} \end{matrix}

\begin{matrix} + (- 10 + [- \frac{67759}{840} + \frac{41}{64} π^{2} + 22 ln (\frac{r_{12}}{r_{0}^{'}}) + \frac{44}{3} λ] ν + \frac{19}{2} ν^{2} + ν^{3}) γ^{3}} \end{matrix}

\begin{matrix} + O (\frac{1}{c^{8}}) . \end{matrix}

(180)

The length scale

r_{0}^{'}

is given in terms of the two gauge-constants

r_{1}^{'}

and

r_{2}^{'}

\begin{matrix} ln r_{0}^{'} = \frac{m_{1}}{m} ln r_{1}^{'} + \frac{m_{2}}{m} ln r_{2}^{'} . \end{matrix}

(181)

As for the energy, it is immediately obtained from the circular-orbit reduction of the general result ( 165 ).

We have

\begin{matrix} E & = & - \frac{μ c^{2} γ}{2} {1 + (- \frac{7}{4} + \frac{1}{4} ν) γ + (- \frac{7}{8} + \frac{49}{8} ν + \frac{1}{8} ν^{2}) γ^{2} \end{matrix}

\begin{matrix} + (- \frac{235}{64} + [\frac{106301}{6720} - \frac{123}{64} π^{2} + \frac{22}{3} ln (\frac{r_{12}}{r_{0}^{'}}) - \frac{22}{3} λ] ν \end{matrix}

\begin{matrix} + \frac{27}{32} ν^{2} + \frac{5}{64} ν^{3}) γ^{3}} + O (\frac{1}{c^{8}}) . \end{matrix}

(182)

This expression is that of a physical observable

E

; however, it depends on the choice of a coordinate system, as it involves the post-Newtonian parameter

γ

defined from the harmonic-coordinate separation

r_{12}

. But the numerical value of

E

should not depend on the choice of a coordinate system, so

E

must admit a frame-invariant expression, the same in all coordinate systems. To find it we re-express

E

with the help of a frequency-related parameter

x

instead of the post-Newtonian parameter

γ

. Posing

\begin{matrix} x \equiv {(\frac{G m ω}{c^{3}})}^{2 / 3} = O (\frac{1}{c^{2}}), \end{matrix}

(183)

we readily obtain from Eq. ( 180 ) the expression of

γ

in terms of

x

at 3PN order,

\begin{matrix} γ & = & x {1 + (1 - \frac{ν}{3}) x + (1 - \frac{65}{12} ν) x^{2} \end{matrix}

\begin{matrix} + (1 + [- \frac{10151}{2520} - \frac{41}{192} π^{2} - \frac{22}{3} ln (\frac{r_{12}}{r_{0}^{'}}) - \frac{44}{9} λ] ν + \frac{229}{36} ν^{2} + \frac{1}{81} ν^{3}) x^{3} \end{matrix}

\begin{matrix} + O (\frac{1}{c^{8}})}, \end{matrix}

(184)

that we substitute back into Eq. ( 182 ), making all appropriate post-Newtonian re-expansions. As a result, we gladly discover that the logarithms together with their associated gauge constant

r_{0}^{'}

have cancelled out.

Therefore, our result is

\begin{matrix} E & = & - \frac{μ c^{2} x}{2} {1 + (- \frac{3}{4} - \frac{1}{12} ν) x + (- \frac{27}{8} + \frac{19}{8} ν - \frac{1}{24} ν^{2}) x^{2} \end{matrix}

\begin{matrix} + (- \frac{675}{64} + [\frac{209323}{4032} - \frac{205}{96} π^{2} - \frac{110}{9} λ] ν - \frac{155}{96} ν^{2} - \frac{35}{5184} ν^{3}) x^{3}} \end{matrix}

\begin{matrix} + O (\frac{1}{c^{8}}) . \end{matrix}

(185)

The constant

λ

is the one introduced in Eq. ( 160 ). For circular orbits one can check that there are no terms of order

x^{7 / 2}

in Eq. ( 185 ), so our result for

E

is actually valid up to the 3.5PN order. In the test-mass limit

ν \to 0

, we recover the energy of a particle with mass

μ

in a Schwarzschild background of mass

m

, i.e.

E_{t e s t} = μ c^{2} [(1 - 2 x) (1 - 3 x)^{- 1 / 2} - 1]

, when developed to 3.5PN order.

^$22$ Actually, in the present computation we do not need the radiation-reaction 2.5PN terms in these relations; we give them only for completeness.

10 Gravitational Waves from Compact Binaries

We pointed out that the 3PN equations of motion, Eqs. ( 179 , 180 ), are merely Newtonian as regards the radiative aspects of the problem, because with that precision the radiation reaction force is at the lowest 2.5PN order. A solution would be to extend the precision of the equations of motion so as to include the full relative 3PN or 3.5PN precision into the radiation reaction force, but, needless to say, the equations of motion up to the 5.5PN or 6PN order are quite impossible to derive with the present technology. The much better alternative solution is to apply the wave-generation formalism described in Part aaa , and to determine by its means the work done by the radiation reaction force directly as a total energy flux at future null infinity. In this approach, we replace the knowledge of the higher-order radiation reaction force by the computation of the total flux

ℒ

, and we apply the energy balance equation as in the test of the

\dot{P}

of the binary pulsar (see Eqs. ( 4 , 5 )):

\begin{matrix} \frac{d E}{d t} = - ℒ . \end{matrix}

(186)

Therefore, the result ( 185 ) that we found for the 3.5PN binary's center-of-mass energy

E

constitutes only “half ” of the solution of the problem. The second “half ” consists of finding the rate of decrease

d E / d t

, which by the balance equation is nothing but finding the total gravitational-wave flux

ℒ

at the 3.5PN order.

