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.
Categories: gr-qc hep-th