Einstein’s 1905 paper ON THE ELECTRODYNAMICS OF MOVING BODIES 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.
The understanding of the transformation, confirmed by Electromagnetic Field Invariants analysis, 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 electron moves along the purely electric field lines in the plates rest frame system but the “hidden”, as unexpected, up to now not recognized torquing due to acceleration on a curved trajectory generated by the electron spin-orbit interaction adds uncertainty component to the electron motion in the moving inertial reference frame system.
Two inertial reference frame systems in a relative motion predict different Lorentz 4-forces acting on the charge particle between the plates because of a disagreement on spin and orbit 4-torque evolutions. The Lorentz 4-force is pure 4-force, orthogonal to the charge particle 4-velocity. 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.
Einstein defines rest inertial reference frame system (grid of inertial observers with synchronized clocks) 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 is
where is the Lorentz factor
The purely electric field
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. The initial electron velocity is equal to the velocity in the moving frames. 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…” point towards and being the magnetic field and being electric field in the equation assuming Gaussian units of measure. The SI units equation is .
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.
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 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. Note: The reference frame systems , have different camera angles.
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.
The time of electron crossing the distance between the plates in the rest frame is calculated from the acceleration where and is electron mass. The rest frame time multiplied by the Lorentz factor gives us the time it takes to cross the plates in the moving reference frame systems. Having said that the rest frame time calculation can be repeated in the moving frames with the similar approach through the electric and magnetic fields, forces, and acceleration.
The magnetic field in is in the direction and the magnetic field in is in the direction. The outcome is an agreement on the direction of force in both moving reference systems, the has direction in and . The electric force has direction, the magnetic force ‘works’ against the electric force. The relativistic acceleration equation is
The electron velocity vector is initially zero in the rest frame and degrees to the field in the moving frames, meaning . The calculation:
The straight is the same result. The force and acceleration decrease with an increase of the Lorentz factor and the time to cross the plates would increase as well. The analysis is aligned with Einstein’s 1905 paper.
Conservation of momentum
The rest frame momentum analysis requires closer look at the event of the electron emission, the event of the electron absorption and also what is happening during the electron flight acceleration. The plates and electron form an isolated system with its barycenter. The electron speed at the red plate is that is approximately .
The electron is becoming relativistic and more accurate calculation is to integrate the motion trajectory using , , and distance equations considering the velocity provides a feedback loop to the Lorentz factor . The magnetic component is . The table below shows the calculation for iterations and the last line is the end-result for iterations.
The velocity magnitude calculated in one step is higher than velocity magnitude calculated in step iteration , this is an expected result. The force acting on the electron through the EM field is continuous as well as the force acting on the plates from the electron. The conservation of momentum holds in every tiny moment. The velocity magnitude increments at the electron initial acceleration are bigger compared to the velocity magnitude changes caused by the acceleration at the end, but the relativistic mass is increasing with the growing velocity magnitude, it appears these values might offset. Still the total electron momentum changes are decreasing over time. Nevertheless, the barycenter of the isolated system is inertial and located at the origin of the rest frame. The red plate has bigger mass with one extra electron at the end, that means the plates shifted a small in the direction to compensate and to keep the barycenter inertial.
Conservation of momentum in moving frames
The moving systems and observe electron curved trajectories. The electron has two components and at the time of emission but just before the absorption. The component is obvious and undeniable. The consequence based on the electromagnetism and electromagnetic nature of things is deflection of the electron trajectory. As it was already shown the electromagnetic force works against the electric force in the moving systems and . As the force rotates, no initial cause exists for the magnetic component to change, due to the assumption is constant. The magnetic component increases because the electron acceleration increases . Suddenly it appears a new cause arises to deflect the electron in direction in reference system and in direction in reference system. Does the magnetic component break the relativity? The answer is in the electric component of the electromagnetic force , specifically the component is equal in magnitude but opposite in direction to the magnetic counterpart and they form a constraint pair of forces cancelling any effect.
Figure 6: The curved electron trajectory x’,y’ profile at of and reference frame systems. The rotation of the magnetic component of the electromagnetic force, the blue arrow. The electric component of the electromagnetic force, the black arrow.
The conclusion is there is no momentum change along the axes in the moving systems and .
Spin evolution of spin particles has two attributes, 4-torque on the spin and orbital 4-torque. Moving charge creates an electromagnetic field around the charge. The shape of the field is a function of velocity. The electric field lines point away from the positive charge or towards the negative charge, the magnetic field lines are perpendicular to the electric field lines creating a ‘rotating’ velocity dependent ellipsoid because the particle EM field undergoes a ‘length contraction’ with increasing speed. The apparent rotation comes from the charge acceleration increasing the magnetic field. The rotating ellipsoid represented by a relativistic wheel was used as an electron model investigating the Relativistic Hall Effect. Observational evidence points towards a rotating flywheel being a truthful electron model as a spin particle in predicting spin evolution utilizing spin and orbital 4-torques. The spin does not describe the charge mass rotation but the field rotation. If there is an electron charge mass rotation underneath the field, then the electron radius could be much smaller so the charge mass rotation does not exceed the speed of light.
The Figure 7 shows the initial orientation of the electron flywheel. The electron velocity along its trajectory is the main input factor in aligning the flywheel rotational axis with axis in the moving frames and axis in the rest frame .
Figure 7: The electron flywheel orientation at t=t’=0s. The left hand rule applies to the electron because of the negative charge.
The Figure 8 shows the orientation of the electron flywheel just before hitting the anode plate in the moving and reference frame systems. The flying electron spin is being torqued around axis. The electron is undergoing spin-flip transition similar to the famous Hydrogen Line. The torquing creates rotation of the isolated system in the moving frames but no rotation is being predicted by the rest frame system .
Figure 8: The electron flywheel orientation at s of and reference frame systems.
The spin and orbit 4-torque evolutions are hidden from the rest reference system, and we are left with a statistical approach to understand our world. All the inertial reference systems are wrong till the preferred reference system is recognized and the relativity is anchored in the absolute space and time.
To the anonymous physicist, his gracious and noble approach to our email correspondence is highly admirable. Thank you for your invaluable feedback!
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 Éric Gourgoulhon, Special Relativity in General Frames, From Particles to Astrophysics, page 556, ISBN 978-3-642-37275-9, 2013.
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 Éric Gourgoulhon, Special Relativity in General Frames, From Particles to Astrophysics, page 554, ISBN 978-3-642-37275-9, 2013.
 Éric Gourgoulhon, Special Relativity in General Frames, From Particles to Astrophysics, page 568, ISBN 978-3-642-37275-9, 2013.
 David J. Griffiths, Introduction to Electrodynamics, Fourth Edition, page 549, ISBN-13: 978-0-321-85656-2, 2013.
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 Konstantin Y. Bliokh, Franco Nori, Relativistic Hall Effect, from https://arxiv.org/abs/1112.5618v
 An anonymous physicist.