On the Motion of a Point Electric Charge in the Field of Wu-Yang Monopoles in Bertotti-Robinson Spacetimes

Abstract

We study the $SU(2)$ Einstein-Yang-Mills system with non-minimal $RF^2$-type couplings of Yang-Mills fields to curvature, formulated with exterior differential forms and a Lagrange-multiplier constraint for the Levi-Civita connection. We show that the conformally flat Bertotti-Robinson geometry $AdS_2\times S^2$ carries the Wu-Yang magnetic pole as an exact solution for arbitrary non-minimal couplings; the couplings only renormalise the radius of the $S^2$. The non-conformally-flat branch of solutions known for this system is recalled. The Hopf fibration $S^3/S^2\cong S^1$ of the gauge group manifold projects the Wu-Yang field on a uniform Dirac-type magnetic field, and the motion of a point electric charge in it is solved in closed form: the charge moves on a cone and oscillates between the horizon and a maximal radius.