clarification on Goldberg-Sachs theorem

We will have to shut down our server temporarily for maintenance. The downtime will start at Wednesday, 27. January 2021 at 12:00 GMT and have a duration of about two hours. Please save your edits before this time. Thanks for your patience and your understanding.

+ 2 like - 0 dislike
82 views

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
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\varnothing$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.