A nonlinear relativistic approach to mathematical representation of vacuum electromagnetism based on extended Lie derivative

TL;DR

This paper introduces a novel nonlinear relativistic framework for vacuum electromagnetism using extended Lie derivatives, emphasizing local energy-momentum dynamics in classical electrodynamics.

Contribution

It develops a new mathematical formulation based on extended Lie derivatives to describe vacuum electromagnetism, offering a dynamical perspective on the differential equations involved.

Findings

New extended Lie derivative-based equations for vacuum electromagnetism

Reconceptualization of Maxwell's equations as local energy-momentum relations

Potential for improved understanding of nonlinear electromagnetic phenomena

Abstract

This paper presents an alternative {\it relativistic nonlinear} approach to the vacuum case of classical electrodynamics. Our view is based on the understanding that the corresponding differential equations should be dynamical in nature. So, they must represent local energy-momentum balance relations. Formally, the new equations are in terms of appropriately extended Lie derivative of $\mathbb{R}^2$-valued differential 2-form along a $\mathbb{R}^2$-valued 2-vector on Minkowski space-time.