Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 54 / 3 / e

In the ingoing Eddington-Finkelstein coordinates, the stationary Killing vector field is $k={\partial }_v$ with norm $k^2=-f$. The normal vector to a level set of $r$ isAt $r=r_{+}$ it equals $k^a$, which is null there. Thus this null hypersurface is a Killing horizon of $k$. Its surface gravity is fixed by ${\nabla }_a(k^2)=-2\kappa k_a$ on the horizon. Since $k_a=(dr{)}_a$ there and ${\nabla }_a(k^2)=-f^{\prime }(r_{+})(dr{)}_a$,The positive sign follows from the future outer-horizon generator, normalized by the static time at infinity. It vanishes as the two radii approach the extremal limit.