<figure><img src="image_1.png" /></figure>

HAL
open science

# A relativistic interpretation of the spin

Stefan Catheline

► To cite this version:

Stefan Catheline. A relativistic interpretation of the spin. 2025. inserm-04924923

HAL Id: inserm-04924923

https://inserm.hal.science/inserm-04924923v1

Preprint submitted on 1 Feb 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
II-A relativistic
interpretation of the
spin

Stefan Catheline

Affiliation : LabTAU, INSERM, Centre Léon Bérard, Université Lyon 1,
Univ Lyon, F-69003, LYON, France

stefan.catheline@inserm.fr

**Abstract**

This paper aims at studying the spin once again. The departure point is thus the Stern and Gerlach experimental results that can be described in a coherent way in the frame of quantum mechanics only. Instead, the relativistic mechanics point of view is proposed here following the work presented in a previous article about rigid body rotation. Indeed, a careful study of the horizon of this relativistic rigid body rotation appears to be invariant from any observation angle in full agreement with the spin property.

**Introduction**

Quantum mechanics is intimately related to the concept of spin introduced
in the experiment of Stern and Gerlach in1922 [1]. Indeed, no classical
interpretation ever described in a coherent manner the quantification of magnetic
moment of particle. The concept of “spin” as pure quantum entity was born. It
justified in itself all abstract and counterintuitive concepts of quantum mechanics.
According to Richard Feynman [2], “we can never have the comfortable feeling”
that the concept of spin is “natural”. The aim of this section is not to give a
classical explanation of the spin but rather a relativistic one. The electron is the
lightest charged particle. It is defined entirely by its elementary charge, mass and
spin. It is this spin-bearing particle which, in the silver atoms of Stern and
Gerlach's original experiments, steers their trajectory. We sometimes take the short path in the text to replace silver atoms in the original experiment of Stern and Gerlach by electrons. Since 1820 and the Arago-Ampère experiment, magnetism is described as the consequence of current loops. The classical point of view of quantum mechanics pioneers is adopted: the electron is a rotating sphere with a charge e at a constant distance r_e from the center and traveling at the speed of light, thus with an intensity $I = \frac{ec}{\pi r_e}$. This charge rotation at the origin of the spin is thus also dependent on the rigid body rotation as introduced in a previous publication [3]. This spin-rotation connection, obsolete for some or heretical for others, is the starting point of this manuscript, which of course remains to be demonstrated. This will be the subject of discussion in a future publication. For the moment, let's focus our attention on solid rotation in relativity, which has been the subject of numerous articles [4,5,6,7]. The originality of the approach presented in this manuscript lies in its object of study, which focuses exclusively on the behavior of the limit of solid rotation, i.e. its horizon. In particular, we aim to show that its geometric properties allow us to revisit Stern and Gerlach's experiment.

## Background of relativistic rigid body rotation

The relativistic rigid body rotation has already been described in a previous publication [3]. Some usefull properties are quickly reminded here. First, as any rigid body rotation, it is composed of an angular rotation $\vec{\omega}$, a velocity $\vec{v}$ and a distance radius $\vec{r}_0$ from the center O. These three vectors are connected by the equation $\vec{v} = \vec{\omega} \wedge \vec{r}_0$, figure 2. The entire originality lies in the fact that the physical path followed by a light ray emitted from O is not the straight line $\vec{r}_0$. Instead, this radial path imposed by the Coriolis acceleration follows a simple circle called “space circle”. The arc of space circle from the center O to the satellite position is deduced from the angle rotation $\Omega$ and the half radius vector of the rotation $\frac{\vec{R}}{2}: \vec{r} = \vec{\Omega} \wedge \frac{\vec{R}}{2}$. Using relativistic transverse Doppler shift, the space circle is described since 1963 [8]. It was later studied through a spatial metrical tensor borrowed to general relativity [9]. When the satellite velocity is weak compared to speed of light, the classic rotation is recovered. On the contrary, when this speed is close to the speed of light, the satellite reaches its maximum radius R. That is the horizon. At last, the relativistic rotation fully satisfies the space-time Lorentz transformation. The angular distortion $\theta$ between the direction of the velocity and of the space circle tangent modifies the perception of space and time
from the inertial frame of central observer O. Relativistic parameters are defined
by the geometry of Figure 1 according to $\cos\theta = \frac{r_0}{R} = \frac{v}{c}$ and $\sin\theta = \sqrt{1 - \frac{v^2}{c^2}}$.
Lorentz contraction of length aligned with the velocity are such that from O :
$L_{\phi} = L_0 \sin\theta$. As opposed to classic rotation, the relativistic rigid body rotation
only involves a finite region of space. By construction, it allows a finite value of
velocity: the first of all universal constant of physics c, thus follows the finite
geometrical nature of relativistic rigid body rotation. Although major, these
distinctions do not justify the necessity to employ the relativistic rigid body
rotation rather than special relativity since these points of view are fully
compatible [10,11]. However, the first and novel approach conveys a simple and
straightforward interpretation of the spin using Euclidian geometry as the
expression of horizon circle.

