Dear Stefano,

In the file http://people.exeter.ac.uk/sh481/shockley-james.html

the author asks not on conservation of the momentum but on violation of Newton's third law in the classical electrodynamics.

This problem had been explained in detail in L. Page and N. Adams, ``Action and Reaction Between Moving Charges'', Am. J. Phys. Vol. 13, 141--147, 1945.

Page and Adams showed that Newton's third law in the electrodynamics is not fulfilled (no action-reaction principle). Instead the law of the total momentum conservation holds.

In the electrodynamics, for example, the center-of-mass of the closed (isolated) system can move only due to internal forces of EM origin, which formally contradicts to Noether's theorem.

Dear Vladimir Onoochin ,

Dear Stefano,

Unfortunately, In Wolfgang Engelhardt's example additional angular momentum cannot exist. I explain it in the attached L_rocket file.

But in another example (the inventor is Georgy Ivanov, not me - I only told to McDonald on this 'device') the center-of-mass of the closed system is able to move due to the internal forces. McDonald derived the expression for trajectory of this motion - Eq. (30), p. 7 of the second file.

So magnetic force still contains some enigmas.

With the best wishes,

Vladimir

Dear Vladimir Onoochin ,

have you got some details?

Dear Stefano,

> have you got some details?

The device described in onoochin.pdf file is the 'thought device'. Dr. Ivanov suggested it to show that internal forces of some isolated system can give non-compensated force which would able to provide one-directional motion.

Such a motion should not break the law of the energy conservation but should break the law of the momentum conservation.

Actually such forces are too weak (in the devices like Dr. Ivanov offered), about 10^(-10) Newtons. But now the question is 'if it is possible to have devices breaking the law of the momentum conservation?'

Dear Vladimir Onoochin ,

Ampere died in 1836, Lorentz was not even born. How could Ampere be in disagreement about something which did not exist?

Dear Stefano,

> Ampere died in 1836, Lorentz was not even born. How could Ampere be in disagreement about something which did not exist?

Unfortunately, I don't know how to treat these words of McDonald. We discussed only calculations - if only magnetic force acts in Ivanov's thought device, this device would demonstrate one-directional internal force.

Meanwhile, one paradox of the Lorentz force has been detected experimentally. In work of:

W.F. Edwards, C.S. Kenyon and D.K. Lemon. Continuing investigation into possible electric field arising from steady conducting currents. Phys. Rev. D, 14, 922 (1976),

it is reported on detection of the electric field which cannot be. So if we place some charged body near the coil loaded by superconducting current, the coil will act on the body but the body cannot act on the coil. Paradox.

With the best wishes,

Vladimir

Dear Vladimir Onoochin ,

it seems that such strange effect has been clarified...

Article Investigation of possible electric potential arising from a ...

Dear Stefano,

I see that different designs of the experiments were used by Edwards et. al. and Shishkin et.al. In Edwards, teflon is used as a basis of the bifilial coil. In Shishkin, the coil was put inside the teflon cylinder.

So it is unclear point how the current in the coil can polarize the teflon basis. In Shishkin's experiments, the stress in teflon cylinder is caused by magnetic force created by the coil. What factor can cause the stress in teflon unit of Edwards experiments?

It is difficult to estimate what data are reliable after two series of the experiments made under not identical conditions.

I see some problem in the measurement of the effect. In both experiments, the potential of the wire connecting to the coil is measured but not the electric field of the coil itself. Additional charge do not acccumulate in the coil. The effect should be created expecially by the retrdation effect (the number of negative and positive charges are the same but one type of charges moves and other the type is fixed).

With the best wishes,

Vladimir

"Several scientists pointed out a paradoxical consequence of the application of the Lorentz Force as an addendum to Maxwell's equations in the form given by Heaviside. There is at least one case where the momentum is not conserved..."

Maxwell's equations focus on electromagnetic waves in free space and not on the near interaction between charged objects such as protons, electrons and photons. The solution is provided by Coulomb laws, electrodynamics, electrostatics: Research Proposal Coulomb interactions and EM statics

"Maxwell's equations focus on electromagnetic waves in free space and not on the near interaction between charged objects such as protons, electrons and photons. The solution is provided by Coulomb laws, electrodynamics, electrostatics:"

The Maxwell equations describe not only the EM waves in the empty space. Yes, if they are written in causal form, as Jefimenko did it, they are not able to describe the forces.

But some scientists think that the Lorentz force, at least its magnetic component, can be derived from the Maxwell equations:

A. Yaghjian. 'Maxwell's derivation of the Lorentz force from Faraday's law'. http://arxiv.org/abs/1911.04605

Dear Vladimir Onoochin ,

to my understanding, Maxwell equations in the Heaviside notations are enough to explain electromagnetic phenomena, including Lorentz Force.

Considering how the Maxwell equations were originally derived in a "rest frame of vortices", overriding Galilean relativity (replaced by an absolute mean of transport), It is not strange at all that such a feature does not comply with the action and reaction between charges, since the field itself plays a big role as a material rest frame.....

Dear Stefano ~

The center of mass of an isolated object (ie, an object not subjected to external forces) moves with uniform motion in a straight line. This is Newton’s first law and is essentially equivalent to the conservation of linear momentum.

However, consider a complex object with internal moving parts. For definiteness, think for example of a box in “free space” containing a hidden internal mechanism, whose moving parts are governed by internal forces. This internal motion can quite naturally cause the box to shake, so that an external (inertial) observer will see the box not “moving with uniform motion in a straight line” - it will move in a jerky way, though no violation of Newtons first law is taking place. Nor is there any violation of Newton’s third law - the varying forces (forces of the internal machinary acting on the box and vice-versa) remain always “equal and opposite”. The centre of gravity of the total ensemble (box plus internal moving parts) moves with uniform velocity in a straight line.

The momentum of the box pB and that of the internal machine pM are not separately conserved, but their sum pB + pM remains constant. (We are dealing with a many-body problem).

This “thought experiment” resolves Engelhardts Lorentz Rocket paradox. It doesn’t matter whether or not the forces governing the internal mechanism are electromagnetic. If the “rocket” is “at rest” (relative to an inertial frame) when the ion is emitted horizontally, it will start to move downward as the ion moves upward under the influence of the magnetic field. This downward motion will cease when the ion hits the “cup” - causing a compensating upward motion - restoring the status quo. No law of physics, electromagnetic or otherwise, is violated. The “rocket” will just vibrate - jiggling up and down. It won’t take off!!

Engelhardt’s confusion seems to be arising from a failure to clearly distinguish between linear momentum and velocity. I used to be similarly puzzled by a related question - the well known phenomenon that a cat, dropped from an upside-down orientation, can right itself and land on its feet. Naively, this looks like a violation of angular momentum conservation. The cat’s remarkable in-flight manoever modifies the angular velocities of its front and rear - its overall angular momentum remains zero throughout.

Incidentally, the “Lorentz force” is hardly an “addendum” to Maxwell’s equations. The electrostatic "law" F = qE is, essentially, the experimental definition of E. The Lorentz transformations are the symmetries of Maxwell’s equations. Hence the transformation laws of E and B are prescribed. The Lorentz force law F = q(E + vxB) then readily follows by applying a Lorentz transformation to the electrostatic “law”.

Dear Eric Lord

„The Lorentz force law F = q(E + vxB) then readily follows by applying a Lorentz transformation to the electrostatic “law”.“

If you do so you will have an additional gamma factor in front of the transformed field…

Best regards

Jörn

Dear Eric Lord,

> The center of mass of an isolated object (ie, an object not subjected to external forces) moves with uniform motion in a straight line. This is Newton's first law and is essentially equivalent to the conservation of linear momentum.

1. Page and Adams showed by direct calculations that the law of the total momentum conservation and the Newton's third and first laws are not equivalent in the electrodynamics.

2. McDonald derived the expression for the law of motion of the center-of-mass of the isolated system (Ivanov's thought device). So in the electrodynamics the center-of-mass of isolated system can move.

> The electrostatic "law" F = qE is, essentially, the experimental definition of E. The Lorentz transformations are the symmetries of Maxwell's equations. Hence the transformation laws of E and B are prescribed.

1. I mentioned only magnetic component of the Lorentz force. The Maxwell equations written in not causal form can be used to 'derive' the magnetic force.

2. E and B fields are connected by the Lorentz transformations only for the uniformly moving charge. It is no so for other types of the charge motion.

