Uniqueness properties of the Kerr metric

We obtain a geometrical condition on vacuum, stationary, asymptotically hat spacetimes which is necessary and sufficient for the spacetime to be local ly isometric to Kerr. Namely, we prove a theorem stating that an asymptotic ally flat, stationary, vacuum spacetime such that the so-called Killing for m is an eigenvector of the self-dual Weyl tensor must be locally isometric to Kerr. Asymptotic flatness is a fundamental hypothesis of the theorem, as we demonstrate by writing down the family of metrics obtained when this re quirement is dropped. This result indicates why the Kerr metric plays such an important role in general relativity. It may also be of interest in orde r ro extend the uniqueness theorems of black holes to the non-connected and to the non-analytic case.