Because the orbit of inspiralling binaries is circular, the balance equation for the energy is sufficient (no need of a balance equation for the angular momentum). This all sounds perfect, but it is important to realize that we shall use the equation ( 186 ) at the very high 3.5PN order, at which order there are no proofs (following from first principles in general relativity) that the equation is correct, despite its physically obvious character.

Nevertheless, Eq. ( 186 ) has been checked to be valid, both in the cases of point-particle binaries [85, 86] and extended weakly self-gravitating fluids [5, 9] , at the 1PN order and even at 1.5PN (the 1.5PN approximation is especially important for this check because it contains the first wave tails).

Obtaining

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 aaa ).

10.1 The binary's multipole moments

The general expressions of the source multipole moments given by Theorem 6 (Eqs. ( 88 )) are first to be worked out explicitly for general fluid systems at the 3PN order. For this computation one uses the formula ( 98 ), and we insert the 3PN metric coefficients (in harmonic coordinates) expressed in Eq. ( 133 , 134 , 135 ) by means of the retarded-type elementary potentials ( 140 , 143 , 146 ). 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^{α β}

, the standard expression made out of Dirac delta-functions. The infinite self-field of point-particles is removed by means of the Hadamard regularization (see Section 8 ). This computation has been performed by Blanchet and Schäfer [30] at the 1PN order, and by Blanchet, Damour and Iyer [18] at the 2PN order; we report below the most accurate 3PN results obtained by Blanchet, Iyer and Joguet [26] .

The difficult part of the analysis is to find the closed-form expressions, fully explicit in terms of the particle's positions and velocities, of many non-linear integrals. We refer to [26] for full details; nevertheless, let us give a few examples of the type of technical formulas that are employed in this calculation. Typically we have to compute some integrals like

\begin{matrix} \overset{(n, p)}{Y_{L}} (y_{1}, y_{2}) = - \frac{1}{2 π} ℱ P \int d^{3} x {\hat{x}}_{L} r_{1}^{n} r_{2}^{p}, \end{matrix}

(187)

where

r_{1} = | x - y_{1} |

and

r_{2} = | x - y_{2} |

. When

n > - 3

and

p > - 3

, this integral is perfectly well-defined (recall that the finite part

ℱ P

deals with the bound at infinity). When

n \leq - 3

p \leq - 3

, our basic ansatz is that we apply the definition of the Hadamard partie finie provided by Eq. ( 151 ). Two examples of closed-form formulas that we get, which do not necessitate the Hadamard partie finie, are (quadrupole case

l = 2

)

\begin{matrix} \overset{(- 1, - 1)}{Y_{i j}} & = & \frac{r_{12}}{3} [y_{1}^{〈 i j 〉} + y_{1}^{〈 i} y_{2}^{j 〉} + y_{2}^{〈 i j 〉}], \end{matrix}

(188)

\begin{matrix} \overset{(- 2, - 1)}{Y_{i j}} & = & y_{1}^{〈 i j 〉} [\frac{16}{15} ln (\frac{r_{12}}{r_{0}}) - \frac{188}{225}] + y_{1}^{〈 i} y_{2}^{j 〉} [\frac{8}{15} ln (\frac{r_{12}}{r_{0}}) - \frac{4}{225}] \end{matrix}

\begin{matrix} + y_{2}^{〈 i j 〉} [\frac{2}{5} ln (\frac{r_{12}}{r_{0}}) - \frac{2}{25}] . \end{matrix}

(189)

We denote for example

y_{1}^{〈 i j 〉} = y_{1}^{〈 i} y_{1}^{j 〉}

; the constant

r_{0}

is the one pertaining to the finite-part process (see Eq. ( 39 )). One example where the integral diverges at the location of the particle 1 is

\begin{matrix} \overset{(- 3, 0)}{Y_{i j}} = [2 ln (\frac{s_{1}}{r_{0}}) + \frac{16}{15}] y_{1}^{〈 i j 〉}, \end{matrix}

(190)

where

s_{1}

is the Hadamard-regularization constant introduced in Eq. ( 151 )^$23$ .

The crucial input of the computation of the flux at the 3PN order is the mass quadrupole moment

I_{i j}

, since this moment necessitates the full 3PN precision. The result of Ref. [26] for this moment (in the case of circular orbits) is

\begin{matrix} I_{i j} = μ (A x_{〈 i j 〉} + B \frac{r_{12}^{3}}{G m} v_{〈 i j 〉} + \frac{48}{7} \frac{G^{2} m^{2} ν}{c^{5} r_{12}} x_{〈 i} v_{j 〉}) + O (\frac{1}{c^{7}}), \end{matrix}

(191)

where we pose

x_{i} = x^{i} \equiv y_{12}^{i}

and

v_{i} = v^{i} \equiv v_{12}^{i}

. The third term is the 2.5PN radiation-reaction term, which does not contribute to the energy flux for circular orbits. The two important coefficients are

