What if there was a merry-go-round without inertial forces?

[Eggs Ackley]

Imagine if you saw children playing on a merry-go-round, but that they weren't experiencing any centrifugal or Coriolis forces due to the rotation. If they toss a ball back and forth, the ball goes in a straight line path in a coordinate system that is rotating with the merry-go-round. As the merry-go-round spins faster and faster, the children continue to play without being thrown off by what they would normally experience as a centrifugal force. What would we be forced to conclude?

Watching from the park bench, we would be "forced" (ha ha) to conclude that there are forces acting on the merry-go-round that are countering the missing inertial forces. For example, it is a centripetal force that keeps a body in a closed orbit, be it gravity in the case of a satellite or the tension in a string holding a yo-yo. There would have to be a sort of centripetal force present on the merry-go-round preventing the children from being flung off.

It turns out, there is a relativistic effect that has been known to exist for almost 100 years, and is a certain consequence of the mathematics of relativity, that does exactly what our magic merry-go-round does. This effect is called the Thomas precession. It happens to reference frames that are simultaneously translating and cross-wise accelerating relative to an inertial reference frame. These frames must rotate, but if the idea of a reference frame rotating is to have any non-trivial meaning, it seems to me it must mean an absense of rotational pseudoforces due to the rotation. It is a most fantastic thing that is apparently being almost completely overlooked. I think, the missing Coriolis force is what we see as the magnetic force.

It is not very difficult to show that the anti-Coriolis force of the Thomas precession can be of the same magnitude as the magnetic force, if the interacting particles have the same rest masses. It is not exactly the same however, even then, except in certain restrictive cases. Perhaps our description of the magnetic force is incomplete.

The correspondent of the missing centrifugal force, on the other hand, is not to be found in electrodynamics at all. Yet, it arguably belongs there as much as the magnetic force. It acts only in very extreme situations, however. While the magnetic force is a "beta squared" (where beta is the fraction of the speed of light with which the interacting particles are traveling) correction to the electrostatic force, the anti-centrifugal force is a beta^4 correction. So it is entirely negligible in electrodynamics down to the atomic and even nuclear scale. But if the quarks moving in a proton were traveling at the speed of light, and electrostatically repelled by their electric charges, then it would begin to become significant. Since it would be opposite the centrifugal force, it would always be attractive, even though the quarks were being repelled electrostatically. This begins to sound like the strong force.

In turns out to be essentially beyond the state of the art to calculate exactly what the classical electrodynamics of motion at this scale is. (There is some other theory, I forget what it is called at the moment, usually used for this.) I don't possess state-of-the art electrodynamics skills anyhow. I could only do a very rough calculation, which obtained that for a particle of the observed mass of the proton and the quarks moving at the speed of light, the anticentrifugal force of the Thomas precession equals Coulomb repulsion at one one-hundredth of the proton measured radius. But it is also an established fact that the exact relativistic electric force will actually fall off a bit at this scale, meaning this is only a lower bound on the proton size. So I am sitting here thinking I probably have an explanation for the strong force as not a separate fundamental force, but rather a natural consequence of electrodynamics and relativity.

I wrote this up and sent it to three physics journals so far, but have been unable to get it reviewed. I gave them an out I guess by calling it just a plausibility argument. I have a stronger argument in work that I hope will be ready by year end, that I will argue makes the magnetic force correspondence, at least, air tight. The strong force willl have to be addressed by others, probably, or a much later date if by me. But I think it's quite easy to see that covariance (that is, the relativistic requirement that we be able to write a description of the same physics in all reference frames) demands these types of forces (and an anti-Euler force as well) so I'm disappointed that it isn't being better received. I will welcome comments from physicists and non-physicists alike, but, especially from the non-physicists or physicist wanabees such as myself. Doesn't this idea seem attractive at all? Doesn't any plausible explanation for so important a thing as the strong force deserve at least a review?

 

[This is The End]

Did this seriously get to page 2 without anyone letting Eggs know just how correct it is?

 

[Eggs Ackley]

Thanks for the kind words, OnlyTellsTruths, I appreciate them a lot.

