E. Cartan's attempt at bridge-building between Einstein and the Cosserats -- or how translational curvature became to be known as {\em torsion}

TL;DR

Elie Cartan's 1922 work unified concepts of curvature and torsion in differential geometry, linking Einstein's gravity and elasticity theories, and introducing torsion as a rotational measure before quantum spin was known.

Contribution

The paper highlights Cartan's pioneering idea of associating torsion with translational curvature, bridging gravity and elasticity theories before the advent of quantum spin.

Findings

Cartan's torsion predates quantum spin concepts.

Unified geometric framework for gravity and elasticity.

Introduction of translational curvature as torsion.

Abstract

\'Elie Cartan's "g\'en\'eralisation de la notion de courbure" (1922) arose from a creative evaluation of the geometrical structures underlying both, Einstein's theory of gravity and the Cosserat brothers generalized theory of elasticity. In both theories groups operating in the infinitesimal played a crucial role. To judge from his publications in 1922--24, Cartan developed his concept of generalized spaces with the dual context of general relativity and non-standard elasticity in mind. In this context it seemed natural to express the translational curvature of his new spaces by a rotational quantity (via a kind of Grassmann dualization). So Cartan called his translational curvature "torsion" and coupled it to a hypothetical rotational momentum of matter several years before spin was encountered in quantum mechanics.