Magnetic Monopole as a Three-Vortex Structure: A Vortex-Based Framework for Magnetic Force Generation

Nader Butto

Independent Researcher, Petah Tikva, Israel.

DOI: 10.4236/jhepgc.2026.121030                

Abstract

The magnetic monopole—an isolated magnetic charge—has long been proposed to restore the symmetry of Maxwell’s equations, yet it has never been observed as a free particle. This paper presents a new theoretical model describing the magnetic monopole as a tri-vortex excitation of a superfluid vacuum. Building on the Quark Vortex Theory and the Mushroom Model of the nucleon, the proton and neutron are shown to represent confined monopoles of opposite magnetic polarity, formed by three coupled vortices: two centrifugal and one centripetal. The same geometry, when unconfined, produces an open magnetic flux corresponding to a free monopole. Within this hydrodynamic framework, magnetic permeability μ₀ is reinterpreted as the shear response of the vacuum, and magnetic charge arises from a circulation imbalance quantified by $g=\left({\Gamma }_{\text{eff}}/2\pi \right)\sqrt{\rho /{\mu }_0}$ , linking magnetism directly to vacuum density ρ and vortex dynamics. The pressure gradient between interacting vortices yields the inverse-square magnetic force, identical in form to gravitation, which is shown to originate from large-scale vortex structures in the same medium. Energy analysis demonstrates that monopole self-energies range from sub-MeV for nucleon-sized cores to eV for causality-limited vortices, eliminating singularities inherent in point-charge models. The tri-vortex monopole satisfies all theoretical requirements for magnetic charge—quantized flux, finite energy, topological stability, and dynamic equilibrium—while unifying magnetic, electric, gravitational, and nuclear forces under a single physical principle: the self-organized rotation and pressure balance of the superfluid vacuum.

Keywords

Magnetic Monopole, Superfluid Vacuum, Tri-Vortex Structure, Vacuum Vortex Dynamics, Unified Field Theory

1. Introduction

The concept of the magnetic monopole has fascinated physicists for nearly a century. Since Dirac’s 1931 proposal, it has been regarded as the missing counterpart of the electric charge, required to restore the full symmetry of Maxwell’s equations and to explain the quantization of electric charge [1]. Despite extensive experimental efforts, no isolated magnetic pole has ever been observed, suggesting that magnetism may not originate from discrete magnetic charges but rather from dynamic vortex configurations within a structured vacuum.

Recent theoretical developments indicate that the vacuum behaves as a superfluid medium endowed with density, viscosity, and elasticity. In this framework, electromagnetic fields are interpreted as organized motions of the vacuum, and the classical electromagnetic constants—permittivity ε₀ and permeability μ₀—represent, respectively, the elastic and shear properties of this medium [2] [3]. The magnetic permeability μ₀, in particular, has been derived hydrodynamically as the shear stress per unit time of the vacuum, demonstrating that it is not a fixed fundamental constant but an observable parameter describing the vacuum’s internal frictionless flow [3]. Its invariance across scales reflects the homogeneity of the superfluid vacuum.

Building upon this understanding, the present work introduces a vortex-based model describing the magnetic monopole as a composite tri-vortex structure within the superfluid vacuum. This model extends the author’s previous studies on the proton and neutron, where both particles are represented by three coupled vortices—two centrifugal (outward-flowing) and one centripetal (inward-flowing)—connected through helical gluon arms that confine the flux within a mushroom-like geometry [4].

When the magnetic flux of this tri-vortex structure is confined internally, the system manifests as a proton. When the same geometry is considered without confinement, it produces an open magnetic flux—the monopole. The orientation of this flux determines the magnetic polarity: in the proton, the inward (centripetal) flux creates a negative magnetic pole that attracts the electron; in the neutron, the outward (centrifugal) flux forms a positive magnetic pole that repels electrons. Therefore, the proton and neutron represent the two opposite magnetic monopoles, identical in shape but reversed in magnetic flow direction.

This symmetry explains the strong attraction between nucleons as the coupling between opposite magnetic monopoles. The magnetic force within nuclei thus emerges as the hydrodynamic interaction between the inward and outward flux systems of neighboring tri-vortex structures, mediated by the superfluid shear response of the vacuum represented by μ₀.

Moreover, monopoles can exist in various scales of size and energy while maintaining the same three-vortex morphology. The variation in vortex circulation and radius determines the magnitude of magnetic charge and magnetic moment, whereas the geometric configuration remains invariant. This suggests that magnetic monopoles are a universal structure appearing at different levels of matter organization, from subatomic nucleons to cosmic plasma formations [2] [7].

By integrating the concept of magnetic permeability into this vortex-based framework, a unified description of magnetic charge, vacuum density, and vortex circulation emerges. In this view, the magnetic monopole, the proton, and the neutron are three expressions of the same fundamental structure within the superfluid vacuum. The resulting model establishes a direct correspondence between magnetism, mass, and vacuum dynamics, leading toward a unified hydrodynamic interpretation of the fundamental interactions [2]-[4] [6].

2. Current Theoretical Background and Limitations

The magnetic monopole was first introduced by Dirac in 1931 to restore the symmetry of Maxwell’s equations and to explain the quantization of electric charge [1]. His condition, expressed as

$eg=n\hslash c/2$ ,

relates the electric charge e and the hypothetical magnetic charge g through a topological constraint. This quantization ensures that the electromagnetic field remains single-valued under a rotation of 4π. However, the Dirac model does not describe any physical mechanism for the formation, structure, or stability of such a monopole.

Subsequent theoretical developments, including grand unified and supersymmetric models, predicted monopoles as topological solitons resulting from spontaneous symmetry breaking in the early universe [8]. These monopoles were expected to have enormous masses (~10¹⁶ GeV), rendering them unobservable in laboratory conditions. Moreover, none of these frameworks provided a concrete geometric representation or hydrodynamic mechanism capable of generating an isolated magnetic pole within the observable energy range.

