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I am calculating surface gravity of Kerr Black Hole following 'A Relativist's toolkit' which uses the definition

$$ \left(-\xi^\beta \xi_\beta\right);_\alpha=2\kappa \xi_\alpha$$

where $\kappa$ is the surface gravity. However in the calculation it is used that on the horizon $$\xi_\alpha=\left(1-a\Omega_H\sin^2\theta\right)\partial_\alpha r$$

Can anyone point out how to get this expression for $\xi_\alpha$?

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Nearly 7 years later,I faced the same question and its easy to solve.

The coordinate $r$ is not well defined on the horizon, so we have to use other coordinate such as $v$ and $\psi$ in (5.61).

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