Extending General Covariance: A Unified Framework for Electromagnetic Interactions and Geometric Unification

Abstract

This paper develops a geometrically symmetric framework that unifies gravity and electromagnetism by extending general covariance to incorporate electromagnetic interactions through an "electromagnetic tetrad." These tetrads, constructed from the electromagnetic four-potential in flat spacetime (absent gravity), preserve general covariance while providing a novel geometric formulation of electromagnetism. Within this framework, the Lorentz force equation naturally emerges from the geodesic equation, and the electromagnetic field’s Lagrangian is derived using the Ricci scalar R, establishing a direct geometric analogy with gravity. In the quantum regime, a novel formulation of the electromagnetic interaction is shown to be consistent with translation gauge invariance and the Weitzenböck connection derived from the electromagnetic tetrad. This extends a principle, previously utilized in gravity, to the domain of electromagnetism. By providing a geometric foundation for electromagnetism, this framework offers new insights into the interplay between fundamental interactions.