Extremal Isolated Horizons: A Local Uniqueness Theorem

TL;DR

The paper addresses a local, quasi-local uniqueness problem for extremal isolated horizons in vacuum and electrovac settings. By deriving and solving the horizon constraint equations for the geometry $(q,\mathbf{D})$ and the pulled-back EM field under symmetry assumptions, the authors show that the axisymmetric extremal electrovac IH data on the horizon coincide with those of the monopolar extremal Kerr–Newman horizon on the event horizon. They also classify symmetric IHs, prove at most one extremal IH structure on a given null surface, and establish that, in the axisymmetric case, the unique realization is the extremal Kerr–Newman horizon (monopolar when the magnetic charge vanishes). Together, these results provide a local, quasi-local characterization of the Kerr–Newman horizon and highlight the decisive role of extremal axisymmetric horizon data in reproducing Kerr–Newman geometry.

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.

Theorems & Definitions (10)

• Lemma 1
• Lemma 2
• Lemma 3
• proof : Proof
• Proposition 1
• Corollary 1
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 1