An Electrodynamic Lifting of the Toda Monopoles in Curved 3+1 Spacetime

Abstract

Every solution of the hyperbolic Toda equation determines a classical Abelian monopole on a (2+1)-dimensional Einstein--Weyl space. We determine when its connection is also a source-free electromagnetic potential on a curved Lorentzian 3+1 spacetime with a spacelike Killing field. On the generic branch, Maxwell's equation classifies the admissible Killing-fibre scale and horizontal twist through one characteristic of the Toda field. A separated Liouville branch gives purely magnetic fields with considerably freer fibre geometry. The round spherical solution yields a monopole of unit charge with conserved flux on a contracting screen. The background metric here is neither assumed to be Einstein nor self-dual.