Cartan's spiral staircase in physics and, in particular, in the gauge theory of dislocations

TL;DR

This paper explores Cartan's concept of torsion and the helical staircase in differential geometry, connecting it to physical theories like continuum mechanics, gravity, and dislocation gauge field theory, highlighting interdisciplinary applications.

Contribution

It provides a detailed geometric description of Cartan's helical staircase and demonstrates its realization in dislocation gauge field theory, linking geometry to physical models.

Findings

Cartan's torsion corresponds to a continuum with constant pressure and torque.

The helical staircase is realized in dislocation gauge theories with screw dislocations.

Connections between differential geometry and physical theories of gravity and materials are established.

Abstract

In 1922, Cartan introduced in differential geometry, besides the Riemannian curvature, the new concept of torsion. He visualized a homogeneous and isotropic distribution of torsion in three dimensions (3d) by the "helical staircase", which he constructed by starting from a 3d Euclidean space and by defining a new connection via helical motions. We describe this geometric procedure in detail and define the corresponding connection and the torsion. The interdisciplinary nature of this subject is already evident from Cartan's discussion, since he argued - but never proved - that the helical staircase should correspond to a continuum with constant pressure and constant internal torque. We discuss where in physics the helical staircase is realized: (i) In the continuum mechanics of Cosserat media, (ii) in (fairly speculative) 3d theories of gravity, namely a) in 3d Einstein-Cartan gravity - this is Cartan's case of constant pressure and constant intrinsic torque - and b) in 3d Poincare gauge theory with the Mielke-Baekler Lagrangian, and, eventually, (iii) in the gauge field theory of dislocations of Lazar et al., as we prove for the first time by arranging a suitable distribution of screw dislocations. Our main emphasis is on the discussion of dislocation field theory.