A complete physical model must satisfy four essential requirements for the existence of a magnetic monopole:

1) Field topology—It must produce a divergence of the magnetic field, ∇·B ≠ 0, corresponding to a non-zero magnetic charge density.

2) Energy finiteness—The internal structure must confine field energy within a finite core to prevent divergence at r → 0.

3) Quantization—Circulation or flux through any closed surface must be quantized, ensuring stability against continuous deformation.

4) Dynamic consistency—It must respect Maxwell’s equations and the hydrodynamic analogs of momentum and energy conservation in the vacuum medium.

The present hydrodynamic formulation satisfies all these requirements by embedding the monopole within a superfluid vacuum that supports quantized vortex motion. In this view, magnetic flux is not a static field entity but a manifestation of circulating vacuum flow. The divergence of magnetic flux appears wherever circulation symmetry is broken, producing an effective magnetic charge [6].

In the author’s earlier studies, the vacuum was characterized as a superfluid medium possessing measurable density ρ, viscosity, and elasticity [3]. Within this medium, the magnetic permeability μ₀ represents the vacuum’s shear response per unit time. The product (ρ, μ₀) defines the two mechanical constants governing all magnetic phenomena: ρ establishes the inertial mass density of the vacuum, while μ₀ sets the resistance of the medium to transverse deformation.

Hydrodynamic analogs such as quantized vortices in superfluid helium demonstrate that field structures resembling magnetic lines can emerge from rotational singularities [7]. These analog systems reproduce several features expected for monopoles, including quantized circulation, topological stability, and discrete flux quanta. The same principles can be extended to the vacuum itself, where the velocity field v(r) = Γ/(2πr) describes an irrotational vortex whose circulation Γ is quantized by Γ = nh/m_v, with m_v the effective mass associated with vacuum density.

In this framework, the magnetic monopole is modeled as a tri-vortex system in the superfluid vacuum, identical in geometry to the proton and neutron configurations described previously [4]. The proton corresponds to a negative monopole with inward (centripetal) flux; the neutron represents the opposite, positive monopole with outward (centrifugal) flux. Both satisfy Dirac’s quantization naturally because the total circulation around the tri-vortex is discrete, determined by the quantized circulation of each constituent vortex [1].

The hydrodynamic model also addresses the requirement of finite self-energy. In a superfluid medium, the kinetic energy density associated with vortex motion, (½ρv²), decreases with 1/r², while the pressure field counterbalances this gradient according to Bernoulli’s equation. The resulting total energy remains finite when integrated from the core radius a to the external boundary R, yielding a stable equilibrium between rotational and pressure forces.

Unlike Dirac’s point monopole, which introduces a singular potential at the origin, the tri-vortex configuration provides a continuous, differentiable field structure in which the flux emerges or converges smoothly through the vacuum. This ensures that both field energy and angular momentum remain finite and quantized, fulfilling the necessary mathematical and physical conditions for monopole stability.

A critical limitation of earlier monopole theories was their reliance on purely topological or gauge-field descriptions, which lacked an explicit physical medium to carry magnetic flux. The superfluid-vacuum approach overcomes this by supplying a mechanical substrate in which circulation, pressure, and shear are well-defined quantities [2] [3]. In this description, the magnetic charge g becomes a measurable function of vortex circulation Γ, vacuum density ρ, and magnetic permeability μ₀, all of which can be related to experimentally accessible parameters.

While this model provides a clear physical mechanism for monopole formation and stability, it also introduces new challenges. Experimental confirmation requires detecting manifestations of monopolar fields or confined monopole pairs, such as proton-neutron magnetic asymmetries or monopole analogs in superfluid or Bose-Einstein condensate systems [7]. Moreover, the superfluid nature of the vacuum, though supported by several theoretical and cosmological arguments, remains to be confirmed by direct measurement.

Nevertheless, the tri-vortex formulation meets all theoretical requirements for monopole existence and bridges classical electromagnetism, quantum field theory, and vacuum hydrodynamics. It replaces the abstract concept of magnetic charge with a concrete, measurable quantity derived from the geometry and circulation of the superfluid vacuum [2]-[4].

3. The Proton and Neutron as Confined Monopoles

In the tri-vortex model, both the proton and the neutron are interpreted as confined magnetic monopoles within the superfluid vacuum. Each nucleon is composed of three interacting vortices—two forming the upper cap and one forming the central stem—embedded within a continuous, frictionless medium characterized by vacuum density ρ and magnetic permeability μ₀. These parameters determine the energy balance and field confinement that maintain the stability of the vortex system.

The difference between the proton and the neutron lies in the direction of the vacuum flow within their vortices. In the proton, the two up-type vortices that form the cap are centrifugal, generating outward flow, while the single down-type vortex forming the stem is centripetal, drawing vacuum flux inward. In the neutron, this configuration is inverted: the two down-type vortices in the cap are centripetal, and the single up-type vortex in the stem is centrifugal. Thus, both nucleons share the same geometric structure—a mushroom-like tri-vortex—but their internal flow orientations are reversed, producing opposite magnetic polarity [4].

In the proton, the inward flux along the central stem corresponds to a converging magnetic field. The centripetal down vortex concentrates vacuum energy toward the core, establishing a region of flux convergence equivalent to a negative magnetic pole. The two centrifugal up vortices in the cap produce outward flow that balances the inward suction of the stem, creating a self-confined magnetic circuit. The resulting configuration behaves as a confined negative magnetic monopole whose field lines close internally. The magnetic moment of the proton (μ_p = +2.79μ_N) arises naturally from the rotation of these vortices and the asymmetry between the inflowing and outflowing flux components [4].

In the neutron, the situation is reversed. The two down vortices in the cap are centripetal and dominate the inward flow, while the single up vortex in the stem is centrifugal and drives flux outward from the core. The result is a net divergence of magnetic flux through the outer surface, corresponding to a positive magnetic pole. The neutron thus behaves as a confined positive magnetic monopole. Its magnetic moment (μ_n = −1.91μ_N) reflects this opposite circulation orientation and confirms its reversed polarity relative to the proton.

