Geometric Characterizations of the Kerr Isolated Horizon

We formulate conditions on the geometry of a nonexpanding horizon Δ which are sufficient for the spacetime metric to coincide on Δ with the Kerr metric. We introduce an invariant which can be used as a measure of how different the geometry of a given nonexpanding horizon is from the geometry of the Kerr horizon. Directly, our results concern the spacetime metric at Δ at the zeroth and the first orders. Combined with the results of Ashtekar, Beetle and Lewandowski, our conditions can be used to compare the spacetime geometry at the nonexpanding horizon with that of Kerr to every order. The results should be useful to numerical relativity in analyzing the sense in which the final black hole horizon produced by a collapse or a merger approaches the Kerr horizon.