Jan Onderco

August 2022

## Abstract

Einstein’s 1905 paper ON THE ELECTRODYNAMICS OF MOVING BODIES^{[1]} is the foundation of the Relativity. The paper second part related to the electromagnetism starts with the transformation of the Maxwell-Hertz Equations for Empty Space and on the Nature of Electromotive Forces occurring in a Magnetic Field during motion.^{[2]}

The understanding of the transformation, confirmed by Electromagnetic Field Invariants analysis^{[3]}, leads to a conclusion the “purely electric” or “purely magnetic” field is observer-dependent. Purely electric field between an anode and a cathode parallel plates in the plates inertial rest frame generates magnetic field in any other moving inertial reference frame system based on the transformation of the Maxwell-Hertz equations. An electric charge, proton or electron, moves along the purely electric field lines in the plates rest frame system but the “hidden”, as unexpected, torquing due to acceleration on a curved trajectory generated by the magnetic field adds transverse drift to the charge particle motion in the moving inertial reference frame system.

Two inertial reference frame systems in a relative motion predict different Lorentz 4-forces^{[4][5]} acting on the charge particle between the plates. The Lorentz 4-force is pure 4-force, orthogonal to the charge particle 4-velocity.^{[6]} The disagreement on the Lorentz 4-force translates to different prediction of the charge particle 4-velocity change, manifesting in different analysis of the charge particle wordline, leading to disagreement on physics of the charge particle.

## Introduction

Einstein defines rest inertial reference frame system (grid of inertial observers with synchronized clocks^{[7]}) where denotes vector of the electric force, and magnetic force vector. The moving inertial reference frame system (grid of inertial observers with synchronized clocks) , renamed to in order to keep the notation up to date, characterized by velocity along the axis, ( moves at in the (-) minus direction along the axis), where is vector of the electric force, and is the vector of magnetic force.. The transformation of the Maxwell-Hertz equations^{[8]} is

where is the Lorentz factor

The purely electric field

transforms to

The vector of magnetic force emerges in the moving reference frame system . The lower case axes notation characters are used throughout this document. The document thought experiment analyses will be done in three inertial reference systems; – the rest frame reference system, – the moving reference system with a constant velocity along the axis in the plus (+) direction, – the moving reference system with a constant velocity along the axis in the minus (-) direction.

Even though Einstein talks about electric and magnetic forces we will assume they represent electric and magnetic fields, and notation will be used in this document. Einstein’s electric force would assume charge is being accounted for . The magnetic force direction would not be aligned with expected direction. Einstein’s words: *“…in addition to the electric force, an ‘electromotive force’… is equal to the vector-product of the velocity of the charge and the magnetic force, divided by the velocity of light…”*^{[9]} point towards and being the magnetic field and being electric field in the equation even though units of measure do not add up. The correct equations^{[10]} is . It has been suggested has been *“lost in a translation”* to give us correct ^{[11]} units of measure.

## Pure electric field

A thought experiment of an isolated system in the rest frame . Two plates, electric potential field **E**, the top plate is cathode, the bottom plate is anode. The plates are apart. The plates dimensions are , having said that they could be larger to ensure uniform field **E**.

*Figure 1: Electric plates with 500V electric potential form a uniform pure electric field E in the plates rest reference frame system K.*

## Electric field and magnetic field

The moving reference frame systems and observe magnetic fields in the minus (-) direction and in the plus (+) direction along the axis of their respective reference systems.

*Figure 2: Electric plates with 502.5V electric potential form a uniform electric fields , and a uniform magnetic fields , in the moving inertial reference frame systems (left) and (right) after Maxwell-Hertz equations transformation*.

The Figure 2 shows only a couple of electric field lines for simplified view even though the electric field is stronger in the moving reference frame systems and compared to the rest frame reference system being multiplied by the Lorentz factor in the Lorentz transformation. The relative velocity along the axis leads to . The plates undergo small Lorentz contraction.

