Einstein-Cartan-Dirac equations in the Newman-Penrose formalism

TL;DR

This paper reformulates the Einstein-Cartan-Dirac equations using the Newman-Penrose formalism, clarifying the effects of torsion and presenting solutions including a solitonic one that addresses unphysical behaviors.

Contribution

It introduces a detailed Newman-Penrose formulation of Einstein-Cartan-Dirac equations and provides explicit solutions, including a novel solitonic solution on Minkowski space.

Findings

Explicit Newman-Penrose form of Einstein-Cartan-Dirac equations

Presentation of a solitonic solution that mitigates unphysical Dirac behavior

Foundation for future studies in Poincare gauge gravity

Abstract

We formulate the Einstein-Cartan-Dirac equations in the Newman-Penrose (NP) formalism, thereby presenting a more accurate and explicit analysis of previous such studies. The equations show in a transparent way how the Einstein-Dirac equations are modified by the inclusion of torsion. In particular, the Hehl-Datta equation is presented in NP notation. We then describe a few solutions of the Hehl-Datta equation on Minkowski space-time, and in particular report a solitonic solution which removes the unphysical behavioiur of the corresponding Dirac solution. The present work serves as a prelude to similar studies for non-degenerate Poincare gauge gravity.