Jan Onderco
July 2023
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 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[4][5] 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.[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
(1)
(2)
(3)
(4)
(5)
(6)
where is the Lorentz factor
(7)
The purely electric field
(8)
transforms to
(9)
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…”[9] point towards
and
being the magnetic field and
being electric field in the
equation assuming Gaussian units of measure. The SI units equation[10] 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. The isolated system floats in an intergalactic space.
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[11] 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.
Time dilation
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[12]
(10)
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.[13]
Conservation of linear 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
(11)
(12)
(13)
relativistic 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 linear 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. 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
cause a linear momentum change along the
and
axes? The answer is no because 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-orbit interaction
Spin evolution of spin particles has two attributes, 4-torque on the spin and orbital 4-torque.[14] Moving charge creates an electromagnetic field around the charge.[15] 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.[16] 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.
Conclusions
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.
Acknowledgement
To the anonymous physicist[17], his gracious and noble approach to our email correspondence is highly admirable. Thank you for your invaluable feedback!
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] Éric Gourgoulhon, Special Relativity in General Frames, From Particles to Astrophysics, page 568, ISBN 978-3-642-37275-9, 2013.
[12] David J. Griffiths, Introduction to Electrodynamics, Fourth Edition, page 549, ISBN-13: 978-0-321-85656-2, 2013.
[13] ON THE ELECTRODYNAMICS OF MOVING BODIES by Albert Einstein, page 23 from ffn.ub.es
[14] Éric Gourgoulhon, Special Relativity in General Frames, From Particles to Astrophysics, page 345 ISBN 978-3-642-37275-9, 2013.
[15] Éric Gourgoulhon, Special Relativity in General Frames, From Particles to Astrophysics, page 559 ISBN 978-3-642-37275-9, 2013.
[16] Konstantin Y. Bliokh, Franco Nori, Relativistic Hall Effect, from https://arxiv.org/abs/1112.5618v
[17] An anonymous physicist.