The interaction between these two opposite monopoles gives rise to the fundamental nucleon-nucleon attraction. The inward (negative) flux of the proton couples naturally with the outward (positive) flux of the neutron, forming a closed double-loop magnetic circuit that minimizes the vacuum’s total shear energy. The binding force between them emerges as a hydrodynamic consequence of pressure equilibration within the superfluid vacuum. This mechanism represents the physical basis of the strong nuclear force as the coupling of opposite magnetic monopoles confined within the same continuous medium [2]-[4].

The stability of each nucleon arises from the dynamic equilibrium between magnetic and hydrodynamic pressures. The magnetic pressure is given by P_m = B²/(2μ₀), while the opposing vacuum flow pressure is ΔP = ρv²/2. At equilibrium, these two pressures balance at the boundary radius r = r₀, leading to:

$\rho v^2/2=B^2/\left(2{\mu }_0\right)$ .

This guarantees that total energy remains finite and that the vortices maintain stable confinement. The proton, whose central vortex is inward, maintains closed flux and perfect energetic equilibrium; the neutron, whose central vortex is outward, allows limited flux leakage through the superfluid medium, explaining its finite lifetime [4].

From a geometric perspective, the proton and neutron are structurally identical, both exhibiting a mushroom-like shape formed by a cap and stem, but their flow polarities are opposite. The proton’s cap (two centrifugal vortices) drives expansion, while its stem (one centripetal vortex) draws flux inward. The neutron’s cap (two centripetal vortices) drives convergence, while its stem (one centrifugal vortex) drives expansion. This mirror symmetry defines their opposite magnetic polarities and explains their mutual magnetic attraction within nuclei.

The existence of these two confined monopoles—the proton as the negative magnetic monopole and the neutron as the positive one—demonstrates that the magnetic dipole observed in atomic nuclei is not a separate phenomenon but the direct result of vortex polarity pairing in the vacuum. The nucleon pair acts as a bound monopole-antimonopole system, stabilizing through mutual flux coupling and energy minimization [4].

4. Tri-Vortex Configuration of the Magnetic Monopole

The magnetic monopole represents the unconfined state of the same tri-vortex system that forms the proton and neutron. The three vortices preserve the nucleon geometry and circulation structure, but the magnetic flux is not internally balanced; it extends through the surrounding vacuum, producing a net magnetic field with open field lines.

As shown in Figure 1, two like-polarized magnetic vortices spinning outward (black ascendent arrows) produce a divergent magnetic flux analogous to twin tornadoes.

Figure 1. Artistic illustration of the three-vortex structure of the magnetic negative monopole. Two like-polarized magnetic vortices spinning outward (ascendent arrows) producing a divergent magnetic flux analogous to twin tornadoes. a third, oppositely polarized vortex (descendent arrows) rotating inward, channeling the return flow through the vacuum like a whirlpool.

Figure 2. Artistic illustration of the three-vortex structure of the magnetic positive monopole. Two like-polarized magnetic vortices spinning inward (red acsendent arrows) producing a divergent magnetic flux analogous to twin whirlpooles. A third, oppositely polarized vortex (ascendening red arrows) rotating inward, channeling the return flow through the vacuum like a tornado.

As shown in Figure 2, two like-polarized magnetic vortices spinning inward (red discendent arrows) produce a convergent magnetic flux analogous to twin whirlpools, while a third, oppositely polarized vortex (red ascendent arrows).

This configuration generates a net magnetic flux in one direction only—outward for the north monopole and inward for the south monopole—representing the magnetic equivalent of the electric charge in vortex dynamics [4]-[6].

The superposition of the two upper vortices produces a net radial magnetic flux, while the stem closes the circulation through the vacuum, forming a self-sustained monopolar magnetic field. The balance between these flows sets the magnetic polarity. Dominance of the two centrifugal vortices yields an outward-directed flux (positive monopole); dominance of the single centripetal vortex yields inward-directed flux (negative monopole).

Each vortex carries quantized circulation in the superfluid vacuum. The effective circulation of the triad, Γ_eff = 2Γ₊ − Γ₋, determines net magnetic flux and field strength. Because circulation is quantized, magnetic charge is discrete, fulfilling Dirac quantization [1]. The tri-vortex possesses non-zero helicity—velocity and magnetic field lines are topologically linked—ensuring topological stability [4] [7].

5. Derivation of Magnetic Force from Vortex Principles

In the superfluid-vacuum model, the magnetic force originates from the pressure gradients and velocity fields generated by vortex motion. When a vortex rotates within a fluid of density ρ, it produces a centrifugal velocity field, and an associated pressure decrease near the core. This pressure imbalance generates the same inverse-square dependence observed in magnetic interactions. Thus, magnetism can be described as a hydrodynamic phenomenon arising from vacuum circulation and shear response governed by the magnetic permeability μ₀.

5.1. Circulation Quantization and Magnetic Flux

The circulation Γ of each vortex in a superfluid vacuum is quantized according to

$\Gamma =nh/m_v$ ,

where m_v is the effective mass density of the vacuum and n is the circulation quantum number. The total magnetic flux associated with the monopole results from the differential circulation between the two centrifugal vortices (Γ₊) and the single centripetal vortex (Γ₋):

$\Phi ={\displaystyle \oint v\cdot \text{d}l}=2{\Gamma }_{+}-{\Gamma }_{-}$ .

A non-zero Φ indicates that the superposition of the three vortices generates a net magnetic flux through the surrounding vacuum, which corresponds to a macroscopic manifestation of vacuum circulation imbalance. The associated magnetic charge is then expressed as

$g=\Phi /\left(4\pi \right)$ .

This relation shows that magnetic charge emerges as the large-scale effect of microscopic differences in vortex circulation. Because circulation is quantized, magnetic flux and magnetic charge are discrete quantities, fulfilling the Dirac quantization condition naturally within the hydrodynamic model [1].

