Torsion as Dislocation density

Torsion as Dislocation density Torsion as Dislocation Density

Einsteins theory is based on the tensor L he defined in eq.

L_abⁿ =

(D_abⁿ -D_baⁿ).

This tensor, introduced by E. Cartan already in 1922, is nowadays called torsion tensor T_abⁿ or torsion two-form Tⁿ (two indices for the surface are surpressed).

Due to the discoveries of Kondo (1952), Bilby, Bullough and Smith (1955) this tensor can be seen as a density of dislocations piercing through a surface element. For an excellent review, see Kröner (1980, literature).

Examples of an edge dislocation (left) and a screw dislocation (right) in a crystal. Suppose direction 1, 2, 3 point to the right, backwards and up as indicated. Then in the left picture, after surrounding a surface element in the 1-3-plane one gets shifted in direction 1, therefore this gives a contribution to the T₁₃¹ component of the torsion tensor. In the right picture, after surrouding a surface element in the 1-2 plane the shift is in direction 3, therefore this contributes to the T₁₂³ component. Note that the singularity line of the dislocation in the left case goes in direction 2 and is perpendicular to the shift (Burgers vector), and in the right case parallel to the shift (both in direction 3). Torsion is just a continuous version of disloaction density, that means one lets the lattice spacing go to zero while maintaining the quantity shift/surface element.

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On 17 Aug 2000, 11:16.