In the nonlinear terms we can simply use

We begin by focusing on the portion of the righthandside 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 lefthand 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

we readily determine 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 reexpressed 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 momentumspin 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).
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Living Rev. Relativity 15, (2012), 1
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