The Relativistic Equilibrium: How Electron Velocity Shifts the Balance of Time and Charge to Define Atomic Structure

J. Rogers, SE Ohio, 31 Jul 2025, 1805

Abstract

A model of particle stability is proposed, positing that matter forms at the geometric equilibrium of dissimilar fields. This framework is extended to include relativistic effects. The foundational postulate—that a particle's stable radius r is determined by an equilibrium between its mass-generated time field and its charge-generated electric field, m/r ~ α/r²—is updated to account for the relativistic mass increase of a moving particle. The equilibrium thus becomes (γm₀)/r ~ α/r², where m₀ is the rest mass and γ is the Lorentz factor. The solution to this relativistic equilibrium, r ~ α/(γm₀), demonstrates that a particle's stable radius must contract as its velocity increases. This single, simple principle provides a first-principles explanation for the well-documented relativistic effects in heavy atoms, such as the color of gold and the liquidity of mercury. This framework suggests that relativistic quantum chemistry is not a separate, more complex theory, but a natural and necessary consequence of a fundamental geometric balance between the time field and the charge field.

1. Introduction: From Static Balance to Dynamic Equilibrium

In a previous paper, it was demonstrated that the classical electron radius could be derived from the principle that a stable particle forms at the specific radius r where its linearly-scaling time-field potential ( finds equilibrium with its inverse-square scaling charge-force field (. This equilibrium, m/r ~ α/r², leads to the structural property m*r ~ α, which correctly projects to the classical electron radius formula.

However, this initial model treated the mass m as a static quantity. In reality, especially within the inner orbitals of heavy atoms, electrons can reach significant fractions of the speed of light. According to Special Relativity, the effective mass of a particle is not constant but increases with velocity.

This paper investigates the consequences of incorporating this relativistic mass increase into the foundational equilibrium postulate. It will be shown that this simple refinement not only preserves the model but enhances its predictive power, naturally explaining complex phenomena in atomic physics and chemistry that are otherwise treated as separate relativistic "corrections."

2. The Relativistic Refinement of the Foundational Postulate

The core of this theory is the equilibrium between two fields generated by the particle itself.

• The Time Field: Its potential is proportional to the particle's mass.

• The Charge Field: Its force is proportional to the particle's dimensionless charge.

A crucial insight is that the "mass" that generates the time field is not the invariant rest mass (m₀), but the total relativistic mass (, where γ = 1 / √(1-v²/c²). The time-field, being a manifestation of the particle's interaction with spacetime, must be sourced by its full energy content, which includes its kinetic energy.

The foundational postulate of field equilibrium must therefore be updated to its dynamic, relativistic form:

Postulate: The Relativistic Field Equilibrium
(γm₀)/r ~ α/r²

This is the more complete statement of the principle. It asserts that the stable radius of a particle is determined by the balance between its charge field and the time field generated by its velocity-dependent effective mass.

3. The Consequence: Relativistic Contraction of the Equilibrium Radius

The power of this refined postulate becomes immediately apparent when one solves for the equilibrium radius r.

From the proportionality (γm₀)/r ~ α/r², we can multiply both sides by r² and divide by γm₀ to find the scaling for r:

r ~ α / (γm₀)

This result is profoundly significant and leads to a clear, testable prediction:

1. The Inverse Relationship with  The stable radius r is now inversely proportional to the Lorentz factor γ.

2. The Lorentz Factor ( Since γ is always ≥ 1 and increases as velocity approaches c, the equilibrium radius r must decrease as the particle's velocity increases.

3. The Prediction (Relativistic Contraction): High-velocity electrons must necessarily occupy smaller, more tightly bound stable radii than their low-velocity counterparts.

This model has thus derived the phenomenon of relativistic orbital contraction from a single, fundamental principle of field equilibrium.

4. Experimental Validation: Explaining the Quirks of Heavy Elements

This derived principle is not a mere theoretical curiosity. It is the direct explanation for several well-known, empirically verified phenomena that have long been the domain of complex relativistic quantum chemistry.

• The Color of Gold: In a gold atom (atomic number 79), the inner-shell electrons move at over half the speed of light (v ≈ 0.58c), resulting in a significant Lorentz factor (γ ≈ 1.2). This increased effective mass γm₀ causes the inner orbitals to contract dramatically. This contraction shifts the energy levels of the outer electrons, changing the atom's light absorption spectrum. The atom absorbs blue light more strongly, reflecting the yellow and red light that gives gold its characteristic color. The (γm₀)/r term is the proposed reason gold isn't silvery like its neighbors on the periodic table.

• The Liquidity of Mercury: Mercury (atomic number 80) experiences even stronger relativistic contraction. This pulls its valence electrons into a very tight, stable shell, resulting in unusually weak metallic bonds between mercury atoms. These weak bonds are not strong enough to form a solid crystal lattice at room temperature, explaining why mercury is a liquid.

• The Lanthanide Contraction: This trend across the periodic table, where elements become unexpectedly smaller than predicted by simpler models, is directly explained by the progressive increase in electron velocities and the corresponding relativistic contraction of their orbitals as the nuclear charge grows.

5. The Significance: Relativity is Not an "Add-On"

In the standard pedagogical and historical approach to quantum mechanics, these effects are treated as "relativistic corrections." One first learns the simple, non-relativistic Schrödinger model of the atom, which gets the basics right for light elements. Then, to explain heavy elements, one must introduce a much more complex and less intuitive theory, like the Dirac equation, which bolts relativity onto quantum mechanics.

This framework demonstrates that this separation is artificial. Relativity is not an add-on; it is an intrinsic and necessary part of the foundational equilibrium.

The m in the original m/r was never meant to be a static constant. It was always the effective, dynamic mass. By simply acknowledging this, the entire suite of "relativistic effects" emerges naturally and gracefully from the core principle, without the need for a separate, more complex theory.

6. Conclusion: A Deeper, More Dynamic Unity

The inclusion of the Lorentz factor transforms this model from a static picture of stability into a dynamic one. The equilibrium point r is not fixed; it shifts in response to the particle's kinetic state.

This demonstrates the profound unifying power of the geometric field equilibrium principle:

1. It is Inherently Relativistic: The model naturally incorporates Special Relativity by treating mass as dynamic, not static.

2. It is Causally Explanatory: It provides a clear physical mechanism for relativistic contraction, explaining it as a necessary shift in the balance point between the time field and the charge field.

3. It is Simple and Unified: It replaces the need for separate, complex "relativistic quantum" theories with a single, elegant principle that covers all cases, from a stationary electron to one moving at nearly the speed of light.

The structure of matter, from the lightest hydrogen atom to the heaviest superheavy elements, appears to be dictated by a single, dynamic dance: the search for equilibrium between a time field that grows with velocity and a charge field that does not. The shifting balance point of this dance may be what paints the universe with the colors of gold and the liquidity of mercury.