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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,
We also have the linearized reality relations – easily found earlier or from Equation (267):
The associated angle field, , and the Bondi shear,
, are given parametrically by
From Equation (274), the transformed asymptotic Weyl tensor becomes, Equations (275
) – (279
),
The procedure is centered on Equation (276), where we search for and set to zero the
harmonic
in
on an s = constant slice. This determines the complex center-of-mass world line and singles out a
particular GCF referred to as the UCF,
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with the real version,
for the gravitational field in the general asymptotically-flat case. (For the case of the Einstein–Maxwell fields, in general, there will be two complex world lines, one for
the center of charge, the other for the center of mass and the two associated UCFs. 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 Equation (52
), that
,
Equation (276
), to second order, is
Using the spherical harmonic expansions (see Equations (272) and (273
)),
This equation, though complicated and unattractive, is our main source of information concerning the
complex center-of-mass world line. Extracting this information, i.e., determining at constant values of
by expressing
and
as functions of
and setting it equal to zero, takes considerable
effort.
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