About Twistor Spinors with Zero in Lorentzian Geometry

Details About Twistor Spinors with Zero in Lorentzian Geometry About Twistor Spinors with Zero in Lorentzian Geometry

2004-06-15

arxiv.org/abs/math/0406298v2

SIGMA 5 (2009), 079, 12 pages

Mathematics

Differential Geometry

Scientific paper

10.3842/SIGMA.2009.079

We describe the local conformal geometry of a Lorentzian spin manifold $(M,g)$ admitting a twistor spinor $\phi$ with zero. Moreover, we describe the shape of the zero set of $\phi$. If $\phi$ 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 $\phi$, which is a conformal Killing vector field, plays an important role for our discussion as well.

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