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7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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The Magnetic Force as a
Kinematical Consequence of theThomas Precession
David C. Lush
Presented at PIERS 2013 Stockholm
14 August 2013
quantumskeptic.blogspot.com
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Work in Progress
• The following describes a work in progress,
so is subject to revision
• As updates become available they will be
posted as replacements at arxiv.org.
(Search author: Lush, same title as for
PIERS talk and paper)
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Why a kinematics point of view?
• Can start with a Coulomb force, in a
reference frame where the field-source
charge is stationary, and then get magnetic
force present in other reference frames,without reference to electromagnetic fields
• Predicts new forces?
• Unifying principle for non-gravitational
fundamental forces?
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Field-Source and Test Charged Particles
(r,
t ) = (r
t -r
s,
t )
Lab frame
observer
Source particleTest particle
qt , mt
qs , ms
r t (t )
v t (t )
a t (t )
Test particle
position,
velocity and
acceleration
r s (t )
v s (t )
a s (t ) = 0
Source particle
position and
velocity
= Charge, mass
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Magnetic force between two charges
The magnetic force is formally
similar to a Coriolis force
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Rotational pseudoforces review
“Effective” force in a coordinate system rotating with
angular velocity w is the force applied in the non-rotating
system, plus Coriolis, centrifugal, and Euler pseudoforces.
Coriolis centrifugal Euler
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Thomas precession
• Translating and cross-accelerating reference
frame appears to rotate
• A consequence of noncommutativity of
non-colinear Lorentz boosts
• Invoked by Thomas to explain spin-orbit
coupling anomaly
• Observer dependent
• Has to be observed mutually between two
observers if seen by either
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Angular Velocity of the Thomas
Precession
Angular velocity of the rest frame of a particle with
acceleration a and velocity v in the laboratory
frame, as observed from the laboratory frame.
(Jackson, Classical Electrodynamics, 2nd ed, Eq.
(11.119))
for v/c << 1
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Thomas precession cannot
generate rotational pseudoforces
• An accelerating observer sees Thomas
precession of inertial coordinate frames
– Will present proof – The accelerated observer sees a lack of
pseudoforces as having dynamical effects (i.e.,
real forces acting)
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Sign of the Angular Velocity of Thomas Precession,
as Seen by Different Observers
Jackson Eq. (11.107) :
, with
More generally:
So if (as seen from lab frame):
Then (as seen from rest frame):
The sign of the angular
velocity of the Thomas precession of the lab
frame is opposite that
of the rest frame. (See
also Malykin 2006.)
(see, e.g., Goldstein)
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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The Magnetic force as a Coriolis Force seen
by the Test Particle Co-Moving Observer
• The test particle co-moving observer (TPCMO) sees Thomas precession of both
the source particle and laboratory frames
• Since the TPCMO knows that the test particle acceleration is purely radial
(Coulombic) in the (inertial) source particle
rest frame, he expects a Coriolis-like forceto exist in the lab frame based on the
relative angular velocity of the lab frame
compared to the source frame
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Relative Thomas Precession Angular Velocity of
Lab Frame Compared to Source Frame, as seen
by the Test Particle Co-Moving ObserverAngular velocity of Thomas precession
of an inertial frame, as seen from an
accelerated frame.
Lab frame angular velocity as seen from
test particle rest frame.
Source particle rest frame (SRF) angular
velocity as seen from test particle rest
frame.
Relative angular velocity of lab frame
compared to SRF as seen from test
particle rest frame.
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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The Magnetic Force as the Expected Coriolis
Force Due to Relative Angular Velocity
(per chart 7)
expected Coriolis force in lab frame
relative angular velocity of lab frame
compared to SRF as seen from test
particle rest frame (from previous chart).
The test particle acceleration is approximately Coulombic, obtaining
and had
Substituting for gives
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Is the Magnetic Force a Coriolis or an Anti-
Coriolis Force?
• The above derivation characterizes the
magnetic force as a Coriolis force, but it is more
properly an anti-Coriolis force
• The magnetic force is ultimately a consequence
of the lack of Coriolis force in the source particle
rest frame (an inertial frame), despite its
appearance of rotation to the TPCMO
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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A More Rigorous Analysis of the Kinematics of
the Two-Body Electromagnetic Problem
• An exact relativistic-kinematic description of the two-body electromagnetic interaction is needed to check for
consistency of physical law with relativistic covariance
• Starting in the source particle rest frame where the
motion is easy to describe, obtain motion in lab frameby making a series of Lorentz transformations of the test
particle position, first to the test particle rest frame,
then to the lab frame
– The expected result is a description of the motion in the lab
frame, as predicted by the TPCMO
• Today will present an overview of first part
– Contend the results shows the necessity of anti-Euler and anti-centrifu al forces in addition to the anti-Coriolis force
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Coordinate Systems for Derivation of Thomas Precession
(as Seen by Test-Particle Co-Moving Observer)
(r,t )
(r,t )
(s,x )
Source particle rest
frame (equivalent to
lab frame with a
stationary source
particle, for thisanalysis)
Test particle
momentary rest
frames
Test particle (t.p.) and
co-located, co-moving observer
t is the t.p. proper time,
r is the t.p. displacement
from the field-source
particle, as observed by
the observer co-moving
with the test particle
(s,x ) = (0,0) at (r,t)= ( 0,t 0)
(s’,x ’) = (0,0) at (r,t)= ( 0,t 0+ d t )
(s’,x ’)
= test particle
position in source
rest frame
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Significance of the Accelerated Observer
• Same legitimacy as an inertial observer (the
principle of relativity)
• Must be able to describe physics in inertial
frames based on observations in his frame
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Prior Work: Bergstrom
• Similarities: Accelerated observer, mass factor cancels
• Differences: "mosaic" avoids expectation of
centrifugal/Euler forces
• Mosaic argument disproved by sequential Lorentz
transformations of displacement?
