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
Equations (27
), and on four supplementary functions
parametrizing the gauge
vector (28
).
The proof is quite easy. With Equation (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 Equations (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 Equations (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
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 Equations (28
). So, at the linearized level, we have only the two types of moments
and
, parametrizing
by the same formulas as in Equations (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. [16
] and Equation (96
) 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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