### 8.3 A specific construction for the Kerr spacetime

Logically, this specific construction perhaps would have to be presented only in Section 12, but the
technique that it is based on may justify its placing here.
By investigating the propagation law (59, 60) of Ludvigsen and Vickers, for the Kerr spacetimes
Bergqvist and Ludvigsen constructed a natural flat, (but non-symmetric) metric connection [67]. Writing
the new covariant derivative in the form , the ‘correction’ term
could be given explicitly in terms of the GHP spinor dyad (adapted to the two principal null directions), the
spin coefficients , and , and the curvature component . admits a
potential [68]: , where .
However, this potential has the structure appearing in the form of the metric
for the Kerr-Schild spacetimes, where is the flat metric. In fact, the flat connection
above could be introduced for general Kerr-Schild metrics [170], and the corresponding
‘correction term’ could be used to find easily the Lánczos potential for the Weyl
curvature [10].

Since the connection is flat and annihilates the spinor metric , there are precisely two
linearly independent spinor fields, say and , that are constant with respect to and form a
normalized spinor dyad. These spinor fields are asymptotically constant. Thus it is natural to choose the
spin space to be the space of the -constant spinor fields, independently of the 2-surface
.

A remarkable property of these spinor fields is that the Nester-Witten 2-form built from them is closed:
. This implies that the quasi-local energy-momentum depends only on the homology class
of , i.e. if and are 2-surfaces such that they form the boundary of some hypersurface in ,
then , and if is the boundary of some hypersurface, then . In particular, for
two-spheres that can be shrunk to a point the energy-momentum is zero, but for those that can
be deformed to a cut of the future null infinity the energy-momentum is that of Bondi and
Sachs.