5.2. Pressure Gradient and Magnetic Interaction

The velocity field around an irrotational vortex is v(r) = Γ/(2πr). Applying Bernoulli’s law to this motion gives a pressure difference between the core and the periphery:

$\Delta P=\frac{1}{2}\rho v^2=\rho {\Gamma }^2/\left(8{\pi }^2{r}^2\right)$ .

This pressure gradient acts radially and represents the vacuum’s magnetic tension. When two vortices interact, their overlapping pressure fields produce a net force that depends on the product of their circulations and decays with the square of their separation:

$F_m\propto \rho {\Gamma }_1{\Gamma }_2/r^2$ .

The same law describes the magnetic interaction between two monopoles carrying charges g₁ and g₂:

$F_m={\mu }_0{g}_1{g}_2/\left(4\pi r^2\right)$ .

Equating these two expressions gives the hydrodynamic origin of magnetic charge:

$g=\left(\Gamma /2\pi \right)\sqrt{\rho /{\mu }_0}$ .

This relation shows that magnetic charge depends directly on the circulation of the vortex, the density of the vacuum, and its permeability. Magnetic interaction is thus a mechanical manifestation of overlapping pressure gradients in the superfluid vacuum [2]-[4] [7].

5.3. Magnetic Energy and Equilibrium

The energy density of the magnetic field is u_B = B²/(2μ₀), where B corresponds to the rotational field of the vacuum and is proportional to Γ/(2πr²). Substituting yields u_B = μ₀g²/(32π²r⁴). Integrating from the vortex-core radius a to an external cutoff R gives the total monopole energy

$U_{\text{mono}}=\left({\mu }_0{g}^2/8\pi \right)\left(1/a-1/R\right)\approx {\mu }_0{g}^2/\left(8\pi a\right)$ .

The energy remains finite because the centrifugal kinetic term ½ρv² is balanced by the vacuum pressure gradient ΔP [3] [4]. This equilibrium ensures that the monopole is stable and that its field energy is confined within a finite region of space.

5.4. Physical Interpretation

The correspondence between hydrodynamic and electromagnetic quantities is direct and fundamental.

The vacuum pressure gradient ΔP corresponds to the magnetic stress, given by:

$\Delta P=\left({\mu }_0\cdot B^2\right)/2$

The circulation Γ represents the analogue of the vector potential A, defining the rotational motion of the vacuum:

$\Gamma ={\displaystyle \oint v\cdot \text{d}l}\iff A=\Gamma /\left(2\pi r\right)$

The magnetic flux Φ corresponds to the total angular momentum of the rotating superfluid medium:

$\Phi ={\displaystyle \int B\cdot \text{d}S}\iff L=\rho \cdot \Gamma \cdot r$

The magnetic permeability μ₀ expresses the vacuum’s resistance to transverse shear deformation, linking the hydrodynamic and electromagnetic pressures through the same mechanical law.

From this correspondence, magnetic phenomena are understood as emergent properties of the rotating superfluid vacuum: the magnetic field arises from vortex circulation, the magnetic force results from pressure interaction between neighboring vortices, and the constant μ₀ reflects the vacuum’s mechanical response to shear, just as viscosity and compressibility define fluid behavior.

The quantization of circulation ensures discrete magnetic charge:

${\Gamma }_n=n\cdot h/m$

while the finite density and permeability of the vacuum prevent singularities in field energy.

Thus, the magnetic monopole represents the elementary open-flux configuration of the superfluid vacuum—its field, flux, and charge all emerging from a single mechanism: the imbalance of vortex circulation within a continuous and quantized medium.

In the same hydrodynamic framework, gravitation follows an identical principle.

When large-scale vortices form in the vacuum, the resulting pressure gradient produces an attractive force proportional to the inverse square of the distance between masses:

$F_g=G\cdot m_1\cdot m_2/r^2$

This mechanism, previously developed in the unified vortex theory of gravitation [2], shows that gravitational and magnetic forces share the same physical origin—both arise from vortex-induced pressure gradients in the vacuum.

Gravitation corresponds to low-frequency, large-scale vortices whose pressure gradients act on mass distributions, while magnetism corresponds to high-frequency, small-scale vortices influencing charge and spin.

In both regimes, the force emerges from the same fundamental law of vacuum vorticity and pressure equilibrium.

The gravitational constant G, like the magnetic permeability μ₀, represents a macroscopic property of the vacuum, characterizing its elastic response to rotational motion.

Together, G and μ₀ link mass attraction and magnetic interaction as complementary expressions of one unified vortex dynamic law governing the superfluid vacuum [2] [3].

6. Magnetic Field, Energy, and Force for a Tri-Vortex Monopole

The tri-vortex monopole represents the simplest configuration that produces a net divergence of magnetic flux in the superfluid vacuum. Its field structure, energy distribution, and mutual interactions arise directly from the quantized circulation of the three coupled vortices forming the monopole core.

The magnetic field associated with a monopole follows the same inverse-square law observed in classical magnetostatics, but in this model it emerges from hydrodynamic rotation rather than from an intrinsic charge. The field at distance r from the monopole center is

$B\left(r\right)={\mu }_{0}g/\left(4\pi r^2\right)\widehat{r}$ ,

where g is the magnetic charge derived from vortex circulation according to $g=\left({\Gamma }_{\text{eff}}/2\pi \right)\sqrt{\rho /{\mu }_0}$ . This field corresponds to the net outward or inward vacuum flux depending on the sign of g, reflecting the dominance of either centrifugal or centripetal vortices in the triad.

6.1. Energy Density and Field Distribution

The energy density of the magnetic field in the vacuum is

$u_B=B^2/\left(2{\mu }_0\right)={\mu }_0{g}^2/\left(32{\pi }^2{r}^4\right)$ .

This energy density decreases as r⁻⁴, ensuring finite total energy when integrated over all space. The field intensity is strongest near the core, where the effective circulation is maximal, and diminishes gradually with distance.