## Electron worldline/trajectory

The thought experiment of the isolated system in the rest reference frame system consists of electron located at event of reference frame system coordinates . The event is synchronized with the event of moving reference system coordinates and the event of moving reference system coordinates . The electron is released from the cathode plate and being accelerated to the anode plate. The Figure 3 shows the initial events , and .

*Figure 3: The initial electron trajectory direction along y axis in (middle) reference frame system and almost straight along the x’ axis in (left) and (right) reference frame systems at time t=t’=0s.*

The electron is the center piece of the analysis. The electron straight line acceleration^{[12]} in the system is mapped into a curved acceleration in the and systems. The first basic assumption is the electron’s positions along the and axes are equal , because if the electron accelerates along the axis in the system then no position change is expected according to the relativity. The Figure 4 shows the electron acceleration trajectories up to the system event ; the system event and the system event when the electron reaches the anode (red) plate.

*Figure 4: Middle: The straight line electron trajectory at and of reference frame system. Left: The curved electron trajectory at when origin of is at of (left) reference frame system. Right: The curved electron trajectory at when origin of is at of reference frame system.*

*Figure 5: The curved electron trajectory x’,y’ profile at of (left) and (right) reference frame systems.*

Figure 5 shows the curved electron trajectory profile. It seems there is no problem with the relativity.

## Angular velocity change – torquing

The moving systems and observe electron curved trajectories. The exists for and systems but the angular velocity change is not predicted by the rest frame system .

*Figure 6: Left: The curved electron trajectory with a transverse drift caused by torque in the z’ axis (+) plus direction. Right: The curved electron trajectory with a transverse drift caused by torque in the z’ axis (-) minus direction.*

## Discussion

Only one electron left the cathode therefore only one electron hits the anode. Assuming the rest frame system analysis is correct and the electron hits the anode plate at the event then this would violate the uncertainty principle of the quantum mechanics. We would know the momentum and the position with a higher precision than expected by the principle, an infinite precision.

The electron trajectory is observer dependent and every different moving system predicts different value. It is logical to conclude that **only one preferred system** can be correct in determining the torque value that helps to predict exact position where the electron hits the anode. Together with Einstein we can say only God knows the preferred frame and He does not play dice. It appears the preferred frame and consequently only one possible 4-torque evolution acting on the electron are the hidden variables Einstein was looking for.

## Spin/Relativistic Hall effect

Hall effects arise from rotational and linear motions of particles. The spin Hall effect and related torque are being studied^{[13]}. A relativistic flywheel simulation of the Hall effect is very intriguing^{[14]} pointing to one of the best, if not the best, electron mechanical model ever conceived.

A wheel of radius rotating with angular velocity forms an isolated system and the centre of inertia of the isolated system is in the middle on the wheel axle when observed from the rest reference system . Other names used are centre of mass or barycentre and it is an intrinsic centroid to the system, comoving with the system.^{[15]} Any other inertial observer grids moving in relation^{[16]} to the rest frame system do not recognize this centroid as shown in Figure 7. The moving system centroids and are frame dependent where the relative velocity is a primary factor that determines the relativistic flywheel deformation.

*Figure 7: Left: A relativistic flywheel of radius R rotating with angular velocity , in the rest frame . Right: Deformations of the wheel shape in the frame moving with velocity . The dots indicate the positions of the geometric and the energy centroids, and . ^{[17]}*

## Relativistic flywheel acceleration

We will accelerate the relativistic flywheel in our next thought experiment. An external force acting on an isolated system will generate a uniform acceleration, the originally inertial system barycentre observer becomes a uniformly accelerated observer. The external force acts through the barycentre, the flywheel axle as per Figure 8 Left in the original rest frame system . The flywheel ‘free body diagram’ is split into two parts the first above the axle and the second below the axle. Both sections have their own centroids top, bottom, and arms, top, bottom. The force does not propagate instantaneously through the arms in the rest frame . Nevertheless and act at the same time on the top and bottom centroids in the original rest frame system and all additional comoving inertial observers. Due to the agreement on the simultaneity there would be no change.

