Lorentz transformations on linked and knotted Maxwell fields

Abstract

A topological study of electromagnetic theory was initially done by Ranada in 1989, leading to the discovery of linked and knotted solutions to the Maxwell's equations in vacuum. In the most general form of the solution, the field lines of the electric and magnetic fields appear closed only when t=0. The Hopfion solution, however, a special case of the solution which has linking numbers n=m=l=s has field lines which are closed for all time due to the constant electric and magnetic helicities. Under a Lorentz transform, however, we find that the field lines for the Hopfion solution remain closed, albeit distorted, whereas the field lines of the knotted solutions no longer appear closed at t' = 0. This result implies that in general, the closedness and knottedness of the field equations are simply artifacts of the choice of the coordinate system.