= Black-hole rigidity theorem
For a regular analytic stationary asymptotically flat vacuum black hole with a connected nondegenerate horizon, the stationary <Killing vector field> either generates the horizon or the spacetime admits an additional commuting axial <Killing vector field>. In the rotating case the horizon generator is $K+\Omega_H\Phi$, where $\Phi$ has closed rotational orbits. Together with the appropriate staticity and <black-hole uniqueness theorem>, this identifies an equilibrium uncharged collapse remnant with the <Kerr metric>. The theorem concerns an equilibrium geometry; it does not itself prove that arbitrary dynamical collapse settles to equilibrium.
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