Black-hole uniqueness theorem

In four-dimensional asymptotically flat general relativity, a regular stationary vacuum black hole is a Kerr black hole under the standard connectedness and analyticity hypotheses. The electrovacuum extension is the Kerr--Newman family.

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.

The black-hole no-hair principle says that radiation carries away the higher multipole structure of a collapsing isolated body, leaving a stationary black hole specified by conserved asymptotic charges such as mass, angular momentum, and electric charge.