I have been going through the Chandrasekhar's "The Mathematical Theory of Black holes", in particular the chapter on Newman Penrose formalism.

I have a question about what he calls a "class III transformation", where, given a null tetrad $\lbrace l,n,m,\overline{m} \rbrace$ we can rescale the null directions and rotate the other as

$\lbrace Al,A^{-1}n,e^{i\phi}m,e^{-i\phi}\overline{m} \rbrace$

where $A$ is a positive function. Such transformation indeed preserves the underlying orthogonality and normalization conditions. Furthermore, the direct consequence (I also don't know how to prove that) of such choice of scaling is that some spin coefficients vanish, in particular $\epsilon+\overline{\epsilon}=0$.

My question is: how do we know that the function $A$ *exists*?

**Attempt at the solution**:

I started by writing out the covariant derivative

$\nabla_{l}l=Dl^{a}=(\epsilon+\overline{\epsilon})l^{a}-\overline{\kappa}m^{a}-\kappa\overline{m}^{a}$.

What I want to show is that there exists a choice of scaling such that the right hand side vanishes. I then substituted $l\rightarrow Al$ but it didn't get me anywhere. I suppose the real solution is more complicated than that because I have to show the existence of a solution to a PDE but I still have no idea how to start.

This is not homework.