• Also notable: observes effect applies to gravity too;
implies gravitomagnetism
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Counter-argument to Bergstrom’s claim of
impossibility of centrifugal effects of Thomas
precession
• Sequential Lorentz transformations of
displacement should obtain directly anti-
Euler and anti-Centrifugal forces (work still
in progress)
• Both have explicit dependence on inter-
particle displacement
• Have preliminary result for test particle rest
frame: have obtained anti-Euler force
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Velocity of Test Particle Relative to Source Particle as Seen by
Observer Co-Moving and Co-Located with the Test Particle
(see Moller, Theory of
Relativity, 1952 edition,
Eq. (146), page 257)(or
3rd chart following)
where
test particle velocity relative to source particle, as seen by TPCMO
and
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Thomas precession of source rest frame as seen by TPCMO derived by
successive Lorentz transformations of displacement
Lorentz Transformation of
Test Particle Position from
Source Rest Frame to Test
Particle Momentary Rest
Frame at time t = t 0
Lorentz Transformation of Test Particle Position from
Source Rest Frame to Test
Particle Momentary Rest
Frame at time t = t 0 + dt
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Variation of interparticle displacement as observed by
TPCMO with respect to time in source rest frame
where
Find that:
so
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Time flow at the Field Source Particle as Seen by Test
Particle Co-Moving Observer
LT of time in TP momentary
rest frame to SRF:
Evaluate at location of
field source particle :
Then:
To order
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Test Particle Velocity Relative to Source Particle as Seen From
Test Particle (Fermi-Walker) Rest Frame
where
Reduces to (to order b 2)
W
as expected
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Relative Acceleration as Seen by Test Particle Co-
Moving Observer
Had
so
Apply chain rule as
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Thomas Precession Angular Velocity Relations
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Finding the Anti-Coriolis and Anti-Euler Accelerations in
the Test Particle Rest Frame Acceleration
Can’t find this term
Only have half of this
term
Anti-Euler terms are found
exactly
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Finding the Anti-Coriolis and Anti-Euler
Accelerations in the TPRF Acceleration
(Expected anti-
Coriolis acceleration)
Based on relativistic law of inertia, need a Lorentz factor in derivative,
with respect to proper time, which obtains
Can re-write with anti-Coriolis force as
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Finding the Anti-Coriolis and Anti-Euler
Accelerations in the TPRF Acceleration
Relative acceleration in
the test particle
momentary rest frame
Continuing effort is to interpret and refine this result, then
complete kinematic analysis to obtain description of motion
in laboratory frame.
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Anti-centrifugal force
• Implication of lack of rotational
pseudoforces: a missing centrifugal force
predicts an always-attractive force (evenfor like charges)
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Strength of Anti-centrifugal force
compared to Coulomb repulsion
(for b 1)
For circular motion (and neglecting delay):
Assuming equal masses and equating with relativistic Coulomb force, then
solving for R obtains :
for equal to the proton mass. This is about 100 times smaller than themeasured proton size.
Expect
Exact T.P. formula per Jackson:
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Strong magnetic force
Exact magnetic force in Lienard-Wiechert magnetic
acceleration field:
Evaluate for particle 1 accelerated by Coulomb field of
particle 2 and assuming circular motion to obtain:
This magnetic force is always attractive, even for like charges.
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Completeness of Lorentz force?
• Missing magnetic-like force?
– If the field-source particle is free to accelerate, then
expect an additional magnetic-like force, with mass
ratio factor
• Predicts an increase in strength of spin-orbit coupling in
hydrogen by factor (1 + 1836)/1836
• Unable to find basis in L-W acceleration fields
• Anti-Euler force? (Atomic scale dynamical
effects of terms at order v^2/c^2?)
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Next steps
• Full relativistic kinematical proofs: order v^2/c^2 givesmagnetic, part of anti-Euler force
• Look for more T.P.-related forces in Maxwell-LorentzED
• Full relativistic kinematical proofs: order v^4/c^4 givesanti-centrifugal force, more of anti-Euler force
• Study time-symmetric dynamics of strong magnetic
force
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Conclusion
• Relativistic-kinematical approach is a new
way to think about and describe
electromagnetic forces• Relativistic-kinematical approach may
reveal new electromagnetic forces
• Relativistic-kinematical approach ispossibly a new explanation for
fundamental forces
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Backup/Extra
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Submittal History
• Submitted to Physica Scripta Sept 2010
– Title: “Does Thomas Precession Cause
Rotational Pseudoforces in Particle Rest
Frames?” – Sent out for review, then review was cancelled
due to (my) statement that the submittal was
only a plausibility argument for the magnetic
force being caused by T.P.
• It is (still) only an order v^2/c^2 analysis
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Pseudoforces of Rotation Do Not Occur
in Thomas Precessing Reference Frames
• For example, an accelerated observer sees
Thomas Precession of inertial frames
• See also UC Berkeley tutorial article by R. G.
Littlejohn linked in Wikipedia article on
Thomas precession: (page 8), ``... if we fix
ourselves to the parallel transported frame,we will feel no centrifugal forces.''
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Schild
• Full relativistic ED description of two-particlecircular motion; time symmetric ED
• No negative relativistic mass bound state
(Sternglass mentioned)• No anti-centrifugal force found, but did not
invoke Coulomb acceleration; assumed onlycircular-centripetal acceleration
• Also, anti-centrifugal force wants to vanish undertime-symmetry
7/27/2019 Symposium Talk on Origin of Magnetic and Strong Force in Relativistic Kinematics, David C Lush
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Problem 11.3 from Jackson