Exact Solutions for Extreme Black Hole Magnetospheres

TL;DR

This paper derives and classifies exact force-free electrodynamics solutions near extremal Kerr black holes, revealing hidden symmetries and generating a broad family of solutions with various symmetries and time dependencies.

Contribution

It introduces a complete classification of exact FFE solutions in NHEK, uncovers a hidden SL(2,C) symmetry, and develops a superposition principle for more general solutions.

Findings

Classified solutions forming highest-weight representations.

Discovered a hidden SL(2,C) symmetry in FFE equations.

Generated a family of solutions with diverse symmetries and time dependence.

Abstract

We present new exact solutions of Force-Free Electrodynamics (FFE) in the Near-Horizon region of an Extremal Kerr black hole (NHEK) and offer a complete classification of the subset that form highest-weight representations of the spacetime's SL(2,R) x U(1) isometry group. For a natural choice of spacetime embedding of this isometry group, the SL(2,R) highest-weight conditions lead to stationary solutions with non-trivial angular dependence, as well as axisymmetry when the U(1)-charge vanishes. In addition, we unveil a hidden SL(2,C) symmetry of the equations of FFE that stems from the action of a complex automorphism group, and enables us to generate an SL(2,C) family of (generically time-dependent) solutions. We then obtain still more general solutions with less symmetry by appealing to a principle of linear superposition that holds for solutions with collinear currents and allows us to resum the highest-weight primaries and their SL(2,R)-descendants.