Integrating the energy density from the core radius a to an external boundary R yields the total magnetic self-energy of the monopole:

$U_{\text{mono}}={\displaystyle {\int }_a^R{u}_B\cdot 4\pi r^2\text{d}r}=\left({\mu }_0{g}^2/8\pi \right)\left(1/a-1/R\right)$

Since R is much larger than a, the second term can be neglected, giving

$U_{\text{mono}}\approx {\mu }_0{g}^2/\left(8\pi a\right)$ .

The total energy thus depends on the magnetic charge and on the vortex-core radius. Smaller cores yield more intense fields and higher energy densities, while larger cores correspond to weaker fields and reduced energy concentration.

At the equilibrium radius, the centrifugal kinetic energy of the rotating vacuum flow (½ρv²) is balanced by the magnetic pressure B²/(2μ₀). This equality stabilizes the monopole and prevents unbounded energy growth.

6.2. Field Topology and Polarity

The sign of the magnetic field depends on the orientation of the central vortex. When the central vortex is centripetal, drawing vacuum flux inward, field lines converge toward the core, representing a negative magnetic pole. When it is centrifugal, expelling flux outward, field lines diverge, representing a positive pole.

In both cases, field lines are continuous and free of singularities. They originate from or terminate at the monopole core but close through the vacuum, conserving energy and circulation quantization. The tri-vortex structure therefore provides a physical realization of magnetic polarity without requiring a point-charge singularity.

6.3. Magnetic Interaction between Monopoles

Two monopoles interact through the superposition of their magnetic-pressure fields. The resulting force follows directly from the hydrodynamic equivalence:

$F_m\left(r\right)={\mu }_0{g}_1{g}_2/\left(4\pi r^2\right)$ .

The interaction is attractive when g₁ and g₂ have opposite signs and repulsive when they share the same sign. From the hydrodynamic viewpoint, this behavior arises from the coupling of pressure gradients generated by the rotating vacuum around each monopole. The superposition of these gradients creates a net flow field that draws the monopoles together or pushes them apart, reproducing magnetic attraction and repulsion.

6.4. Energy Balance and Finite Self-Energy

The total energy of the monopole includes both magnetic and hydrodynamic components. The magnetic part comes from the field energy stored in the vacuum, while the hydrodynamic part arises from the kinetic energy of vortex circulation. Stability requires equality between the two pressures at the equilibrium radius:

$\rho v^2/2=B^2/\left(2{\mu }_0\right)$ .

At this radius, the inward vacuum pressure exactly compensates the outward magnetic stress, keeping the energy finite and ensuring dynamic stability.

This hydrodynamic structure removes the singularities that appear in point-charge models. The finite core radius acts as a natural cutoff, so both field and energy density remain bounded at the origin. The continuous distribution of vacuum velocity and pressure replaces the abstract notion of a point magnetic charge with a physically extended, quantized structure.

6.5. Energy-Scale Estimate

The characteristic energy of a free monopole can be estimated from its magnetic self-energy,

$U_{\text{mono}}\approx {\mu }_0{g}^2/\left(8\pi a\right)$ ,

where a is the vortex-core (stem) radius and $g=\left({\Gamma }_{\text{eff}}/2\pi \right)\sqrt{\rho /{\mu }_0}$ .

Substituting gives

$U_{\text{mono}}\approx \rho {\Gamma }_{\text{eff}}^2/\left(32{\pi }^{3}a\right)$ ,

showing that the monopole energy depends quadratically on the effective circulation Γ_eff and inversely on the core radius a.

Taking vacuum density ρ ≈ 10⁻²⁶ kg∙m⁻³, magnetic permeability μ₀ = 4π × 10⁻⁷ H∙m⁻¹, and a core radius comparable to the nucleon stem (a ≈ 0.8 - 1.0 fm), the near-core magnetic field is expected to reach 10¹² - 10¹³ T, consistent with fields inferred from proton magnetic moments.

The magnetic-energy density u_B = B²/(2μ₀) ≈ 10²⁴ - 10²⁶ J∙m⁻³, integrated over a core volume V ≈ 4/3πa³ ≈ 4 × 10⁻⁴⁵ m³, gives a monopole self-energy of roughly

$U_{\text{mono}}\approx \left(1\text{-}10\right)\times {10}^{-14}\text{J}\approx 0.06\text{-}0.6\text{MeV}$ .

Smaller cores or larger circulation numbers increase this value, whereas larger or partially confined geometries reduce it to the keV range.

Hence, monopoles sharing the nucleon’s mushroom geometry fall naturally within the sub-MeV to MeV band—many orders of magnitude below the 10¹⁶ GeV energies predicted by grand-unified monopole theories.

This result supports the interpretation that magnetic monopoles are open-flux states of the same tri-vortex configuration that forms stable nucleons rather than ultra-massive topological relics.

7. Monopole Requirements and Consistency Conditions

To validate any theoretical model of the magnetic monopole, certain fundamental requirements must be satisfied. These include mathematical consistency with Maxwell’s equations, compatibility with quantization conditions, finite energy density, topological stability, and physical plausibility within the known parameters of the universe. The tri-vortex monopole model developed here fulfills all of these criteria within the framework of a superfluid vacuum characterized by density ρ and magnetic permeability μ₀.

7.1. Predictive Coherence and Experimental Plausibility

The tri-vortex model not only satisfies the theoretical requirements but also aligns with physical observables:

• The magnetic moments of the proton and neutron are direct consequences of their confined monopole configurations, supporting the validity of the model at measurable scales.

• The finite energy of the monopole corresponds to mass-energy values compatible with baryonic particles, rather than unattainable grand-unified energy levels.

• The absence of singularities ensures compatibility with general relativity and eliminates the need for artificial renormalization.

• The model predicts that analog monopoles could be reproduced in laboratory superfluid systems or Bose-Einstein condensates, where quantized circulation and flux imbalance mimic the same hydrodynamic mechanisms.

• The identification of the proton and neutron as confined monopoles of opposite polarity provides a natural explanation for nuclear binding and magnetic asymmetry without invoking separate fundamental forces.

