The relation between the model of a crystal with defects and Plebanski's theory of gravity

TL;DR

This paper explores the analogy between spacetime geometry in general relativity and crystal defects, relating Kleinert's crystal defect model to Plebanski's gravity, highlighting gauge symmetries and the role of torsion.

Contribution

It establishes a connection between crystal defect models and Plebanski's formulation of gravity, showing torsion can be viewed as a gauge choice and is not a separate dynamical variable.

Findings

4D crystal defects correspond to Riemann-Cartan spacetime with torsion.

Torsion can be eliminated as a dynamical variable in Plebanski's gravity.

The phase of gravity with torsion is equivalent to topological gravity.

Abstract

In the present investigation we show that there exists a close analogy of geometry of spacetime in GR with a structure of defects in a crystal. We present the relation between the Kleinert's model of a crystal with defects and Plebanski's theory of gravity. We have considered the translational defects - dislocations, and the rotational defects - disclinations - in the 3- and 4-dimensional crystals. The 4-dimensional crystalline defects present the Riemann-Cartan spacetime which has an additional geometric property - "torsion" - connected with dislocations. The world crystal is a model for the gravitation which has a new type of gauge symmetry: the Einstein's gravitation has a zero torsion as a special gauge, while a zero connection is another equivalent gauge with nonzero torsion which corresponds to the Einstein's theory of "teleparallelism". Any intermediate choice of the gauge with nonzero connection A^{IJ}_\mu is also allowed. In the present investigation we show that in the Plebanski formulation the phase of gravity with torsion is equivalent to the ordinary or topological gravity, and we can exclude a torsion as a separate dynamical variable.