A teleparallel effective geometry for Einstein's unified field theory

TL;DR

This paper explores a teleparallel geometric framework to unify gravitation and electromagnetism, deriving Maxwell's equations and Einstein's equations within this approach, highlighting new ways to interpret electric fields and potential black hole horizons.

Contribution

It introduces a teleparallel effective geometry that unifies gravity and electromagnetism, providing novel definitions of electric fields and deriving fundamental equations.

Findings

Effective electric field defined via torsion in teleparallelism

Maxwell equations derived from the teleparallel framework

Potential for effective electrodynamic black hole horizons

Abstract

Riemannian and teleparallel geometrical approaches to the investigation of Maxwell electrodynamics shown that a unified field theory of gravitation and electromagnetism a la Einstein can be obtained from a stationary metric. This idea contrasts with the recently proposed pre-metric electrodynamics by Hehl and Obukhov. In the teleparallel case the definition of the electric field is obtained straightforward from the spacetime metric and the orthonormal basis frame of teleparallelism. In this case the only nonvanishing component of Cartan torsion is defined as the effective electric field. In this approach the gravitational potentials or metric coefficients are expressed in terms of the effective or analogous electric and magnetic potentials. Thefore the Maxwell equations in vacuum can be obtained by derivation of this electric field definition as usual. In the Riemannian case we consider an electrostatic spacetime where the Einstein equations in vacuum in the approximation of linear fields. The constraint of Einstein equations in vacuum are shown to lead or to the Coulomb equation or to a singular behaviour on the metric which would represent a kind of effective electrodynamic black hole event horizon.