We shall provide the answer to that problem in the case of a post-Newtonian source for which the post-Newtonian parameter defined by Eq. (1*) is small. The fundamental fact that permits the connection of the exterior field to the inner field of the source is the existence of a “matching” region, in which both the multipole expansion and the post-Newtonian expansion are valid. This region is nothing but the exterior part of the near zone, such that (exterior) and (near zone); it always exists around post-Newtonian sources whose radius is much less than the emitted wavelength, . In our formalism the multipole expansion is defined by the multipolar-post-Minkowskian (MPM) solution; see Section 2. Matching together the post-Newtonian and MPM solutions in this overlapping region is an application of the method of matched asymptotic expansions, which has frequently been applied in the present context, both for radiation-reaction [114*, 113*, 7, 58*, 43*] and wave-generation [59*, 155, 44*, 49*] problems.54*). In the matching region, where both the multipolar and post-Newtonian expansions are valid, we write the numerical equality
We now transform Eq. (102*) into a matching equation, by replacing in the left-hand side by its near-zone re-expansion , and in the right-hand side by its multipole expansion . The structure of the near-zone expansion () of the exterior multipolar field has been found in Theorem 3, see Eq. (53*). We denote the corresponding infinite series with the same overbar as for the post-Newtonian expansion because it is really an expansion when , equivalent to an expansion when . Concerning the multipole expansion of the post-Newtonian metric, , we simply postulate for the moment its existence, but we shall show later how to construct it explicitly. Therefore, the matching equation is the statement thatfunctional identities, valid , 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. (102*), which is valid only when belongs to some limited spatial domain. The matching equation (103*) 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 , 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 Eq. (53*): 104*) –, or the singular re-expansion of the post-Newtonian series when – the 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. As we shall detail in Sections 4.2 and 5.2, the matching equation (103*), supplemented by the condition of no-incoming radiation [say in the form of Eq. (29*)], permits determining all the unknowns of the problem: On the one hand, the external multipolar decomposition , i.e., the explicit expressions of the multipole moments therein (see Sections 4.2 and 4.4); on the other hand, the terms in the inner post-Newtonian expansion that are associated with radiation-reaction effects, i.e., those terms which depend on the boundary conditions of the radiative field at infinity, and which correspond in the present case to a post-Newtonian source which is isolated from other sources in the Universe; see Section 5.2.
Proof (see Refs. [44*, 49*]): First notice where the physical restriction of considering a post-Newtonian source enters this theorem: The multipole moments (106*) depend on the post-Newtonian expansion of the pseudo-tensor, rather than on itself. Consider , namely the difference between , which is a solution of the field equations everywhere inside and outside the source, and the first term in Eq. (105*), namely the finite part of the retarded integral of the multipole expansion :30*), is given by the retarded integral of the pseudo-tensor . So, 108*), as it stands, is well-defined because we are considering only some smooth field distributions: . There is no need to include a finite part 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 in the first term is that we can re-write Eq. (108*) into the more interesting form 109*) is that 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 [by the same equation as Eq. (102*)]. Therefore, 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. , yields the second term in Eq. (105*), 106*); instead,29 111*) has a compact support limited to the domain of the source. In consequence, we can replace the integrand in Eq. (111*) by its post-Newtonian expansion, valid over all the near zone: 112*). Actually, this term is a bit curious, because the object it contains is only known in the form of the formal series whose structure is given by the first equality in Eq. (104*) (indeed and 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. (112*) is in fact identically zero. The proof is based on the properties of the analytic continuation as applied to the formal structure (104*) of . Each term of this series yields a contribution to Eq. (112*) that takes the form, after performing the angular integration, of the integral , and multiplied by some function of time. We want to prove that the radial integral is zero by analytic continuation (). First we can get rid of the logarithms by considering some repeated differentiations with respect to ; thus we need only to consider the simpler integral . We split the integral into a “near-zone” integral and a “far-zone” one , where is some constant radius. When is a large enough positive number, the value of the near-zone integral is , while when is a large negative number, the far-zone integral reads the opposite, . Both obtained values represent the unique analytic continuations of the near-zone and far-zone integrals for any except . The complete integral is equal to the sum of these analytic continuations, and is therefore identically zero (, including the value ). At last we have completed the proof of Theorem 5:
The latter proof makes it clear how crucial the analytic-continuation finite part is, which we recall is the same as in our iteration of the exterior post-Minkowskian field [see Eq. (45*)]. Without a finite part, the multipole moment (113*) would be strongly divergent, because the pseudo-tensor has a non-compact support owing to the contribution of the gravitational field, and the multipolar factor behaves like when . The latter divergence has plagued the field of post-Newtonian expansions of gravitational radiation for many years. In applications such as in Part B of this article, we must carefully follow the rules for handling the operator.
