Surface gravity and affine parametrization

Question

Consider a Killing vector $\chi^\mu$ with the Killing Horizon $\Sigma$. From Carroll's book (pg 245), along the Killing horizon, the Killing vector obeys the geodesic equation $$\chi^\mu\,\nabla_\mu\,\chi^\nu = -\kappa\,\chi^\nu$$ where the right-hand side is non-zero because the integral curves of the Killing vector $\chi^\mu$ may not be affine-parametrized. The constant $\kappa$ is called surface gravity which is uniquely defined if the Killing vector is properly normalized.

My question is that we know the surface gravity is non-zero in general (e.g. non-extremal Schwarzschild, Kerr BH). What ensures that the integral curves of $\chi^\mu$ will not be affinely parametrized so that the right-hand side of the above equation is non-zero giving rise to a non-zero $\kappa$ ?

Also, for a general geodesic equation like the above one, isn't it true that it is always possible to choose an affine parametrization such that the RHS can be made zero. What happens to the surface gravity then?

Answer 1

The proof of (12.19) is surprisingly tricky. The reason for this is that although (12.19) is evaluated on the Killing horizon $\mathcal N$ , the fact that the expression involves derivatives of $\sigma$ means that one must first carry out manipulations that are valid away from $\mathcal N$, and only move onto the horizon after the derivatives are taken (Chris Pope, Gravitational Physics, p. 154)

http://people.physics.tamu.edu/pope/GravPhys2019/grav-phys.pdf