[Paper Review] Unification of gravity and electromagnetism revisited

Ahmad Borzou, Hamid Reza Sepangi|arXiv (Cornell University)|Apr 8, 2009

Cosmology and Gravitation Theories6 references3 citations

TL;DR

This paper proposes a 5D unified theory of gravity and electromagnetism where Einstein's equations and modified Maxwell equations emerge naturally from a geometric framework. By introducing a five-dimensional metric with a dynamical fifth dimension, the theory derives the homogeneous Maxwell equations from the torsion-free condition and includes a curvature-dependent correction term in the inhomogeneous Maxwell equations, potentially explaining anomalously strong galactic magnetic fields without new physics beyond the standard model.

ABSTRACT

Within the context of a $5D$ space-time, we construct a unified theory of gravity and electromagnetism from which the Einstein field equations and Maxwell equations emerge, with homogenous Maxwell equations appearing naturally. We also introduce a well-defined five dimensional energy-momentum tensor consistent with our unification scheme. A correction term appears in Maxwell equations which can be used to explain the recently discovered galactic magnetic fields.

Motivation & Objective

• To unify gravity and electromagnetism within a 5D geometric framework that naturally reproduces both Einstein's and Maxwell's equations.
• To resolve the issue in Kaluza-Klein theory where homogeneous Maxwell equations are not derived but only implicitly satisfied.
• To introduce a physically meaningful five-dimensional energy-momentum tensor consistent with the unification scheme.
• To explain the origin of recently observed strong galactic magnetic fields through a curvature-dependent correction in Maxwell's equations.

Proposed method

• Construct a 5D metric tensor with a dynamical fifth dimension, parameterized by a scalar field φ and a 4D gauge potential Aα.
• Derive the 5D Ricci tensor and Einstein tensor from the metric, leading to field equations that decompose into 4D Einstein and Maxwell equations.
• Use the torsion-free condition on the 5D connection to derive the homogeneous Maxwell equations (dF = 0) as a geometric consequence.
• Identify the inhomogeneous Maxwell equations with a correction term involving the 4D scalar curvature R, arising from the 5D field equations.
• Apply coordinate transformations that preserve the 4D hypersurface structure, ensuring consistency with 4D observations.
• Introduce a well-defined 5D energy-momentum tensor that couples gravity and electromagnetism consistently.

Experimental results

Research questions

• RQ1 Can the homogeneous Maxwell equations be derived geometrically from a 5D unification framework, rather than being imposed as constraints?
• RQ2 How does the inclusion of a dynamical fifth dimension modify the standard Maxwell equations in 4D?
• RQ3 What is the physical origin of the correction term in the inhomogeneous Maxwell equations, and how does it affect electromagnetic field strength?
• RQ4 Can the curvature-dependent correction term explain the observed strength of galactic magnetic fields, which exceed standard model predictions?
• RQ5 Is it possible to define a consistent 5D energy-momentum tensor that unifies gravitational and electromagnetic energy-momentum in a single geometric structure?

Key findings

• The homogeneous Maxwell equations emerge naturally from the torsion-free condition of the 5D connection, providing a geometric derivation rather than an imposed constraint.
• The inhomogeneous Maxwell equations are modified by a term proportional to the 4D scalar curvature R, resulting in the equation ∇αFαλ − g4λR = 2κT4λ.
• In regions of large scalar curvature, the correction term enhances the effective electromagnetic field strength beyond standard Maxwell predictions.
• The theory reproduces the standard 4D Einstein equations with a source term that includes the electromagnetic energy-momentum tensor.
• The 5D energy-momentum tensor is explicitly constructed and shown to be consistent with the unification scheme.
• The model provides a geometric explanation for anomalously strong galactic magnetic fields, suggesting that curvature effects may account for their observed intensity.