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A Relativistic Correction to the 1925 Lorentz-Pauli Calculation of Electron Spin

This paper identifies a historical oversight in the 1925 Lorentz-Pauli objection to electron spin by demonstrating that incorporating relativistic field transformations allows a near-light-speed surface velocity to generate the observed magnetic moment, thereby correcting the classical calculation without validating a mechanical origin for spin.

Original authors: Frank Wang

Published 2026-07-22

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CC BY 4.0

Original authors: Frank Wang

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Spin-Off Story: Why Electrons Might Not Be Breaking the Speed Limit

Imagine you are trying to understand how a tiny, invisible magnet works. In the world of physics, this is the story of the electron, a fundamental particle that carries a negative electric charge and acts like a tiny bar magnet. For over a century, scientists have been fascinated by a specific property called "spin." It's a bit like the electron is spinning on its own axis, creating a magnetic field, but it's not actually a ball of clay rotating in space; it's a quantum property that just is.

To make sense of this, let's look at two big ideas that act as the stage for our story. First, there's electromagnetism. Think of electricity and magnetism as two sides of the same coin. If you have a stationary electric charge, it just sits there with an electric field. But if you move that charge, or if you look at it from a moving perspective, that electric field transforms and creates a magnetic field. It's like how a stationary tree looks different to you when you are driving past it at high speed; the motion changes how you see the world around it.

Second, there's Special Relativity, Einstein's famous theory which says nothing can travel faster than the speed of light ( $c$ ). This is the universe's ultimate speed limit. If you try to push an object with mass to light speed, it would require infinite energy, which is impossible. So, for decades, physicists have been stuck on a puzzle: if the electron is spinning to create its magnetism, how fast is it spinning? If you do the math using old-school, non-relativistic rules, the answer seems to be that the electron's surface would have to spin 137 times faster than light. That's a clear violation of the speed limit, suggesting the "spinning ball" idea must be wrong. But is the math really that simple, or did we miss a trick?

The Paper's Twist: Fixing a Century-Old Math Mistake

This paper by Frank Wang from the University of Kent takes a fresh look at a famous argument from 1925. Back then, two giants of physics, Lorentz and Pauli, tried to explain why the electron couldn't be a spinning ball of charge. They calculated that to produce the electron's observed magnetic strength (called the Bohr magneton), the surface of the electron would need to race around at a speed of $137c$ —137 times the speed of light. Since this is impossible, they concluded that the electron's spin couldn't be a physical rotation at all.

Wang's paper suggests that Lorentz and Pauli made a specific, overlooked error: they forgot to apply the rules of relativity to the fields themselves, not just the speed.

Here is the core of the new idea: When an object moves very fast, close to the speed of light, its electric and magnetic fields don't just stay the same; they get squashed and concentrated. Imagine a flashlight beam that is normally wide and soft. If you zoom past it at nearly light speed, that beam gets squeezed into a super-thin, super-intense pancake of light right in front of you. The paper argues that the electron's magnetic field does exactly this. Because the electron is spinning so fast, its magnetic field gets "Lorentz contracted" or squeezed into a tight jet along its axis of rotation.

This squeezing acts like a magnifying glass. In the old calculation, the magnetic field was spread out, so the electron needed to spin impossibly fast to generate enough strength. But in Wang's corrected model, the field is so concentrated that the electron doesn't need to spin at $137c$ . Instead, it only needs to spin at about $0.99997c$ (which is just a tiny bit slower than the speed of light). At this speed, the relativistic squeezing boosts the magnetic field strength by a factor of roughly 137. This boost perfectly compensates for the fact that the electron is spinning just under the speed limit, allowing it to generate the exact magnetic strength we observe without breaking the laws of physics.

The paper suggests that the "impossible" speed of $137c$ was just an artifact of using the wrong math (non-relativistic math) for a fast-moving object. When you fix the math to include how fields behave at high speeds, the paradox disappears. The electron could physically be spinning, and it wouldn't need to break the speed limit to do it.

However, the author is careful to point out that this doesn't prove the electron is definitely a spinning ball. It just shows that the old argument against it was flawed. The paper also notes that while this classical model works for the basic magnetic strength, it doesn't fully explain other complex details, like why the electron's "gyromagnetic ratio" (a specific measure of how its spin relates to its magnetism) is exactly 2, a number that usually requires quantum mechanics to explain. The author suggests that this classical relativistic view might be a stepping stone to understanding the deeper connection between electricity, magnetism, and the geometry of space itself, but it remains a theoretical correction rather than a final, proven fact.

In short, the paper argues that we shouldn't have dismissed the idea of a spinning electron so quickly. By remembering that fast-moving fields get squeezed and amplified, the electron can spin at a "normal" (though still incredibly fast) speed and still create the magnetism we see, resolving a mystery that has puzzled physicists for a hundred years.

