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.
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