Force-free Electrodynamics in Arbitrary Geometries

TL;DR

This paper introduces a geometric foliation method to find exact solutions of Force-Free Electrodynamics equations in various spacetimes, enhancing understanding of magnetized plasma dynamics near compact astrophysical objects.

Contribution

It extends the foliation formalism to new spacetimes, providing novel exact solutions and a class of solutions transitioning between different electromagnetic regimes.

Findings

Developed new exact solutions in multiple spacetimes.

Constructed vacuum degenerate fields in axisymmetric geometries.

Presented solutions transitioning between magnetic and electric dominance.

Abstract

The dynamics of highly magnetized plasmas in extreme astrophysical environments are effectively modeled by Force-Free Electrodynamics (FFE), a framework essential for studying objects like neutron stars and accreting black holes. The inherently nonlinear nature of the FFE equations makes finding exact solutions a challenging task. This paper explores an innovative approach to solving these equations by foliating spacetime into two-dimensional surfaces, specifically tailored to the geometry of electromagnetic fields. The foliation approach exploits the fact that the kernel of the force-free field defines an involutive distribution, naturally lending itself to a geometric decomposition. This method has previously been applied with great success in Kerr and FLRW spacetimes. By extending this formalism, we develop new exact solutions to the FFE equations in several distinct spacetimes, including cases where the metric remains partially undetermined. Additionally, we present a novel class of solutions that smoothly transition between magnetically dominated and electrically dominated regimes over time. Finally, we construct a pair of vacuum degenerate fields in arbitrary axisymmetric spacetimes, further demonstrating the utility of the foliation approach. These results offer a new perspective on the dynamical evolution of electromagnetic fields and their geometric underpinnings in various spacetime geometries, broadening the applicability of the framework for understanding the dynamics of highly magnetized plasmas in general relativistic astrophysical contexts.