A

and

B

, whose expressions through 3PN order are

\begin{matrix} A & = & 1 + γ (- \frac{1}{42} - \frac{13}{14} ν) + γ^{2} (- \frac{461}{1512} - \frac{18395}{1512} ν - \frac{241}{1512} ν^{2}) \end{matrix}

\begin{matrix} + γ^{3} (\frac{395899}{13200} - \frac{428}{105} ln (\frac{r_{12}}{r_{0}}) + [\frac{139675}{33264} - \frac{44}{3} ξ - \frac{88}{3} κ - \frac{44}{3} ln (\frac{r_{12}}{r_{0}^{'}})] ν \end{matrix}

\begin{matrix} + \frac{162539}{16632} ν^{2} + \frac{2351}{33264} ν^{3}), \end{matrix}

(192)

\begin{matrix} B & = & γ (\frac{11}{21} - \frac{11}{7} ν) + γ^{2} (\frac{1607}{378} - \frac{1681}{378} ν + \frac{229}{378} ν^{2}) \end{matrix}

\begin{matrix} + γ^{3} (- \frac{357761}{19800} + \frac{428}{105} ln (\frac{r_{12}}{r_{0}}) + [- \frac{75091}{5544} + \frac{44}{3} ζ] ν + \frac{35759}{924} ν^{2} + \frac{457}{5544} ν^{3}) . \end{matrix}

\begin{matrix}  \end{matrix}

(193)

These expressions are valid in harmonic coordinates via the post-Newtonian parameter

γ

given by Eq. ( 178 ).

As we see, there are two types of logarithms in the moment: One type involves the length scale

r_{0}^{'}

related by Eq. ( 181 ) to the two gauge constants

r_{1}^{'}

and

r_{2}^{'}

present in the 3PN equations of motion; the other type contains the different length scale

r_{0}

coming from the general formalism of Part aaa – indeed, recall that there is a

ℱ P

operator in front of the source multipole moments in Theorem 6 . As we know, that

r_{0}^{'}

is pure gauge; it will disappear from our physical results at the end. On the other hand, we have remarked that the multipole expansion outside a general post-Newtonian source is actually free of

r_{0}

, since the

r_{0}

's present in the two terms of Eq. ( 74 ) cancel out. We shall indeed find that the constants

r_{0}

present in Eqs. ( 193 ) are compensated by similar constants coming from the non-linear wave “tails of tails”. More seriously, in addition to the harmless constants

r_{0}

and

r_{0}^{'}

, there are three unknown dimensionless parameters in Eqs. ( 193 ), called

ξ

κ

and

ζ

. These parameters reflect some incompleteness of the Hadamard self-field regularization (see the discussion in Section 8.2 ).

Besides the 3PN mass quadrupole ( 191 , 193 ), we need also the mass octupole moment

I_{i j k}

and current quadrupole moment

J_{i j}

, both of them at the 2PN order; these are given by [26]

\begin{matrix} I_{i j k} & = & μ \frac{δ m}{m} {\hat{x}}_{i j k} [- 1 + γ ν + γ^{2} (\frac{139}{330} + \frac{11923}{660} ν + \frac{29}{110} ν^{2})] \end{matrix}

\begin{matrix} + μ \frac{δ m}{m} x_{〈 i} v_{j k 〉} \frac{r_{12}^{2}}{c^{2}} [- 1 + 2 ν + γ (- \frac{1066}{165} + \frac{1433}{330} ν - \frac{21}{55} ν^{2})] \end{matrix}

\begin{matrix} + O (\frac{1}{c^{5}}), \end{matrix}

(194)

\begin{matrix} J_{i j} & = & μ \frac{δ m}{m} ɛ_{a b 〈 i} x_{j 〉 a} v_{b} [- 1 + γ (- \frac{67}{28} + \frac{2}{7} ν) + γ^{2} (- \frac{13}{9} + \frac{4651}{252} ν + \frac{1}{168} ν^{2})] \end{matrix}

\begin{matrix} + O (\frac{1}{c^{5}}) . \end{matrix}

(195)

Also needed are the 1PN mass

2^{4}

-pole, 1PN current

2^{3}

-pole (octupole), Newtonian mass

2^{5}

-pole and Newtonian current

2^{4}

-pole:

\begin{matrix} I_{i j k l} & = & μ {\hat{x}}_{i j k l} [1 - 3 ν + γ (\frac{3}{110} - \frac{25}{22} ν + \frac{69}{22} ν^{2})] \end{matrix}

\begin{matrix} + \frac{78}{55} μ x_{〈 i j} v_{k l 〉} \frac{r_{12}^{2}}{c^{2}} (1 - 5 ν + 5 ν^{2}) + O (\frac{1}{c^{3}}), \end{matrix}

(196)

\begin{matrix} J_{i j k} & = & μ ɛ_{a b 〈 i} x_{j k 〉 a} v_{b} [1 - 3 ν + γ (\frac{181}{90} - \frac{109}{18} ν + \frac{13}{18} ν^{2})] \end{matrix}

\begin{matrix} + \frac{7}{45} μ (1 - 5 ν + 5 ν^{2}) ɛ_{a b 〈 i} v_{j k 〉 b} x_{a} \frac{r_{12}^{2}}{c^{2}} + O (\frac{1}{c^{3}}), \end{matrix}

(197)

\begin{matrix} I_{i j k l m} & = & μ \frac{δ m}{m} (- 1 + 2 ν) {\hat{x}}_{i j k l m} + O (\frac{1}{c}), \end{matrix}

