I have two questions about this:
Surface gravity is defined on the Killing horizon by $\xi^\mu \nabla_\nu \xi^\nu = \kappa \xi^\nu$ for the Killing vector $\xi$. Why can we interpret this as the force required at infinity to hold a unit mass at rest on the horizon?
Is there an obvious reason why $\nabla^\mu \xi^2$ is orthogonal to the horizon? Clearly, I can expand this and get $\nabla^\mu \xi^2 = - 2 \xi^\rho \nabla_\rho \xi^\mu =-2 \kappa \xi^\mu$ which is orthogonal since $\xi$ is orthogonal and $\kappa$ is just a constant. But, I want to know if there is an obvious, intuitive way to see this without doing any calculations (the author says it's "obvious") or if the person that wrote the notes was expecting me to do the above calculation to see this?