In the nonlinear terms we can simply use
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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
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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
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we readily determine that
Feeding (6.32) into Eq. (6.31
) and performing the relevant Clebsch–Gordon expansions, we find:
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
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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
:
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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 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,
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).
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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