I have a question within the framework of spin-coefficient formalism / Newman-Penrose formalism. Suppose I have a space-time metric whose components with respect to some coordinate basis are known. I can then easily construct a null tetrad (to be more precise: the vector components of the tetrad vectors to the coordinate basis).
Since many calculations are easier if the corresponding spinor basis is a normalized spin frame $o_A\iota^A=1$ I would like to know if the spinor basis that corresponds to my null tetrad is such a normalized spin frame.
I am a bit lost at that point. Can I find a spinor basis in spinor space that corresponds to the null tetrad basis in tensor space and will it be a normalized spin frame? Or is there a multitude of possible spinor bases that can be normalized through some procedure? Does it depend on the specific null tetrad (for a given metric there is no unique null tetrad, but a number of independent possibilities)?