About Twistor Spinors with Zero in Lorentzian Geometry

We describe the local conformal geometry of a Lorentzian spin manifold (M, g) admitting a twistor spinor φ with zero. Moreover, we describe the shape of the zero set of φ. If φ has isolated zeros then the metric g is locally conformally equivalent to a static monopole. In the other case the zero set consists of null geodesic(s) and g is locally conformally equivalent to a Brinkmann metric. Our arguments utilise tractor calculus in an essential way. The Dirac current of φ, which is a conformal Killing vector field, plays an important role for our discussion as well.