### 6.1 A brief summary – Before continuing

Very briefly, for the purpose of organizing the many strands so far developed, we summarize our procedure for finding the complex center of mass. We begin with the gravitational radiation data, the Bondi shear, and solve the good-cut equation,

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

Assuming slow motion and the gauge condition (see Section 4), we have
(Though the world line is arbitrary, the quadrupole term, , and higher harmonics, are determined by both the Bondi shear and the world line.)

The inverse function,

can be found by the following iteration process [27]; writing Equation (262) as
with
the iteration relationship, with the zeroth-order iterate, , is
Though the second iterate easily becomes
For most of our calculations, all that is needed is the first iterate, given by
This relationship is important later.

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

and
using , while the asymptotically shear-free NGC is given (again) by the null rotation

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,

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

where has been replaced by the mass aspect, Equation (54), .

Using the spherical harmonic expansions (see Equations (272) and (273)),

Remembering that is zeroth order, Equation (281), becomes
or
Note that the right-hand side of Equation (287) depends initially on both and , with .

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.