There is one calculation in Chandrasekhar's "Mathematical Theory of Black Holes" that I cannot understand. Here is the setup:


We want to show that Petrov type D (i.e. two principal null directions) corresponds to the only non-vanishing Weyl scalar $\Psi\_{2}$. To see this, we first rotate the null tetrad $\lbrace l,n,m,\overline{m}\rbrace$ by a class II rotation with parameter $b$ (complex function), i.e. 


$n\rightarrow n, m\rightarrow m+bn, \overline{m}\rightarrow \overline{m}+b^{\*}n$ and $l\rightarrow l+b^{\*}m+b\overline{m}+bb^{\*}n$.


He then writes down an expression for $\Psi\_{0}$ for the transformed tetrad, and then says that *the values of the remaining scalars can be obtained by successively differentiating the expression for* $\Psi\_{0}$ *and normalizing at each stage to have the same coefficient for the highest power of* $b$.


Now, I'm very confused by this statement. I know what the definitions of the Weyl scalars are, as contractions of the Weyl tensor. What I don't understand, how is it that they are the successive *derivatives* of each other?


Many thanks!



Petrov classification/Weyl scalars
You should re-read the immediately previous section on Petrov Type II first. There he did the computation is a tiny bit more detail. 


Then you will realize that since we are dealing with a fourth order polynomial in $b$ which, by assumption, has two roots each with multiplicity 2, the notion of derivative here is **differentiation with respect to $b$**.


The whole point is that, e.g., the derivative relative to $b$ of the transformation law for $\Psi\_0$ is (4 times) the transformation law for $\Psi\_1$, etc. So each time you have a principal null direction with algebraic multiplicity $k$, you can make $\Psi\_0$ up to $\Psi\_{k-1}$ vanish simultaneously with that choice of $b$.


