Affine Dynamics with Torsion

TL;DR

This paper thoroughly analyzes affine gravity theories with torsion, exploring their equations of motion, solutions, and the relationships between curvature and torsion tensors, revealing non-local correlations.

Contribution

It introduces new affine gravity models based on determinants of curvature and torsion tensors, providing detailed solutions and insights into their geometric structure.

Findings

Curvature tensors are correlated via non-local exponential factors.

Solutions reduce complex equations to simpler forms.

Constructs affine connection from curvature and torsion tensors.

Abstract

In this study, we give a thorough analysis of a general affine gravity with torsion. After a brief exposition of the affine gravities considered by Eddington and Schr\"{o}dinger, we construct and analyze different affine gravities based on the determinants of the Ricci tensor, the torsion tensor, the Riemann tensor and their combinations. In each case we reduce equations of motion to their simplest forms and give a detailed analysis of their solutions. Our analyses lead to the construction of the affine connection in terms of the curvature and torsion tensors. Our solutions of the dynamical equations show that the curvature tensors at different points are correlated via non-local, exponential rescaling factors determined by the torsion tensor.