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),Update
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,
with the real version,
(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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