The Non-Metricity Formulation of General Relativity

After recalling the differential geometry of non-metric connections in the formalism of differential forms, we introduce the idea of a Non-Metricity (NM) connection, whose connection $1$--forms coincides with the non-metricity $1$--forms for a class of cobase fields. Then we formulate a theory of gravitation (equivalent to General Relativity (GR)) which admits a geometrical interpretation in a flat torsionless space where the gravitational field is completely manifest in the non-metricity of a NM connection. We define and then apply the non-metricity gauge to a gravitational Lagrangian density discovered by Wallner and which is equivalent to the Einstein-Hilbert Lagrangian density. The Einstein equations coupled to the matter currents $\left( \mathcal{J}_{\alpha}\right) $ thus becomes $\delta dg_{\alpha}=\mathcal{T}_{\alpha}+\mathcal{J}_{\alpha}$, where $\left( \mathcal{T}_{\alpha}\right) $ is identified as the gravitational energy-momentum currents, to which we shall find a relatively simple and physically appealing form. It is also shown that in the gravitational analogue of the Lorenz gauge, our field equations can be written as a system of Proca equations, which may be of interest in the study of propagation of gravitational-electromagnetic waves.