Petrov classification/Weyl scalars

Question

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!

Answer 1

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 $b$.