The example of deviation of the EM fields from this 'rule', the fields of uniformly accelerated charge:

- Born in paper of 1909 showed that when uniformly accelerated charge is at rest (v = 0), its H field is equal to zero in all space - this fact was used by Pauli to state that this charge does not radiate;

- if H = 0 for v =0, then in the frame where the velocity of the charge is V, one should have according to the Lorentz transformations H(V) = [V x E(V)]/c ;

- Let's check this relation using well known expressions for these field by Schott,

H(V) is the angular component of the magnetic fields; the others = 0 if the charge moves in the x axis. E(V) is the radial component in the cylindrical coordinat system. Then (latex symbols; Schott's notation)

\[

H_{\phi}=\frac{8k^2 ct\omega }{s^{3/2}}

E_{\omega}=\frac{8k^2 x\omega }{s^{3/2}}

\]

where q = 1, \omega = \sqrt{y^2+z^2}, V = c^2t /\sqrt{k^2+c^2t^2}

s = \left [ (x^2+\omega ^2-\xi ^2)^2+4k^2\omega ^2\right] ;

- calculations

\[

\frac{8k^2 ct\omega }{s^{3/2}}= \frac{c^2t 8k^2 x\omega}{\sqrt{k^2+c^2t^2}s^{3/2}}

\]

The last equation is not correct - one has the x variable in numerator of the rhs and hasn't this variable in the lhs.

No 'relativistic' relation H = V*E/c

Dear Eric Lord ,

I only quite marginally referred to Enghelardt.

Einstein A and Laub J "Über die im elektromagnetischenFelde aus ruhende Körper ausgeubten pondermotorischeKräfte"

Ann. Phys. 26 541 (1911)

--------------

Coleman, S. and Van Vleck, J. H. "Origin of Hidden Momentum Forces on Magnets"

Phys. Rev. 171 1370 (1968)

------------

Shockley W "Hidden linear momentum related to the α,E term for a Dirac-electron wave packet in an electric field"

Phys. Rev. Lett. 20 3434 (1968)

-------------

Furry, W. H. "Examples of Momentum Distributions in the Electromagnetic Field and in Matter",

Am. J. Phys. 37 621 (1969)

--------------

Boyer, T. H. "Concerning hidden momentum",

Am. J. Phys. 76 190 (2008)

---------------

Babson, D., Reynolds, S. P., Bjorkquist, R. and Griffiths, D. J. "Hidden momentum, field momentum, and electromagnetic impulse",

Am. J. Phys. 77 826 (2009)

-----------------------------

Mansuripur M. Trouble with the Lorentz law of force: incompatibility with special relativity and momentum conservation.

Phys. Rev. Lett. 108 193901 (2012)

------------------------------

I referred instead to several papers of several authors pointing out such a "strange feature"

which by itself does not invalidate the formula which has been tested experimentally, but shows quite peculiar properties.

Thank you

Hi Eric, The Lorentz force law has actually got nothing to do with the Lorentz transformation. In fact it was already one of Maxwell's original equations when Lorentz was only a young boy.

As regards applying a Lorentz transformation to the electric field that already acts on stationary particles so as to produce a vxB term, the vxB term results from the 4D space-time and not from the relativistic gamma factor.

I demonstrated this by deliberately omitting the gamma factor from the derivation.

Also, there is the issue of the physical interpretation of vxB in the context. It is not referring to the force on a charged particle moving in a magnetic field.

See equation (1), and also sections VI and VII in this article,

Article The Lorentz Aether Theory

Like the Faraday paradox, the source of this measurement difference is that There are 2 different types of magnetic field - electric induced and permanent magnets.

Secondly, the strengths of the N pole and S pole are slightly different.

https://www.researchgate.net/publication/327158807_Two_different_types_of_magnetic_field

https://www.researchgate.net/publication/337544299_Magnetic_Field_Evolves_to_Gravity_Field_Part5_Final

Dear Stefano ~

Yes, I realized that “Engelhardt’s rocket” was only peripheral to the apparent anomalies under discussion. But it presented a puzzling paradox that, like all alleged “paradoxes” of Relativity, was fun to resolve!

I do not agree that the Lorentz force law violates any fundamental laws of Newtonian dynamics, or that it's “wrong”, as claimed by many of the authors whose papers you cited. I haven’t of course studied all these papers in any detail, but taking quick glance at a few of them I see no cause for concern! The confusion is obviously closely related to furore and controversy that arose from my simple question about the Poynting vector in static fields. The mathematics of electrodynamics does indeed imply “hidden momentum” which has to be taken into consideration to achieve logical consistency, even if it turns out be in principle experimentally undetectable and not measurable. I don’t find that counter-intuitive, and I’m no longer troubled by it!

It’s obvious that Maxwell’s theory and the Lorentz force law cannot be in conflict with SR, because their Minkowski space formulations

∂iFik= Jk, ∂iFjk + ∂jFki + ∂kFij= 0 and

Fi = JkFik

are manifestly Lorentz-covariant.

Here, the ‘matter’, consisting of moving charges, is considered in the continuum limit. Jk = ρUk, where ρ is charge density and Uk is the four-velocity field of the charged matter. Charge conservation is ∂iJi = 0.

Fi is the force density expressing the forces exerted on the moving charges by the electromagnetic fields. It must (by Newton’s third law) be equal and opposite to the density of forces exerted on the elecromagnetic field by the ‘matter’:

∂kTki= −Fi

where Tki is the energy-momentum tensor density of the electromagnetic field. It can be readily verified that the canonical symmetric electromagnetic energy-momentum tensor density satisfies this relation. Indeed, the form of Tki can be deduced from this expression (as in the attached page from my 1976 book). In any inertial frame, the three components that correspond to momentum density of the electromagnetic field are the components of the Poynting vector.

Eric Lord , You are not considering the difference between the classical Lorentz force law as per Maxwell, listed as equation (1) in this article, on the one hand,

Article The Lorentz Aether Theory

and the relativistic equation (34), on the other hand. In the latter, I have not yet added the relativistic factor, but the physical context is clearly not the same as in the case of equation (1). In equation (34), the term vxB is operating on a different scale and in a different physical context, and as the moving source approaches the speed of light, vxB, becomes related to the radial pressure in the disc-like magnetic field which arises as the electrostatic field flattens along its direction of motion.

And see appendix D regarding how centrifugal force fits in with Gauss's law.

Dear Dear Frederick and Jörn ~

Maxwell’s equations are Lorentz-invariant. This means that their familiar form is unchanged when we transform between two inertial frames K and K′ with relative constant velocity v.

However, I now see, from what you are saying, that it was naive and hasty of me to jump to the conclusion that the Lorentz force law follows trivially from the electrostatic law F = qE by naively applying a Lorentz transformation. It’s more subtle than that. That would tell us how the measurements of the various physical quantities (E, B, the velocity u of a charged point particle and the force F exerted on the particle by the electromagnetic field) are related across two different inertial frames K and K′. But that is not what is being asked here. The question is, given a single chosen inertial frame and given electromagnetic fields E and B at the location of the particle, how does F depend on the the velocity u of the particle. The correct approach is evidently to apply a Lorentz transformation to u and F but not to E and B. The Lorentz force F = q(E + uxB) then is indeed seen to be a consequence of the electrostatic law F = qE. The troubling gamma factor anomaly does not arise.

Dear Eric Lord

thanks for admitting that the issue is troubling! You are right, Lorentz transformation makes sense if and only if you speak about a moving particle. But as soon if you ask: what is a particle? you have to leave Lorentz transformation behind, as it is just an oversimplification of reality and particles are doomed to be pointlike. But we know they are wavelike…

To support my position I cite Max Plank 1906:

“Also as regards the concerns ​that according to the relativity principle a moving electron would be subject to a specific deformation work, I would attach no decisive importance, because in general we can add this work to the kinetic energy of the electron. However, the question of an electrodynamic explanation of inertia remains open; but instead there arises, on the other hand, the advantage that it's not necessary to ascribe to the electron neither a spherical form nor even any other form in order to arrive at a certain dependence of inertia on speed.“

Source: https://en.m.wikisource.org/wiki/Translation:The_Principle_of_Relativity_and_the_Fundamental_Equations_of_Mechanics

Best regards

Jörn

Eric Lord , There are two issues here. Firstly, while the maths is fine, it is meaningless until you identify the physical context in which it applies, along with the associated physical interpretation.

The application of a Lorentz transformation to the non-convectively induced electric field, E = −∂A/∂t −∇φ, results in a morass of confusion and paradoxes until such times as you identify the physical meaning of the terms A, E, and H within the context of a physical medium which serves as the rest frame for the velocity v, along with a recognition of what is actually moving with velocity v.

