Elie Cartan's torsion in geometry and in field theory, an essay

TL;DR

This paper reviews the concept of torsion in differential geometry and its applications in field theory, including gravity models like Einstein-Cartan theory and teleparallelism, highlighting its physical interpretations and mathematical foundations.

Contribution

It provides a comprehensive overview of torsion's role in geometry and field theory, connecting classical differential geometry with modern gravitational theories and physical models.

Findings

Torsion relates to translations in differential geometry.

Dislocation density in crystals equals torsion in a connection.

Torsion is essential in Einstein-Cartan and teleparallel gravity theories.

Abstract

We review the application of torsion in field theory. First we show how the notion of torsion emerges in differential geometry. In the context of a Cartan circuit, torsion is related to translations similar as curvature to rotations. Cartan's investigations started by analyzing Einsteins general relativity theory and by taking recourse to the theory of Cosserat continua. In these continua, the points of which carry independent translational and rotational degrees of freedom, there occur, besides ordinary (force) stresses, additionally spin moment stresses. In a 3-dimensional continuized crystal with dislocation lines, a linear connection can be introduced that takes the crystal lattice structure as a basis for parallelism. Such a continuum has similar properties as a Cosserat continuum, and the dislocation density is equal to the torsion of this connection. Subsequently, these ideas are applied to 4-dimensional spacetime. A translational gauge theory of gravity is displayed (in a Weitzenboeck or teleparallel spacetime) as well as the viable Einstein-Cartan theory (in a Riemann-Cartan spacetime). In both theories, the notion of torsion is contained in an essential way. Cartan's spiral staircase is described as a 3-dimensional Euclidean model for a space with torsion, and eventually some controversial points are discussed regarding the meaning of torsion.