<figure><img src="image_3.png" /><figcaption>Figure 1: Relativistic rigid body rotation. Its maximal size is bounded by the horizon circle. The horizon properties are at the origin of the spin.</figcaption></figure>

**Failure of the classical interpretation of the Stern and Gerlach experiments**

Let us consider a silver atoms beam going through the Stern and Gerlach
experimental set-up, figure 2. For sake of simplification, let us concentrate on the
single spin property of an electron. Inspired by a classical model of a spinning
sphere of charge density, the electron consists of a charge e traveling at the speed
of light c on an orbit located at an electron radius r_e. In a classic interpretation, the
current loop I = $\frac{ec}{\pi r_e}$ undergoes a force when immersed in a magnetic field gradient
and is deviated from its uniform rectilinear motion. It is worthy to recall how
pertinent is the classic calculus of the spin from a current loop. Indeed, the kinetic
moment of relativistic rotation is known to be the Planck constant: m_e r_e c = $\hbar$.
Therefore, the product intensity-surface is IS = er_e c = $2 \frac{e\hbar}{2m_e}$. Finally, given the
gyromagnetic ratio of electron, the spin magnetic moment straightly comes as
$\mu_B = \frac{e\hbar}{2m_e}$, sometimes called Bohr-Procopiu magneton. By virtue of the classical
induction force of a circuit immersed in a magnetic field and described by the
following equation: $\vec{F} = - \overrightarrow{grad} (I \vec{B} \cdot \vec{S}) = IS \cos\theta \overrightarrow{grad} B$, this loop will migrate
according to its orientation towards regions of more or less dense magnetic field,
seeking to optimize the flux lines across its surface.

Prior its crossing through the experimental device, all the possible
orientations of the current loop offer the same probability. Therefore, the current
loop surface involved in the flux is a continuous function S = S₀ cosθ, figure 3.
The perspective effect is the same for a circular current loop than for a square
current loop shown here for clarity purpose. The side length L₀ enlightened by a fat
grey line becomes shorter for an off-axis observer by the quantity L = L₀ cosθ.
As a consequence, a flux distribution and finally a force distribution obeys the
exact same perspective rule. The initial angle θ being random, the expected
statistic deviations of an electron beam by a gradient of magnetic field is a
continuous spatial distribution on a screen placed outside the experimental set-up.
The experiment shows a splitting of the beam in two parts which invalidates the
latter classic reasoning, figure 1. Many physicists claim that any mechanic
explanation of Stern and Gerlach experimental results is vain [12], any intuitive
approach of spin is devoted to failure.
<figure><img src="image_4.png" /><figcaption>Figure 2: Stern and Gerlach experimental set-up. The electron beam splitting visible on the screen out of the set-up is not described by classical physics in a coherent manner.</figcaption></figure>

## Emergence of the spin

Let us examine more carefully the magnetic flux through the circuit and the relativistic effects of its surface. Known since the antiquity but studied during the Italian renaissance as point-projection perspective in paint and architecture, this effect describes the size shrinking of objects as the distance increases. Within the frame of the rigid body rotation, figure 2, a satellite of proper length L is located at a distance $r_0 = R \cos\theta$. Its size is thus reduced in proportion of $\frac{L}{R\cos\theta}$. When a measure is made, the meter-ruler chosen by the central observer O in an inertial frame, has a proper length $L_0$ defined when the velocity is minimum, $v_{min} \approx 0$, $\theta \approx \frac{\pi}{2}$, figure 2. This ruler is placed at the exact same distance as the satellite and is thus reduced in the same proportion $\frac{L_0}{R\cos\theta}$. As a consequence, length estimations are not impacted by distance. However, in relativistic rigid body rotation, a distortion appears because distance and velocity are connected by a rotation field. For the observer O, the measurement direction is perpendicular to the space circle, this latter being the radial direction of light. From a classic point of view of weak velocity, the alignment of the measurement can be approximated with the velocity direction. But an angular distortion appears. A direct effect of this angular distortion is to shrink the ruler by projection of $L_0$ according to the equation $L_{\phi} = L_0\sin\theta$: this is the Lorentz contraction of lengths. If this contracted ruler is used in the estimation of a circle perimeter P then divided by its radius, the following
non-Euclidian relation arises $\frac{P}{r_0} > 2\pi$. A more quantitative relation is extracted from the geometric construction $\frac{P}{r_0} = \frac{2\pi}{\sin\theta}$, in agreement with relativity literature [13]. When the satellite travels at its maximum velocity $v_{max}$ very close to the speed of light, this horizon perimeter is maximum and defines the geometric contour of the spin.

