I am reading An Introduction to General Relativity Spacetime and Geometry by Sean Carroll, but simple calculations stop me.
At page 245, a formula for the surface gravity is given $$\kappa^2=-\frac{1}{2}(\nabla_\mu\chi_\nu)(\nabla^\mu\chi^\nu) \tag{6.9}$$ where $\kappa$ is a parameter called surface gravity, $\chi$ is the Killing vector with a Killing horizon.
Can you derive the above formula step by step using Killing's equation $\nabla_{(\mu}\chi_{\nu)}=0$ and the fact that $\chi_{[\mu}\nabla_\nu\chi_{\sigma]}=0$ ? The starting point is the geodesic equation : $\chi^\mu\nabla_\mu\chi^\nu=-\kappa\chi^\nu$.