(198)

\begin{matrix} J_{i j k l} & = & μ \frac{δ m}{m} (- 1 + 2 ν) ɛ_{a b 〈 i} x_{j k l 〉 a} v_{b} + O (\frac{1}{c}) . \end{matrix}

(199)

These results permit one to control what can be called the “instantaneous” part, say

ℒ_{i n s t}

, of the total energy flux, by which we mean that part of the flux that is generated solely by the source multipole moments, i.e. not counting the “non-instantaneous” tail integrals. The instantaneous flux is defined by the replacement into the general expression of

ℒ

given by Eq. ( 67 ) of all the radiative moments

U_{L}

and

V_{L}

by the corresponding (

l

th time derivatives of the) source moments

I_{L}

and

J_{L}

. Actually, we prefer to define

ℒ_{i n s t}

by means of the intermediate moments

M_{L}

and

S_{L}

. Up to the 3.5PN order we have

\begin{matrix} ℒ_{i n s t} & = & \frac{G}{c^{5}} {\frac{1}{5} M_{i j}^{(3)} M_{i j}^{(3)} + \frac{1}{c^{2}} [\frac{1}{189} M_{i j k}^{(4)} M_{i j k}^{(4)} + \frac{16}{45} S_{i j}^{(3)} S_{i j}^{(3)}] \end{matrix}

\begin{matrix} + \frac{1}{c^{4}} [\frac{1}{9072} M_{i j k m}^{(5)} M_{i j k m}^{(5)} + \frac{1}{84} S_{i j k}^{(4)} S_{i j k}^{(4)}] \end{matrix}

\begin{matrix} + \frac{1}{c^{6}} [\frac{1}{594000} M_{i j k m n}^{(6)} M_{i j k m n}^{(6)} + \frac{4}{14175} S_{i j k m}^{(5)} S_{i j k m}^{(5)}] \end{matrix}

\begin{matrix} + O (\frac{1}{c^{8}})} . \end{matrix}

(200)

The time derivatives of the source moments ( 191 , 193 , 195 , 199 ) are computed by means of the circular-orbit equations of motion given by Eq. ( 179 , 180 ); then we substitute them into Eq. ( 200 ) (for circular orbits there is no difference at this order between

I_{L}

J_{L}

and

M_{L}

S_{L}

). The net result is

\begin{matrix} ℒ_{i n s t} & = & \frac{32 c^{5}}{5 G} ν^{2} γ^{5} {1 + (- \frac{2927}{336} - \frac{5}{4} ν) γ + (\frac{293383}{9072} + \frac{380}{9} ν) γ^{2} \end{matrix}

\begin{matrix} + (\frac{53712289}{1108800} - \frac{1712}{105} ln (\frac{r_{12}}{r_{0}}) \end{matrix}

\begin{matrix} + [- \frac{332051}{720} + \frac{123}{64} π^{2} + \frac{110}{3} ln (\frac{r_{12}}{r_{0}^{'}}) + 44 λ - \frac{88}{3} θ] ν \end{matrix}

\begin{matrix} - \frac{383}{9} ν^{2}) γ^{3} \end{matrix}

\begin{matrix} + O (\frac{1}{c^{8}})} . \end{matrix}

(201)

The Newtonian approximation,

ℒ_{N} = (32 c^{5}) / (5 G) \cdot ν^{2} γ^{5}

, is the prediction of the Einstein quadrupole formula ( 4 ), as computed by Landau and Lifchitz [97] . The self-field regularization ambiguities arising at the 3PN order are the equation-of-motion-related constant

λ

and the multipole-moment-related constant

θ = ξ + 2 κ + ζ

(see Section 8.2 ).

^$23$ When computing the gravitational-wave flux in Ref. [26] 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. [22] . Indeed these regularization constants need not neccessarily need to be the same when employed in different contexts.

10.2 Contribution of wave tails

To the “instantaneous” part of the flux, we must add the contribution of non-linear multipole interactions contained in the relationship between the source and radiative moments. The needed material has already been provided in Eqs. ( 101 , 6 ). Up to the 3.5PN level we have the dominant quadratic-order tails, the cubic-order tails or tails of tails, and the non-linear memory integral. We shall see that the tails play a crucial role in the predicted signal of compact binaries. By contrast, the non-linear memory effect, given by the integral inside the 2.5PN term in Eq. ( 101 ), does not contribute to the gravitational-wave energy flux before the 4PN order in the case of circular-orbit binaries (essentially because the memory integral is actually “instantaneous” in the flux), and therefore has rather poor observational consequences for future detections of inspiralling compact binaries. We split the energy flux into the different terms

\begin{matrix} ℒ = ℒ_{i n s t} + ℒ_{t a i l} + ℒ_{(t a i l)^{2}} + ℒ_{t a i l (t a i l)}, \end{matrix}

(202)

where

ℒ_{i n s t}

has just been found in Eq. ( 201 );

ℒ_{t a i l}

is made of the quadratic (multipolar) tail integrals in Eq. ( 6 );

ℒ_{(t a i l)^{2}}

is the square of the quadrupole tail in Eq. ( 101 ); and

ℒ_{t a i l (t a i l)}

is the quadrupole tail of tail in Eq. ( 101 ). We find that

ℒ_{t a i l}

contributes at the half-integer 1.5PN, 2.5PN and 3.5PN orders, while both

ℒ_{(t a i l)^{2}}

and

ℒ_{t a i l (t a i l)}

appear only at the 3PN order. It is quite remarkable that so small an effect as a “tail of tail” should be relevant to the present computation, which is aimed at preparing the ground for forthcoming experiments.

