The asymptotic structure of algebraically special spacetimes

Abstract

It is shown that the only vacuum algebraically special spacetime that is asymptotically simple is the Minkowski space. The structure of null infinity for algebraically special spacetimes satisfying a subset of the vacuum equations is shown to be particularly simple in terms of the coordinates used in Kerr's original reduction of the field equations. With the assumption of the existence of a global future null infinity, (in the sense of having topology ), past null infinity can be constructed with the same topology and canonically identified with future null infinity. This identifies the corresponding asymptotic data. The Bondi momentum on corresponding cuts of future null infinity and past null infinity sums to zero. In particular, the Bondi energy is negative or zero on one of future or past null infinities, thus implying singularities or negative energy densities in the interior if the spacetime is non-trivial. In appendix A the shear and radiation fields at null infinity are all derived. In appendix B the Cauchy - Riemann structure on the space of geodesics of the congruence underlying these spacetimes is considered and the auxiliary structures required to determine the spacetime are characterized intrinsically with respect to the Cauchy - Riemann structure.