= Solution
A <Killing horizon> is a <null hypersurface> with a <Killing vector field> $\xi$ as its normal and generator. Write $\xi=d/d\alpha$ along its orbits. These are <null geodesics>, but their Killing parameter need not be affine:
$$
\nabla_\xi\xi=\kappa\xi.
$$
The coefficient is the <surface gravity> for the chosen normalization; nondegeneracy means $\kappa\ne0$. If $\lambda$ is affine and $q=d\lambda/d\alpha$, then $\xi=q\,d/d\lambda$ and $d(\log q)/d\alpha=\kappa$. For constant $\kappa$, $\lambda=\lambda_0+C e^{\kappa\alpha}$.
The <Killing equation> yields $\partial_\mu(\xi^2)=2\xi^\nu\nabla_\mu\xi_\nu=-2\xi^\nu\nabla_\nu\xi_\mu$. On the horizon, \b[$\boxed{\partial_\mu\xi^2=-2\kappa\xi_\mu}$.] Multiplying $\xi$ by a positive constant multiplies $\kappa$ by that constant; physical temperature therefore needs a stated time normalization.
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