Technical Summary: A Relativistic Correction to the 1925 Lorentz-Pauli Calculation of Electron Spin

Problem Statement
The paper addresses a historical paradox in physics dating back to 1925, when Goudsmit and Uhlenbeck proposed the concept of electron spin. This hypothesis was immediately rejected by Hendrik Lorentz and Wolfgang Pauli based on a classical non-relativistic calculation. Their argument posited that if an electron were modeled as a rigid, rotating charged sphere, the equatorial surface velocity required to generate the observed magnetic moment (the Bohr magneton, $\mu_B$ ) would be approximately $137c$ (where $c$ is the speed of light). This superluminal velocity violates the constraints of Einstein's theory of special relativity, leading to the dismissal of any classical mechanical origin for electron spin. The paper identifies a fundamental oversight in the original Lorentz-Pauli computation: the neglect of relativistic field transformations.

Methodology
The study reconstructs the classical electron model by incorporating relativistic corrections that were absent in the 1925 analysis. The methodology proceeds in three stages:

Relativistic Field Transformation: The authors analyze the electromagnetic fields of a moving charge using Lorentz transformations. They demonstrate that as a charge approaches the speed of light, its electric and magnetic fields undergo Lorentz contraction, concentrating field lines into a "pancake" shape perpendicular to the direction of motion. This results in a relativistic amplification factor, $\gamma$ (the Lorentz factor), for the field strength in the transverse plane.
Revised Classical Model: The electron is modeled as a charged ring of radius $r_e$ rotating at angular velocity $\omega$ with tangential velocity $v$ . The authors apply the relativistic amplification factor to the current loop. Specifically, they argue that for a field point on the symmetry axis, the magnetic field contributions from infinitesimal charge elements are amplified by $\gamma$ due to the relativistic focusing of the magnetic field lines.
General Relativistic Comparison: To further validate the model, the paper compares the calculated magnetic moment with the prediction derived from the Kerr–Newman metric in general relativity. This metric describes the spacetime of a rotating charged mass and naturally yields a gyromagnetic ratio ( $g$ -factor) of 2, matching the Dirac electron.

Key Results
The application of relativistic corrections yields the following results:

Resolution of Superluminality: The original non-relativistic calculation yielded $v \approx 137c$ . By introducing the relativistic amplification factor $\gamma$ , the equation for the magnetic moment becomes $\mu = \frac{evr_e}{2}\gamma$ . Equating this to the Bohr magneton ( $\mu_B = e\hbar/2m_e$ ) leads to the condition $v\gamma \approx 137c$ .
Subluminal Velocity: Solving for $v$ with a Lorentz factor of $\gamma \approx 137$ (corresponding to $\beta \approx 0.99997$ ) results in a surface velocity of $v \approx c$ . Thus, the required velocity is strictly subluminal, resolving the contradiction with special relativity.
Field Geometry: The corrected model predicts that the magnetic field lines are tightly concentrated along the symmetry axis, forming an "axial magnetic jet" distribution analogous to relativistic jets from Kerr–Newman black holes.
Gyromagnetic Ratio: The paper notes that the Kerr–Newman solution for a rotating charged sphere naturally produces a $g$ -factor of 2, coinciding with the Dirac electron and experimental observations, whereas a simple non-relativistic classical sphere predicts $g=1$ .

Significance and Claims
The paper claims to correct a century-old algebraic and conceptual error in physics education and literature. Its primary contributions are:

Pedagogical Correction: It argues that the "proof" that classical rotation cannot explain electron spin is based on a non-relativistic treatment of electromagnetic fields. By correctly applying Lorentz transformations, the mechanical model becomes physically consistent with relativity.
Re-evaluation of Spin's Origin: The work suggests that the intrinsic magnetic moment of an electron may be a natural consequence of combining rotation, charge, and relativity, rather than being purely a quantum mechanical artifact. It posits that magnetism is fundamentally a relativistic effect of moving electric charges.
Historical Irony: The authors highlight that the formalism necessary to resolve the paradox (Lorentz transformations) was available to Lorentz himself in 1904 but was omitted from his 1925 calculation, leading to a persistent misconception.

Limitations and Future Directions
The paper acknowledges that while the superluminal paradox is resolved, the classical model still faces significant challenges, including:

The Anomalous g-Factor: The classical model predicts $g=1$ (or $g=2$ via Kerr–Newman geometry), but experimental values are $g \approx 2.0023$ , attributed to vacuum fluctuations in Quantum Electrodynamics (QED).
Point-Particle Divergence: High-energy scattering experiments treat the electron as a point particle ( $r \le 10^{-18}$ m), conflicting with the finite radius required for the classical model to avoid infinite self-energy.
Textbook Reform: The authors advocate for updating physics curricula to include this relativistic correction, presenting the classical electron model as a valid first-order approximation that fails only when relativistic field transformations are ignored.

The study concludes that while quantum field theory treats the electron as a point particle, the success of this relativistic correction invites a re-evaluation of the boundary between classical electrodynamics and quantum mechanics regarding the origin of spin.

The Spin-Off Story: Why Electrons Might Not Be Breaking the Speed Limit

The Paper's Twist: Fixing a Century-Old Math Mistake

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