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

Abstract

A modified gravitational action is considered which involves the quantity $F_{\mu \nu }={\partial }_{\mu }{\Gamma }_{\nu }-{\partial }_{\nu }{\Gamma }_{\mu }$, where ${\Gamma }_{\mu }={\Gamma }_{\mu \alpha }^{\alpha }$. 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.

arXiv comment9 pages

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