### 6.2 The complex center-of-mass world line

Before trying to determine the harmonics of Eq. (6.28), several comments and repetitions (for emphasis) are in order:

1. As previously noted, Eq. (6.28) is a function of both (via the , ) and (via the and ). The extraction of the part of must be taken on the constant cuts. In other words must be eliminated by using Eq. (6.2).
2. This elimination of (or ) is done in the linear terms via the expansion:

In the nonlinear terms we can simply use

3. In the Clebsch–Gordon expansions of the harmonic products, though we need both the and terms in the calculation, we keep at the end only the terms for the . (Note that there are no terms since is spin weight .)
4. For completeness, we have included into the calculations Maxwell fields with both a complex dipole (electric and magnetic), and complex quadrupole (electric and magnetic) fields .

We begin by focusing on the portion of the right-hand-side of Eq. (6.28) in the complex center of mass frame. Assuming that , we see that all remaining terms on this side of the equation are nonlinear, so we can simply make the replacement . On the left-hand side of the equation, extracting the component of on a constant slice is more complicated though; using Eq. (6.30), we have that

Using the Bianchi identity (2.53) and inserting the proper factor of to account for the retarded Bondi time, we have that (to second order)

As enters Eq. (6.31) only in nonlinear terms, we only need to extract the linear portion of this Bianchi identity. Recalling (suppressing factors of for the time being) that

Feeding (6.32) into Eq. (6.31) and performing the relevant Clebsch–Gordon expansions, we find:

Here, we have implicitly used the fact that, to our level of approximation, .

We can now incorporate this into Eq. (6.28) to obtain the full complex center of mass equation as a function of . The components of the mass aspect are replaced by the expressions

we insert and the appropriate factors of elsewhere, at which point Eq. (6.28) can be re-expressed in a manner that determines the complex center of mass, with all terms being functions of :

Note that the linear term

coincides with the earlier results in the stationary case, Eq. (5.44).

Now, we recall our identification for the complex gravitational dipole,

as well as the identification between the harmonic coefficient of the UCF and the gravitational quadrupole:
Feeding these into (6.34), we can separate out the real and imaginary parts via (6.35) to obtain expressions for the mass dipole and angular momentum, our primary results:

Though these results are discussed at greater length later, we point out that Eqs. (6.37) and (6.38) already contains terms of obvious physical interest. Note that the first two items in are the spin,

(identified via the special case of the Kerr–Newman metric) and the orbital angular momentum
The mass dipole has the conventional term and a momentum-spin coupling term (which appears to be new):

We will see shortly that there is also a great deal of physical content to be found in the nonlinear terms of Eq. (6.34).