The results follow from the reduction to the case of circular compact binaries of the general formulas ( 101 , 6 ). Without going into accessory details (see Ref. [10] ), let us give the two basic technical formulas needed when carrying out this reduction:

\begin{matrix} \int_{0}^{+ \infty} d τ ln τ e^{- σ τ} & = & - \frac{1}{σ} (C + ln σ), \end{matrix}

(203)

\begin{matrix} \int_{0}^{+ \infty} d τ {ln}^{2} τ e^{- σ τ} & = & \frac{1}{σ} [\frac{π^{2}}{6} + (C + ln σ)^{2}], \end{matrix}

(204)

where

σ \in C

and

C = 0.577 . . .

denotes the Euler constant [78] . The tail integrals are evaluated thanks to these formulas for a fixed (non-decaying) circular orbit. Indeed it can be shown [31] that the “remote-past” contribution to the tail integrals is negligible; the errors due to the fact that the orbit actually spirals in by gravitational radiation do not affect the signal before the 4PN order. We then find, for the quadratic tail term stricto sensu, the 1.5PN, 2.5PN and 3.5PN amounts

\begin{matrix} ℒ_{t a i l} & = & \frac{32 c^{5}}{5 G} γ^{5} ν^{2} {4 π γ^{3 / 2} + (- \frac{25663}{672} - \frac{109}{8} ν) π γ^{5 / 2} \end{matrix}

\begin{matrix} + (\frac{90205}{576} + \frac{3772673}{12096} ν + \frac{32147}{3024} ν^{2}) π γ^{7 / 2} + O (\frac{1}{c^{8}})} . \end{matrix}

(205)

For the sum of squared tails and cubic tails of tails at 3PN, we get

\begin{matrix} ℒ_{(t a i l)^{2} + t a i l (t a i l)} & = & \frac{32 c^{5}}{5 G} γ^{5} ν^{2} {(- \frac{116761}{3675} + \frac{16}{3} π^{2} - \frac{1712}{105} C \end{matrix}

\begin{matrix} + \frac{1712}{105} ln (\frac{r_{12}}{r_{0}}) - \frac{856}{105} ln (16 γ)) γ^{3} \end{matrix}

\begin{matrix} + O (\frac{1}{c^{8}})} . \end{matrix}

(206)

By comparing Eqs. ( 201 ) and ( 206 ) we observe that the constants

r_{0}

cleanly cancel out. Adding together all these contributions we obtain

\begin{matrix} ℒ & = & \frac{32 c^{5}}{5 G} γ^{5} ν^{2} {1 + (- \frac{2927}{336} - \frac{5}{4} ν) γ + 4 π γ^{3 / 2} \end{matrix}

\begin{matrix} + (\frac{293383}{9072} + \frac{380}{9} ν) γ^{2} + (- \frac{25663}{672} - \frac{109}{8} ν) π γ^{5 / 2} \end{matrix}

\begin{matrix} + (\frac{129386791}{7761600} + \frac{16 π^{2}}{3} - \frac{1712}{105} C - \frac{856}{105} ln (16 γ) \end{matrix}

\begin{matrix} + [- \frac{332051}{720} + \frac{110}{3} ln (\frac{r_{12}}{r_{0}^{'}}) + \frac{123 π^{2}}{64} + 44 λ - \frac{88}{3} θ] ν - \frac{383}{9} ν^{2}) γ^{3} \end{matrix}

\begin{matrix} + (\frac{90205}{576} + \frac{3772673}{12096} ν + \frac{32147}{3024} ν^{2}) π γ^{7 / 2} + O (\frac{1}{c^{8}})} . \end{matrix}

(207)

The gauge constant

r_{0}^{'}

has not yet disappeared because the post-Newtonian expansion is still parametrized by

γ

instead of the frequency-related parameter

x

defined by Eq. ( 183 ) – just as for

E

when it was given by Eq. ( 182 ). After substituting the expression

γ (x)

given by Eq. ( 184 ), we find that

r_{0}^{'}

does cancel as well. Because the relation

γ (x)

is issued from the equations of motion, the latter cancellation represents an interesting test of the consistency of the two computations, in harmonic coordinates, of the 3PN multipole moments and the 3PN equations of motion. At long last we obtain our end result:

\begin{matrix} ℒ & = & \frac{32 c^{5}}{5 G} ν^{2} x^{5} {1 + (- \frac{1247}{336} - \frac{35}{12} ν) x + 4 π x^{3 / 2} \end{matrix}

\begin{matrix} + (- \frac{44711}{9072} + \frac{9271}{504} ν + \frac{65}{18} ν^{2}) x^{2} + (- \frac{8191}{672} - \frac{535}{24} ν) π x^{5 / 2} \end{matrix}

\begin{matrix} + (\frac{6643739519}{69854400} + \frac{16}{3} π^{2} - \frac{1712}{105} C - \frac{856}{105} ln (16 x) \end{matrix}

\begin{matrix} + [- \frac{11497453}{272160} + \frac{41}{48} π^{2} + \frac{176}{9} λ - \frac{88}{3} θ] ν - \frac{94403}{3024} ν^{2} - \frac{775}{324} ν^{3}) x^{3} \end{matrix}

\begin{matrix} + (- \frac{16285}{504} + \frac{176419}{1512} ν + \frac{19897}{378} ν^{2}) π x^{7 / 2} + O (\frac{1}{c^{8}})} . \end{matrix}

(208)

In the test-mass limit

ν \to 0

for one of the bodies, we recover exactly the result following from linear black-hole perturbations obtained by Tagoshi and Sasaki [137] . In particular, the rational fraction

6643739519 / 69854400

comes out exactly the same as in black-hole perturbations.