*Figure 8: Left: A relativistic flywheel acceleration analysis in the rest frame system . Right: A relativistic flywheel acceleration analysis in the moving frame system .*

Figure 8 Right shows the external force acting on the axle in the moving system . The force magnitude keeps the flywheel uniformly accelerated. The relativistic flywheel deformation determines the different position and the mass of the top, bottom centroids when compared to the rest frame system . The top centroid represents more mass than the bottom centroid and the top arm is longer compared to the bottom arm . The force propagates faster towards the bottom centroid and slower towards the top centroid. The bottom centroid will be accelerated by the bottom force sooner than the top centroid . The simultaneity delta will cause leading towards torquing of the relativistic flywheel.

## Discussion

Both thought experiments, the electromagnetic and the mechanical, demonstrate a necessity to anchor the relativity in a preferred reference frame in order to determine the correct and consequently 4-torque evolution. The electron model as the relativistic flywheel represents a gyroscope. The gyroscope straight line uniform acceleration predicts different behaviour when compared to the gyroscope on a curved or circular trajectory and the following analysis shows the acceleration and torquing relationship.

## References

[1] *ON THE ELECTRODYNAMICS OF MOVING BODIES* by Albert Einstein, from ffn.ub.es

[2] *ON THE ELECTRODYNAMICS OF MOVING BODIES* by Albert Einstein, page 12 from ffn.ub.es

[3] Éric Gourgoulhon, *Special Relativity in General Frames, From Particles to Astrophysics*, page 556, ISBN 978-3-642-37275-9, 2013.

[4] Éric Gourgoulhon, *Special Relativity in General Frames, From Particles to Astrophysics*, page 313, ISBN 978-3-642-37275-9, 2013.

[5] Éric Gourgoulhon, *Special Relativity in General Frames, From Particles to Astrophysics*, page 545, ISBN 978-3-642-37275-9, 2013.

[6] Éric Gourgoulhon, *Special Relativity in General Frames, From Particles to Astrophysics*, page 312, ISBN 978-3-642-37275-9, 2013.

[7]* ON THE ELECTRODYNAMICS OF MOVING BODIES *by Albert Einstein, page 3 from ffn.ub.es

[8] *ON THE ELECTRODYNAMICS OF MOVING BODIES* by Albert Einstein, page 14 from ffn.ub.es

[9] *ON THE ELECTRODYNAMICS OF MOVING BODIES* by Albert Einstein, page 15 from ffn.ub.es

[10] Éric Gourgoulhon, *Special Relativity in General Frames, From Particles to Astrophysics*, page 554, ISBN 978-3-642-37275-9, 2013.

[11] An anonymous physicist.

[12] Éric Gourgoulhon, *Special Relativity in General Frames, From Particles to Astrophysics*, page 568, ISBN 978-3-642-37275-9, 2013.

[13] Wenqiang Wang, Kaiyuan Zhou, Xiang Zhan, Zui Tao, Qingwei Fu, Like Liang, Zishuang Li, Lina Chen, Chunjie Yan, Haotian Li, Tiejun Zhou, Ronghua Liu, *Enhancement of spin-orbit torque efficiency by tailoring interfacial spin-orbit coupling in Pt-based magnetic multilayers*, from https://arxiv.org/abs/2209.02282v1

[14] Konstantin Y. Bliokh, Franco Nori, *Relativistic Hall Effect*, from https://arxiv.org/abs/1112.5618v2

[15] Éric Gourgoulhon, *Special Relativity in General Frames, From Particles to Astrophysics*, page 332 ISBN 978-3-642-37275-9, 2013.

[16] Éric Gourgoulhon, *Special Relativity in General Frames, From Particles to Astrophysics*, page 330 ISBN 978-3-642-37275-9, 2013.

[17] The plotting software code provided by Konstantin Y. Bliokh. Thank you!

[18] Éric Gourgoulhon, *Special Relativity in General Frames, From Particles to Astrophysics*, page 423, ISBN 978-3-642-37275-9, 2013.