Let's now examine the classic perspective effect on such a contour, starting with a square circuit. The segment of the side of length $L_0$ highlighted in grey in the top view of figure 3a is traversed by a particle animated by velocity $c$. When the circuit is rotated by an angle $\theta$ along its vertical axis of symmetry, this segment appears contracted from the side by the projection of the rotation, $L = L_0\cos\theta$. By the same perspective effect, the projection of the particle's velocity onto this portion gives $v = c \cos\theta$. This is also the definition of the angle of relativistic distortion [3]. This coincidence of $\theta$ as the angle of perspective rotation and the angle of relativistic velocity distortion is crucial to the demonstration, as it allows the effect of perspective on the horizon of square and circular circuits to be represented on the Euclidean plane of Fig. 3. This is precisely the aim of this article. Let's start with the front view of the black square circuit. Perspective contracts its horizontal sides only, and it becomes a rectangle with a red border. The lengths of the sides travelled at apparent speed $v$ cannot, by definition, be the boundaries of the horizon. To re-establish the horizon's own maximum speed $c$, we need to apply the law of solid rotation through a homothety of reason $\frac{1}{\cos\theta} = \frac{c}{v}$ from circuit center O. After dilation, we obtain the new red dotted-line rectangle. This is the horizon we're looking for. Applying the same steps to the black circular circuit in figure 3b produces the red-edged ellipse from perspective, then the dotted-line ellipse after applying homothety to find the horizon. Both cases lead to the following observation: the surface $S = LL' = \frac{v}{c}L_0\frac{c}{v}L_0 = L_0^2$ of the horizon rectangle is also that of the square circuit and the surface $S = \pi ab = \pi\frac{v}{c}r_e\frac{c}{v}r_e = \pi r_e^2$ of the horizon ellipse is also that of the circular circuit. In other words, perspective effects change the shape of the horizon without changing its surface. The immediate consequence is that, since the surface available to the magnetic flux line does not vary with perspective, the modulus of the magnetic force applied to the circuit is the same whatever its orientation. Only the sign of the force depends on the orientation of the current lines in relation to the magnetic field. A single modulus with two possible signs: that's exactly what's expected from spin.
<figure><img src="image_5.png" /><figcaption>Figure 3 : a) Top and front views of the θ angle rotation of a square circuit used for educational purposes. In the front view, classic perspective effects transform the square into a rectangle with a red border. The horizon of this circuit is represented by the dotted rectangle. b) Perspective effect on a circular circuit. Rotating a circle along its vertical diameter produces an ellipse with a red border. On its horizon, traversed by particles travelling at speed c, the dotted ellipse is dilated by the quantity $\frac{1}{\cos\theta} = \frac{c}{v}$.</figcaption></figure>

In short, the angular perspective effects of a relativistic solid rotation for an observer moving around its center O modifies the shape of the horizon, which is transformed from a circle into a longer ellipse with an invariant area, as in the classical case: the surface circumscribed by the horizon of a relativistic solid rotation is the same from all directions of observation. It follows that the induction force resulting from the flow of the stationary magnetic field across this invariant surface is constant. It depends only on its orientation, parallel or antiparallel to the magnetic field gradient, causing the beam to split in two in the Stern and Gerlach experiment. This is the spin effect. The secret of spin therefore lies entirely in the invariance of the surface of a relativistic solid rotation with respect to the viewing angle. This fundamental idea of angular invariance of the surface bounded by the horizon of a relativistic solid rotation thus brings one of the founding concepts of quantum mechanics back into the properties of relativity.
**Experimental test of the relativistic spin**