I note from your posting above, that even within the context of the maths alone, you left out the −∂A/∂t term and considered only the electrostatic term.

The −∂A/∂t term, which is the E field induced by a time-varying magnetic field, is crucial if we are going to produce a vxB term through a Lorentz transformation.

The paper which I wrote, goes through the Lorentz transformation of a non-convectively induced electric field, illustrating that the vxB term is a consequence of 4D space time, and not of the relativistic gamma factor, and that in the circumstances it relates to the the radial constricting pressure coming from the disc-like magnetic field which forms around the moving charge.

This is not the vxB that acts on a charge that is moving in an external magnetic field, so as to deflect the path of motion, as is the case in the classical Lorentz force law, although the commonality of form of the term vxB is ultimately indicative of the underlying hydrodynamics in a sea of vortices which underlies both case scenarios.

Dear Frederick ~

You are quite correct: the gamma factor does indeed imply something seriously amiss with the “Lorentz force law”. I recall struggling with this peculiar anomaly very many years ago, but had forgotten the details. I’ll try to reconstruct my analysis and offer my present conclusion.

The equation

Fi = qUjFji (1)

is self-evidently Lorentz covariant if q is an invariant, Fi and Uj are four-vectors and Fji is a tensor, in Minkowskian four-dimensional spacetime. That is a purely mathematical statement. It becomes a statement in physics when the symbols are assigned meanings as physical quantities.

Let q be the charge on a point particle, Ui the particle’s four-velocity γ(u, 1) where γ = 1/√(1 − u2) (taking c = 1 for convenience) and Fij contains the electromagnetic fields E and B in the usual way. (Note that u is not to be confused with the relative velocity v between two inertial frames that enters into a Lorentz tranformation. We are dealing here with a single chosen inertial frame).

The components of the right-hand side of equation (1) are readily seen to be

γq(E + u×B, u·E).

The left-hand side is trickier. Let’s begin from Newton’s first law f = ma, a = du/dt, where m is the “restmass” of a moving point particle − an invariant. The Newtonian acceleration readily generalizes to the Lorentz-covariant four-acceleration

Ai = dUi/dτ

where τ is the proper time associated with the particle’s trajectory:

dτ2 = dt2 − (dx)2 and hence dt/dτ= γ.

The components of Ai turn out to be

(γ2a + γ4u(u.a), γ2u.a)

and, correspondingly, the relativistic four-force Fi = mAi associated with a Newtonian force f acting on a moving point particle is given by

(γ2f + γ4u(u.f), γ2u.f)

This surprising (and disturbing) result is quite independent of electrodynamics. It applies to any force acting on a moving point particle.

The unavoidable implication for electrodynamics, if we accept Newton’s second law and Einstein’s Relativity, is that q(E + u×B) is not the Newtonian force f exerted on a charged point particle unless the particle is “sufficiently slow”. It is

γf + γ3u(u.f).

This is not a "paradox", it's just a complication that needs to be accepted!

(I’m finding it frustratingly laborious and time-consuming to put equations into these comment boxes. Please forgive me if I don’t continue with the discussion...)

Eric Lord , The gamma factor is likely to be correct. It leads to an asymptotic effect at high speeds close to the speed of light, and the physical reason for it will undoubtedly be due to the interaction between the moving object and the background luminiferous medium.

As regards the classical q(E + u×B) where no gamma factor is involved, the u×B term applies to the force on a charged particle that is moving in a magnetic field.

All I was saying was that when we derive this expression from a Lorentz transformation, where it then includes the gamma factor, the u×B term has a different physical meaning in the context. This is explained in sections VI and VII here, where the A vector pertains to the tiny electron-positron dipoles themselves. The context is best understood by beginning with the Biot-Savart Law,

Article The Lorentz Aether Theory

Thanks, Frederick (-:

I downloaded 'The Lorentz Aether Theory' and will look into it ...

Returning to Stefano's question: 'Were some scientists right in showing that the Lorentz Force brings to a paradox??', the only fact can be stated with sure - the Lorentz force do not lead to paradoxes if this force acts between two closed current circuits, and the currents in both circuits are constant with time. Then the Lorentz forces acting between these circuits are equal and have opposite direction

F_12 + F_21 = 0

Any statements that the Lorentz force, created by free charges, does not violate the action-reaction principle (Newton's third law) and the law of the total momentum conservation are only statements, without firm math background. The firm math background means that one should make rigorous calculations of the EM fields created by these charges. These fields are retarded fields. The effect of retardation cannot be ignored even for the stationary systems. For example, Boyer (Stefano cites his work) introduced the retardation correction via the Darwin Lagrangian.

But even Darwin Lagrangian is some approximation. The rigorous calculations of the EM fields created by moving classical charges can be made only in two cases, the uniform motion of the charge and its motion with constant acceleration (along one axis). In the stationary systems described in papers cited by Stefano, there is no such a behavior of the charges.

Thus, any calculations can be made, if can be made, with some approximations. These approximations 'kill' any opportunity to prove that the Lorentz force does not give violation of the action-reaction principle and the law of the total momentum conservation.

Any arguments of general type, including relativistic, do not improve situation. The rigorous calculations are needed, but they cannot be made.

So new paradoxes based on the Lorentz force, i.e. on non-central type of this force, will appear in future.

Eric Lord , OK. I could put it all more simply by saying that the vxB term that emerges from a Lorentz transformation relates to the magnetic field that forms around a moving charge, while the vxB term in the classical Lorentz force law arises in connection with the force acting on a charged particle that is moving through an already existing magnetic field.

And so long as you have a physical medium relative to which the v term is measured, and with respect to which the vxB force is induced due to a mutual physical interaction, then there will be no paradoxes.

Frederick David Tombe

Indeed, the field in the vicinity of a moving charged point particle is a superposition two electromagnetic fields: the “ambient” field through which it “swims”, and it’s own contribution as a “source” of electromagnetism. The former exerts a force on it; the latter does not. If I understand you correctly, the error in the naive “derivation” of the Lorentz force by appealing to the Lorentz transformation is its failure to take account of this distinction.

I remain skeptical, however, about your claim that these considerations necessitate the Aether hypothesis. I’ve never encountered any problem with my own conception of the electromagnetic field as some kind of "medium” in its own right, that needs nothing to “support” or “explain” it. But I’m quite willing to adopt your viewpoint heuristically and try to better understand it. I need to give the matter more thought...

Eric Lord , It's the physical interpretation of vxB in the textbooks, as it emerges from a Lorentz transformation, which fails to take account of the distinction mentioned above.

But none of it makes any sense at all until the vectors A, E, and B are given physical significance within the context of one of Maxwell's tiny molecular vortices, whereby A is the circumferential momentum density, and B is the vorticity.

The radial component of E is the electrostatic field, while the circumferential component of E, as in E = −∂A/∂t, is what arises when a magnetic field is time-varying. It's only the latter, which is actually the Coulomb Gauge where ∇·A = 0, that is involved in wireless EM radiation and the Poynting vector. Hence the Coulomb gauge is alternately known as the radiation gauge or the transverse gauge.

But when it comes to the Lorentz transformation of fields, we use the Lorenz gauge to get the desired end result. That's the radial gauge, hence implying that the Lorentz transformation is more a longitudinal effect.

Dear Stefano,

Lorentz equation is a very limited equation that can be applied easily out of its limits of validity. Let me to write some of them:

1. It assumes space-time translations, where the mechanical energy-linear momentum leaves to be conserved and therefore doesn't follow the above symmetry for the canonical energy-momentum tensor.

2. In a curved spacetime to glue translations and geodesic motion needs special care.

3. The vectorial Lorentz law that you wrote for one electric charge is nothing more than a projection of the electromagnetic equation. The same for this equation, notice that outside of the geodesics there is an extra electromagnetic radiation energy-momentum.

4. Maxwell equations are independent of Lorentz equations and they do not need symmetries to follow for being obtained from the electromagnetic action. The conservation of the charge U(1) enters in a very subtle form related with the gauge transformation of the potentials or the degree of freedom thanks to enlarge the Ampere's law by Maxwell.

