Chaos of charged particles in quadrupole magnetic fields under Schwarzschild backgrounds

TL;DR

This paper investigates how external quadrupole magnetic fields induce chaos in charged particle dynamics around Schwarzschild black holes, highlighting the dominant role of quadrupole fields over dipole fields in chaotic behavior.

Contribution

It demonstrates that quadrupole magnetic fields significantly enhance chaos in charged particle motion near black holes, using explicit symplectic integrators and Lyapunov indicators.

Findings

Quadrupole magnetic fields induce more chaos than dipole fields.

Lorentz forces from quadrupole fields are larger and act attractively.

Chaos depends on the strength and configuration of magnetic fields.

Abstract

A four-vector potential of an external test electromagnetic field in a Schwarzschild background is described in terms of a combination of dipole and quadrupole magnetic fields. This combination is an interior solution of the source-free Maxwell equations. Such external test magnetic fields cause the dynamics of charged particles around the black hole to be nonintegrable, and are mainly responsible for chaotic dynamics of charged particles. In addition to the external magnetic fields, some circumstances should be required for the onset of chaos. The effect of the magnetic fields on chaos is shown clearly through an explicit symplectic integrator and a fast Lyapunov indicator. The inclusion of the quadrupole magnetic fields easily induces chaos, compared with that of the dipole magnetic fields. This result is because the Lorentz forces from the quadrupole magnetic fields are larger than those from the dipole magnetic fields. In addition, the Lorentz forces act as attractive forces, which are helpful to bring the occurrence of chaos in the nonintegrable case.