The Non-metricity Formulation of General Relativity

Abstract

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 (Acta Phys Austr 54:165–189, 1981) (proved in Appendix A to be equivalent to Einstein–Hilbert). 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.