Non-relativistic limit of Einstein-Cartan-Dirac equations

TL;DR

This paper derives the Schrödinger-Newton equation from Einstein-Dirac and Einstein-Cartan-Dirac equations in the non-relativistic limit, relaxing symmetry assumptions and highlighting the role of spin-torsion coupling.

Contribution

It generalizes previous derivations by removing symmetry constraints and incorporates torsion effects in the non-relativistic limit of Einstein-Cartan-Dirac equations.

Findings

Schrödinger-Newton equation derived from Einstein-Dirac equations.

Generalization to Einstein-Cartan-Dirac equations including torsion.

Relaxation of spherical symmetry assumption.

Abstract

We derive the Schr\"{o}dinger-Newton equation as the non-relativistic limit of the Einstein-Dirac equations. Our analysis relaxes the assumption of spherical symmetry, made in earlier work in the literature, while deriving this limit. Since the spin of the Dirac field couples naturally to torsion, we generalize our analysis to the Einstein Cartan-Dirac (ECD) equations, again recovering the Schr\"{o}dinger-Newton equation.