Surface gravity from the Killing derivative (${\kappa }^2=-\frac{1}{2}({\nabla }_a{K}_b)({\nabla }^a{K}^b)$)

At a regular Killing horizon, the generator is hypersurface orthogonal and satisfies ${\nabla }_{K}K=\kappa K$. The Killing derivative contraction identity gives the displayed squared surface gravity, including through a bifurcation surface by continuity. The generator's normalization is part of this formula.