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?

thespin coefficients of a given spacetime is an ill-posed question. Those coefficients are only defined once a tetrad is chosen and that's a huge additional set of free parameters. $\endgroup$1more comment