Dear Daniel,

very happy to see you here again...

so there is indeed such an issue of linear momentum non-conservation...

but as you know such a formula has been tested in accelerators

F= qE + v x B

in circular accelerators the force

F= qvB gives the centripetal acceleration γv2/r to a mass m0

qv|B| = γv2/|r| m0 is the equilibrium in circular motion of a charge

q|B| = m0γ*|v|/|r| --> |B| = (γm0 |v|)/q|r|

the magnetic field necessary to bend a charge q of mass m0, along a circular path at speed v

Dear Stefano,

Yes, you have chosen a good example. In this system you can have cyclotron radiation (photons) with energies

E=heB/m

Therefore you can see that the Lorentz equation doesn't give us all the information about this system. There are much more examples as synchroton, etc, where we could play to find new electromagnetic behavours, but this one is very clear.

Dear Daniel,

yes for sure, in that case one has to consider the momentum which was lost by radiation recoil. But if speed is sufficiently low, the radiation is negligible and that formula holds...

Stefano Quattrini , The cyclotrons are circular which means that the centripetal force must be constant for an entire circuit. Is it therefore calculated from F = qvxB on the basis of the particles having reached a certain fixed speed?

Dear Stefano,

All that I tried to tell you is that the Lorentz formula is right, but with strong limits of applicability when it is written in the usual vectorial calculus, as you have made. Therefore, it is possible to find many different cases were it might be seem to arise a paradoxical behaviour as the ones that you show in this question. That is all !!!

Stefano Quattrini , There is also the issue that applying the gamma factor to the F = qvxB force will cancel with the gamma factor in front of the mass, resulting in the acceleration being unchanged from the classical result.

My thoughts on electrodynamics are based only on the following simple hypotheses:

(1) The fundamental laws of Nature are identical in every inertial frame.

(2) Newton’s second law is a “fundamental law of Nature”;

(3) So are the electromagnetic laws encoded in Maxwell’s equations.

It follows from (3), provided that the quantities ε0 and μ0 are accepted as universal constants, that

(4) The velocity of light, measured relative to any inertial frame, is a universal constant c.

It does not concern me when people when take issue with these hypotheses. All that concerns me is that they are mutually consistent. I am interested only in their logical consequences.

Hypothesis (2) means to me that the force f on a body and its acceleration a = du/dt (u = dx/dt), measured by any inertial observer, satisfies f = ma where m is an invariant (defined as the mass of the body measured by any inertial observer for which it is at rest).

As I showed in an earlier comment, the hypotheses lead inexorably to the electrodynamic law that, for a moving charged particle in an electromagnetic field,

γ2f+γ4u(u.f)/c2 = q(E + u×B)

where γ = 1/√(1 - u2/c2).

This is not a “paradox”!! It’s simply a recognition that the “Lorentz force law”, as stated in far too many textbooks and published articles, is an approximation, valid for a “sufficiently slow moving” charged particle. It is valid only in the limit (u/c)2

Eric Lord :

Hello Eric! It is so good to see you commenting in RG again (to my knowledge) after a long break! Although I was aware of this forum for some time, your comment above (~8 hrs. ago) caught my attention. It is typical of your clear, short, concise and articulate expression of ideas that I greatly appreciated and from which I learnt a lot of physics, clarifying my own ideas; and tremendously enjoyed our long exchanges before!

Due to lack of my strong grounding in electrodynamics and Maxwell’s equations, I do not wish to be an active participant in this scholarly discussion, but I have few concerns regarding the points and the final equation you have listed in your comment above and hope for some further clarification.

1) I have no problem with your items (1), (2) and (3), but only with (4) and the final equation. In your assumption of the universal constant c (SR), you assume that “ε0 and μ0 are accepted as universal constants”. My dialectical approach to QED and the space and time dynamics (virtual particles) tells me that the universal constancy of ε0 and μ0 may not be valid and hence also the constancy of c. It is because the nature of space near high mass concentration (near the galactic centre for example) may be different from intergalactic space; with differing ε0 and μ0 values and hence also c. If c is the limiting value of any motion, then the bending of the light beam by strong gravitational field and the reported superluminal speed of ejected quasars would be inconsistent with a constant c and hence also with SR. Superluminal ejection from AGN is no longer a heretic idea, but now seems to have become mainstream as my following publication would show: Article QUASARS – RETROSPECT, PROSPECT AND A POINT of DEPARTURE

2) Another concern I have is with γ (= 1/√(1 - u2/c2)) and related “spacetime” – a fundamental conceptual difference. I have tried to show in a relatively recent work that Lorentz’s transforms and “spacetime” are abstract and artificial (not even mathematically strong) mental constructs, which are highly unlikely to be valid representations of objective reality, from a dialectical perspective. So, its use in any scientific formulation is open to question; as you have done in your equation: γ2f+γ4u(u.f)/c2 = q(E + u×B).

Please see my following article on this issue:

"The Mystery of the Lorentz Transform: A Reconstruction and Its Implications for Einstein's Theories of Relativity and cosmology" :

Article The Mystery of the Lorentz Transform: A Reconstruction and I...

3) γ is a purely electrodynamics concept, where the mass of a photon is supposed to be 0. So, if we try to relate electrodynamics with Newtonian dynamics with tangible mass > 0, as in the above equation, I wonder how compatible (conceptually) that would be. I learnt a lot from my (now Late) friend Wolfgang Engelhardt when writing my article on LTs; but we had un-resolvable difference on Planck’s use of γ as signifying variable mass idea of accelerated massive charged particles. Wolfgang rejected relativistic mass of SR; but (inconsistently, in my view) strongly asserted the idea of the “variable mass” implied by γ. I have a very different explanation in relation to the workings of particle accelerators; because I question the validity of the role of γ or LTs! This is also the reason I have concern about your equation: γ2f+γ4u(u.f)/c2 = q(E+ u×B).

Please also see: Planck “On Principle of Relativity & Mechanics:

https://en.m.wikisource.org/wiki/Translation:The_Principle_of_Relativity_and_the_Fundamental_Equations_of_Mechanics

I hope you would take the time and the pain to attend to my concerns.

Thanking you.

Best regards, Abdul

Dear Eric, Eric Lord

although you are right that the Lorentz formula is only a first-order approximation in v/c , the discrepancy at high speeds can be very remarkable. Nevertheless the issue does not depend on the speed as you can see from the post, that occurs at low speeds.

Although conservation of energy for example is preserved in inelastic scattering thanks to Special relativity, considering the case I brought to our attention, higher order terms do not help.

Stefano Quattrini , Does the gamma factor, as applied to the F = qvxB force, cancel with the gamma factor as applied to the mass, so as to make the net acceleration dependent only on the vxB factor?

Dear @ Eric Lord

I didn't check yet your formula

γ^2f+γ^4u(u.f)/c^2 = q(E + u×B)

from the point of view of your postulates. I will do this as soon as my duties make we free for a while. Now I would like to mention that a similar one (with additional gamma factor on the RHS) can be obtained by assumption that the proper acceleration is a result of action of linear operation upon the proper velocity. Thus I came to a problem whether it is just a random coincidence or a justified similarity perturbed by severe difference of meaning of our different fashions just at one simple point. I will try to see what is the core of the difference.

Could you also try to compare the reasons. For my details please, consult the derivation of formulas 7 and 8 of my preprint

https://www.researchgate.net/publication/331559958

Any comment is very welcome.

Best regards,

Joachim Domsta

Frederick David Tombe ,

since F=dp/dt =d(mvγ)/dt , in a steady rotational motion without tangential acceleration the force F is only radial, or let's say : the force (exerted by a magnet) necessary to bend the trajectory is only radial (otherwise a work would be performed which is not the case).

By deriving that formula, the radial component of F = m (t*at*γ3 + n*an*γ ) is

Fradial=mγ*n*an

Yes, if the gamma factor were present in the F= γ(qvxB), it would cancel out.

But according to Eric it is γ2F+γ4u(u.F)/c2 = (qvxB)+ qE,

in a configuration of a synchrotron where E = 0,

in circular motion F and u are perpendicular, F is only radial, since again it does not perform work unless it has to counter-ract to the emitted radiation which for now we neglect: u.F =0 in a circular motion, setting u=v, normal to B considering v=|v|

Fnγ2 =(qv |B|),

meaning that if the magnetic field is constant and the charge constant, the acceleration gets smaller and smaller, with increasing speed, after a certain speed... 1/γ3 approaches to 0 at high speeds.

That is the opposite of what occurs in accelerators!!!!!!! The force gets incredibly strong with high speed, far from being proportional to the speed.

mγan = (qv |B|) /γ2 ,

an = (qv |B|) /(mγ3) which is not what occurs in experiments.

