Imposing time independence on the asymptotic Bianchi identities, Eqs. (2.52) – (2.54
),
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we have, using Eqs. (2.47) and 2.48
) with
, that
From Eq. (5.37), we find (after a simple calculation) that the imaginary part of
is determined by
the ‘magnetic’ [60
] part of the Bondi shear (spin-weight
) and thus must contain harmonics only of
. But from Eq. (5.35
), we find that
contains only the
harmonic. From this it follows
that the ‘magnetic’ part of the shear must vanish. The remaining part of the shear, i.e., the ‘electric’ part,
which by assumption is time independent, can be made to vanish by a supertranslation, via the Sachs
theorem:
From the mass identification, becomes
Our procedure for the identification of the complex center of mass, namely setting in the
transformation, Eq. (4.7
),
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leads, after using Eqs. (5.39), (5.34
) and (5.42
), to
From the time independence, , the spatial part of the world line is a constant vector. By a (real)
spatial Poincaré transformation (from the BMS group), the real part of
can be made to vanish, while
by ordinary rotation the imaginary part of
can be made to point in the three-direction. Using the the
gauge freedom in the choice of
we set
. Then pulling all these items together, we
have for the complex world line, the UCF,
and the angular momentum,
:
These results for the lower multipole moments, i.e., , are identical to those of the Kerr metric
presented earlier! The higher moments are still present (appearing in higher
terms in the Weyl tensor)
and are not affected by these results.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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