2 Non-linear Iteration of the Vacuum Field Equations

2.1 Einstein’s field equations

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

They form a system of ten second-order partial differential equations obeyed by the metric, where the Einstein curvature tensor is generated, through the gravitational coupling constant , by the stress-energy tensor of the matter fields . Among these ten equations, four govern, via the contracted Bianchi identity, the evolution of the matter system, The matter equations can also be obtained by varying the matter action in (17*) with respect to the matter fields . The space-time geometry is constrained by the six remaining equations, which place six independent constraints on the ten components of the metric , leaving four of them to be fixed by a choice of the coordinate system.

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

where denotes the contravariant metric (satisfying ), where is the determinant of the covariant metric, , and where represents an auxiliary Minkowskian metric . The harmonic-coordinate condition, which accounts exactly for the four equations (19*) corresponding to the conservation of the matter tensor, reads¹³ Equation (21*) introduces into the definition of our coordinate system a preferred Minkowskian structure, with Minkowski metric . Of course, this is not contrary to the spirit of general relativity, where there is only one physical metric without any flat prior geometry, because the coordinates are not governed by geometry (so to speak), but rather can be chosen at convenience, depending on physical phenomena under study. The coordinate condition (21*) is especially useful when studying gravitational waves as perturbations of space-time propagating on the fixed background metric . This view is perfectly legitimate and represents a fruitful and rigorous way to think of the problem using approximation methods. Indeed, the metric , originally introduced in the coordinate condition (21*), does exist at any finite order of approximation (neglecting higher-order terms), and plays the role of some physical “prior” flat geometry at any order of approximation.

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

where . The source term can rightly be interpreted as the stress-energy pseudo-tensor (actually, is a Lorentz-covariant tensor) of the matter fields, described by , and the gravitational field, given by the gravitational source term , i.e., The exact expression of in harmonic coordinates, including all non-linearities, reads¹⁴

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

These various terms can be straightforwardly computed from expanding Eq. (24); for instance the leading quadratic piece is explicitly given by¹⁵

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

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

1. The matter stress-energy tensor is of spatially compact support, i.e., can be enclosed into some time-like world tube, say , where is the harmonic-coordinate radial distance. Outside the domain of the source, when , the gravitational source term, according to Eq. (27*), is divergence-free,
   
2. The matter distribution inside the source is smooth: .¹⁶ We have in mind a smooth hydrodynamical fluid system, without any singularities nor shocks (a priori), that is described by some Euler-type equations including high relativistic corrections. In particular, we exclude from the start the presence of any black holes; however, we shall return to this question in Part B when we look for a model describing compact objects;
3. The source is post-Newtonian in the sense of the existence of the small parameter defined by Eq. (1*). For such a source we assume the legitimacy of the method of matched asymptotic expansions for identifying the inner post-Newtonian field and the outer multipolar decomposition in the source’s exterior near zone;
4. The gravitational field has been independent of time (stationary) in some remote past, i.e., before some finite instant in the past, namely
   

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

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

containing the usual retarded inverse d’Alembertian integral operator, given by extending over the whole three-dimensional space .

2.2 Linearized vacuum equations

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

where the subscript “ext” reminds us that the solution is valid only in the exterior of the source, and where we have introduced Newton’s constant as a book-keeping parameter, enabling one to label conveniently the successive post-Minkowskian approximations. Since is a dimensionless variable, with our convention the linear coefficient in Eq. (32*) has the dimension of the inverse of (which should be a mass squared in a system of units where ). In vacuum, the harmonic-coordinate metric coefficient satisfies

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

The multipolar-post-Minkowskian (MPM) method will be implemented systematically, using symmetric-trace-free (STF) harmonics to describe the multipole expansion [403*], and looking for a definite algorithm for the approximation scheme [57*]. The solution of the system of equations (33) takes the form of a series of retarded multipolar waves¹⁷

where , and where the functions are smooth functions of the retarded time [i.e., ], which become constant in the past, when , see Eq. (29*). Since a monopolar wave satisfies and the d’Alembertian commutes with the multi-derivative , it is evident that Eq. (34*) represents the most general solution of the wave equation (33a); but see Section 2 in Ref. [57*] for a rigorous proof based on the Euler–Poisson–Darboux equation. The gauge condition (33b), however, is not fulfilled in general, and to satisfy it we must algebraically decompose the set of functions , , into ten tensors which are STF with respect to all their indices, including the spatial indices , . Imposing the condition (33b) reduces the number of independent tensors to six, and we find that the solution takes an especially simple “canonical” form, parametrized by only two moments, plus some arbitrary linearized gauge transformation [403*, 57*].

