I've read the following formulation of the Goldberg-Sachs theorem in Chandrasekhar's Mathematical theory of black holes:
Spacetime is Petrov type D iff $\kappa=\sigma=\lambda=\nu=0$.
Now, I was wondering about the following: which other spin coefficients are allowed to vanish for the space-time still to be Type D? I know that, for example, one could require $\epsilon=0$ (so that $\nabla_{l}l=0$) and that doesn't affect the space-time.
What about the rest of the spin coefficients?
I'll be grateful for any hints.
This post imported from StackExchange MathOverflow at 2015-08-08 15:48 (UTC), posted by SE-user GregVoit