Spinorial characterisations of rotating black hole spacetimes

Abstract

In this thesis, the implications of the existence of Killing spinors in a spacetime are investigated. In particular, it is shown that in vacuum and electrovacuum spacetimes a Killing spinor, along with some assumptions on the associated Killing vector in an asymptotic region, guarantees that the spacetime is locally isometric to a member of the Kerr or Kerr-Newman family. It is shown that the characterisation of these spacetimes in terms of Killing spinors is an alternative expression of characterisation results of Mars (Kerr) and Wong (Kerr-Newman) involving restrictions on the Weyl curvature and matter content. In the next section, the construction of a geometric invariant characterising initial data for the Kerr-Newman spacetime is described. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr-Newman initial data, and so characterises this type of data. First, the characterisation of the Kerr-Newman spacetime in terms of Killing spinors is illustrated. The space spinor formalism is then used to obtain a set of four independent conditions on an initial Cauchy hypersurface that guarantee the existence of a Killing spinor on the development of the initial data. Following a similar analysis in the vacuum case, the properties of solutions to the approximate Killing spinor equation are studied, and used to construct the geometric invariant. Finally, the problem of Killing spinor initial data in the characteristic problem is investigated. It is shown that data need only be speci ed on the bifurcation surface of the two intersecting null hypersurfaces in order to guarantee the existence of a Killing spinor in a neighbourhood of the bifurcation surface. This characterises the class of spacetimes known as distorted black holes, which include but is strictly larger than the Kerr family of spacetimes.