Geometric Characterizations of the Kerr Isolated Horizon

We formulate conditions on the geometry of a non-expanding horizon $\mathrm{\Delta }$ which are sufficient for the space-time metric to coincide on $\mathrm{\Delta }$ with the Kerr metric. We introduce an invariant which can be used as a measure of how different the geometry of a given non-expanding horizon is from the geometry of the Kerr horizon. Directly, our results concern the space-time metric at $\text{IH}$ at the zeroth and the first orders. Combained with the results of Ashtekar, Beetle and Lewandowski, our conditions can be used to compare the space-time geometry at the non-expanding 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.