Time evolving potentials for electromagnetic knots

Abstract

Electromagnetic knots are electromagnetic fields in which the magnetic and electric lines are level curves of two complex scalar fields. If these scalar fields are chosen so that they can be interpreted as maps between the three-sphere and the two-sphere every time, then the electromagnetic helicity is proportional to the sum of the Hopf indices of both maps, that are topological invariants. An important example of this kind of electromagnetic fields with topological properties is often called the Hopfion. It is an electromagnetic knot in which the scalar fields are built from Hopf maps. In this work, we study the conditions for these electromagnetic fields to be null, i.e. to have both Lorentz invariant quantities [Formula: see text] and [Formula: see text] equal to zero. We derive from these fields explicit vector potentials [Formula: see text] and [Formula: see text] satisfying force-free like conditions for every time. In particular, equations [Formula: see text], [Formula: see text] are derived for them. As a consequence, the energy, which is discretized, is proportional to the electromagnetic helicity. This relation between Physics and Topology is intriguing and worth of future research.