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?



Minkowski spacetime in Newman Penrose formalism
The one thing which I think you are missing is that



> 
> Unlike the Riemann curvature, the spin coefficients are not tensors (they are not co- or contra-variant). 
> 
> 
> 


In particular, the spin coefficients heavily depend on the choice of the tetrad. 


In Minkowski space you can construct "accelerated frames"; these (non-inertial) tetrads in which the spin coefficients do not identically vanish are somewhat important in terms of the interpretation of the equivalence principle. What you can observe, however, is that with regards to *any* tetrad, Minkowski space satisfies the condition that the curvature scalars $\Psi\_0, \ldots, \Psi\_4$ all vanish identically. This fact uniquely characterizes Minkowski space among Ricci-flat solutions, and if you also include conditions on the Ricci curvature scalars (that they all vanish), this uniquely (locally) characterizes Minkowski space. 


In general, however, if you are willing to pick very wild tetrads, you can get very wild values of the spin coefficients. 




---


You mentioned Golberg Sachs. Pay attention that it does not say "Algebraically special iff such and such coefficients vanish." It says "Algebraically special iff *there exists a tetrad* in which such and such coefficients vanish." One can certainly consider tetrads for Type D spacetimes which are not geodesic!


