 Surface gravity of a static spherical horizon

For a static spacetime with $g_{tt}=-A(r)$, $g_{rr}=1/B(r)$ and simple simultaneous horizon zeros, the squared surface gravity of ${\partial }_t$ is $\lim B{A}^{\prime 2}/(4A)=A^{\prime }(r_{+})B^{\prime }(r_{+})/4$. Choose the positive value for the outer horizon. A rescaling of the time generator rescales its surface gravity.