Einstein-Maxwell and Einstein-Proca theory from a modified gravitational action

TL;DR

This paper derives Einstein-Maxwell and Einstein-Proca theories from a modified gravitational action involving a gauge-like quantity, showing how adding specific terms yields these well-known field equations.

Contribution

It introduces a modified gravitational action involving a gauge-like quantity that leads to Einstein-Maxwell and Einstein-Proca equations through a Palatini variation.

Findings

Derivation of Einstein-Proca equations from the modified action.

Recovery of Einstein-Maxwell equations under specific scalar density conditions.

Demonstration of gauge invariance in the modified gravitational framework.

Abstract

A modified gravitational action is considered which involves the quantity $F_{\mu\nu}=\partial_{\mu}\Gamma_{\nu}-\partial_{\nu}\Gamma_{\mu}$, where $\Gamma_{\mu}=\Gamma^{\alpha}_{\mu\alpha}$. Since $\Gamma_{\mu}$ transforms like a U(1) gauge field under coordinate transformations terms such as $F^{\mu\nu}F_{\mu\nu}$ are invariant under coordinate transformations. If such a term is added to the usual gravitational action the resulting field equations, obtained from a Palatini variation, are the Einstein-Proca equations. The vector field can be coupled to point charges or to a complex scalar density of weight $ie$, where $e$ is the charge of the field. If this scalar density is taken to be $g^{-ie/2}$ and the overall factor of the scalar density Lagrangian takes on a particular value the resulting field equations are the Einstein-Maxwell equations.