10.3 Orbital phase evolution

We shall now deduce the laws of variation with time of the orbital frequency and phase of an inspiralling compact binary from the energy balance equation ( 186 ). The center-of-mass energy

E

is given by Eq. ( 185 ) and the total flux

ℒ

by Eq. ( 208 ). For convenience we adopt the dimensionless time variable^$24$

\begin{matrix} Θ \equiv \frac{ν c^{3}}{5 G m} (t_{c} - t), \end{matrix}

(209)

where

t_{c}

denotes the instant of coalescence, at which the frequency tends to infinity (evidently, the post-Newtonian method breaks down well before this point). We transform the balance equation into an ordinary differential equation for the parameter

x

, which is immediately integrated with the result

\begin{matrix} x & = & \frac{1}{4} Θ^{- 1 / 4} {1 + (\frac{743}{4032} + \frac{11}{48} ν) Θ^{- 1 / 4} - \frac{1}{5} π Θ^{- 3 / 8} \end{matrix}

\begin{matrix} + (\frac{19583}{254016} + \frac{24401}{193536} ν + \frac{31}{288} ν^{2}) Θ^{- 1 / 2} \end{matrix}

\begin{matrix} + (- \frac{11891}{53760} + \frac{29}{1920} ν) π Θ^{- 5 / 8} \end{matrix}

\begin{matrix} + (- \frac{10052469856691}{6008596070400} + \frac{1}{6} π^{2} + \frac{107}{420} C - \frac{107}{3360} ln (\frac{Θ}{256}) \end{matrix}

\begin{matrix} + [\frac{15335597827}{3901685760} - \frac{451}{3072} π^{2} - \frac{77}{72} λ + \frac{11}{24} θ] ν \end{matrix}

\begin{matrix} - \frac{15211}{442368} ν^{2} + \frac{25565}{331776} ν^{3}) Θ^{- 3 / 4} \end{matrix}

\begin{matrix} + (- \frac{113868647}{433520640} - \frac{141389}{483840} ν + \frac{275201}{3870720} ν^{2}) π Θ^{- 7 / 8} \end{matrix}

\begin{matrix} + O (\frac{1}{c^{8}})} . \end{matrix}

(210)

The orbital phase is defined as the angle

φ

, oriented in the sense of the motion, between the separation of the two bodies and the direction of the ascending node

N

within the plane of the sky, namely the point on the orbit at which the bodies cross the plane of the sky moving toward the detector. We have

d φ / d t = ω

, which translates, with our notation, into

d φ / d Θ = - 5 / ν \cdot x^{3 / 2}

, from which we determine

\begin{matrix} φ & = & - \frac{1}{ν} Θ^{5 / 8} {1 + (\frac{3715}{8064} + \frac{55}{96} ν) Θ^{- 1 / 4} - \frac{3}{4} π Θ^{- 3 / 8} \end{matrix}

\begin{matrix} + (\frac{9275495}{14450688} + \frac{284875}{258048} ν + \frac{1855}{2048} ν^{2}) Θ^{- 1 / 2} \end{matrix}

\begin{matrix} + (- \frac{38645}{172032} - \frac{15}{2048} ν) π Θ^{- 5 / 8} ln (\frac{Θ}{Θ_{0}}) \end{matrix}

\begin{matrix} + (\frac{831032450749357}{57682522275840} - \frac{53}{40} π^{2} - \frac{107}{56} C + \frac{107}{448} ln (\frac{Θ}{256}) \end{matrix}

\begin{matrix} + [- \frac{123292747421}{4161798144} + \frac{2255}{2048} π^{2} + \frac{385}{48} λ - \frac{55}{16} θ] ν \end{matrix}

\begin{matrix} + \frac{154565}{1835008} ν^{2} - \frac{1179625}{1769472} ν^{3}) Θ^{- 3 / 4} \end{matrix}

\begin{matrix} + (\frac{188516689}{173408256} + \frac{140495}{114688} ν - \frac{122659}{516096} ν^{2}) π Θ^{- 7 / 8} \end{matrix}

\begin{matrix} + O (\frac{1}{c^{8}})}, \end{matrix}

(211)

where

Θ_{0}

is a constant of integration that can be fixed by the initial conditions when the wave frequency enters the detector's bandwidth. Finally we want also to dispose of the important expression of the phase in terms of the frequency

x

. For this we get

\begin{matrix} φ & = & - \frac{x^{- 5 / 2}}{32 ν} {1 + (\frac{3715}{1008} + \frac{55}{12} ν) x - 10 π x^{3 / 2} \end{matrix}

\begin{matrix} + (\frac{15293365}{1016064} + \frac{27145}{1008} ν + \frac{3085}{144} ν^{2}) x^{2} \end{matrix}

\begin{matrix} + (\frac{38645}{1344} + \frac{15}{16} ν) π x^{5 / 2} ln (\frac{x}{x_{0}}) \end{matrix}

\begin{matrix} + (\frac{12348611926451}{18776862720} - \frac{160}{3} π^{2} - \frac{1712}{21} C - \frac{856}{21} ln (16 x) \end{matrix}

\begin{matrix} + [- \frac{15335597827}{12192768} + \frac{2255}{48} π^{2} + \frac{3080}{9} λ - \frac{440}{3} θ] ν \end{matrix}

\begin{matrix} + \frac{76055}{6912} ν^{2} - \frac{127825}{5184} ν^{3}) x^{3} \end{matrix}

\begin{matrix} + (\frac{77096675}{2032128} + \frac{1014115}{24192} ν - \frac{36865}{3024} ν^{2}) π x^{7 / 2} \end{matrix}

\begin{matrix} + O (\frac{1}{c^{8}})}, \end{matrix}

(212)

where

x_{0}

is another constant of integration.

