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 . 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 : , 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 , and the corresponding ‘correction term’ could be used to find easily the Lánczos potential for the Weyl curvature .
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.
© Max Planck Society and the author(s)