A unified view of curvature and torsion in metric-affine gauge theory of gravity through affine-vector bundles

TL;DR

This paper introduces a new affine-vector bundle framework in metric-affine gravity, clarifying the geometric relationship between curvature and torsion, and suggests a potential measurable physical effect related to torsion.

Contribution

It develops a rigorous affine-vector bundle approach to unify curvature and torsion in metric-affine gravity without ad hoc assumptions.

Findings

Derived the covariant derivative on affine-vector bundles.

Established a natural parallel between curvature and torsion.

Proposed a measurable kinematical effect of torsion similar to Aharonov-Bohm.

Abstract

One of the most appealing results of metric-affine gauge theory of gravity is a close parallel between the Riemann curvature two-form and the Cartan torsion two-form: While the former is the field strength of the Lorentz-group connection one-form, the latter can be understood as the field strength of the coframe one-form. This parallel, unfortunately, is not fully established until one adopts Trautman's idea of introducing an affine-vector-valued zero-from, the meaning of which has not been satisfactorily clarified. This paper aims to derive this parallel from first principles without any ad hoc prescriptions. We propose a new mathematical framework of an associated affine-vector bundle as a more suitable arena for the affine group than a conventional vector bundle, and rigorously derive the covariant derivative of a local section on the affine-vector bundle in the formal Ehresmann-connection approach. The parallel between the Riemann curvature and the Cartan torsion arises naturally on the affine-vector bundle, and their geometric and physical meanings become transparent. The clear picture also leads to a conjecture about a kinematical effect of the Cartan torsion that in principle can be measured \`{a} la the Aharonov-Bohm effect.