[Paper Review] The Einstein-Cartan-Dirac theory

Swanand Khanapurkar|arXiv (Cornell University)|Mar 28, 2018

Cosmology and Gravitation Theories1 references3 citations

TL;DR

This paper proposes a generalized Einstein-Cartan-Dirac (ECD) theory incorporating a new mass-dependent length scale $L_{cs}$, unifying quantum and gravitational effects. Using WKB-like expansion and Newman-Penrose formalism, it shows that in the low-mass limit, gravity vanishes at leading order, while in the high-mass limit, it reduces to Poisson’s equation with a delta-source, suggesting a curvature-torsion duality between small and large masses.

ABSTRACT

There are various generalizations of Einstein's theory of gravity (GR); one of which is the Einstein-Cartan (EC) theory. It modifies the geometrical structure of manifold and relaxes the notion of affine connection being symmetric. The theory is also called $U_4$ theory of gravitation; where the underlying manifold is not Riemannian. The non-Riemannian part of the space-time is sourced by the spin density of matter. Here mass and spin both play the dynamical role. We consider the minimal coupling of Dirac field with EC theory; thereby calling the full theory as Einstein-Cartan-Dirac (ECD) theory. In the recent works by T.P Singh titled "A new length scale in quantum gravity", the idea of new unified; mass dependent length scale $L_{cs}$ in quantum gravity has been proposed. We discuss this idea and formulate ECD theory in both - standard length scales as well as this new length scale. We found the non-relativistic limit of ECD theory using WKB-like expansion in $\sqrt{\hbar}/c$ of the ECD field equations with both the length scales. At leading order, ECD equations with standard length scales give Schrödinger-Newton equation. With $L_{cs}$, in the low mass limit, it gives source-free Poisson equation and for higher mass limit, it reduces to Poisson equation with delta function source. Based on this, a falsifiable test of the idea of $L_{cs}$ has been proposed. Next, we formulate ECD theory with both the length scales (especially the Dirac equation (Hehl-Datta equation) and Contorsion spin coefficients) in Newman-Penrose (NP) formalism. The idea of $L_{cs}$ suggests a symmetry between small and large masses. Formulating ECD theory with $L_{cs}$ in NP formalism is desirable because NP formalism happens to be the common vocabulary for the description of low masses (Dirac theory) and high masses (gravity theories). A duality between Curvature and torsion has also been discussed.

Motivation & Objective

• To investigate the non-relativistic limit of the Einstein-Cartan-Dirac (ECD) theory using a WKB-like expansion in $\sqrt{\hbar}/c$.
• To formulate the ECD theory in the Newman-Penrose (NP) formalism to unify descriptions of low-energy (Dirac) and high-energy (gravitational) physics.
• To test a conjecture of curvature-torsion duality, where torsion sources small masses and curvature sources large masses via the unified length scale $L_{cs}$.
• To explore the implications of $L_{cs}$ on the non-linear Dirac equation and field equations in $U_4$ spacetime with torsion.
• To establish a theoretical framework where quantum and gravitational regimes are dual through the $L_{cs}$ scale.

Proposed method

• Applying a WKB-like expansion in $\sqrt{\hbar}/c$ to derive the non-relativistic limit of ECD field equations with both standard and $L_{cs}$ length scales.
• Using the Newman-Penrose formalism to express the ECD field equations in spinor-tetrad variables, enabling a unified description of spin-1/2 and gravitational fields.
• Deriving the Hehl-Datta equation as the modified Dirac equation in $U_4$ spacetime with non-symmetric affine connections and torsion.
• Introducing the unified length scale $L_{cs} = \sqrt{\hbar c / (G m)}$ to interpolate between quantum and classical gravity regimes.
• Analyzing the non-linear Dirac equation on Minkowski space with torsion to test the curvature-torsion duality conjecture.
• Using the Belinfante-Rosenfeld symmetrization procedure to construct a conserved energy-momentum tensor for the Dirac field in curved spacetime with torsion.

Experimental results

Research questions

• RQ1 How does the inclusion of the $L_{cs}$ length scale modify the non-relativistic limit of the ECD theory compared to the standard length scale?
• RQ2 What is the behavior of the ECD equations in the low-mass limit when using $L_{cs}$, and does it lead to a source-free Poisson equation?
• RQ3 Can the Newman-Penrose formalism consistently describe the ECD theory with $L_{cs}$, and does it reveal a duality between torsion and curvature?
• RQ4 Does the $L_{cs}$-based ECD theory support a curvature-torsion duality conjecture, where small masses source torsion and large masses source curvature?
• RQ5 What are the implications of solving the non-linear Dirac equation on Minkowski space with torsion for the proposed duality?

Key findings

• In the low-mass limit with $L_{cs}$, the non-relativistic limit of ECD theory yields a source-free Poisson equation, indicating that small masses do not contribute to gravity at leading order.
• In the high-mass limit with $L_{cs}$, the non-relativistic limit reduces to Poisson’s equation with a delta-function source, recovering the classical Newtonian potential.
• With standard length scales, the non-relativistic limit of ECD theory reproduces the Schrödinger-Newton equation, linking quantum mechanics and gravity.
• The Newman-Penrose formalism successfully embeds the ECD equations with $L_{cs}$, enabling a unified description of spinor dynamics and gravitational fields in a torsioned spacetime.
• Solutions to the non-linear Dirac equation on Minkowski space with torsion were found, supporting the curvature-torsion duality conjecture.
• The $L_{cs}$-based ECD theory suggests a deep symmetry between small masses (via torsion) and large masses (via curvature), formalized as the curvature-torsion duality conjecture.