In summary, the tri-vortex monopole model fulfills all theoretical, topological, and energetic criteria for monopole existence. It provides a self-consistent, experimentally plausible, and mathematically complete description of magnetic charge as a macroscopic expression of vacuum circulation. By unifying hydrodynamic and electromagnetic principles, it bridges the gap between quantum field theory, general relativity, and observable particle physics, establishing a coherent foundation for a unified field framework.

7.2. Satisfaction of Monopole Conditions by the Tri-Vortex Model

The tri-vortex monopole satisfies all fundamental monopole conditions as follows:

1) Non-zero field divergence:

The differential circulation between the two centrifugal vortices and the central centripetal vortex generates a net flux Φ = 2Γ₊ − Γ₋. This circulation imbalance produces an open magnetic field with measurable divergence, realizing ∇·B = 4πg ≠ 0.

2) Quantization:

Because vortex circulation in the superfluid vacuum is quantized, the magnetic flux and charge are also quantized. The magnetic charge is expressed as $g=\left({\Gamma }_{\text{eff}}/2\pi \right)\sqrt{\rho /{\mu }_0}$ , and since Γ_eff = nh/m_v, magnetic charge and flux take discrete values consistent with Dirac’s quantization condition.

3) Finite energy:

The total magnetic energy is U_mono = (μ₀g²/8πa), which remains finite due to the finite core radius a. The internal pressure gradient ½ρv² balances the magnetic stress B²/(2μ₀), ensuring energy convergence and stability.

4) Topological stability:

The tri-vortex structure possesses quantized helicity, meaning that the magnetic field lines and velocity field lines are topologically linked. This helicity protects the configuration from decay, making it a stable soliton-like excitation of the vacuum.

5) Dynamic equilibrium:

The monopole achieves mechanical equilibrium when the inward vacuum pressure equals the outward magnetic stress. This condition guarantees that the structure neither collapses nor expands indefinitely, maintaining constant total energy over time.

6) Continuity and differentiability:

Unlike the Dirac monopole, which introduces singularities, the tri-vortex configuration provides continuous and differentiable field and pressure distributions, satisfying the differential form of Maxwell’s equations throughout space.

7) Gauge and symmetry compatibility:

The superfluid vacuum behaves as an effectively Abelian medium on macroscopic scales but can accommodate non-Abelian structures at microscopic levels through vortex coupling and helicity exchange. Thus, the tri-vortex monopole remains consistent with gauge invariance and field quantization principles.

8. Equivalence between Magnetic and Strong Nuclear Forces

8.1. Statement of the Problem

We consider two neighboring nucleons, each modeled as a confined tri-vortex (monopolar) object. We show that their interaction force at a typical nuclear separation r ~ 1 fm equals the strong (nuclear) force in both magnitude and scaling, and that the monopole (magnetic-charge) expression and the hydrodynamic (circulation) expression are strictly equivalent.

8.2. Two Equivalent Force Laws in the Tri-Vortex Framework

1) Monopole (magnetic-charge) form

$F_m\left(r\right)=\frac{{\mu }_0{g}_1{g}_2}{4\pi r^2}$

2) Hydrodynamic (circulation) form

$F_h\left(r\right)=\frac{\rho }{32{\pi }^3{r}^2}{\Gamma }_{\text{eff},1}{\Gamma }_{\text{eff},2},g=\frac{{\Gamma }_{\text{eff}}}{2\pi }\sqrt{\frac{\rho }{2{\mu }_0}}.$

Substituting g(Γ_eff) into F_m reproduces F_h, proving both are the same force written in different variables.

8.3. Calibration at the Nuclear Scale (r = 1 fm)

A standard strong-interaction benchmark is the string tension

$\sigma \simeq 0.9\text{GeV}/\text{fm}$ .

Converting units,

$1\text{GeV}=1.602\times {10}^{-10}\text{J},1\text{fm}={10}^{-15}\text{m}\Rightarrow F_{\sigma }=\frac{0.9\times 1.602\times {10}^{-10}}{{10}^{-15}}\simeq 1.44\times {10}^5\text{N}$

Required monopole charge at $r={10}^{-15}\text{m}$ .

Impose $F_m=F_{\sigma }$ :

$g_{\ast }=\sqrt{\frac{4\pi F_{\sigma }{r}^2}{{\mu }_0}}=\sqrt{\frac{4\pi \times {10}^{-7}\left(1.44\times {10}^5\right){\left({10}^{-15}\right)}^2}{4\pi }}\simeq 1.2\times {10}^{-9}\text{A}\cdot \text{m}$

8.4. Finite Core Energy in the MeV-GeV Hadronic Range

The monopole self-energy for a finite core a is

$U_{\text{mono}}=\frac{{\mu }_0{g}^2}{8\pi a}\left(R\gg a\right).$

With $g=g_{\ast }$ and a = 0.8 fm,

$U_{\text{mono}}=\frac{4\pi \times {10}^{-7}{\left(1.2\times {10}^{-9}\right)}^2}{8\times \pi \times 0.8\times {10}^{-15}}\simeq 9.0\times {10}^{-11}\text{J}\approx 5.6\times {10}^8\text{eV}=0.56\text{GeV}$

For $a=1.0\text{fm}$ , $U_{\text{mono}}\approx 0.45\text{GeV}$

This hadronic (hundreds of MeV) scale is exactly what one expects for short-range nuclear dynamics inside nucleons; the value depends on the core radius aaa and is finite (no point-singularity).

8.5. Hydrodynamic Consistency Check

To test internal consistency of the tri-vortex framework, we compare the interaction force obtained in the magnetic-charge description, $F_{\sigma }$ , with the force predicted by the hydrodynamic circulation form. Assuming symmetry among the three vortices ${\Gamma }_{\text{eff,1}}={\Gamma }_{\text{eff,2}}={\Gamma }_{\text{eff,3}}\equiv {\Gamma }_{\text{eff}}$ , we use the hydrodynamic scaling

With the vacuum mass density adopted in the model, $\rho \simeq 9.5\times {10}^{-27}\text{kg}\cdot {\text{m}}^{-3}$ , evaluated at $r=1.0\times {10}^{-15}\text{m}$ and $F_{\sigma }=1.44\times {10}^5\text{N}$ , this yields

i.e., a circulation scale consistent with a femtometer-range interaction of strength $\sim {10}^5\text{ N}$ .

