Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 52 / 3 / b / iv

A generator of the Killing horizon is an orbit of $k+{\Omega }_{H}m$. In ingoing Kerr coordinates it has constant $r=r_{+}$ and $\theta$, with $d\chi /dv={\Omega }_H$. Since the coordinate shifts depend only on $r$, this is also $d\varphi /dt={\Omega }_H$ in the limiting Boyer-Lindquist coordinates description.

Thus ${\Omega }_H$ is the angular velocity of the horizon relative to the nonrotating stationary frame at infinity. The stationary Killing vector field $k$ is normalized to unit time translation there, while the axial Killing vector field has $2\pi$-periodic orbits. This normalization makes the Kerr horizon angular velocity physically definite. It describes the rotation of the null generators and the dragging of inertial frames, not a material solid surface rotating through space. In the Schwarzschild black hole limit $a=0$, ${\Omega }_H=0$.