The two terms in the right-hand side of Eq. (105*) depend separately on the length scale that we have introduced into the definition of the finite part, through the analytic-continuation factor introduced in Eq. (42*). However, the sum of these two terms, i.e., the exterior multipolar field itself, is independent of . To see this, the simplest way is to differentiate formally with respect to ; the differentiations of the two terms of Eq. (105*) cancel each other. The independence of the field upon is quite useful in applications, since in general many intermediate calculations do depend on , and only in the final stage does the cancellation of the ’s occur. For instance, we have already seen in Eqs. (93*) – (94*) that the source quadrupole moment depends on starting from the 3PN level, but that this is compensated by another coming from the non-linear “tails of tails” at the 3PN order.
Will & Wiseman [424*] (see also Refs. [422, 335]), extending previous work of Epstein & Wagoner  and Thorne , 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. (105*),truncated, as indicated by the subscript , to extend only in the “far zone”: i.e., in the notation of Eq. (31*), where is a constant radius enclosing the source (). The near-zone part of the retarded integral is thereby removed, and there is no problem with the singularity of the multipole expansion at the origin. The multipole moments are then given, in contrast with our result (106*), by an integral extending over the “near zone” only:
Let us show that the two different formalisms are equivalent. We compute the difference between our moment defined by Eq. (106*), and the moment given by Eq. (115*). For the comparison we split into far-zone and near-zone pieces corresponding to the radius . Since the finite part present in deals only with the bound at infinity, it can be removed from the near-zone piece, which is then seen to reproduce exactly. So the difference between the two moments is simply given by the far-zone piece:103*). At this stage, we recall from our reasoning right after Eq. (112*) that the finite part of an integral over the whole space of a quantity having the same structure as is identically zero by analytic continuation. The main ingredient of the present proof is made possible by this fact, as it allows us to transform the far-zone integration in Eq. (116*) into a near-zone one , at the price of changing the overall sign in front of the integral. So, 114*), so that the “complete” retarded integral as given by the first term in our own definition (105*) is exactly reconstituted. In conclusion, the formalism of Ref. [424*] is equivalent to the one of Refs. [44*, 49].
In principle, the bridge between the exterior gravitational field generated by the post-Newtonian source and its inner field is provided by Theorem 5; however, we still have to make the connection with the explicit construction of the general multipolar and post-Minkowskian metric in Section 2. Namely, we must find the expressions of the six STF source multipole moments , parametrizing the linearized metric (35*) – (37) at the basis of that construction.30
To do this we first find the equivalent of the multipole expansion given in Theorem 5, which was parametrized by non-trace-free multipole functions , in terms of new multipole functions that are STF in all their indices . The result is21*) onto the multipole decomposition (118*), and to decompose the multipole functions into STF irreducible pieces with respect to both and their spatial indices contained into . This technical part of the calculation is identical to the one of the STF irreducible multipole moments of linearized gravity . The formulas needed in this decomposition read
where the ten tensors are STF, and are uniquely given in terms of the ’s by some inverse formulas. Finally, the latter decompositions yield the following.
In these formulas the notation is as follows: Some convenient source densities are defined from the post-Newtonian expansion (denoted by an overbar) of the pseudo-tensor by
(where ). As indicated in Eqs. (123) all these quantities are to be evaluated at the spatial point and at time .
For completeness, we give also the formulas for the four auxiliary source moments , which parametrize the gauge vector as defined in Eqs. (37):
As discussed in Section 2, one can always find two intermediate “packages” of multipole moments, namely the canonical moments and , which are some non-linear functionals of the source moments (123) and (125), and such that the exterior field depends only on them, modulo a change of coordinates. However, the canonical moments , do not admit general closed-form expressions like (123) – (125).31
These source moments are physically valid for post-Newtonian sources and make sense only in the form of a post-Newtonian expansion, so in practice we need to know how to expand the -integrals as series when . Here is the appropriate formula:3*).32
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 ; this necessitates solving the field equations inside the matter, which we shall investigate in the next Section 5. Then is to be inserted into the source moments (123) and (125), where the formula (126*) permits expressing all the terms up to that post-Newtonian order by means of more tractable integrals extending over . Given a specific model for the matter source we then have to find a way to compute all these spatial integrals; this is done in Section 9.1 for the case of point-mass binaries. Next, we must substitute the source multipole moments into the linearized metric (35*) – (37), and iterate them until all the necessary multipole interactions taking place in the radiative moments and are under control. In fact, we have already worked out these multipole interactions for general sources in Section 3.3 up to the 3PN order in the full waveform, and 3.5PN order for the dominant mode. Only at this point does one have the physical radiation field at infinity, from which we can build the templates for the detection and analysis of gravitational waves. We advocate here that the complexity of the formalism simply reflects the complexity of the Einstein field equations. It is probably impossible to devise a different formalism, valid for general sources devoid of symmetries, that would be substantially simpler.