[Paper Review] The Dirac field and the possible origin of gravity

A. Makhlin|ArXiv.org|Aug 8, 2004

Algebraic and Geometric Analysis3 citations

TL;DR

This paper proposes that gravity arises from an axial vector field minimally coupled to the axial current of the Dirac field, which induces auto-localization of the field into stable, compact objects. These objects obey Einstein’s field equations when energy-momentum is self-adjoint, and the long-range effect of the axial field reproduces Newtonian gravity, revealing the microscopic origin of gravitational mass and linking it to the peak amplitude of Dirac field localization.

ABSTRACT

The spin connections of the Dirac field have three ingredients that are connected with the Ricci rotations, the Maxwell field, and an axial field which minimally interacts with the axial current. I demonstrate that the axial field provides an effective mechanism of auto-localization of the Dirac field into compact objects. The condition that these objects are stable (the energy-momentum is self-adjoint) leads to Einstein's field equations. The Dirac field with its spin connection seem to be a natural material carrier of the space-time continuum in which compact objects are moving along geodesic lines. The long distance effect of the axial field is indistinguishable from Newton's gravity, which reveals the microscopic nature of gravity and the origin of the gravitational mass.

Motivation & Objective

• To investigate the conditions under which the Dirac field forms stable, compact objects that move along geodesics.
• To identify the origin of gravity as arising from a spin connection component—specifically an axial vector field—interacting with the axial current of the Dirac field.
• To demonstrate that the self-adjointness of energy-momentum leads to Einstein’s field equations.
• To show that the long-distance effect of the axial field is indistinguishable from Newtonian gravity, revealing the microscopic nature of gravitational mass.

Proposed method

• Uses the tetrad formalism and Fock’s method to define spinor parallel transport via a spin connection with three components: geometric (Ricci rotation), electromagnetic (Maxwell), and axial (new field).
• Derives the general form of the spin connection Γₐ as a combination of gauge fields: Γₐ = -ieAₐ - igρ₃ℵₐ + Ωₐ, where ℵₐ is the axial field.
• Imposes the condition that the covariant derivative of observables must vanish under parallel transport, ensuring stability of the spinor object.
• Applies the self-adjointness condition on the differential operator iDₘu to derive the Einstein field equations from the dynamics of the Dirac field.
• Uses a spherically symmetric ansatz for the axial potential and solves the Dirac equation exactly in radial form, showing localized bound states with constant probability density near r=0.
• Demonstrates that the gravitational mass is proportional to the peak amplitude of the axial field, R(0), and relates it to the energy scale via R(0) ∼ |E|, suggesting universality of inertial and gravitational mass.

Experimental results

Research questions

• RQ1 Can the Dirac field form stable, compact objects through self-interaction via a spin connection component?
• RQ2 Does the axial field in the spin connection reproduce the long-range behavior of Newtonian gravity?
• RQ3 Under what conditions does the self-adjointness of the energy-momentum operator lead to Einstein’s field equations?
• RQ4 How is the gravitational mass of a compact Dirac object related to the localization of the field amplitude?
• RQ5 Why are only Dirac fields capable of forming physically meaningful solutions to Einstein’s equations in the context of singularities?

Key findings

• The axial field ℵₐ in the spin connection induces auto-localization of the Dirac field into compact, stable objects that move along geodesics.
• The condition that energy-momentum is self-adjoint leads directly to Einstein’s field equations, establishing a dynamical derivation of general relativity from the Dirac field.
• The long-range effect of the axial field is indistinguishable from Newtonian gravity, with the potential g₀₀ = 1 + 2Υgrav, where Υgrav ∝ (m𝒫/T₀₀), confirming the Newtonian limit.
• The gravitational mass is proportional to the peak amplitude of the probability density R(0) = ∫(F² + G²)r²dr at the origin, linking it directly to field localization.
• The inertial mass (from energy E) and gravitational mass (from ∫𝒫dV) are proportional via R(0) ∼ |E|, suggesting universality of free fall and a possible unification of gravitational and inertial mass.
• Only solutions of Einstein’s equations that have a corresponding compact Dirac field solution with the axial field in the spin connection are physically meaningful, implying a deep compatibility between gravity and the Dirac field.