期刊
FOUNDATIONS OF PHYSICS
卷 40, 期 9-10, 页码 1298-1325出版社
SPRINGER
DOI: 10.1007/s10701-010-9440-4
关键词
Cartan's torsion; Differential geometry; Dislocations; Cosserat continuum; Einstein-Cartan theory; 3-dimensional theories of gravitation
资金
- Deutsche Forschungsgemeinschaft [La1974/1-2, La1974/1-3]
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 Poincar, 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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