Helmholtz theorem for antisymmetric second-rank tensor fields and electromagnetism with magnetic monopoles

Abstract

A generalized Helmholtz’s theorem is proved, which states that an antisymmetric second-rank tensor field in 3+1 dimensional space-time, which vanishes at spatial infinity, is determined by its divergence and the divergence of its dual. When the divergence of the antisymmetric electromagnetic field strength tensor is equal to the electric charge-current density and the divergence of the dual of the electromagnetic field strength tensor is equal to the magnetic charge-current density, the equations of electromagnetism are obtained. As a convenience, in the solution of the equations of electromagnetism two different four-vector potentials can be used, one of which couples to the electric charge-current density and the other to the magnetic charge-current density.