9.3 A specific construction for the Kerr spacetime
The angular momentum of Bergqvist and Ludvigsen [68] for the Kerr spacetime is based on their special
flat, non-symmetric 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 [170].)
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 4-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 1 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 2-surface
. Since this
is closed,
(similarly 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 1-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.