= Solution
A generator of the <Killing horizon> is an orbit of $k+\Omega_Hm$. 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 $d\phi/dt=\Omega_H$ in the limiting <Boyer-Lindquist coordinates> description.
Thus \b[$\Omega_H$ 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 $a=0$, $\Omega_H=0$.
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