i'm losing my goddamned mind. i'm trying to show that the surface gravity of a black hole with metric \( g = -f(r)\,\text{d}t^{2}+f(r)^{-1}\,\text{d}r^{2}+g_{\mathbb{S}^{2}} \) is simply \( \kappa = f'(r_{s}) / 2 \), where \( r_{s} \) is the radius of the black hole. it's simple enough to pass to eddington-finkelstein coordinates, where the metric has the form \( g = -f(r)\,\text{d}v^{2}+2\,\text{d}v\,\text{d}r+g_{\mathbb{S}^{2}} \) to eliminate the coordinate singularity. i know that \( \partial_{v} \) is killing, so it should just be a matter of observing that \( (\nabla^{\alpha}g_{vv})\partial_{\alpha} = -2\kappa\partial_{v} \). it should be a simple computation from there, but i just can't seem to get it. it's driving me completely insane