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

A modified gravitational action is considered which involves the quantity $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{\mathrm{\Gamma }}_{\nu }-{\mathrm{\partial }}_{\nu }{\mathrm{\Gamma }}_{\mu }$, where ${\mathrm{\Gamma }}_{\mu }={\mathrm{\Gamma }}_{\mu \alpha }^{\alpha }$. Since ${\mathrm{\Gamma }}_{\mu }$ 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 $g^{-ie\mathrm{/}2}$ and the overall factor of the scalar density Lagrangian takes on a particular value the resulting field equations are the Einstein-Maxwell equations.