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 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 $\delta d{g}_{\alpha }={\mathcal{T}}_{\alpha }+{\mathcal{J}}_{\alpha }$, 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.

35 pages, no figures; improved version in Adv. Appl. Clifford Algebras 2017

References in corpus (3)

• A Solution to Symmetric Teleparallel Gravity
• The Symmetric Teleparallel Gravity
• Revisiting the Schwarzschild and the Hilbert-Droste Solutions of Einstein Equation and the Maximal Extension of the Latter

Cited by in corpus (37)

• Coincident General Relativity
• $f(Q,T)$ gravity
• Coupling matter in modified $Q$-gravity
• Teleparallel Palatini theories
• Weyl type $f(Q,T)$ gravity, and its cosmological implications
• Spherically symmetric configuration in $f(Q)$ gravity
• Observational constraints on cosmological solutions of $f(Q)$ theories
• General covariant symmetric teleparallel cosmology
• The spectrum of symmetric teleparallel gravity
• Geodesic deviation, Raychaudhuri equation, Newtonian limit, and tidal forces in Weyl-type $f(Q,T)$ gravity
• Post-Newtonian limit of generalized symmetric teleparallel gravity
• Family of scalar-nonmetricity theories of gravity
• Modified gravity: a unified approach
• A class of static spherically symmetric solutions in $f(Q)$-gravity
• Complete dark energy scenario in $f(Q)$ gravity
• Variational Principles in Teleparallel Gravity Theories
• Minimal coupling in presence of non-metricity and torsion
• Transit cosmological models with observational constraints in f(Q, T) gravity
• Neutron Star in Covariant $f(Q)$ gravity
• $f(Q,L_m)$ gravity, and its cosmological implications
• Conformally invariant proper time with general non-metricity
• Nonminimally coupled Weyl gravity
• DGP and DGPish cosmologies from $f(Q)$ actions
• Polarized gravitational waves in the parity violating scalar-nonmetricity theory
• Parametrized post-Newtonian limit of generalized scalar-nonmetricity theories of gravity
• $f(Q,T)$ gravity: From early to late-time cosmic acceleration
• Spatially covariant gravity with nonmetricity
• From GR to STG -- Inheritance and development of Einstein's heritages
• Energy-momentum squared symmetric Teleparallel gravity: $f(Q,T_{\mu \nu }{T}^{\mu \nu })$ gravity
• Cosmological implications of non-minimally coupled f(Q) gravity
• Quintessence like behavior of symmetric teleparallel dark energy: Linear and nonlinear model
• Warm Inflation in $f(Q)$ gravity
• A Cosmological Holographic Reconstruction of f(Q) Theory
• Charged Gravastar Solutions in the Finch-Skea Framework with f(\mathbb{Q}) Gravity
• Degrees of freedom of a quadratic scalar-nonmetricity theory
• Static plane symmetric solutions in $f(Q)$ gravity
• Equations of Motion and Energy-Momentum 1-Forms for the Coupled Gravitational, Maxwell and Dirac Fields