Geometric Characterizations of the Kerr Isolated Horizon

TL;DR

This paper establishes geometric conditions on non-expanding horizons that ensure the local spacetime metric matches the Kerr black hole solution, providing tools to compare actual horizons with Kerr in numerical relativity.

Contribution

It introduces invariant measures and conditions to determine how closely a non-expanding horizon resembles the Kerr horizon, extending previous results to higher orders.

Findings

Derived conditions for Kerr metric coincidence at the horizon

Introduced an invariant to measure deviation from Kerr geometry

Applicable to numerical relativity for black hole horizon analysis

Abstract

We formulate conditions on the geometry of a non-expanding horizon $\Delta$ which are sufficient for the space-time metric to coincide on $\Delta$ 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 $\IH$ 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.