Imposing time independence on the asymptotic Bianchi identities, Equations (56) – (58),
From Equation (252), we find (after a simple calculation) that the imaginary part of is determined by the ‘magnetic’  part of the Bondi shear (spin-weight ) and thus must contain harmonics only of . But from Equation (250), 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, Equation (276),
leads, after using Equations (254), (249) and (257), 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. The higher moments are still present (appearing in higher terms in the Weyl tensor) and are not affected by these results.
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