Dear @Stefano Quattrini

Am I right that in circular motion of highly accelerated particles a equals almost c^2/r and therefore this quantity is independent of B. Due to other constants, only gamma varies. Accordingly

-- if there is gamma^0 on the RHS in the Eric Lord formula, then gamma^2 is proportional to B

-- if there is gamma^1 (as it comes from my preprint) then gamma is proportional to B.

However in NONE case of such theoretical considerations, v is proportional to B for suitably strong fields.

Regards,

Joachim Domsta

Stefano Quattrini , It seems to me, that whatever is observed to happen in practice in particle accelerators, that this has as yet been correctly matched to any theory. We know that for a given applied force, the acceleration slows down as the speed increases. Some interpret this in terms of mass increase. Others play around with Newton's second law of motion.

But whatever, there is an asymptotic effect, which some (like myself) attribute to the aether.

Now, when we move into the realms of the curved path motion caused by the centripetal magnetic force, F = qvxB, the situation becomes even more confusing, because this force is not doing any work, but merely changing the direction of motion. So do we use a relativistically modified version of Newton's second law, for force in general, on top of a relativistically modified version of qvxB? I can't see that there can be a theoretical argument that definitively answers this question, because it all degenerates into a morass of confusion.

So, the first question that I would ask is, "does the radius of the cyclotron dictate any particular value of v in qvXB"?

I would then take the matter from there.

PS. Stefano, let me note that in the implied by you @Eric Lord's eqn.

F_n γ^2 =q v |B|,

F_n is the INERTIAL force assumed equal to m a_n , where m is the invariant mass.

On the other hand you are concerned with the SOURCE force. Hence the question by @Frederick David Tombe leads to the following conclusions:

IF the source force is gamma q v |B| and we apply your formula for the inertial radial force i.e. gamma m a_n THEN the gamma factor cancels.

IF the source force is gamma q v |B| and we apply Eric's formula for the inertial radial force i.e. gamma^2 m a_n THEN the gamma factor does not cancel.

IF the source force is like in Eric's q v |B| and we apply Eric's formula for the inertial force i.e. gamma^2 m a_n THEN the gamma factor acts in the quadratic power.

My point is that the covariant equations lead to the second option.

Abdul Malek

Dear Abdul ~

It was nice to see your name pop up again. Thank you for your kind words about the way I express my conception of Physics. I always found your viewpoint refreshingly different and thought-provoking and I very much enjoyed our discussions.

Yes, I was at one time very active on Researchgate. Eventually it all became too much; my inbox was bombarded every day by “notifications” and eventually my addiction to Researchgate discussions was taking up far too much of my time (often argueing with people who simply didn’t not understand what I was talking about!). I had to break the habit and get out.

I was drawn back in recently by the word “paradox” in the question that initiated this discussion thread. Whenever that word crops up in scientific discourse it indicates that something is seriously wrong with current thinking, that needs to be sorted out.

You, along with many others, are not comfortable with the “constancy of the speed of light”. Note that I don’t insist on it, I see it as a reasonable hypothesis. I can go further and claim that it is not even a “hypothesis” at all. It can be regarded merely as an artifactof the way physicists have chosen to describe Nature, not an attribute of Nature at all.

Physics is the science of measurement. The result of a measurement is a ratio, a dimensionless number expressing an attribute of a thing or process observed in terms of a unit. These units are not inherent in Nature’s laws, they are chosen by scientists for maximal stability.

When the hypothesis that “the speed of light is a constant” was proposed at the beginning of the 20th century the unit of time, “one second”, was defined as a subdivision (1/86,400) of the period of rotation of the Earth and the unit of length, “one metre” was defined as the length of a precisely manufactured platinum rod made in 1889 and kept at a carefully controlled temperature at the International Bureau of Weights and Measures at Sèvres near Paris. Clearly, based on these choices, the hypothesis was indeed open to doubt! Moreover, the stability of these units became totally inadequate for the extreme precision that would be required by later develpments in physics.

In 1960 the internationally accepted definitions of the unit of time, “one second” was redefined in terms of a precise frequency in the emission spectrum of a Cesium-133 atom. (Note that the assumed stability and reliability of this definition is based on faith. Faith that the behaviours of unperturbed Cesium-133 atoms are forever unchanging and all identical; faith that the operations of Nature (“reality”), are fundamentally rational. Without that faith we could only throw up our hands in despair and abandon “science” altogether! As expressed by Einstein: “Raffiniert ist der Herrgott, aber boshaft ist er nicht.”)

Since 1983 the internationally accepted definition of the unit of length, “one meter” is

“the length of the path travelled by light in a vacuum during a time interval of 1/299,792,458 of a second. “

The speed of light is then 299,792,458 meters per second by definition. This is not a fact of Nature it is a feature of the way physicists have chosen to express their observations of Nature.

Just a brief final word about ε0 and μ0. If they are not universal constants, then of course they are physical properties of “something” that, of course, is traditionally called “the Aether”. It would require some compelling observational evidence to convince me of that. I see no such evidence. I see only speculation that, to my mind, only serves to complicate things unnecessarily.

I'm not an expert in this field but when -- years ago -- I looked at energy and momentum conservation in the ponderable matter I learned this: The issue is much more complicated than you have described. In particular, one must carefully treat both the electrical and mechanical energies and momenta of the charges and currents.

Stefano Quattrini "the principle of action and reaction is violated, what you say necessarily implies the violation of the momentum conservation unless one admits that there is an active background participating in the configuration."

No. The principle of action and reaction may be violated and momentum conservation still hold. Momentum conservation follows from the homogeneity of space via Noether's theorem. So it holds in special relativity, even though the non-absoluteness of simultaneity implies a violation of the principle of action and reaction (for forces at a distance: if two equal and opposite forces arise at one point in time in one system, they will arise at different times in a system moving at constant velocity with respect to the first, voilà a violation of the principle; that is why in relativity there is no action at a distance, force transmission requires fields which transport forces locally, and locally the principle still holds).

Since total momentum is conserved in special relativity, it is also conserved in electromagnetism, which is a relativistic theory to begin with. Adding in mechanical degrees of freedom, violations may seem to appear, if the Newtonian version of mechanics is used, simply because that theory is incorrect. If the entire system is treated relativistically, no violation of conservation of momentum will arise (except in incorrect papers, of course :-)). This holds in the absence of gravitation, i.e., as long as the spacetime is flat, and, hence, the space homogeneous.

K. Kassner , There is no way that electromagnetism is a relativistic theory to begin with. The so-called relativistic effects are an add-on to classical electromagnetism and only become significant at speeds close to the speed of light. And if electromagnetism is aether based, then the so-called relativistic effects are simply an asymptotic consequence of the nature of the elasticity in the aether.

Dear @Eric Lord,

I am sorry for bothering you for the second time but only now I realised that the equation of your last but one post possesses just a misprint (lack of gamma on the RHS). Indeed, for consistency with your post no. 25, it should read as follows

γ^2 f+γ^4 u (u.f)/c^2 = γ q (E + u×B).

Thank you for the contribution since it allowed me to confirm results of my efforts for obtaining this covariant form of the relativistic counterpart of the Newtonian 2. law for em forces acting on a point-like charged particle with positive [rest-]mass.

REMARK. In this fashion the tensor F_{i,j} under LT transforms in the way used already in Planck's '1906 paper. This gives additional reason for propagating the formula as being consistent with the core of SRT.

Dear Joachim Domsta ,

it is according to that γ F+γ^3 u (u.F)/c^2 =  q (E + u×B)

In a circular motion of a synchrotron for example with an electric charge u.F = 0 in a magnetic field with no electric potential (only vector potential).

γ F = q u×B

considering that u and B are perpendicular, (B is given by the bending magnet), |u|=vt

F| = γmvt2/R = q vt*|B|

|B| = γmvt/(qR)

yes, the magnetic field, necessary to bend the trajectory of a fermion, increases non-proportionally with speed, according to γ. It diverges while approaching to c.

The only way to increase the speed of fermions in accelerators, by maintaining the same magnetic field, is to enlarge their radius, which is what happened for the "Large" Hadron Collider (LHC).

Actually all this makes sense and it is quite interesting, although still not close to answer the initial question.

Dear @Stefano Quattrini

Yes, this only one of necessary step that we should agree for talking about the same issues. The problem is for me too hard to handle due to lack of full understanding the wave equations for em fields with sources (charges and currents). Thus probably no more contribution from my side should be expected.

Good luck in further discussing this interesting question you have posted! Perhaps some points will be available for me. Then I'll try to join this post again.

Joachim

Stefano Quattrini Joachim Domsta , So what speed is used in the calculations for a given radius?

