A physical classification of Killing magnetic fields in Thurston geometries

Abstract

In recent years, numerous studies have appeared that considered Killing vectors of three-dimensional Riemannian manifolds as magnetic fields, since these vector fields are divergenceless by definition. The existence of adivergenceless vector field modeled as a magnetic field does not imply that it is physically realizable. In this study, we propose a physical classification scheme based on the divergences of the integral that defines the energy of a Killing magnetic field. We consider all the Killing magnetic fields of Thurston geometries studied in the literature and classify them as either physical or nonphysical.