1
$\begingroup$

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!

$\endgroup$

1 Answer 1

3
$\begingroup$

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 $b$.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.