Extremal Isolated Horizons: A Local Uniqueness Theorem

TL;DR

This paper classifies all axi-symmetric, extremal isolated horizons in vacuum and electrovac conditions, showing they are uniquely characterized by the extremal Kerr-Newman solution's horizon properties.

Contribution

It provides a comprehensive local uniqueness theorem for axi-symmetric extremal isolated horizons, linking them to the extremal Kerr-Newman solution.

Findings

All axi-symmetric extremal isolated horizons match the extremal Kerr-Newman horizon data.

The induced metric, rotation potential, and electromagnetic pullback are uniquely determined.

The classification extends to symmetric, extremal horizons with two null symmetries.

Abstract

We derive all the axi-symmetric, vacuum and electrovac extremal isolated horizons. It turns out that for every horizon in this class, the induced metric tensor, the rotation 1-form potential and the pullback of the electromagnetic field necessarily coincide with those induced by the monopolar, extremal Kerr-Newman solution on the event horizon. We also discuss the general case of a symmetric, extremal isolated horizon. In particular, we analyze the case of a two-dimensional symmetry group generated by two null vector fields. Its relevance to the classification of all the symmetric isolated horizons, including the non-extremal once, is explained.