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 . It depends on two sets of arbitrary
STF-tensorial functions of time and (satisfying the conservation laws) defined by
Eqs. (27), and on four supplementary functions , , parametrizing the gauge
vector (28).

The proof is quite easy. With Eq. (43) we obtained a particular solution of the system of equations (30, 31). To it we should add the most general solution of the corresponding homogeneous system of equations, which is obtained by setting into Eqs. (30, 31). But this homogeneous system of equations is nothing but the linearized vacuum field equations (23, 24), for which we know the most general solution given by (26, 27, 28). 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. 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. (28). So, at the linearized level, we have only the two types of moments and , parametrizing by the same formulas as in Eqs. (27). We must be careful to denote these moments with some 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 (39, 40, 41, 42, 43) 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 is isometric to the original one if and only if and are related to the moments 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 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, , , , ? The reason is that we shall determine (in Theorem 6 below) the explicit closed-form expressions of the six moments , , , , but that, by contrast, it seems to be impossible to obtain some similar closed-form expressions for 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 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 and differ from the corresponding and only by high-order post-Newtonian terms like 2.5PN; see Ref. [7] and Eq. (90) below. Indeed, this is to be expected because the physical difference between both types of moments stems only from non-linearities.

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