Gravitation and Electromagnetism as Geometrical Objects of a Riemann-Cartan Spacetime Structure

TL;DR

This paper demonstrates that gravitation and electromagnetism can be unified as geometric objects within a Riemann-Cartan spacetime, deriving fundamental equations from a single geometric Lagrangian.

Contribution

It introduces a geometric framework where both fields are encoded as objects of a Riemann-Cartan structure, unifying their equations from a common scalar curvature-based Lagrangian.

Findings

Maxwell and Einstein equations derived from a geometric scalar curvature.

Contorsion tensor encodes electromagnetic spin density information.

The theory is gauge invariant and models electromagnetic interactions with matter.

Abstract

In this paper we first show that any coupled system consisting of a gravitational plus a free electromagnetic field can be described geometrically in the sense that both Maxwell equations and Einstein equation having as source term the energy-momentum of the electromagnetic field can be derived from a geometrical Lagrangian proportional to the scalar curvature R of a particular kind of Riemann-Cartan spacetime structure, where those fields are identified as geometrical objects of the structure. We show moreover that the contorsion tensor of the particular Riemann-Cartan spacetime structure of our theory encodes the same information as the one contained in Chern-Simons term that is proportional to the spin density of the electromagnetic field. Next we show that by adding to the geometrical Lagrangian a term describing the interaction of a electromagnetic current with a general electromagnetic plus the gravitational field and a term describing the matter carrier of the current we get Maxwell equations with source term and Einstein equation having as source term the sum of the energy-momentum tensors of the electromagnetic and matter terms. Finally modeling by dust charged matter the carrier of the electromagnetic current we get the Lorentz force equation. Moreover, we prove that our theory is gauge invariant. We also briefly discuss our reasons for the present enterprise.