J. Rogers, SE Ohio
Abstract
Maxwell's Equations, the foundation of classical electromagnetism, are conventionally formulated within a system requiring four fundamental dimensions: mass [M], length [L], time [T], and electric charge [Q]. This paper demonstrates that by eliminating charge as a fundamental dimension and re-conceptualizing it as a dimensionless count of discrete quantum events, Maxwell's Equations can be fully reformulated within a purely mechanical M-L-T framework. In this new system, validated by a re-derivation of the Lorentz Force Law and the constitutive relations, the electric field [E] assumes the dimension of force, and the magnetic field [B] that of mass-frequency. This reformulation reveals Maxwell's Equations not as laws governing a unique "electromagnetic" substance, but as a set of fundamental principles governing the dynamics of force and mass-frequency fields in spacetime, sourced by the geometry of discrete events. The consistency of this approach is critically confirmed by showing that the speed of light, c, remains an invariant velocity derived from the new geometric scaling constants.
1. Introduction
The development of Maxwell's Equations in the 19th century necessitated the introduction of electric charge as a new, fundamental substance, distinct from mass. This led to a system of units in which mechanical and electromagnetic phenomena were dimensionally separate. This paper applies a geometric framework—in which charge is a dimensionless count (e=1) and its effects are mediated by a "natural charge density" constant (ncd) with units of [M·L]—to demonstrate a deeper, underlying mechanical nature of electromagnetism.
2. A Dictionary of Geometric-Mechanical Translation
The translation from the standard SI system to the proposed three-dimensional (M-L-T) framework is predicated on the axiom that charge is a dimensionless count. The dimensional properties of all other electromagnetic quantities are then derived from this premise via established physical laws. The dimensional unit for the B-field, [M·T⁻¹], invites several physical interpretations. While 'Mass-Frequency' is the most direct translation, it is equivalent to Momentum-per-unit-length [(M·L·T⁻¹)/L] or Action-per-unit-area [(M·L²·T⁻¹)/L²]. For consistency, we will use 'Mass-Frequency', while acknowledging these other rich geometric interpretations.
Table 1: Translation of Electromagnetic Quantities
| Charge (q) | Coulomb [C] | [1] (dimensionless) | A pure count of discrete quantum events |
| Charge Density (ρ) | [C/m³] | [m⁻³] | Number density of quantum events |
| Current Density (J) | [A/m²] = [C/(m²·s)] | [m⁻²·s⁻¹] | Frequency density of quantum events |
| Electric Field (E) | [N/C] = [V/m] | [N] = [M·L·T⁻²] | A pure Force field |
| Magnetic Field (B) | Tesla [T] = [kg/(C·s)] | [kg/s] = [M·T⁻¹] | A Mass-Frequency or Momentum-Density field |
| Permittivity (ε₀) | [F/m] | [T²·M⁻¹·L⁻³] | A geometric scalar for Force fields |
| Permeability (μ₀) | [H/m] | [M·L] (4π·ncd) | A geometric scalar for Mass-Frequency fields |
3. The Lorentz Force Law: The Foundational Definition of Fields
The Lorentz Force Law is the operational definition of the E and B fields. Its consistency in the new framework is a primary test.
Standard Form: F = qE + q(v × B)
New Framework Form: F_net = n_count · E_force + n_count · (v × B_massfreq)
Here, n_count is the dimensionless integer count of elementary charges. A dimensional analysis of the new form confirms its validity:
[Force] = [1]·[Force] + [1]·[Velocity] × [Mass-Frequency]
[M·L·T⁻²] = [M·L·T⁻²] + [L·T⁻¹] × [M·T⁻¹]
[M·L·T⁻²] = [M·L·T⁻²] + [M·L·T⁻²]
The equation is dimensionally consistent. The interpretation is powerful: the total force on a particle is the sum of the static force field (E_force) at that location, plus a dynamic, motional force arising from its velocity through the mass-frequency field (B_massfreq). This provides a direct, mechanical justification for the dimensions assigned in Table 1.
4. The Constitutive Relations: Redefining the Vacuum
The constitutive relations (D = ε₀ E, B = μ₀ H) describe how fields behave in a medium, with D and H accounting for the medium's response. In a vacuum, they reveal the geometric nature of the scaling constants.
D-Field: From D = ε₀ E, the dimensions of D become [T²·M⁻¹·L⁻³] × [M·L·T⁻²] = [L⁻²]. The D-field is an inverse area, representing a flux density or "number of field lines per unit area." The relation shows that ε₀ is the scaling factor that converts a Force field into an area-density of its effect.
H-Field: From B = μ₀ H, the dimensions of H become [M·T⁻¹] / [M·L] = [L⁻¹·T⁻¹]. This is consistent with its role in Ampere's law, ∇ × H = J_free. The H-field can be interpreted as a "Circulation Source," and μ₀ (4π·ncd) is the fundamental scaling factor that converts this source circulation into the resulting mass-frequency field B.
This confirms that ε₀ and μ₀ are not properties of the vacuum, but are fundamental geometric constants that define the coupling strength between force, mass, and time in this M-L-T spacetime.
5. Reformulation of Maxwell's Equations
5.1 Gauss's Law for Electricity: ∇ ⋅ E_force = ρ_count / ε₀_geom
A mechanical law stating that the spatial density of quantum events sources the divergence of a force field.
5.2 Gauss's Law for Magnetism: ∇ ⋅ B_massfreq = 0
A geometric constraint stating that the mass-frequency field is continuous and has no sources or sinks.
5.3 Faraday's Law of Induction: ∇ × E_force = -∂B_massfreq/∂t
A mechanical coupling law stating that a time-varying mass-frequency field sources a curling force field.
5.4 Ampère-Maxwell Law: ∇ × B_massfreq = μ₀_geom (J_freq + ε₀ ∂E/∂t)
A mechanical source law stating that the frequency density of real or effective quantum events sources the curl in the mass-frequency field.
6. Discussion and Key Validations
6.1 Validation via the Speed of Light
A critical test of any reformulation of electromagnetism is the recovery of the speed of light. In the standard system, c = 1/√(μ₀ε₀). We test this using our new geometric units:
c = 1 / √([μ₀]·[ε₀])
c = 1 / √([M·L] · [T²·M⁻¹·L⁻³])
c = 1 / √([M·L·T²·M⁻¹·L⁻³]) = 1 / √([T²·L⁻²])
c = 1 / (T·L⁻¹) = L·T⁻¹
The result is a velocity, [m/s]. This demonstrates that the fundamental relationship between the scaling constants, space, and time is preserved. The speed of light emerges not as a uniquely "electromagnetic" phenomenon, but as an invariant speed inherent to the geometric and mechanical properties of this M-L-T spacetime itself.
6.2 Unification with Mechanics
By eliminating charge, the framework grounds electromagnetism in the familiar mechanical dimensions of kg, m, and s. The equations now describe how force and mass-frequency fields are generated and interact, making them a natural extension of Newtonian and relativistic mechanics. The "field" is the collective mechanical response of spacetime to discrete quantum events.
7. Conclusion
The successful reformulation of Maxwell's Equations, the Lorentz Force Law, and the constitutive relations within a three-dimensional M-L-T framework validates the hypothesis that electric charge is not a fundamental physical dimension. This new perspective reveals Maxwell's Equations as a set of laws for Spacetime Mechanics, describing how force fields and mass-frequency fields are sourced by the density and frequency of discrete quantum events. This work eliminates the conceptual barrier between mechanical and electromagnetic phenomena, suggesting a deeper geometric unity in the fundamental laws of nature.
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