The Lorentz force equation as Fermi-Walker transport in geometrodynamics

TL;DR

This paper presents a geometric derivation of the Lorentz force equation as a form of Fermi-Walker transport within a new tetrad framework in curved spacetimes, emphasizing gauge invariance and Frenet-Serret analysis.

Contribution

It introduces a novel tetrad in geometrodynamics that simplifies electromagnetic fields and provides the first purely geometric proof of the Lorentz force's structure.

Findings

Derived the Lorentz force equation from Riemannian geometry.

Expressed the Lorentz force as a generalized Fermi-Walker transport.

Provided a geometric proof of the force's necessary structure.

Abstract

A new tetrad introduced within the framework of geometrodynamics for non-null electromagnetic fields allows for the geometrical analysis of the Lorentz force equation and its solutions in curved spacetimes. When expressed in terms of this new tetrad, the electromagnetic field displays explicitly maximum simplification, and the degrees of freedom are manifestly revealed. In our manuscript we are deducing the Lorentz force equation on purely Riemannian geometrical grounds. The equation arises on the basis of Frenet-Serret analysis through the use of our new tetrads and gauge invariance arguments only. The force is deduced through a geometrical construction that precludes any other mathematical form other than the one already accepted. Therefore, a significant and fundamental result such as the first geometrical proof on the necessity of the force in the equation to have the structure already accepted in physics and not any other, is given. Through the use of the Frenet-Serret formulae and gauge invariance arguments we are also able to express in terms of the new tetrad vectors the Lorentz force equation as a generalized form of Fermi-Walker transport.