While the surface swept by the horizon of the relativistic solid rotation does not vary with the angle of observation, the same cannot be said of its shape. This latter property can be tested as follows: when observed from its edge, the circular rotation becomes an ellipse that lengthens until it reaches and exceeds the size of the region in which the magnetic field gradient prevails. As a result, the surface area available to the line of flux diminishes, as does the resulting magnetic force. Electron deflection decreases and tends towards zero. Although the majority of electrons congregate at two points on the screen at the exit of Stern and Gerlach's experimental set-up, there are a few electrons that appear on their edge at the entrance to the set-up and therefore classically deviate between these two points according to a uniform distribution. More quantitatively, the viewing angle at which the major axis of the ellipse stretches over the entire magnetic field region $\Delta l$ is given by the relation $\frac{r_e}{\cos\theta} = \Delta l$ or equivalently $\theta_{lim} = \arcos\frac{r_e}{\Delta l}$. Assuming a homogeneous angular distribution of the spins in the incident beam, and assuming a millimeter-wide region of the magnetic field gradient $\Delta l = 10^{-3}m$, the particles seen on the slice, $\theta > \theta_{lim}$, represent a proportion of all particles of approximatively $1 - \frac{\theta_{lim}}{\frac{\pi}{2}} \approx 10^{-10}$. Measuring the spatial distribution of these particles in a Stern and Gerlach set-up as a function of the number of incident particles or the geometric characteristics of the magnetic field would represent a test of the thesis defended in this manuscript.

**Conclusion**

To revisit spin through relativistic mechanics, we have adopted the original point of view of a charge rotating at the speed of light in a circular orbit called the horizon. The demonstration in this manuscript focuses exclusively on the surface delimited by this relativistic solid spin horizon as a function of the angle of observation. It turned out that, despite the change in shape of the horizon line, its surface was invariant. Consequently, in an experiment by Stern and Gerlach, the flux of magnetic field lines crossing such a surface is constant. This explains why the modulus of the induction force applied to the circuit no longer depends on its orientation. This is the spin property, which draws its essence from relativistic mechanics. An experimental test is proposed.
<footnote n="1">¹ W. Gerlach, O. Stern, "Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld", Zeitschrift für Physik. 9 (1): 349–352, 1922.</footnote>

<footnote n="2">² Le cours de physique, Electromagnétisme 2, Richard Feynman, Ed. Dunod Paris, p. 274, Chapitre 35, 1998.</footnote>

<footnote n="3">³ S. Catheline, A mechanistic interpretation of relativistic rigid body rotation, in press, Scientific Report, 2023.</footnote>

<footnote n="4">⁴ A. Papapetrou, “Spinning test-particles in general relativity. I,” Proc. R. Soc. Lond. 209, 248 (1951).</footnote>

<footnote n="5">⁵ W. G. Dixon, “Dynamics of extended bodies in general relativity. I. Momentum and angular momentum,” Proc. R. Soc. Lond. A 314, 499 (1970).</footnote>

<footnote n="6">⁶ J. Ehlers and E. Rudolph, “Dynamics of extended bodies in general relativity center-of-mass description and quasirigidity,” Gen. Rel. Grav. 8, 197 (1977).</footnote>

<footnote n="7">⁷ L.Filipe, O.Costa, J.Natario, “Center of mass, spin supplementary conditions and the momentum of spinning particles”, arXiv:1419.6443v2 (2015).</footnote>

<footnote n="8">⁸ “Ray paths in a rotating system” R.C. Jennison, Nature 199, 739, 1963.</footnote>

<footnote n="9">⁹ “Relativistic description of a rotating disk with angular acceleration”, Ø. Grøn, Found. of Phys. 9, 353, 1979.</footnote>

<footnote n="10">¹⁰ Relativity in Rotating Frames, Eds. G. Rizzi and M.L. Ruggiero, Kluwer Academic Publishers, 2005.</footnote>

<footnote n="11">¹¹ Special Relativity in General Frames by E. Gourgoulhon, Ed. Springer 2010.</footnote>

<footnote n="12">¹² Mathematiques Speciales, Electromagnétisme 4, M. Bertin, J.P.Faroux, J.Renault, Ed. Dunod Paris, p.154, Chap 7, 1984.</footnote>

<footnote n="13">¹³ Ø. Grøn, “Relativistic description of a rotating disk”, Am. J. Phys. 43, 869, 1975.</footnote>