Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 56 / 1 / e

In advanced coordinates, $K={\partial }_v$ is a Killing vector field and $K^2=-f$. The normal covector to $r=2m$ is $dr$; raising it gives $g^{ar}{\partial }_a={\partial }_v+f{\partial }_r$, which equals $K$ on the horizon. Thus $r=2m$ is a null hypersurface normal to $K$, and hence a Killing horizon.

The surface gravity is defined by ${\nabla }_{K}K=\kappa K$ on this horizon, equivalently ${\nabla }_a(K^2)=-2\kappa K_a$. Here $K_a=(-f,1,0,0)$, so the radial component gives $\kappa =f^{\prime }(2m)/2$. Because $F(2m)=0$,It is positive for all finite real $\alpha$ and $m>0$; the stationary Killing vector field retains its asymptotic normalization.