Magnetic geodesics on Brinkmann spacetimes from parallel spinors

Abstract

The presence of spinorial structures in spacetime constrains its geometry and, consequently, influences the motion of particles. To describe this relation, we review the necessary elements of spin geometry, from Clifford algebras to spinC structures and parallel spinors. We then consider a four-dimensional supersymmetric Einstein–Maxwell background arising from Freedman’s gauged supergravity. In this setting, the existence of a parallel spinor determines a null parallel Dirac current, which forces the spacetime to be of Brinkmann type. To probe this spacetime, we study the dynamics of a charged test particle. Starting from the minimally coupled Lagrangian and exploiting the cyclic coordinates, we perform a Routh reduction, obtaining an effective Lagrangian on the twosphere. The resulting dynamics contains not only the electromagnetic interaction but also an effective contribution induced by the spacetime geometry itself. Moreover, the symmetries of the system allow us to reduce it further to a one-dimensional problem. We integrate the resulting equations to obtain the magnetic geodesics for charged particles in this background. We exhaustively classify the admissible solutions across the parameter space. The resulting branches describe constant, oscillatory, and asymptotic trajectories