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4.2 Generality of the solution

We have a solution, but is that a general solution? The answer, yes, is provided by the following result [14Jump To The Next Citation Point]:

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 sum +o o hab = n=1 Gnhabn [IL, JL,...,ZL]. It depends on two sets of arbitrary STF-tensorial functions of time IL(u) and JL(u) (satisfying the conservation laws) defined by Eqs. (27View Equation), and on four supplementary functions WL(u), ..., ZL(u) parametrizing the gauge vector (28View Equation).

The proof is quite easy. With Eq. (43View Equation) we obtained a particular solution of the system of equations (30View Equation, 31View Equation). To it we should add the most general solution of the corresponding homogeneous system of equations, which is obtained by setting ab /\ n = 0 into Eqs. (30View Equation, 31View Equation). But this homogeneous system of equations is nothing but the linearized vacuum field equations (23View Equation, 24View Equation), for which we know the most general solution hab1 given by (26View Equation, 27View Equation, 28View Equation). Thus, we must add to our “particular” solution habn a general homogeneous solution that is necessarily of the type hab [dI ,...,dZ ] 1 L L, where dI L, ..., dZ L denote some “corrections” to the multipole moments at the nth 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 IL,...,ZL by posing

Inew = I + Gn -1dI , (44) L L L ... new n-1 ZL = ZL + G dZL. (45)
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 nth 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 IL(u),...,ZL(u) 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, ML(u) and SL(u), for parametrizing the most general isolated source as well. The route for constructing such a general solution is to get rid of the moments W ,X ,Y ,Z L L L L at the linearized level by performing the linearized gauge transformation a a dx = f 1, where a f 1 is the gauge vector given by Eqs. (28View Equation). So, at the linearized level, we have only the two types of moments ML and SL, parametrizing ka1b by the same formulas as in Eqs. (27View Equation). We must be careful to denote these moments with some names different from IL and JL because they will ultimately correspond to a different physical source. Then we apply exactly the same post-Minkowskian algorithm, following the formulas (39View Equation, 40View Equation, 41View Equation, 42View Equation, 43View Equation) as we did above, but starting from the gauge-transformed linear metric ab k1 instead of ab h1. The result of the iteration is therefore some sum kab = +n oo =1 Gnkabn [ML, SL]. 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 kab[M ,S ] L L is isometric to the original one hab[I ,J ,...,Z ] L L L if and only if ML and SL are related to the moments IL,JL,...,ZL 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 ka1b[ML, SL] instead of the more complicated kab[IL,JL] + @afb + @bfa - jab@mfm 1 1 1 1, 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, M L and S L, instead of the six, I L, J L, ..., Z L? The reason is that we shall determine (in Theorem 6 below) the explicit closed-form expressions of the six moments IL, JL, ..., ZL, but that, by contrast, it seems to be impossible to obtain some similar closed-form expressions for ML and SL. The only thing we can do is to write down the explicit non-linear algorithm that computes ML, SL starting from IL, J L, ..., Z L. In consequence, it is better to view the moments I L, J L, ..., Z L as more “fundamental” than ML and SL, 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 IL, JL, ..., ZL as the multipole moments of the source. This being said, the moments ML and SL 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 ML and SL differ from the corresponding IL and JL only by high-order post-Newtonian terms like 2.5PN; see Ref. [7Jump To The Next Citation Point] and Eq. (90View Equation) 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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