- #1

latentcorpse

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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 [itex]l[/itex] is

[itex]l=f g^{vr}|_{r=r_+} \partial_v + f g^{\chi r}|_{r=r_+} \partial_\chi[/itex]

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

[itex] \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}[/itex]

Surely this isn't right - that looks horrific!