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 and the source’s Newtonian potential by

is much less than one. This parameter represents essentially a slow motion estimate , where denotes a typical internal velocity. By a slight abuse of notation, following Chandrasekhar et al. [66, 68, 67], we shall henceforth write , even though is dimensionless whereas has the dimension of a velocity. The small post-Newtonian remainders will be denoted . Thus, in the case of post-Newtonian sources. We have 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. . 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 can be as large as 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 , is the famous
quadrupole formalism of Einstein [105] and Landau and Lifchitz [153]. 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
in a transverse and traceless (TT) coordinate system, covering the far zone of the
source^{2},
as

The multipole expansion is one of the most useful tools of physics, but its use in general relativity is difficult because of the non-linearity of the theory and the tensorial character of the gravitational interaction. In the stationary case, the multipole moments are determined by the expansion of the metric at spatial infinity [120, 129, 201], while, in the case of non-stationary fields, the moments, starting with the quadrupole, are defined at future null infinity. The multipole moments have been extensively studied in the linearized theory, which ignores the gravitational forces inside the source. Early studies have extended the formula (4) to include the current-quadrupole and mass-octupole moments [171, 170], and obtained the corresponding formulas for linear momentum [171, 170, 10, 186] and angular momentum [177, 75]. The general structure of the infinite multipole series in the linearized theory was investigated by several works [191, 192, 181, 210], from which it emerged that the expansion is characterized by two and only two sets of moments: mass-type and current-type moments. Below we shall use a particular multipole decomposition of the linearized (vacuum) metric, parametrized by symmetric and trace-free (STF) mass and current moments, as given by Thorne [210]. The explicit expressions of the multipole moments (for instance in STF guise) as integrals over the source, valid in the linearized theory but irrespective of a slow motion hypothesis, are completely known [159, 65, 64, 89].

In the full non-linear theory, the (radiative) multipole moments can be read off the coefficient of in the expansion of the metric when , with a null coordinate . The solutions of the field equations in the form of a far-field expansion (power series in ) have been constructed, and their properties elucidated, by Bondi et al. [53] and Sachs [193]. The precise way under which such radiative space-times fall off asymptotically has been formulated geometrically by Penrose [175, 176] in the concept of an asymptotically simple space-time (see also Ref. [121]). The resulting Bondi-Sachs-Penrose approach is very powerful, but it can answer a priori only a part of the problem, because it gives information on the field only in the limit where , which cannot be connected in a direct way to the actual behaviour of the source. In particular the multipole moments that one considers in this approach are those measured at infinity - we call them the radiative multipole moments. These moments are distinct, because of non-linearities, from some more natural source multipole moments, which are defined operationally by means of explicit integrals extending over the matter and gravitational fields.

An alternative way of defining the multipole expansion within the complete non-linear theory is that of Blanchet and Damour [26, 12], following pioneering work by Bonnor and collaborators [54, 55, 56, 130] and Thorne [210]. In this approach the basic multipole moments are the source moments, rather than the radiative ones. In a first stage, the moments are left unspecified, as being some arbitrary functions of time, supposed to describe an actual physical source. They are iterated by means of a post-Minkowskian expansion of the vacuum field equations (valid in the source’s exterior). Technically, the post-Minkowskian approximation scheme is greatly simplified by the assumption of a multipolar expansion, as one can consider separately the iteration of the different multipole pieces composing the exterior field (whereas, the direct attack of the post-Minkowskian expansion, valid at once inside and outside the source, faces some calculational difficulties [215, 76]). In this “multipolar-post-Minkowskian” formalism, which is physically valid over the entire weak-field region outside the source, and in particular in the wave zone (up to future null infinity), the radiative multipole moments are obtained in the form of some non-linear functionals of the more basic source moments. A priori, the method is not limited to post-Newtonian sources, however we shall see that, in the current situation, the closed-form expressions of the source multipole moments can be established only in the case where the source is post-Newtonian [15, 20]. The reason is that in this case the domain of validity of the post-Newtonian iteration (viz. the near zone) overlaps the exterior weak-field region, so that there exists an intermediate zone in which the post-Newtonian and multipolar expansions can be matched together. This is a standard application of the method of matched asymptotic expansions in general relativity [63, 62].

