Affine horizon-generator scaling ($\xi =\kappa \lambda k$)

If a Killing horizon generator $\xi$ has constant nonzero surface gravity $\kappa$, write $\xi =F(\lambda )k$ with $k=d/d\lambda$ affine. The equation ${\nabla }_{\xi }\xi =\kappa \xi$ implies $dF/d\lambda =\kappa$, so shifting the affine origin gives $\xi =\kappa \lambda k$.