Dear @ Frederick David Tombe highly accelerated particles v_t is close to c and therefore the quantity gamma=gamma_t is much much more greater than 1.

More precisely in any case we have:

v=v_t is a solution to the equation

|B| = m*v/q*R*sqrt{1-v^2/c^2}

which is

v = V/sqrt{1+V^2/c^2}

where

V := |B|*q*R/m

Regards, Joachim

PS. Thank you for an easy problem :-)

PS2. For very large value of the "classical" V>>c, one can apply an approximation of c-v by

0.5*c*[c/V]^2.

dear K. Kassner ,

yes, this was already mentioned by Eric Lord . The center of mass of a system can oscillate so that the principle of action and reaction is locally violated but the momentum is conserved if we consider a time longer than the period of oscillation (it does not accelerate out of nothing).

The action and reaction principle is a sufficient condition for the conservation of momentum. It is in fact not a necessary condition, meaning that if the AR principle does not hold that does not necessarily mean that the conservation of momentum is infringed.

The feature of oscillation can be a characteristic of a two-body problem, for example, as also mentioned by Eric...

It is peculiar though to stress the fact that , with the EM field, in such case we have a sort of a two-body problem, that is to say that the EM field is an entity acting like a mass with its own dynamics.

This occurs in any case independently of the speed of bodies, at low speeds for example. It could be considered a consequence of SR just on the account that the EM field is a "massive" object, and EM bounded energy is mass.

Joachim Domsta , Yes, but if v is very nearly equal to c, then the gamma factor becomes close to infinity. So I'm not clear about your argument, or where you got this equation here from,

|B| = m*v/q*R*sqrt{1-v^2/c^2}

It looks to me as though you have used the classical solution to the F = qvxB problem of a charged particle moving in a magnetic field along with relativistic mass.

Since |B| is fixed externally, then your equation suggests that R is infinity, i.e. straight line motion.

Dear Joachim Domsta ,

the topic is indeed quite complicated...

even Einstein questioned the Lorentz Force, in a paper with Laub, by saying that it should be replaced with the formula he had proposed.

The paper below from Birula is not easy and brings on the table additional details

Article Ehrenfest theorem in relativistic quantum theory

"We believe that our results might also contribute to thediscussion started by Einstein and Laub [10] and still con-tinued [11, 12] on the incompatibility of the Lorentz forcewith special relativity. In particular, we have shown that inthe case of relativistic quantum particles there is no need totake into account some ‘hidden momentum’ [13]. " (PDF) Ehrenfest theorem in relativistic quantum theory. Available from: https://www.researchgate.net/publication/359838084_Ehrenfest_theorem_in_relativistic_quantum_theory [accessed Nov 26 2022]

dear Frederick David Tombe ,

from |B| = m*v/(q*R*sqrt{1-v^2/c^2}),

|P| = mv/sqrt{1-v^2/c^2} is the relativstic momentum

|B| = |P|/(qR)

the larger the radius the weaker can be the bending magnet.

The larger the charge, the weaker can be the bending magnet.

Stefano Quattrini , OK. So you are going with the idea that we use the standard Newton's second law, F = ma, in conjunction with F = qvxB, albeit with relativistic mass in the latter?

But when v approaches c, and since B is fixed, then R will be nearly infinity, and so we will not be dealing in circular motion to any noticeable degree.

I still think that the gamma factor cancels on both sides of the equation.

Frederick David Tombe ,

I always mention relativistic momentum, there is no relativistic mass for me.

if v approaches c, if the magnets remain the same (same technology), a larger and larger radius is needed in order to keep the trajectory in a circular line, meaning that the trajectory will be close to a straight line locally.

This is what actually happens in LHC, if the magnets were always the same to probe energies/speeds very close to the speed of light the radius would have to increase dramatically.

@Frederick David Tombe

Please read ALL answers above. Also you can consult text books like

http://web.mit.edu/course/22/22.09/ClassHandouts/Charged%20Particle%20Accel/CHAP11.PDF

formula (3.38), where

gamma = 1/sqrt{1-v^2/c^2}

Yes, it looks in such a way, but it is RESULT of a linear formula for 4-vectors of Minkowskian space-time (see @Eric Lord's answer no. 25).

No. Neither v equals c nor gamma equals infinity. Please try own way of solving the quoted equation.

Regards, JoaD

PS.

Sorry, but such thinking is not consistent with SRT. Which does not cancel appropriateness of building another theory, of course.

PS2. The correct link to chapter 3 is as follows

http://web.mit.edu/course/22/22.09/ClassHandouts/Charged%20Particle%20Accel/CHAP03.PDF

Eric Lord > “The speed of light is then 299,792,458 meters per second by definition. This is not a fact of Nature it is a feature of the way physicists have chosen to express their observations of Nature.”

Dear Eric: Thank you for your kind words for me and for your characteristic clear and unambiguous statement that the constancy of the speed of light “is not a fact of Nature it is a feature of the way physicists have chosen to express their observations of Nature.” This makes you a consistent rationalist thinker like Emmanuel Kant. But I am afraid, most prominent official theoretical physicists would disagree with you. Because they would (inconsistently) argue that the constancy of the velocity of light is an attribute of objective reality; meaning that it is not only an epistemological assumption, but an ontological truth!

I do not intend to drag you back to long discussion like before, but i) I would like to make a general statement about “paradox” or contradiction in theoretical physics that Stefano Quattrini and you are talking about – an issue of my fundamental disagreement against modern mainstream theoretical physics and ii) the issue you raised about the constancy of ε0and μ0; You do not have to respond, if you do not wish to.

I) My view of “paradox” or contradiction is that physics can never rid itself of contradictions if it has to deal with objective reality, because, as Hegel's dialectics showed, any physical/material existence (at all), is a contradiction – on the one hand, the ontological contradiction between “Being” and “Nothing” and also the epistemological contradiction between “Being and “Knowing”; on the other. Theology, because it had to deal with the social reality had to accept contradiction of the “Evil” in omnipotent and omniscient God’s Kingdom; this way it tried to avoid contradiction by transferring all contradiction to God Himself! Idealism in the form of (later) rationalism and in the person of the mighty thinker Emmanuel Kant wished to eliminate contradiction in philosophy; but the only way he could do it is by taking resort to subjective idealism of his brain-cooked ‘logical categories’ and by denying knowledge of objective reality as an “unknowable thing-in-itself”; thereby condemning philosophy to its lowest moment in history!

Einstein wished to deny the greatest contradiction in physics, namely the quantum uncertainty. But the only way he could try to do it is by following Kant’s rationalist approach of subjective idealism and by reviving early Greek mathematical idealism; which is supposed to be free of contradictions. But this way Einstein led theoretical physics to be alienated from material reality! All these mean that unlike Kant, whose logical categories were candles to help navigate through “dark” reality; the 'mathematical categories' of modern theoretical physics represent the landscape itself that can be clearly open to knowledge. It means that “the mathematical universe” of modern official theoretical physicists IS the real stuff! This is an illusion that modern theoretical physics shares with all idealism of the past including Hegel’s own “Absolute Idea”.

After Emmanuel Kant’s “Unknowable thing-in-itself” and after he “found it necessary to deny knowledge in order to make room for faith" (Critique of Pure Reason); it fell on Hegel to restore the honour of philosophy as the science of all sciences, the soul of all knowledge; by pulling it out of its intractable problems and the lowest moment it reached by the time of Kant. And strangest of all, this he accomplished by embracing the very same elements, namely the ideas of evolution and contradictions etc., which hitherto known metaphysics (“the view of understanding”) abhorred the most. On the contrary, he put them at the very heart of his new philosophical system – the dialectical method.

Hegel unambiguously rejected the law of non-contradiction of theology, old idealism, rationalism and classical materialism (including natural science), the “excluded middle” of Aristotle and the "thing-in-itself" of Kant. For Hegel absolutely everything depends on “the Identity of identity and non-identity.” Opposites reside together in the very element of a thing or a process in simultaneous unity and opposition to each other and a resolution of this logical contradiction and conflict provides the dynamics for change, motion, evolution, development etc.

In my humble view, theoretical natural science and cosmology has to catch up with Hegel's dialectics to be worthy of any scientific merit: The following link is only meant to justify this extraordinary claim!

