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with solution and the four complex parameters
defining the solution space. Next we
consider an arbitrary complex world line in the solution space,
, so that
, a GCF, which can be expanded in spherical harmonics as
The inverse function,
can be found by the following iteration process [40]: writing Eq. (6.2
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but for most of our calculations, all that is needed is the first iterate, given by
This relationship is, in principle, an important one. We also have the linearized reality relations – easily found earlier or from Eq. (6.7):
The associated angle field, , and the Bondi shear,
, are given parametrically by
As stated in Eqs. (4.6) – (4.10
), under (6.14
) the transformed asymptotic Weyl tensor becomes
The procedure for finding the complex center of mass is centered on Eq. (6.16), where we search for
and set to zero the
harmonic in
on a
= constant slice. This determines the
complex center-of-mass world line and singles out a particular GCF referred to as the UCF,
For the case of the Einstein–Maxwell fields, in general there will be two complex world lines and two
associated UCFs, one for the center of charge, the other for the center of mass. For later use we note that
the gravitational world line will be denoted by , while the electromagnetic world line by
.
Later we consider the special case when the two world lines and the two UCFs coincide, i.e.,
.
From the assumption that and
are first order and, from Eqs. (2.48
) and (2.49
) (e.g.,
), Eq. (6.16
), to second order, is
Using the spherical harmonic expansions (see Eqs. (6.12) and (6.13
)),
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or, re-arranging and performing the relevant Clebsch–Gordon expansions,
Note that though Eq. (6.28 This equation, though complicated and unattractive, is our main source of information concerning the
complex center-of-mass world line. The information is extracted in the following way: Considering only the
harmonics at constant
in Eq. (6.28
), we set the
harmonics of
(with constant
)
to zero (i.e.,
). The three resulting relations are used to determine the three spatial components,
, of
(with
). This fixes the complex center of mass in terms of
,
, and
other data which is readily interpreted physically. Alternatively it allow us to express
in terms of the
.
Extracting this information takes a bit of effort.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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