Horizon angular velocity ($\chi =k+{\Omega }_{H}m$)

In an axisymmetric stationary spacetime, the horizon angular velocity is the constant coefficient in the null horizon generator $\chi =k+{\Omega }_{H}m$, where $k$ is the stationary Killing vector normalized at infinity and $m$ generates axial orbits with period $2\pi$. It measures rotation relative to the stationary frame at infinity. For a Kerr black hole, ${\Omega }_H=a/(r_{+}^2+a^2)$. It enters the rotating physical-process first law through the horizon flux of $E-{\Omega }_{H}J$.