I have a rather basic question for which (surprisingly!) I cannot find a short and clear answer anywhere:
I'm currently looking at the Newman Penrose (NP) formalism (I use primarily Chandrasekhar's "Mathematical Theory of Black Holes").
My question is: what exactly is Minkowski spacetime in NP formalism? That is, which are the vanishing/non-vanishing spin-coefficients?
In particular, is just vanishing of some of those coefficients sufficient to characterize the Minkowski spacetime (in a similar way that Goldberg Sachs theorem characterizes the algebraically special spacetimes).
My thoughts so far:
Clearly, we should have $\kappa=\sigma=\mu=\lambda=0$ since that gives us Type D by Goldberg Sachs theorem, but Minkowski is more than that. Say we also put $\epsilon=0$ for a suitable tetrad scaling. But what about the rest?