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For a Killing horizon associated with a Killing vector $K$, the surface gravity $\kappa$ can be computed by various methods, like $$ \kappa^2 = - \frac{1}{2} \nabla^\mu K^\nu \nabla_\mu K_\nu \ . $$ However Killing vectors multiplying a constant is still a Killing vector, and $c K$ is of course also null on the Killing horizon.

I wonder what is the criteria to fix the constant $c$? For Schwarzschild and Reissner-Nordstrom, I guess one can impose $|K|^2 \to -c^2$ at spacial infinity, but this doesn't seem to be the case for the Kerr black hole, since $$ |\partial_t + \Omega_H \partial_\varphi|^2 \to +\infty \ . $$

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    $\begingroup$ Great question! Of course the answer is hidden in the difference. $\endgroup$ Jun 7 at 16:09
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    $\begingroup$ However in a static, asymptotically free spacetime with time translation Killing vector $K=\partial_t$ one can set the normalisation $ \lim_{r\rightarrow \infty}K^2 =-1 $ and this fixes the surface gravity of any associated Killing horizon. In a stationary asymptotically flat spacetime the Killing horizon is associated with a linear combination of $K=\partial_t$ and $R=\partial_\phi$ and fixing the normalisation of $K$ fixes the linear combination and so also the surface gravity. $\endgroup$ Jun 7 at 17:03

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You're absolutely right. Because the Killing vectors are defined to satisfy Killing's equation $$ 2\nabla_{(a} K_{b)} = 0 \ , $$ a constant scaling is possible, and the value of $\kappa$ changes. You're also right that for the Schwarzschild metric (and for all static asymptotically flat spacetimes), one can fix the constant by demanding $K^a K_a|_{\infty} =-1$.

In general, there's no way to fix this constant. This is covered in a very nice discussion in the following paper https://arxiv.org/abs/0711.1445 where they say

This freedom corresponds to a gauge freedom to rescale the curve parameter along the integral curves of the Killing vector by a constant factor.

In the general case, there is no natural way to fix the parametrization of this generator.

They then go on to discuss some other more general ways to define the surface gravity $\kappa$ in terms of different definitions of horizons/trapped surfaces. But the conclusion is really that, in general, there's no prescribed way of doing this (and depends on the context of the problem you're working with).

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