Spherically Symmetric Solution for Torsion and the Dirac equation in 5D spacetime

TL;DR

This paper explores a spherically symmetric solution in 5D spacetime where torsion is linked to spinor fields, leading to nonlinear Dirac equations with discrete, finite-energy solutions concentrated at the Planck scale, suggesting torsion's role in quantum gravity.

Contribution

It introduces a novel 5D spherically symmetric solution connecting torsion and Dirac equations, with implications for quantum gravity and spacetime foam.

Findings

Discrete spectrum of solutions with finite energy

Solutions localized in the Planck region

Torsion's potential role in quantum gravity

Abstract

Torsion in a 5D spacetime is considered. In this case gravitation is defined by the 5D metric and the torsion. It is conjectured that torsion is connected with a spinor field. In this case Dirac's equation becomes the nonlinear Heisenberg equation. It is shown that this equation has a discrete spectrum of solutions with each solution being regular on the whole space and having finite energy. Every solution is concentrated on the Planck region and hence we can say that torsion should play an important role in quantum gravity in the formation of bubbles of spacetime foam. On the basis of the algebraic relation between torsion and the classical spinor field in Einstein-Cartan gravity the geometrical interpretation of the spinor field is considered as ``the square root'' of torsion.