To translate this circulation into an effective magnetic-charge parameter $g$ , one must specify a dimensionally consistent bridge relation between the hydrodynamic variables $\left(\rho |\text{Γ}\right)$ and the magnetic-like coupling $\left({\mu }_0|g\right)$ . When that bridge is defined within the framework, the resulting $g$ can be compared directly with the independently inferred $g^{*}$ . This establishes whether the magnetic-charge and hydrodynamic descriptions are numerically consistent at femtometer scales.

In this regime, the tri-vortex interaction between nucleons at $r\sim 1\text{ fm}$ reproduces the characteristic strong-force magnitude,

and the corresponding finite-core energy lies in the hadronic range,

$U_{mono}\sim 0.45\text{ - }0.56\text{ GeV fora}=0.8\text{ - }1.0\text{ fm}$

Therefore, within the superfluid-vacuum (tri-vortex) framework, the “magnetic” interaction between nucleons can be interpreted as the strong nuclear force: a pressure-mediated coupling of quantized vacuum vortices, expressible in either hydrodynamic or magnetic-like language once the mapping between parameters is fixed.

9. Implications for Unified Field Theory

The tri-vortex model of the magnetic monopole provides a direct bridge between electromagnetism, the strong interaction, and the inertial properties of matter. Within this framework, all forces and charges arise from the same fundamental property: the circulation of the superfluid vacuum. The vacuum acts as a continuous, quantized medium whose density ρ and permeability μ₀ define the relationship between magnetic, electric, and gravitational phenomena.

In this unified interpretation, magnetic and electric fields are two orthogonal manifestations of vortex motion in the vacuum. The magnetic field represents transverse circulation of the vacuum, governed by its shear response μ₀, while the electric field corresponds to longitudinal compression and expansion, governed by its elastic response ε₀. The speed of light c, expressed as $c=1/\sqrt{{\epsilon }_0{\mu }_0}$ , therefore describes the propagation velocity of perturbations within this medium, connecting both effects through the same underlying substance.

Mass, charge, and spin also emerge from the geometry and dynamics of vacuum vortices. The inertial mass of a particle corresponds to the rotational kinetic energy of the vortex core; electric charge arises from the net inflow or outflow of vacuum flux through the core; and magnetic charge arises from circulation asymmetry between the core and its periphery. The tri-vortex monopole combines these three aspects in one structure: rotational mass from vortex motion, magnetic flux from circulation imbalance, and electric potential from longitudinal displacement of the vacuum.

The proton and neutron, described as confined monopoles of opposite polarity, demonstrate this unification at the nucleonic level. Their mutual attraction is the result of coupling between inward and outward fluxes, which minimizes the total shear energy of the surrounding vacuum. This interaction is not distinct from the magnetic force but rather its confined, short-range form, appearing macroscopically as the strong nuclear force. The nuclear binding energy thus represents the localized reduction of vacuum stress produced by the alignment of opposite monopoles.

The same principle extends naturally to larger physical systems. In atomic and molecular structures, the coupling of electric and magnetic vortex components gives rise to electromagnetic interactions. On the macroscopic and cosmological scale, differences in vacuum density and circulation can manifest as gravitational curvature or large-scale magnetic phenomena. In every case, the dynamics of matter and energy are governed by the same equations of motion describing the superfluid vacuum.

The role of μ₀ in this unified view is fundamental. It is not a constant arbitrarily assigned to free space, but the measurable shear response of the vacuum that links magnetic energy to vortex motion. Together with the vacuum density ρ, it defines the mechanical impedance of the medium and determines the magnitude of magnetic and gravitational effects. Variations in μ₀ or ρ, whether due to local vacuum polarization or cosmic evolution, would therefore affect both electromagnetic and gravitational constants simultaneously, suggesting that all so-called fundamental constants are expressions of the same vacuum state [9]-[11].

The tri-vortex model also clarifies the origin of quantization in nature. Because circulation in a superfluid is quantized, every vortex structure possesses discrete energy and flux levels. This quantization propagates upward through all physical systems, establishing the discrete values of magnetic moment, charge, and energy observed in elementary particles and atomic transitions. Quantum behavior thus emerges as the natural consequence of superfluid-vacuum mechanics rather than as an independent postulate.

From the perspective of unified field theory, the monopole plays a central role as the geometric archetype linking all forms of energy. Its field represents the simplest stable solution of the vacuum’s hydrodynamic equations, combining rotation, flow, and pressure balance in one quantized configuration. When flux is closed within the structure, the system manifests as a stable particle such as the proton or neutron. When flux remains open, it appears as a magnetic monopole. When the same circulation is accelerated, it produces electromagnetic radiation. Each phenomenon corresponds to a different boundary condition of the same fundamental equation governing vacuum motion.

This approach restores continuity between microscopic and macroscopic physics. It unites the quantized properties of subatomic particles with the continuous behavior of fields and waves by identifying both as expressions of the same fluid-dynamic reality. The boundaries between electromagnetism, the strong force, and gravitation dissolve when seen as different manifestations of vortex dynamics within a compressible, superfluid vacuum.

In summary, the tri-vortex monopole model demonstrates that the apparent diversity of fundamental interactions originates from a single universal mechanism: the rotation and deformation of the superfluid vacuum. Electric, magnetic, and nuclear forces correspond to different modes of vacuum motion characterized by the same physical parameters μ₀, ε₀, and ρ. This unification offers a coherent pathway toward a complete field theory in which mass, charge, and magnetism emerge from one continuous hydrodynamic principle governing the structure of space itself.

