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