Killing Horizons: Computing Metric Components in Kerr Coordinates

[latentcorpse]

So in these notes:
http://arxiv.org/PS_cache/gr-qc/pdf/9707/9707012v1.pdf

I'm trying to go from (4.28) to (4.29)

I find that the normal $l$ is

$l=f g^{vr}|_{r=r_+} \partial_v + f g^{\chi r}|_{r=r_+} \partial_\chi$

But then I need to compute these metric components. Since they are evaluated on the outer horizon which is a coordinate singularity if I use Boyer-Lindquist coordinates, I must use Kerr coordinates. However, since they are inverse metric components, and the Kerr metric is not diagonal, it looks like I have to compute the inverse matrix (at least of a 3x3 block) - writing it out, I think I need to find

$\begin{pmatrix} - \frac{\Delta -a^2 \sin^2{\theta}}{\Sigma} & 1 & \frac{-a \sin^2{\theta} ( r^2 + a^2 - \Delta)}{\Sigma} \\ 1 & 0 & -a \sin^2{\theta} \\ \frac{a \sin^2{\theta} ( r^2 + a^2 - \Delta)}{\Sigma} & -a \sin^2{\theta} & \frac{ (r^2 + a^2)^2 - \Delta a \sin^2{\theta}}{\Sigma} \sin^2{\theta} \end{pmatrix} ^{-1}$

Surely this isn't right - that looks horrific!

 

[NateD]

Am I missing something?Yes, you are missing something. The Kerr metric is in fact diagonal, so there is no need to explicitly compute the inverse matrix. The components of the normal l can be found by simply taking the appropriate components of the metric and evaluating them at r=r+.