 Kerr stationary-limit surface is generically timelike

For subextreme Kerr spacetime, the outer stationary-limit surface is $s=r-r_e(\theta )=0$, where $r_e=M+\sqrt{M^2-a^2{\cos }^2\theta }$. Away from the rotation axis, $r_e>r_{+}$ and $\Delta (r_e)=a^2{\sin }^2\theta >0$. Its normal is therefore spacelike and the surface timelike. Although the stationary Killing vector field has zero norm on it, it is not normal to it, and the surface is not a Killing horizon. The actual outer Killing horizon is $r=r_{+}$, generated by $K+{\Omega }_H\Phi$.