Motion of charged particles in spacetimes with magnetic fields of spherical and hyperbolic symmetry

TL;DR

This paper investigates the motion of charged particles in curved spacetimes with magnetic fields of spherical and hyperbolic symmetry, deriving constants of motion and exploring geometric correspondences in various specific spacetimes.

Contribution

It introduces a unified approach to analyze charged particle trajectories in symmetric spacetimes with magnetic fields, revealing geometric correspondences and explicit solutions.

Findings

Constants of motion are derived using gauge-covariant momenta.

Trajectories in spherically symmetric cases relate to geodesics on Poincaré cones.

Explicit solutions are provided for Minkowski, AdS4×S2, and hyperbolic AdS-Reissner–Nordström spacetimes.

Abstract

The motion of charged particles in spacetimes containing a submanifold of constant positive or negative curvature is considered, with the electromagnetic tensor proportional to the volume two-form form of the submanifold. In the positive curvature case, this describes spherically symmetric spacetimes with a magnetic monopole, while in the negative curvature case, it is a hyperbolic spacetime with magnetic field uniform along hyperbolic surfaces. Constants of motion are found by considering Poisson brackets defined on a phase space with gauge-covariant momenta. In the spherically-symmetric case, we find a correspondence between the trajectories on the Poincar\'{e} cone with equatorial geodesics in a conical defect spacetime. In the hyperbolic case, the analogue of the Poincar\'{e} cone is defined as a surface in an auxiliary Minkowski spacetime. Explicit examples are solved for the Minkowski, $\mathrm{AdS}_4\times S^2$, and the hyperbolic AdS-Reissner--Nordstr\"{o}m spacetimes.