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) characterized by velocity in along the axis, ( moves at in 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 magnetic field emerges in the moving reference frame system . The lower case axes notation characters are used throughout this document.

## 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 thought experiment of the isolated system in the moving reference frame system .

*Figure 2: Electric plates with 500V electric potential form a uniform electric field E and a uniform magnetic field B in the moving inertial reference frame system k 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 system 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 . 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 along y axis in K reference frame system and almost straight along the x axis in k reference frame system at time t=t’=0s.*

The electron is the center piece of the analysis. The electron straight line acceleration^{[9]} in the system is mapped into a curved acceleration in the system. The first basic assumption is the electron’s positions along the and axes are equal and 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 and the system event when the electron reaches the anode (red) plate.

*Figure 4: The straight line electron trajectory at t=1.508 ^{-7}s of K reference frame system and the curved electron trajectory at t’=1.516^{-7}s of k reference frame system.*

The classical Lorentz force equation appears to cover the relativity in an orderly fashion. The rest frame does not have any magnetic field **B** so the purely electric force accelerates the electron along the straight line trajectory. The moving frame has/sees some magnetic field **B** and the magnetic part of the Lorentz force is responsible for the rotation, the electron curved trajectory.

*Figure 5: The curved electron trajectory x’,y’ profile at t’=1.516 ^{-7}s of k reference frame system.*

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

## Angular velocity change – torquing

The moving system observes electron curved trajectory. The is real for system but not predicted by the rest frame system .

*Figure 6: The curved electron trajectory with a transverse drift caused by torque T. The electron will not hit the anode plate at the event **. This would violate the uncertainty principle. We would know the momentum and the position with a higher precision than expected by the principle, an infinite precision.*

## Conclusion

The is observer dependent. 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 value of torque acting on the electron are the hidden variables Einstein was looking for.

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