"Dialectics Not Metaphysics Of Nature: From The Quantum To The Cosmic":

https://www.amazon.de/dp/B0BF5W9Q1N?asin=B0BF5W9Q1N&revisionId=9ebf1d7b&format=1&depth=1

[Please note: Only the free preview alone may be enough for a general picture of the book]

ii) On the issue of the constancy of ε0 and μ0, and my view of the quantum vacuum; I have tried to assemble some experimental evidence and scientific/philosophical reasoning to argue for a materialist dialectical approach to theoretical physics and cosmology and for the reality of the “virtual particle/antiparticles” of the quantum, in the following RG question:

"Ex nihilo nihil fit“? Are You Certain Mr. Einstein and Mr. Heisenberg?"

https://www.researchgate.net/post/Ex_nihilo_nihil_fit_Are_You_Certain_Mr_Einstein_and_Mr_Heisenberg2

Through my works, I am trying to demonstrate that the quantum vacuum is full of "virtual particle/antiparticle pairs" that continuously pop in and out of existence and their effect on the spectral lines of atoms known as the "Lamb Shift" and the Lande factor of magnetic moment of electrons, which can be very accurately measured, may be rationalized based on the concept of virtual particles. This effect can also be measured as the Casimir force with much less efficiency. The permittivity and the permeability of the classical vacuum can be attributed to the collective effect of the momentary existence of infinite number of virtual particles of the quantum vacuum.

The "virtual particles" can become "real particles" if sufficient energy equivalent to compensate for their mass is available. There is also a finite probability that a "virtual particle" can become "real" (even without sufficient energy for mass equivalence) through a phenomena known as "quantum tunnelling". In fact, it is possible that this spontaneous creation (and annihilation, ex nihilo) of fundamental particles from the quantum vacuum (and not through the Big Bang) is how the galaxies (along with all other things) evolve, “come into being and pass out of existence” and maintained throughout this infinite and eternal universe as asserted by dialectics.

Stefano Quattrini Joachim Domsta , Whether we use the simple classical equation, F = qvxB, or whether we add the gamma factor, the situation will still be, that the faster the particle moves, the greater will be the radius of curvature. The two alternatives differ only in degrees. But if we use the gamma factor, the radius will nearly be infinity for v close to the speed of light, and hence a virtual straight-line path, whereas with the purely classical formula, there should still be a significant curvature.

What do the experiments say?

Joachim Domsta , OK. Thank you. Equation 3.37 is very interesting. It ties in with what Stefano has said regarding the straight-line path as v approaches c.

I'll just have to try and ascertain the physical implications. Notice that the gamma factor hasn't been applied to the Lorentz force, but only to the inertial force. So once you have confirmed this, I will take back what I said about the gamma factor cancelling on each side of the equation.

@Frederick David Tombe

The curvature is zero if B is zero, only. Approximately straight line is not a straight line.

2. I do not know.

2. Addendum. From books commenting photos of experiments I know that trajectories of charged particles of the cosmic radiation and radiative atoms possess radius the greater the greater is their velocity relative to mass. By observed ionisation like in Wilson chamber one can measure the velocity erc...

@Frederick David Tombe

The SRT dervation the main steps are

-- obtaining a formula for inertial force via the proper acceleration

-- obtaining a formula for the Lorentz 4-force by linear dependence on the 4-velocity (like by @Eric Lord in answer no.35 pf THIS thread)

-- comparing the above formulas

The result is what I have posted in an answer to Erik. There is

-- gamma^2 at the acceleration on the left hand side

-- gamma^1 at the classical formula for the Lorentz force.

ERGO. These factors do not cancel each other.

PS. There is another EQUIVALENT derivation given in the book where in front of a gamma^1 appears and in front of the classical formula for the LForce no gamma factor appears. If one added a gamma factor on the RHS then this would be a coruption since the formulas on the both sides were then taken from different worlds:-(

PS. Please read all my answers carefully and ask unclear places at the first post of appearance though we can omit unnecessary and useless repetitions. Thanks.

Another reason of my last request, I have great trouble with typing formulas by my micro-keybord using macro-fingers.

Joachim Domsta , Your reference to equation 3.37 in this link here http://web.mit.edu/course/22/22.09/ClassHandouts/Charged%20Particle%20Accel/CHAP03.PDF made my day. On the basis of that equation, I believe I have now grasped the situation fully.

What is the point in adding further complications such as an extra gamma factor on each side?

@Frederick David Tombe

I shall omit to discuss your last question since I do not understand how the problem of cancelling of gamma factors appeared.

I can only confirm that

-- the formula of our concern is based CONSISTENTLY on the mathematical structure of SRT

-- the physical interpretation of E and B and the implied validity of the formula is a subject of mathematics only when comes to NUMERICAL verification of its consistency with the observed data.

Oh, oh! My strange answers appeared because I have misunderstood the English meaning of you claim

I thought that you are trying adding back once more the gamma factor :-) I am really sorry for this misleading posts.

Best, JoaD

Joachim Domsta , It's OK. We've moved on now. We're accepting that equation 3.37 in this link is the correct equation.

http://web.mit.edu/course/22/22.09/ClassHandouts/Charged%20Particle%20Accel/CHAP03.PDF

But it then raises an interesting question. Since the gamma factor affects the acceleration, we need to ask why. The only answer I can think of is that the gamma factor is part of the relativistic mass. Some people argue that there is no relativistic mass and that we should simply be talking about relativistic momentum. But it is the inertial mass that determines the acceleration when the particle is acted upon by an external force, such as F = qvxB, hence the only conclusion that I can come to is that mass really does increase as the speed increases, as per the relativistic mass formula.

Dear Abdul Malek ,

I would like to spend some words in regard to the constancy of the speed of light. What can be said is that experiments on an inertial platform P (in a sufficiently small volume) in vacuo, measured that the speed of light is c: emission and detection from P with synchronized clocks in the positions Pe and Pd (emission and detection).

That does not imply at all that it is c for all observers. That means: it is virtually impossible, from within a limited volume, to measure a variation in the speed of light. That is the local Lorentz invariance: it is possible to choose a sufficiently small volume in vacuo such that no variation in the speed of light can be detected.

If the points of emission and detection belong to different platforms P and P' for example, one has to synchronize the relevant clocks (which can occur only when these overlap) and then account for the delay that light has in reaching a moving object at distance x: only as a first approximation it is vx/c2 from which t'=t-vx/c2 (first order Lorentz Transformations)

The speed of light in this case is forced to be c, which as Eric Lord said, is a matter of choice.... from that assumption one builds the space-time an arbitrary construction that has a certain prediction power.

The falsification (working limit) of such an artifact, in sufficiently large volumes, occurs at least where gravitation is present

Preprint SHAPIRO TIME-DELAY, Curved 4D space-time or Variable speed of light

Either bending the space-time according to GR (yet another artifact with quite a good prediction power) making the metric (measurement) different, according to gravitational potentials, or simply considering a euclidean space where the speed of light varies according to (1+2phi/c2) as a first approximation. A light beam crosses a 3D euclidean space with a definite position in the coordinate system of the SUN, that way one does not need to trick the metric to make the speed of light constant, that is to say that the speed of light is in general not constant in nature.

Frederick David Tombe ,

Joachim Domsta is totally right, by replacing m with m*gamma (the relativisic mass) in some classic formulas, in the attempt to extend them to relativstic/high speed ones, the outcome is wrong (for example also 1/2 m*gamma v2 is wrong as a relativistic kinetic energy).

The correct way is to use the relativistic momentum only, which, by the way, with the kinetic energy theorem, provides the right famous relations.

Preprint FROM THE RELATIVISTIC MOMENTUM TO THE MASS ENERGY-EQUIVALENC...

Stefano Quattrini Joachim Domsta , OK, I'll bear in mind what you say about relativistic momentum being a better quantity than relativistic mass, and I'll put it aside for the time being.

I want to return again the relativistic centrifugal force in equation 3.37 in this link,

http://web.mit.edu/course/22/22.09/ClassHandouts/Charged%20Particle%20Accel/CHAP03.PDF

Earlier in this thread I saw something about the relativistic force term d(γp)/dt being expanded into a parallel and a perpendicular component. Can you please remind me what these two components are. Is one of them the centrifugal force on the left-hand-side of equation 3.37?

Frederick David Tombe ,

from F= d(γ(v)mv)/dt, that derivative , by considering the trajectory of a body with speed v, can be seen having two components,

one along the trajectory the other normal to the trajectory

F = d(γ(v)mv)/dt= m (t*at*γ3 + n*an*γ )

Stefano Quattrini Joachim Domsta , What we seem to be learning from this is, that whereas the Lorentz transformation of an E field leads to an E' = γvxB term, there is however no gamma factor involved in the Lorentz force, F = qvxB, when it is acting on a moving charged particle in a magnetic field.