The conservation of the lowest-order moments gives the constancy of the total mass of the source, , center-of-mass position, , total linear momentum ,¹⁸ and total angular momentum, . It is always possible to achieve by translating the origin of our coordinates to the center of mass. The total mass is the ADM mass of the Hamiltonian formulation of general relativity. Note that the quantities , , and 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 and , which thoroughly encode the physical properties of the source at the linearized level (because the other moments 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 of the source: Theorem 1 follows merely from the algebraic and differential properties of the vacuum field equations outside the source.

For completeness, we give the components of the gauge-vector entering Eq. (35*):

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

2.3 The multipolar post-Minkowskian solution

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

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

We substitute the post-Minkowski ansatz (38*) into the vacuum Einstein field equations (21*) – (22*), i.e., with simply given by the gravitational source term , and we equate term by term the factors of the successive powers of our book-keeping parameter . We get an infinite set of equations for each of the ’s: namely, ,

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

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

Let us now proceed by induction. Some being given, we assume that we succeeded in constructing, starting from the linearized solution , the sequence of post-Minkowskian solutions , and from this we want to infer the next solution . The right-hand side of Eq. (39a), , is known by induction hypothesis. Thus the problem is that of solving a flat wave equation whose source is given. The point is that this wave equation, instead of being valid everywhere in , is physically correct only outside the matter source (), and it makes no sense to solve it by means of the usual retarded integral. Technically speaking, the right-hand side of Eq. (39a) is composed of the product of many multipole expansions, which are singular at the origin of the spatial coordinates , 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 , i.e., is expanded into multipole contributions, with a singularity at , and satisfies the d’Alembertian equation as soon as . Such a particular solution can be obtained, following the method of Ref. [57*], by means of a mathematical trick, in which one first “regularizes” the source term by multiplying it by the factor , where is the spatial radial distance and is a complex number, . Let us assume, for definiteness, that is composed of multipolar pieces with maximal multipolarity . This means that we start the iteration from the linearized metric (35*) – (37) in which the multipolar sums are actually finite.¹⁹ The divergences when of the source term are typically power-like, say (there are also powers of the logarithm of ), and with the previous assumption there will exist a maximal order of divergency, say . Thus, when the real part of is large enough, i.e., , the “regularized” source term is regular enough when so that one can perfectly apply the retarded integral operator. This defines the -dependent retarded integral, when is large enough,

where the symbol stands for the retarded integral defined by Eq. (31*). It is convenient to introduce inside the regularizing factor some arbitrary constant length scale in order to make it dimensionless. Everywhere in this article we pose The fate of the constant in a detailed calculation will be interesting to follow, as we shall see. Now the point for our purpose is that the function on the complex plane, which was originally defined only when , admits a unique analytic continuation to all values of except at some integer values. Furthermore, the analytic continuation of can be expanded, when (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 in that expansion, represents the particular solution we are looking for. We write the Laurent expansion of , when , in the form where , and the various coefficients are functions of the field point . When there are poles; and , which depends on , refers to the maximal order of these poles. By applying the d’Alembertian operator onto both sides of Eq. (43*), and equating the different powers of , we arrive at

As we see, the case shows that the finite-part coefficient in Eq. (43*), namely , is a particular solution of the requested equation: . Furthermore, we can prove that this solution, by its very construction, owns the same structure made of a multipolar expansion singular at as the corresponding source.

Let us forget about the intermediate name , and denote, from now on, the latter solution by , or, in more explicit terms,

where the finite-part symbol means the previously detailed operations of considering the analytic continuation, taking the Laurent expansion, and picking up the finite-part coefficient when . The story is not complete, however, because does not fulfill the constraint of harmonic coordinates (39b); its divergence, say , is different from zero in general. From the fact that the source term is divergence-free in vacuum, [see Eq. (28*)], we find instead The factor comes from the differentiation of the regularization factor . So, is zero only in the special case where the Laurent expansion of the retarded integral in Eq. (46*) does not develop any simple pole when . Fortunately, when it does, the structure of the pole is quite easy to control. We find that it necessarily consists of an homogeneous solution of the source-free d’Alembertian equation, and, what is more (from its stationarity in the past), that solution is a retarded one. Hence, taking into account the index structure of , there must exist four STF-tensorial functions of , say , , and , such that

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

Notice the presence of anti-derivatives, denoted e.g., by ; there is no problem with the limit since all the corresponding functions are zero when . The choice made in Eqs. (48) is dictated by the fact that the component involves only some monopolar and dipolar terms, and that the spatial trace is monopolar: . Finally, if we pose

