I came across a statement in Chandrasekhar's "Mathematical Theory of Black Holes" that I don't understand (rather say disagree):

Assume we have a Newman Penrose tetrad $\lbrace l, n,m,\overline{m}\rbrace$. We then arrive at the equation

$l_{i;j}l^{j}=(\epsilon+\overline{\epsilon})l_{i}-\kappa \overline{m_{i}}-\overline{\kappa}m_{i}$

($\kappa$ and $\epsilon$ are the spin coefficients). Then he says that *the $l$-vectors form a congruence of null geodesics if and only if $\kappa=0$; and further, they are affinely parametrized if and only if in addition $\epsilon=0$*.

Now, I don't agree with the second part of the statement. Shouldn't it be when $\epsilon+\overline{\epsilon}=0$, i.e. $\Re(\epsilon)=0$? Of course, if $\epsilon=0$ the affine parametrization holds, but it's a bigger constraint than just $\Re(\epsilon)=0$.

Can anyone tell me what's wrong with my reasoning?

Thank you