Is Kerr the only smooth stationary vacuum black hole?
In plain words
The no-hair theorem says a settled black hole is fully described by its mass and spin. The classic proofs assume the spacetime is analytic, a far stronger smoothness condition than physics justifies, and without it the result is proved only near Kerr.
Precise statement
Prove that every smooth ($C^{\infty}$, not assumed real-analytic), asymptotically flat, stationary vacuum $3+1$ black hole exterior with a connected non-degenerate horizon is isometric to a Kerr exterior with $\mid a \mid < M$. Known results: true under analyticity (Hawking, Carter, Robinson), under conditions on the bifurcate sphere (Ionescu and Klainerman 2009), and for small perturbations of Kerr (Alexakis, Ionescu, Klainerman 2010). An answer is a proof or a smooth non-Kerr counterexample.
What would settle it
A rigidity theorem for smooth stationary vacuum black holes without analyticity or closeness-to-Kerr assumptions.
Status in the literature
Unverified note
Proved near Kerr (2010); the general smooth case is open as of 2026 (moderate confidence).