The discrepancy is probably because the former applies to the magnetic field induced by a charged particle in motion, where energy is accumulating, whereas in the latter case, the force is external and involves no accumulation of energy in the external magnetic field.

Frederick David Tombe ,

E' = γ vxB is correct, it involves a transformation from a primed to unprimed systems

F = q vxB is not exact as explained by Eric Lord..

γ F = q v×B ,as recalculated in the full form by Joachim Domsta, is the exact one in rotational motion where no work (energy accumulation occurs)

Sorry, but this time no recommendation, Sirs :-)

Reason: the gamma of LT is NOT the gamma of moving particle.

Some more detailed explaining of the (more than one) missed points require more preparation. Let me mention only that the forces are covariant wrt LT only if considers as 4-vectors. I will try to follow the SRT consistent fashion in terms of matrix calculus given in my preprint SRTplusGRAVITY03.pdf

Sorry for delay, but writing complex things in compact form needs time.

Stefano Quattrini > "that is to say that the speed of light is in general not constant in nature".

In that case you are denying the scientific validity of SR and also LTs, which are used as a tautology to justify abstract "spacetime", SR and GR and the whole of relativistic physics and cosmology! And what is more, you mainstream theoretical physicists are engaged is nothing but meaningless and endless scholasticism; I am sorry to say!

Stefano Quattrini ,

Before I continue, can you please comment on equation 1.3.11 in this article, as to why he has the gamma factor squared for the perpendicular (normal) acceleration?

http://kestrel.nmt.edu/~raymond/classes/ph321/notes/rel_dyn/rel_dyn.pdf

Abdul Malek ,

I am not a mainstream theoretical Physicst but I strongly appreciate the efforts of them, stressing limitations in the current theories.

that is yet another mistake..

@Eric Lord:

This is not a “paradox”!! It’s simply a recognition that the “Lorentz force law”, as stated in far too many textbooks and published articles, is an approximation, valid for a “sufficiently slow moving” charged particle. It is valid only in the limit (u/c)2

Frederick David Tombe "There is no way that electromagnetism is a relativistic theory to begin with. The so-called relativistic effects are an add-on to classical electromagnetism and only become significant at speeds close to the speed of light."

No. Relativistic effects are no add-on to classical electromagnetism. Nothing has to be changed in electrodynamics to obtain a relativistically correct theory, both at high and low velocities. This can easily be seen by rewriting the Maxwell equations in the form of four tensor and four vector expressions. They already have the correct form and transform covariantly under Lorentz transformations.

When Einstein developed special relativity, he was faced with the dilemma that the two big classical theories of the time, classical mechanics and electrodynamics did not get together well, because the relativity principle of classical mechanics held under Galilean transformations, whereas the only possibility to establish a relativity principle in electrodynamics was to take Lorentz transformations as the basic rule to get from one inertial system to another. Einstein had the choice to either abandon Newtonian mechanics or electrodynamics. He made the right choice to assume that electrodynamics was correct and Newtonian mechanics wrong. This then led to the Lorentz transformations and special relativity.

Your remark about the ether is immaterial. Lorentzian ether theory is empirically equivalent to special relativity, so it is a relativistically correct theory. It can be falsified experimentally only if special relativity is wrong, too. (One might consider it merely an alternative interpretation of special relativity.) However, it is somewhat against the spirit of relativity, because the relativity principle is only phenomenologically satisfied. An ether is introduced and then laws of physics are imposed that make any observation of the motion of the ether impossible... So if you wish to believe in Lorentzian ether theory, you are entitled to do so. It is just not the best theory according to Occam's razor, because it makes the same predictions as special relativity but introduces an entity "praeter necessitatem", beyond necessity -- the ether. Special relativity does not need that entity, so should be preferred.

Stefano Quattrini "The action and reaction principle is a sufficient condition for the conservation of momentum."

Not really. Let me explain the role of the principle in momentum conservation, because that is quite interesting.

If you have particles in an external field of force (say electrons in a static electric field), their momentum will not be conserved, even if their motion is so slow that the principle of action and reaction holds in the volume considered. So the action reaction principle is *not sufficient* for momentum conservation.

The law of momentum conservation can be stated as: If the sum of external forces on a body (system) vanishes, the total momentum of that body (system) remains constant. [In this form, we assume that the only way to disturb homogeneity of space is by external fields (otherwise -- and this would be the case in general relativity -- we have to require homogeneity of space explicitly for momentum conservation to hold).]

The reason we can restrict ourselves in the formulation to *external* forces, is precisely the principle of action and reaction, because it makes sure that the sum of internal forces is zero. If it does not hold, we simply have to replace "external" by "external and internal" in the statement of the law of momentum conservation (or we write "forces" without any specification).

Note that the validity of the principle implies that you need not know the internal forces, if you wish to calculate the change of total momentum under an applied force. This is peculiar to momentum balance (and angular momentum balance, where we have to require the internal forces to be central forces, so the action reaction principle is insufficient for the standard form of angular momentum conservation). On the other hand, if you wish to calculate the total *energy* of a system, it is not sufficient to know the kinetic energy of its pieces and their potential energy in an external field; you also must know the interaction energies of the pieces (typically described by pair potentials V(x1-x2)) -- they contribute to the total energy. There is no analog of actio=reactio for energy balance.

Stefano Quattrini > "I am not a mainstream theoretical Physicst"

I would not wish to bring the discussion to a personal level. I hope everybody (including me) in RG considers you as a (officially) mainstream physicist. If you claim otherwise, then by your own admission, it puts you in the unprincipled middle ground mediating (somewhat opportunistically perhaps) between two mutually exclusive positions! Good luck, my friend, it is your choice!

Dear @Stefano Quattrini

Dear @ Frederick David Tombe

The gamma^2 factor in front of the acceleration appears for considerations of covariant form in terms of proper velocities and accelerations. All this is noted with references in posts

No. 25. by @Eric Lord

No. No. 44 and 52 by me.

-- In this fashion the proper acceleration is the derivative of proper velocity wrt proper time tau

-- whereas the relativistic acceleration is the derivative of the proper velocity with respect to time t.

On the RHS

-- in the covariant form the force is a result of linear operation upon the proper 4-velocity possesses gamma factor (due to the proper velocity)

-- in the relativistic equation the force is a combination of E and B with the ordinary velocity (without gamma)

In the covariant form we have

γ^2 f+γ^4 u (u.f)/c^2 =γ q(E + u×B).

The relativistic form is as in Chapter 03

γ f+γ^3 u (u.f)/c^2 = q (E + u×B)

which both are equivalent, of course.

@Abdul Malek

Your writing to @Stefano Quattrini as follows

is unacceptable.

"Forbidding" anybody to do research for just partial acceptance of some structured knowledge,

and

"ordering" to accept the scientific structure either entirely or to reject entirely is nothing else than just an ugly mockery of the researcher.

@A. Kassner: In his 1905 pioneering article, Einstein begins with the claim that the then-usual representations of electromagnetism do not agree with the equivalence of all inertial reference systems. This claim is not correct. Nevertheless, he comes to the result that nowadays is considered to be correct. However, he assumes the reciprocity property without due justification. Otherwise, I agree with your comment.

@A. Kassner: Newton's 2nd and 3rd axioms indicate that momentum conservation occurs when the space outside a body (Lex II) or a system of bodies (Lex III) is like an empty space. Otherwise, I agree with your comment.

Joachim Domsta , Thank you very much for that clarification. But then, if we return again to equation 3.37 in this link,

http://web.mit.edu/course/22/22.09/ClassHandouts/Charged%20Particle%20Accel/CHAP03.PDF

as in, d(γmov)/dt = qv×B, it suggests to me that there is no gamma factor involved in the Lorentz force in the context where it is applied to a charged particle moving in a magnetic field. After all, it's not like the other context where we are studying the expanding magnetic field around a particle being that is being subjected to a linear acceleration, where the B field being induced by the accelerating particle is rising and hence absorbing energy.

I can see how a gamma factor would be justified in the latter case scenario, but not in the former. In the former, the external B field is not being affected. It is not absorbing any energy, and so a gamma factor wouldn't make any sense in this context.

Joachim Domsta : We are all used with your Inquisition, moral judgments, verdicts etc., Sir; so it is not something new! What is new is that this time I had absolutely nothing to do with you and you volunteered to take the responsibility on yourself for this moral indictment! The problem is; objective truth hurts, but only where, when and to whom it must!