On affine geometrical structure, generalized of Born-Infeld models and Eddington's world conjectures

Abstract

In this work, we give a detailed description and discussion of the dynamic gravitational equations of the model with Lagrangian of the type f√det Rμνd4x as proposed by Eddington time ago but with Rμν being a non-Riemannian generalization of the Ricci tensor with the end to find the geometrical origin of the Eddington and Weyl conjectures concerning Lagrangian densities (generalized volume) and natural gauge. The Ricci tensor in our case is particularly based on an affine geometry with a generalized compatibility condition previously proposed in [B. McInnes, On the geometrical interpretation of 'non-symmetric' space-time field structures, Class. Quantum Grav. 1 (1984) 105-113; D. J. Cirilo-Lombardo, Non-Riemannian geometry, Born-Infeld models and trace-free gravitational equations, J. High Energy Astrophys. 16 (2017) 1-14]. Specifically, we show that: (i) the geometric action can be taken to a BI-type form considering a totally antisymmetric torsion field, (ii) Weyl's proposal considering a universal gauge linked to a cosmological constant λ appears in the model naturally due to the proposed affine geometry, (iii) the Eddington conjecture that establishes a relationship between metric and curvature or fundamental tensor with constant of proportionality λ (natural gauge) is geometrically verified in the model with generalized affine geometry.