# The Non-metricity Formulation of General Relativity

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 (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 $$\left( \mathcal {J}_{\alpha }\right)$$ J α thus becomes $$\delta dg_{\alpha }=\mathcal {T}_{\alpha }+\mathcal {J}_{\alpha }$$ δ d g α = T α + J α , where $$\left( \mathcal {T}_{\alpha }\right)$$ T α 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Applied Clifford Algebras Springer Journals

# The Non-metricity Formulation of General Relativity

, Volume 27 (3) – Jan 3, 2017
32 pages

/lp/springer_journal/the-non-metricity-formulation-of-general-relativity-b0uY6hcFIp
Publisher
Springer International Publishing
Subject
Physics; Mathematical Methods in Physics; Theoretical, Mathematical and Computational Physics; Applications of Mathematics; Physics, general
ISSN
0188-7009
eISSN
1661-4909
D.O.I.
10.1007/s00006-016-0749-8
Publisher site
See Article on Publisher Site

### 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 $$\left( \mathcal {J}_{\alpha }\right)$$ J α thus becomes $$\delta dg_{\alpha }=\mathcal {T}_{\alpha }+\mathcal {J}_{\alpha }$$ δ d g α = T α + J α , where $$\left( \mathcal {T}_{\alpha }\right)$$ T α 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.

### Journal

Advances in Applied Clifford AlgebrasSpringer Journals

Published: Jan 3, 2017

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