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Cartan’s Spiral Staircase in Physics and, in Particular, in the Gauge Theory of Dislocations

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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 Poincaré 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.

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Authors and Affiliations

  1. Emmy Noether Research Group, Department of Physics, Darmstadt University of Technology, Hochschulstr. 6, 64289, Darmstadt, Germany

    Markus Lazar

  2. Department of Physics, Michigan Technological University, Houghton, MI, 49931, USA

    Markus Lazar

  3. Institute for Theoretical Physics, University of Cologne, 50923, Cologne, Germany

    Friedrich W. Hehl

  4. Department of Physics and Astronomy, University of Missouri, Columbia, MO, 65211, USA

    Friedrich W. Hehl

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  1. Markus Lazar
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  2. Friedrich W. Hehl
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Correspondence to Markus Lazar.

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Dedicated to Professor Peter Mittelstaedt on the occasion of his 80th birthday.

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Lazar, M., Hehl, F.W. Cartan’s Spiral Staircase in Physics and, in Particular, in the Gauge Theory of Dislocations. Found Phys 40, 1298–1325 (2010). https://doi.org/10.1007/s10701-010-9440-4

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