Spinning charged test particle dynamics around a Schwarzschild black hole embedded in a homogeneous magnetic field

TL;DR

This paper investigates the complex motion of spinning charged particles around a Schwarzschild black hole in a magnetic field, revealing integrable and chaotic behaviors through numerical and phase space analysis.

Contribution

It provides analytical expressions for conserved quantities and explores the transition from integrable to chaotic dynamics in this astrophysical setting.

Findings

Equatorial motion remains integrable due to symmetries.

Off-equatorial motion exhibits chaos under certain conditions.

Phase space analysis reveals the influence of spin and magnetic field on particle dynamics.

Abstract

We study the dynamics of spinning charged test particles orbiting a Schwarzschild black hole immersed in a test uniform magnetic field. This setup provides a simple but physically relevant framework for modeling particle motion in magnetized astrophysical environments near compact objects, where both spin-curvature coupling and electromagnetic interactions can play a significant role. The particle trajectories are obtained numerically in both equatorial and off-equatorial configurations, allowing us to examine the influence of spin-curvature and Lorentz forces on the motion. In the equatorial plane, assuming the particle's spin vector is orthogonal to the orbital plane, we derive analytical expressions for the conserved energy and angular momentum, as well as for the radial and orbital frequencies as functions of spin parameter and magnetic parameter. We also construct the corresponding effective potential to determine the allowed regions of particle motion. The equatorial dynamics remain integrable due to the existence of conserved quantities associated with the spacetime symmetries and the alignment of the magnetic field. In contrast, the off-equatorial motion constitutes a non-integrable dynamical system. While limiting subcases of the system, i.e., the spinning neutral and non-spinning charged cases, can be analyzed using two-dimensional Poincar\'{e} surface of sections (PSs), the combined system can be reduced only up to three degrees of freedom. Hence, to investigate the resulting complexity, we analyze the phase space using four-dimensional PS along with recurrence analysis, revealing the presence of chaotic behavior for particular choices of parameters and initial conditions. Finally, we compare the dynamics of spinning charged test particles with the limiting cases, thereby distinguishing the respective contributions of spin-curvature and electromagnetic interactions.