[Paper Review] Dynamical torsion and torsion potential

Hong-jun Xie, Takeshi Shirafuji|ArXiv.org|Mar 6, 1996

Experimental and Theoretical Physics Studies3 citations

TL;DR

This paper introduces a dynamical torsion potential as a generalized tetrad field that makes torsion a dynamic quantity in Einstein-Cartan gravity. By treating the metric tensor and torsion potential as independent variables, the authors derive both the Einstein equation and a torsion field equation, showing that the torsion potential acts as a matter field in the gravitational action, with applications to scalar-tensor-like models and local symmetry properties.

ABSTRACT

We introduce a generalized tetrad which plays the role of a potential for torsion and makes torsion dynamic. Starting from the Einstein-Cartan action with torsion, we get two field equations, the Einstein equation and the torsion field equation by using the metric tensor and the torsion potential as independent variables; in the former equation the torsion potential plays the role of a matter field. We also discuss properties of local linear transformations of the torsion potential and give a simple example in which the torsion potential is described by a scalar field.

Motivation & Objective

• To reformulate Einstein-Cartan gravity by introducing a generalized tetrad that acts as a dynamical potential for torsion.
• To derive consistent field equations—Einstein and torsion field equations—by treating the metric and torsion potential as independent variables.
• To explore the role of local linear transformations in the torsion potential and their physical implications.
• To demonstrate that the torsion potential can be modeled as a scalar field in a simple example, linking it to scalar-tensor theories.

Proposed method

• The authors use the Einstein-Cartan action with torsion and perform a variational principle with respect to both the metric tensor and the torsion potential as independent fields.
• They introduce a generalized tetrad field that serves as a potential for the torsion field, enabling a dynamic description of torsion.
• The field equations are derived by varying the action with respect to the metric and the torsion potential, yielding the Einstein equation and a new torsion field equation.
• The torsion potential is shown to couple to gravity as a matter field in the Einstein equation, despite being geometric in origin.
• The paper analyzes local linear transformations of the torsion potential, revealing a gauge-like structure in the field equations.
• A concrete example is constructed where the torsion potential is described by a scalar field, illustrating its dynamical behavior.

Experimental results

Research questions

• RQ1 How can torsion be made dynamically active in Einstein-Cartan gravity through a new field variable?
• RQ2 What are the field equations that emerge when the torsion potential is treated as an independent variable alongside the metric?
• RQ3 How do local linear transformations affect the physical content and symmetries of the torsion potential?
• RQ4 Can the torsion potential be effectively modeled as a scalar field, and what are the implications for gravity-matter coupling?
• RQ5 What is the role of the torsion potential in the gravitational action, and how does it behave like a matter field in the Einstein equation?

Key findings

• The torsion potential acts as a matter field in the Einstein equation, even though it originates from geometry, due to its independent variational treatment.
• The field equations derived include both the standard Einstein equation and a new torsion field equation, ensuring consistency with the variational principle.
• Local linear transformations of the torsion potential are shown to preserve the physical content of the theory, indicating a hidden gauge structure.
• A simple model is constructed where the torsion potential reduces to a scalar field, providing a concrete realization of dynamical torsion.
• The formalism allows for a unified treatment of gravity and torsion, with the torsion potential contributing to the energy-momentum tensor in the Einstein equation.