10. Conclusions

The model presented in this study establishes a unified physical framework in which all fundamental interactions—magnetic, electric, nuclear, and gravitational—emerge from the same hydrodynamic behavior of the superfluid vacuum. In this interpretation, the vacuum is not an empty background but a continuous, compressible medium endowed with density ρ, elasticity ε₀, and shear response μ₀. The dynamic rotation and deformation of this medium give rise to the full spectrum of observable forces in nature.

Within this context, the tri-vortex structure represents the fundamental geometric pattern of the vacuum. When its flux is confined internally, it forms the proton and neutron; when it remains open, it manifests as a magnetic monopole. The proton, with two centrifugal up-type vortices and one centripetal down-type vortex, behaves as a negative magnetic monopole with inward flux. The neutron, with two centripetal down-type vortices and one centrifugal up-type vortex, represents the corresponding positive monopole. Their coupling produces a double-loop flux configuration responsible for the strong nuclear force.

The same pressure-gradient law that governs the interaction between magnetic monopoles also gives rise to the gravitational attraction between masses. As shown in earlier work [3], gravitation originates from large-scale vortices in the superfluid vacuum, where the pressure gradient generated by rotation produces an inward flow that pulls matter toward regions of higher curvature. Magnetism and gravitation are therefore not separate forces but manifestations of the same mechanism operating at different scales: magnetism reflects localized, high-frequency vacuum vortices; gravitation reflects macroscopic, low-frequency vortices acting on mass distributions. Both obey the same inverse-square law arising from the hydrodynamic balance between centrifugal motion and vacuum pressure.

In this unified view, mass, charge, and magnetic moment are emergent quantities that depend on vortex geometry and boundary conditions. The inertial mass corresponds to the kinetic energy of vortex rotation; electric charge arises from longitudinal flux imbalance; magnetic charge from transverse circulation asymmetry. The constants ε₀, μ₀, and G express the vacuum’s mechanical properties at different levels of organization: ε₀ describes its compressibility, μ₀ its shear response, and G its large-scale coupling to curvature. [9]-[11] Each constant is therefore a macroscopic parameter derived from the same vacuum structure rather than a fundamental, independent quantity.

The tri-vortex monopole model thus unifies the microscopic and macroscopic domains under a single physical principle: the self-organized rotation of the superfluid vacuum. At the quantum scale, it explains the quantization of flux, charge, and spin through discrete vortex circulation. At the nuclear scale, it reproduces the strong interaction as the coupling of opposite monopoles. At the cosmic scale, it describes gravitation as the global pressure field produced by coherent vacuum rotation.

The apparent diversity of fundamental forces is therefore an expression of different modes of motion within one universal medium. Electric phenomena correspond to longitudinal oscillations of the vacuum; magnetic phenomena to transverse rotations; gravitational attraction to global curvature produced by large vortices; and nuclear binding to localized coupling between opposite magnetic flows. All follow from the same governing equation of vortex-induced pressure balance in the vacuum.

The monopole energy derived from the mushroom model, ranging from sub-MeV levels for confined nucleon-like configurations to eV levels for causality-limited open flux, fits naturally within this unified description. It confirms that magnetic monopoles are not hypothetical massive relics but open-flux states of the same tri-vortex geometry that forms ordinary matter.

This unified framework eliminates the artificial boundaries separating classical and quantum, electromagnetic and gravitational phenomena. It provides a continuous, quantized description of nature in which every field, particle, and interaction arises from structured motion of the same physical vacuum. The tri-vortex monopole, together with the proton, neutron, and large-scale gravitational vortices, represents different expressions of one universal principle: the dynamic equilibrium between rotation and pressure in the superfluid vacuum.

Future research should formalize this unified field model mathematically by deriving the coupled equations of motion that link ε₀, μ₀, and G through the same hydrodynamic constants and by exploring possible observational evidence of vacuum vorticity in astrophysical and condensed-matter systems. Such work may lead toward an experimentally verifiable theory of everything rooted not in abstract symmetry but in the measurable dynamics of the living vacuum.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Dirac, P.A.M. (1931) Quantized Singularities in the Electromagnetic Field. Proceedings of the Royal Society A, 133, 60-72.
[2]	Butto, N. (2023) A New Theory to Understand the Mechanism of Gravitation. Journal of High Energy Physics, Gravitation and Cosmology, 9, 1-19.
[3]	Butto, N. (2025) Omniom Vacuum Framework: Unifying Vacuum States from Pre-Big Bang to Dark Energy. Journal of High Energy Physics, Gravitation and Cosmology, 11, 612-638.[CrossRef]
[4]	Butto, N. (2025) Unravelling the Proton Mass Puzzle: A Novel Approach through Quark Vortex Dynamics and the Mushroom Model. Journal of High Energy Physics, Gravitation and Cosmology, 11, 951-972.
[5]	Butto, N. (2020) Electron Shape and Structure: A New Vortex Theory. Journal of High Energy Physics, Gravitation and Cosmology, 6, 340-352.[CrossRef]
[6]	Butto, N. (2024) A New Theory Exploring the Internal Structure of Quarks. Journal of High Energy Physics, Gravitation and Cosmology, 10, 1713-1733.[CrossRef]
[7]	Rayfield, G.W. and Reif, F. (1964) Quantized Vortex Rings in Superfluid Helium. Physical Review, 136, A1194-A1208.[CrossRef]
[8]	Preskill, J. (1984) Magnetic Monopoles. Annual Review of Nuclear and Particle Science, 34, 461-530.[CrossRef]
[9]	Butto, N. (2020) New Mechanism and Analytical Formula for Understanding the Gravity Constant G. Journal of High Energy Physics, Gravitation and Cosmology, 6, 357-367.[CrossRef]
[10]	Butto, N. (2020) The Essence and Origin of the Magnetic Constant. Journal of High Energy Physics, Gravitation and Cosmology, 06, 662-669.[CrossRef]
[11]	Butto, N. (2021) Revealing the Essence of Electric Permittivity Constant. Journal of High Energy Physics, Gravitation and Cosmology, 7, 210-217.[CrossRef]