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Abstract
Quarks, as spin-$1/2$ particles, can be described as Dirac spinors, and as massive entities, they are capable of curving their surrounding spacetime. We demonstrate that the general relativistic (GR) gravitation of quarks is sufficient to confine them within hadrons while allowing them to remain asymptotically free. Consequently, we conclude that strong nuclear forces and general relativistic gravitation are one and the same at the femtometer scales of quarks and hadrons.
I. INTRODUCTION
To the best of our knowledge, i) there is currently no established wave function for hadrons or their constituent quarks that parallels, for instance, Dirac waves for electrons, and ii) there is no definitive consensus as to what exactly nuclear forces are, except that they are strong and short-ranged.
In this paper, we assume that quarks, as massive entities, are capable of curving their surrounding spacetime. Furthermore, as spin-$1/2$ particles, they can be represented as Dirac spinors under proper normalization. We formulate the Dirac equation within a spherically symmetric and static spacetime, impose a Yukawa-like normalization, and solve it. Ultimately, we conclude that GR forces at femtometer scales are sufficiently strong to confine quarks within nucleons, while allowing them to be asymptotically free in the inner regions.
The standard Dirac matrices $\gamma^{\mu}$ are invariant under Lorentz transformations in flat spacetime; however, this invariance does not hold in curved spacetimes. In his detailed study [1], Alcubierre addresses this issue by transforming the $\gamma$-matrices into a spherically symmetric spacetime utilizing tetrad analysis to model a Dirac star. By formulating the Dirac equation in a spherically symmetric and static spacetime, he derives four coupled, first-order differential equations: two for the Dirac field and two for the spacetime metric coefficients. A Yukawa-normalizable version of Alcubierre's four coupled equations serves as our point of departure.
Herein, we provide a formal definition of the problem and design an iteration scheme to analyze it. We demonstrate how a spin-$1/2$ quark can be modeled as an Alcubierre-designed and Yukawa-normalized Dirac spinor. Finally, we derive the wave functions for quarks and nucleons and investigate the gravitational their manifestations. We conclude that the asymptotically flat spacetime inside a nucleon guarantees the asymptotic freedom of quarks, while the exponential falloff of the outer spacetime toward flatness ensures their confinement.
Dedicated to the International Year of Quantum Science and Technology (IYQST2025).
REFERENCES
[1] M. Alcubierre, "The Dirac equation in general relativity and the 3+1 formalism" (2025), arXiv:2503.03918 [gr-qc].