To be more precise, we shall show how a systematic multipolar and post-Minkowskian iteration scheme for the vacuum Einstein field equations yields the most general physically admissible solution of these equations [26]. The solution is specified once we give two and only two sets of time-varying (source) multipole moments. Some general theorems about the near-zone and far-zone expansions of that general solution will be stated. Notably, we find [12] that the asymptotic behaviour of the solution at future null infinity is in agreement with the findings of the Bondi-Sachs-Penrose [53, 193, 175, 176, 121] approach to gravitational radiation. However, checking that the asymptotic structure of the radiative field is correct is not sufficient by itself, because the ultimate aim is to relate the far field to the properties of the source, and we are now obliged to ask: What are the multipole moments corresponding to a given stress-energy tensor describing the source? Only in the case of post-Newtonian sources has it been possible to answer this question. The general expression of the moments was obtained at the level of the second post-Newtonian (2PN) order in Ref. [15], and was subsequently proved to be in fact valid up to any post-Newtonian order in Ref. [20]. The source moments are given by some integrals extending over the post-Newtonian expansion of the total (pseudo) stress-energy tensor , which is made of a matter part described by and a crucial non-linear gravitational source term . These moments carry in front a particular operation of taking the finite part ( as we call it below), which makes them mathematically well-defined despite the fact that the gravitational part has a spatially infinite support, which would have made the bound of the integral at spatial infinity singular (of course the finite part is not added a posteriori to restore the well-definiteness of the integral, but is proved to be actually present in this formalism). The expressions of the moments had been obtained earlier at the 1PN level, albeit in different forms, in Ref. [28] for the mass-type moments (strangely enough, the mass moments admit a compact-support expression at 1PN order), and in Ref. [90] for the current-type ones.

The wave-generation formalism resulting from matching the exterior multipolar and post-Minkowskian field [26, 12] to the post-Newtonian source [15, 20] is able to take into account, in principle, any post-Newtonian correction to both the source and radiative multipole moments (for any multipolarity of the moments). The relationships between the radiative and source moments include many non-linear multipole interactions, because the source moments mix with each other as they “propagate” from the source to the detector. Such multipole interactions include the famous effects of wave tails, corresponding to the coupling between the non-static moments with the total mass of the source. The non-linear multipole interactions have been computed within the present wave-generation formalism up to the 3PN order in Refs. [29, 21, 19]. Furthermore, the back-reaction of the gravitational-wave emission onto the source, up to the 1.5PN order relative to the leading order of radiation reaction, has also been studied within this formalism [27, 14, 18]. Now, recall that the leading radiation reaction force, which is quadrupolar, occurs already at the 2.5PN order in the source’s equations of motion. Therefore the 1.5PN “relative” order in the radiation reaction corresponds in fact to the 4PN order in the equations of motion, beyond the Newtonian acceleration. It has been shown that the gravitational wave tails enter the radiation reaction at precisely the 1.5PN relative order, which means 4PN “absolute” order [27]. A systematic post-Newtonian iteration scheme for the near-zone field, formally taking into account all radiation reaction effects, has been recently proposed, consistent with the present formalism [185, 41].

A different wave-generation formalism has been devised by Will and Wiseman [220] (see also Refs. [219, 173, 174]), after earlier attempts by Epstein and Wagoner [107] and Thorne [210]. This formalism has exactly the same scope as ours, i.e. it applies to any isolated post-Newtonian sources, but it differs in the definition of the source multipole moments and in many technical details when properly implemented [220]. In both formalisms, the moments are generated by the post-Newtonian expansion of the pseudo-tensor , but in the Will-Wiseman formalism they are defined by some compact-support integrals terminating at some finite radius enclosing the source, e.g., the radius of the near zone). By contrast, in our case [15, 20], the moments are given by some integrals covering the whole space and regularized by means of the finite part . We shall prove the complete equivalence, at the most general level, between the two formalisms. What is interesting about both formalisms is that the source multipole moments, which involve a whole series of relativistic corrections, are coupled together, in the true non-linear solution, in a very complicated way. These multipole couplings give rise to the many tail and related non-linear effects, which form an integral part of the radiative moments at infinity and thereby of the observed signal.

Part A of this article is devoted to a presentation of the post-Newtonian wave generation formalism. We try to state the main results in a form that is simple enough to be understood without the full details, but at the same time we outline some of the proofs when they present some interest on their own. To emphasize the importance of some key results, we present them in the form of mathematical theorems.

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