With the formula ( 212 ) the orbital phase is complete up to the 3.5PN order, except for a single linear combination of the unknown regularization constants

λ

and

θ

. More work should be done to determine these constants. The effects due to the spins of the particles, i.e. spin-orbit coupling from the 1.5PN order for compact bodies and spin-spin coupling from the 2PN order, can be added if necessary (they are known up to the 2.5PN order [91, 90, 108, 136] ). On the other hand, the contribution of the quadrupole moments of the compact objects, which are induced by tidal effects, is expected to come only at the 5PN order (see Eq. ( 8 )).

^$24$ Notice the strange post-Newtonian order of this time variable: $Θ = O (c^{+ 8})$ .

10.4 The two polarization wave-forms

The theoretical templates of the compact binary inspiral follow from insertion of the previous solutions for the 3.5PN-accurate orbital frequency and phase into the binary's two polarization wave-forms

h_{+}

and

h_{\times}

We shall include in

h_{+}

and

h_{\times}

all the harmonics, besides the dominant one at twice the orbital frequency, up to the 2PN order, as they have been calculated by Blanchet, Iyer, Will and Wiseman [27] .

The polarization wave-forms are defined with respect to two polarization vectors

p = (p_{i})

and

q = (q_{i})

\begin{matrix} h_{+} & = & \frac{1}{2} (p_{i} p_{j} - q_{i} q_{j}) h_{i j}^{T T}, \end{matrix}

(213)

\begin{matrix} h_{\times} & = & \frac{1}{2} (p_{i} q_{j} + p_{j} q_{i}) h_{i j}^{T T}, \end{matrix}

(214)

where

p

and

q

are chosen to lie along the major and minor axis, respectively, of the projection onto the plane of the sky of the circular orbit, with

p

oriented toward the ascending node

N

. To the 2PN order we have

\begin{matrix} h_{+, \times} & = & \frac{2 G μ x}{c^{2} R} {H_{+, \times}^{(0)} + x^{1 / 2} H_{+, \times}^{(1 / 2)} + x H_{+, \times}^{(1)} + x^{3 / 2} H_{+, \times}^{(3 / 2)} + x^{2} H_{+, \times}^{(2)} \end{matrix}

\begin{matrix} + O (\frac{1}{c^{5}})} . \end{matrix}

(215)

The post-Newtonian terms are ordered by means of the frequency-related variable

x

. They depend on the binary's 3.5PN-accurate phase

φ

through the auxiliary phase variable

\begin{matrix} ψ = φ - \frac{2 G m ω}{c^{3}} ln (\frac{ω}{ω_{0}}), \end{matrix}

(216)

where

ω_{0}

is a constant frequency that can conveniently be chosen to be the entry frequency of a laser-interferometric detector (say

ω_{0} / π = 10

Hz). We have, for the plus polarization,

\begin{matrix} H_{+}^{(0)} & = & - (1 + c_{i}^{2}) cos 2 ψ, \end{matrix}

(217)

\begin{matrix} H_{+}^{(1 / 2)} & = & - \frac{s_{i}}{8} \frac{δ m}{m} [(5 + c_{i}^{2}) cos ψ - 9 (1 + c_{i}^{2}) cos 3 ψ], \end{matrix}

(218)

\begin{matrix} H_{+}^{(1)} & = & \frac{1}{6} [19 + 9 c_{i}^{2} - 2 c_{i}^{4} - ν (19 - 11 c_{i}^{2} - 6 c_{i}^{4})] cos 2 ψ \end{matrix}

\begin{matrix} - \frac{4}{3} s_{i}^{2} (1 + c_{i}^{2}) (1 - 3 ν) cos 4 ψ, \end{matrix}

(219)

\begin{matrix} H_{+}^{(3 / 2)} & = & \frac{s_{i}}{192} \frac{δ m}{m} {[57 + 60 c_{i}^{2} - c_{i}^{4} - 2 ν (49 - 12 c_{i}^{2} - c_{i}^{4})] cos ψ \end{matrix}

\begin{matrix} - \frac{27}{2} [73 + 40 c_{i}^{2} - 9 c_{i}^{4} - 2 ν (25 - 8 c_{i}^{2} - 9 c_{i}^{4})] cos 3 ψ \end{matrix}

\begin{matrix} + \frac{625}{2} (1 - 2 ν) s_{i}^{2} (1 + c_{i}^{2}) cos 5 ψ} - 2 π (1 + c_{i}^{2}) cos 2 ψ, \end{matrix}

