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Killing Horizons

So in these notes:

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

I find that the normal  is
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
Surely this isn't right - that looks horrific!
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