The Relativistic Helmholtz Theorem and Solitons

Without reference to fluid mechanics, the necessary and sufficient equations for a three-dimensional vector field with the geometric and kinematic properties of a perfectly conserved vorticity field are derived. The equations are scale-invariant and Lorentz-invariant. The Lorentz limiting velocity varies with the length and time scales. The equations are an underdetermined set which require auxiliary equations. These may define energy and momentum as a function of the fields and require its conservation. One choice leads to the equations given by Einstein and by Pauli as necessary and sufficient for a purely electromagnetic particle (PEP) with internal charge/current density. Exact analytic solutions are presented for PEP solitons, each centered on a three-dimensional subspace filled with a vortexlike recirculating charge/current density. Such discrete conserved flux solitons infinite continuous fields are typical of systems satisfying this relativistic Helmholtz theorem.