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

2.4 Generality of the MPM solution

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

The proof is quite easy. With Eq. (49*) we obtained a particular solution of the system of equations (39). To it we should add the most general solution of the corresponding homogeneous system of equations, which is obtained by setting into Eqs. (39). But this homogeneous system of equations is nothing but the linearized vacuum field equations (33), to which we know the most general solution given by Eqs. (35*) – (37). Thus, we must add to our particular solution a general homogeneous solution that is necessarily of the type , where denote some corrections to the multipole moments at the -th post-Minkowskian order (with the monopole and dipoles , being constant). It is then clear, since precisely the linearized metric is a linear functional of all these moments, that the previous corrections to the moments can be absorbed into a re-definition of the original ones by posing

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 -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 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, and , for parametrizing the most general isolated source as well. The route for constructing such a general solution is to get rid of the moments at the linearized level by performing the linearized gauge transformation , where is the gauge vector given by Eqs. (37). So, at the linearized level, we have only the two types of moments and , parametrizing by the same formulas as in Eqs. (36). We must be careful to denote these moments with names different from and because they will ultimately correspond to a different physical source. Then we apply exactly the same post-Minkowskian algorithm, following the formulas (45*) – (49*) as we did above, but starting from the gauge-transformed linear metric instead of . The result of the iteration is therefore some

Obviously this post-Minkowskian algorithm yields some simpler calculations as we have only two multipole moments to iterate. The point is that one can show that the resulting metric (52*) is isometric to the original one (50*) if and only if the so-called canonical moments and are related to the source moments by some (quite involved) non-linear equations. We shall give in Eqs. (97) – (98) the most up to date relations we have between these moments. Therefore, the most general solution of the field equations, modulo a coordinate transformation, can be obtained by starting from the linearized metric instead of the more complicated , 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, and , instead of the six types, ? The reason is that we shall determine in Theorem 6 below the explicit closed-form expressions of the six source moments , but that, by contrast, it seems to be impossible to obtain some similar closed-form expressions for the canonical moments and . The only thing we can do is to write down the explicit non-linear algorithm that computes , starting from . In consequence, it is better to view the moments as more “fundamental” than and , 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 as the multipole moments of the source. This being said, the moments and are generally very useful in practical computations because they yield a simpler post-Minkowskian iteration. Then, one can generally come back to the more fundamental source-rooted moments by using the fact that and differ from the corresponding and only by high-order post-Newtonian terms like 2.5PN; see Eqs. (97) – (98) below. Indeed, this is to be expected because the physical difference between both types of moments stems only from non-linearities.

2.5 Near-zone and far-zone structures

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

²⁰

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

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

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 with : ,

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

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

where the gauge vector is This gauge transformation is non-harmonic: Its effect is to correct for the well-known logarithmic deviation of the retarded time in harmonic coordinates, with respect to the true space-time characteristic or light cones. After the change of gauge, the coordinate coincides with a null coordinate at the linearized level.²¹ This is the requirement to be satisfied by a linearized metric so that it can constitute the linearized approximation to a full (post-Minkowskian) radiative field [304]. One can easily show that, at the dominant order when , where is the outgoing Minkowskian null vector. Given any , let us recursively assume that we have obtained all the previous radiative post-Minkowskian coefficients , i.e. , and that all of them satisfy From this induction hypothesis one can prove that the -th post-Minkowskian source term is such that To the leading order this term takes the classic form of the stress-energy tensor of massless particles, with being proportional to the power in the massless waves. One can show that all the problems with the appearance of logarithms come from the retarded integral of the terms in Eq. (62*) that behave like : See indeed the integration formula (83*), which behaves like 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 -th post-Minkowskian order. We pose, as a gauge vector, where refers to the same finite part operation as in Eq. (45*). This vector is such that the logarithms that will appear in the corresponding gauge terms cancel out the logarithms coming from the retarded integral of the source term (62*); see Ref. [41*] for the details. Hence, to the -th post-Minkowskian order, we define the radiative metric as Here and denote the quantities that are the analogues of and , which were introduced into the harmonic-coordinates algorithm: See Eqs. (45*) – (48). In particular, these quantities are constructed in such a way that the sum is divergence-free, so we see that the radiative metric does not obey the harmonic-gauge condition, but instead The far-zone expansion of the latter metric is of the type (56*), i.e., is free of any logarithms, and the retarded time in these coordinates tends asymptotically toward a null coordinate at future null infinity. The property of asymptotic simplicity, in the form given by Geroch & Horowitz [220], is proved by introducing the usual conformal factor in radiative coordinates [41]. Finally, it can be checked that the metric so constructed, which is a functional of the source multipole moments (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 , where are the harmonic coordinates and the radiative ones, together with a re-definition of the multipole moments.