### 9.3 A specific construction for the Kerr spacetime

The angular momentum of Bergqvist and Ludvigsen [86] for the Kerr spacetime is based on their special
flat, nonsymmetric but metric, connection explained briefly in Section 8.3. But their idea is not simply the
use of the two -constant spinor fields in Bramson’s superpotential. Rather, in the background of their
approach there are twistor-theoretical ideas. (The twistor-theoretic aspects of the analogous flat connection
for the general Kerr–Schild class are discussed in [234].)
The main idea is that, while the energy-momentum is a single four-vector in the dual of the Hermitian
subspace of , the angular momentum is not only an anti-symmetric tensor over the same space,
but should depend on the ‘origin’, a point in a four-dimensional affine space as well, and should
transform in a specific way under the translation of the ‘origin’. Bergqvist and Ludvigsen defined the affine
space to be the space of the solutions of , and showed that is, in fact,
a real, four-dimensional affine space. Then, for a given , to each -constant spinor field
they associate a primed spinor field by . This turns out to satisfy the
modified valence-one twistor equation . Finally, they form the 2-form

and define the angular momentum with respect to the origin as times the
integral of on some closed, orientable spacelike two-surface . Since this is
closed, (similar to the Nester–Witten 2-form in Section 8.3), the integral depends
only on the homology class of . Under the ‘translation’ of the ‘origin’ by a
-constant one-form , it transforms as , where the components
are taken with respect to the basis in the solution space. Unfortunately, no
explicit expression for the angular momentum in terms of the Kerr parameters and is
given.