(220)

\begin{matrix} H_{+}^{(2)} & = & \frac{1}{120} [22 + 396 c_{i}^{2} + 145 c_{i}^{4} - 5 c_{i}^{6} + \frac{5}{3} ν (706 - 216 c_{i}^{2} - 251 c_{i}^{4} + 15 c_{i}^{6}) \end{matrix}

\begin{matrix} - 5 ν^{2} (98 - 108 c_{i}^{2} + 7 c_{i}^{4} + 5 c_{i}^{6})] cos 2 ψ \end{matrix}

\begin{matrix} + \frac{2}{15} s_{i}^{2} [59 + 35 c_{i}^{2} - 8 c_{i}^{4} - \frac{5}{3} ν (131 + 59 c_{i}^{2} - 24 c_{i}^{4}) \end{matrix}

\begin{matrix} + 5 ν^{2} (21 - 3 c_{i}^{2} - 8 c_{i}^{4})] cos 4 ψ \end{matrix}

\begin{matrix} - \frac{81}{40} (1 - 5 ν + 5 ν^{2}) s_{i}^{4} (1 + c_{i}^{2}) cos 6 ψ \end{matrix}

\begin{matrix} + \frac{s_{i}}{40} \frac{δ m}{m} {[11 + 7 c_{i}^{2} + 10 (5 + c_{i}^{2}) ln 2] sin ψ - 5 π (5 + c_{i}^{2}) cos ψ \end{matrix}

\begin{matrix} - 27 [7 - 10 ln (3 / 2)] (1 + c_{i}^{2}) sin 3 ψ + 135 π (1 + c_{i}^{2}) cos 3 ψ}, \end{matrix}

\begin{matrix}  \end{matrix}

(221)

and, for the cross polarization,

\begin{matrix} H_{\times}^{(0)} & = & - 2 c_{i} sin 2 ψ, \end{matrix}

(222)

\begin{matrix} H_{\times}^{(1 / 2)} & = & - \frac{3}{4} s_{i} c_{i} \frac{δ m}{m} [sin ψ - 3 sin 3 ψ], \end{matrix}

(223)

\begin{matrix} H_{\times}^{(1)} & = & \frac{c_{i}}{3} [17 - 4 c_{i}^{2} - ν (13 - 12 c_{i}^{2})] sin 2 ψ \end{matrix}

\begin{matrix} - \frac{8}{3} (1 - 3 ν) c_{i} s_{i}^{2} sin 4 ψ, \end{matrix}

(224)

\begin{matrix} H_{\times}^{(3 / 2)} & = & \frac{s_{i} c_{i}}{96} \frac{δ m}{m} {[63 - 5 c_{i}^{2} - 2 ν (23 - 5 c_{i}^{2})] sin ψ \end{matrix}

\begin{matrix} - \frac{27}{2} [67 - 15 c_{i}^{2} - 2 ν (19 - 15 c_{i}^{2})] sin 3 ψ \end{matrix}

\begin{matrix} + \frac{625}{2} (1 - 2 ν) s_{i}^{2} sin 5 ψ} - 4 π c_{i} sin 2 ψ, \end{matrix}

(225)

\begin{matrix} H_{\times}^{(2)} & = & \frac{c_{i}}{60} [68 + 226 c_{i}^{2} - 15 c_{i}^{4} + \frac{5}{3} ν (572 - 490 c_{i}^{2} + 45 c_{i}^{4}) \end{matrix}

\begin{matrix} - 5 ν^{2} (56 - 70 c_{i}^{2} + 15 c_{i}^{4})] sin 2 ψ \end{matrix}

\begin{matrix} + \frac{4}{15} c_{i} s_{i}^{2} [55 - 12 c_{i}^{2} - \frac{5}{3} ν (119 - 36 c_{i}^{2}) + 5 ν^{2} (17 - 12 c_{i}^{2})] sin 4 ψ \end{matrix}

\begin{matrix} - \frac{81}{20} (1 - 5 ν + 5 ν^{2}) c_{i} s_{i}^{4} sin 6 ψ \end{matrix}

\begin{matrix} - \frac{3}{20} s_{i} c_{i} \frac{δ m}{m} {[3 + 10 ln 2] cos ψ + 5 π sin ψ \end{matrix}

\begin{matrix} - 9 [7 - 10 ln (3 / 2)] cos 3 ψ - 45 π sin 3 ψ} . \end{matrix}

(226)

We use the shorthands

c_{i} = cos i

and

s_{i} = sin i

for the cosine and sine of the inclination angle

i

between the direction of the detector as seen from the binary's center-of-mass, and the normal to the orbital plane (we always suppose that the normal is right-handed with respect to the sense of motion, so that

0 \leq i \leq π

To conclude, the use of theoretical templates based on the preceding 2PN wave forms, and having their frequency evolution built in via the 3.5PN phase evolution ( 211 , 212 ), 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 ( 211 , 212 ), notably those associated with the specific quadratic and cubic non-linear tails exactly as they are predicted by Einstein's theory [28, 29] . Indeed, we don't know of any other physical systems for which it would be possible to perform such tests.

11 Acknowledgments

The author is very grateful to Sergei Kopeikin for useful remarks on this work. It is a pleasure to thank Silvano Bonazzola, Thibault Damour, Jürgen Ehlers, Gilles Esposito-Farèse, Guillaume Faye, Eric Gourgoulhon, Bala Iyer, Misao Sasaki, Gerhard Schäfer, Bernd Schmidt, Kip Thorne, and Clifford Will for interesting discussions and/or collaborations.

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