Exact Solutions to Force-Free Electrodynamics in Black Hole Backgrounds

TL;DR

This paper derives a broad class of exact, time-dependent solutions to force-free electrodynamics in black hole backgrounds, revealing new wave behaviors and unifying previous solutions through their null current structure.

Contribution

It introduces a large, unified class of exact solutions to force-free electrodynamics in Kerr and Schwarzschild spacetimes, including non-axisymmetric and time-dependent cases, expanding the known solution landscape.

Findings

Solutions include waves propagating without scattering on black hole curvature.

Stationary axisymmetric solutions reduce to known solutions by Menon and Dermer.

In Schwarzschild and flat spacetime, solutions can incorporate magnetic monopoles, leading to magnetically dominated fields.

Abstract

A shared property of several of the known exact solutions to the equations of force-free electrodynamics is that their charge-current four-vector is \textit{null}. We examine the general properties of null-current solutions and then focus on the principal congruences of the Kerr black hole spacetime. We obtain a large class of exact solutions, which are in general time-dependent and non-axisymmetric. These solutions include waves that, surprisingly, propagate without scattering on the curvature of the black hole's background. They may be understood as generalizations to Robinson's solutions to vacuum electrodynamics associated with a shear-free congruence of null geodesics. When stationary and axisymmetric, our solutions reduce to those of Menon and Dermer, the only previously known solutions in Kerr. In Kerr, all of our solutions have null electromagnetic fields ($\vec{E} \cdot \vec{B} = 0$ and $E^2=B^2$). However, in Schwarzschild or flat spacetime there is freedom to add a magnetic monopole field, making the solutions magnetically dominated ($B^2>E^2$). This freedom may be used to reproduce the various flat-spacetime and Schwarzschild-spacetime (split) monopole solutions available in the literature (due to Michel and later authors), and to obtain a large class of time-dependent, non-axisymmetric generalizations. These generalizations may be used to model the magnetosphere of a conducting star that rotates with arbitrary prescribed time-dependent rotation axis and speed. We thus significantly enlarge the class of known exact solutions, while organizing and unifying previously discovered solutions in terms of their null structure.