Teleparallel Lagrange Geometry and a Unified Field Theory

TL;DR

This paper develops a purely geometric unified field theory combining gravity and electromagnetism within an extended absolute parallelism geometry, generalizing previous theories and deriving new field equations from a tangent bundle framework.

Contribution

It introduces a novel geometric framework unifying gravity and electromagnetism, extending the Generalized Field Theory using EAP-geometry and deriving generalized Einstein and Maxwell equations.

Findings

Derived generalized Einstein equations with geometric energy-momentum tensors.

Formulated generalized Maxwell equations with purely geometric electromagnetic fields.

Explored special cases highlighting the nonlinear connection's role in the theory.

Abstract

In this paper, we construct a field theory unifying gravity and electromagnetism in the context of Extended Absolute Parallelism (EAP-) geometry. This geometry combines, within its structure, the geometric richness of the tangent bundle and the mathematical simplicity of Absolute Parallelism (AP-) geometry. The constructed field theory is a generalization of the Generalized Field Theory (GFT) formulated by Mikhail and Wanas. The theory obtained is purely geometric. The horizontal (resp. vertical) field equations are derived by applying the Euler-Lagrange equations to an appropriate horizontal (resp. vertical) scalar Lagrangian. The symmetric part of the resulting horizontal (resp. vertical) field equations gives rise to a generalized form of Einstein's field equations in which the horizontal (resp. vertical) energy-momentum tensor is purely geometric. The skew-symmetric part of the resulting horizontal (resp. vertical) field equations gives rise to a generalized form of Maxwell equations in which the electromagnetic field is purely geometric. Some interesting special cases, which reveal the role of the nonlinear connection in the obtained field equations, are examined. Finally, the condition under which our constructed field equations reduce to the GFT is explicitly established.