### 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 , parametrizing the linearized metric (26, 27, 28) at the basis of that construction.

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 (which follows from Equation (B.14a) in [28]) is

where the STF multipole functions (witness the multipolar factor ) read
Notice the presence of an extra integration variable , ranging from to 1. The -integration involves the weighting function
which is normalized in such a way that
The next step is to impose the harmonic-gauge conditions (12) onto the multipole decomposition (80), and to decompose the multipole functions into STF irreducible pieces with respect to both and their space-time indices . This technical part of the calculation is identical to the one of the STF irreducible multipole moments of linearized gravity [89]. 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 lead to the following theorem.

Theorem 6 The STF multipole moments and of a post-Newtonian source are given, formally up to any post-Newtonian order, by ()

These moments are the ones that are to be inserted into the linearized metric that represents the lowest approximation to the post-Minkowskian field 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 by

(where ). As indicated in Equations (85) 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 Equations (28):

As discussed in Section 4, one can always find two intermediate “packages” of multipole moments, and , which are some non-linear functionals of the source moments (85) and Equations (87, 88, 89, 90), and such that the exterior field depends only on them, modulo a change of coordinates (see, e.g., Equation (96) 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 -integrals as series when . Here is the appropriate formula:

Since the right-hand side involves only even powers of , the same result holds equally well for the “advanced” variable or the “retarded” one . Of course, in the Newtonian limit, the moments and (and also , ) reduce to the standard expressions. For instance, we have
where is the Newtonian mass-type multipole moment (see Equation (3)). (The moments have also a Newtonian limit, but it is not particularly illuminating.)

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, see Section 5.5) before inserting them into the source moments (85, 86, 82, 83, 91, 87, 88, 89, 90). The formula (91) is used to express all the terms up to that post-Newtonian order by means of more tractable integrals extending over . Given a specific model for the matter source we then have to find a way to compute all these spatial integrals (we do it in Section 10 in the case of point-mass binaries). Next, we must substitute the source multipole moments into the linearized metric (26, 27, 28), and iterate them until all the necessary multipole interactions taking place in the radiative moments and 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.