Covariant Formulation of EM-Force

Abstract

We introduce a covariant Lorentz force written in terms of the newly formed 4-potential vector. We show that it satisfies the established constraints: it is a spacelike 4-vector and it is 4-orthogonal to the 4-velocity. The 0th component of the Lorentz 4-force is recognized as describing the energy transfer from the electromagnetic-field to the particle motion. We next seek a covariant variation-principle. Several approaches are seen in literature: A) the non-covariant relativistic generalization of the usual three dimensional action; B) the covariant generalization based on particle proper time as the evolution parameter; C) a variation of the particle proper times, making it a part of dynamics for the kinetic energy only; and D) we also describe an extension of the case to be fully consistent, with both kinetic and potential terms in the action treated in the same fashion. These different approaches attempt to resolve in different fashions the need to implement the constraint $c^{2}=u^{2}$ which reduced the number of independent velocity components. An approach that avoids this difficulty was proposed long ago, but in essence was lost in literature; we reintroduce this Hamiltonian-like 4-dimensional alternative based on a mass-function.