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 [40]: writing Eq. (6.2) as
with
the iteration relationship, with the zeroth-order iterate, , is
To second order, this is

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

and
using the inverse to , Eq. (6.7). The asymptotically shear-free NGC is given by performing the null rotation

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,

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 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

where has been replaced by the mass aspect (2.50): .

Using the spherical harmonic expansions (see Eqs. (6.12) and (6.13)),

and remembering that is zeroth order, Eq. (6.22), becomes

or, re-arranging and performing the relevant Clebsch–Gordon expansions,

Note that though Eq. (6.28) depends initially on both and , with , we will eventually replace all the (or ) by their expressions in terms of , using Eq. (6.2). The transformation equation is then a function only of and (), at least to the given order in our perturbative framework.

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.