Sorry to be remiss in not posting a link to the actual analysis in question. Here it is: http://arxiv.org/abs/1108.4343

Also, I don't want to run the physicists off. I would like to ask them some questions: 1) Do you think my interpretation of the Thomas precession as not giving rise to rotational inertial forces is correct? 2) If not correct, then what is the meaning of it to the observer in the Thomas precessing reference frame? and 3) If there are no rotational pseudoforces in the Thomas precessing frame, then do you agree that there must be corresponding anti forces (ant-Coriolis, anti-Euler, and anticentrifugal) in the inertial frame?

I should probably also mention that of course the merry-go-round in this case is not actually a hospitable place for the children to play. It is being accelerated tremendously, about 10^14 gravities at the atomic scale and far far higher at the sub-nuclear level. The inertial forces due to the linear acceleration are still felt on it. It is only the rotational psuedoforces that would be absent.

 

[sol invictus]

1) Do you think my interpretation of the Thomas precession as not giving rise to rotational inertial forces is correct?

I can't understand it, so I don't know.

2) If not correct, then what is the meaning of it to the observer in the Thomas precessing reference frame? and 3) If there are no rotational pseudoforces in the Thomas precessing frame, then do you agree that there must be corresponding anti forces (ant-Coriolis, anti-Euler, and anticentrifugal) in the inertial frame?

What do you mean?

I think you're talking about the rest frame of an object that continuously accelerates in some direction - is that right? If so, can you be very clear about exactly how that acceleration is defined? For instance, let's say you're in a rocket. You've got a main engine and some smaller engines that allow you to rotate. What do you do to stay at rest in the frame you're interested in?

I haven't a clue what you're talking about with respect to magnetic fields, or why you think this has anything whatsoever to do with them, but let's start with the above.

 

[meow]

any chance of you drawing a pic?

 

[JJM 777]

Imagine if you saw children playing on a merry-go-round, but that they weren't experiencing any centrifugal or Coriolis forces due to the rotation. If they toss a ball back and forth, the ball goes in a straight line path in a coordinate system that is rotating with the merry-go-round.

A metal ball and a strong magnet at center of the carousel? Or simply a gravity center that draws also the children at force equal to the centrifugal force.

 

[Eggs Ackley]

What do you mean?

I think you're talking about the rest frame of an object that continuously accelerates in some direction - is that right? If so, can you be very clear about exactly how that acceleration is defined?

I'm talking about the precession of a reference frame fixed to a body that is translating, and accelerating perpendicularly (say) to the direction of translation, relative to another reference frame that is an inertial reference frame. This is the Thomas precession. Surely this is not something you haven't heard of. Here is a wikipedia article:

http://en.wikipedia.org/wiki/Thomas_precession

and there is a good description of particle rest frames in Jackson Classical Electrodynamics chapter 11 or in the Malykin paper I cite in my paper I linked above.

So then, if we have a Thomas-precessing reference frame, attached to a particle or rocketship or nothing at all, we need to be able to write the laws of physics, let's comsider just the laws of electrodynamics, in it. It is an allowed frame of reference, although not an inertial one, so it is not unreasonable to ask what does electrodynamics look like to an observer who is stationary in such a reference frame.

It is a more general question but I'm considering it at the atomic scale or smaller here. I'm not thinking quantum mechanically, however; I'm thinking of a classical Rutherfordian hydrogen atom model (say) with a point-charge electron. We can call the reference frame where the center of mass of the atom is at rest the laboratory frame, then the Thomas-precessing frame is one in which the electron is at rest. However, this frame is obtained from the lab frame by only pure Lorentz boosts. Start in the lab frame and boost to a frame where the electron is momentartily at rest, then continue applying a sequence of infinitesimal boosts as needed to maintain the electron stationary in the sequence of these inertial frames.

After the first infinitesimal boost, if we want to transform back to the laboratory frame we will have to apply a rotation as well as a boost, and time variation of the rotation needed is the angular velocity of the Thomas precession.

Jackson makes the careful distinction that the rest frame defined this way is actually a sequence of inertial frames. In the rest frame sequence as defined this way, the electron is always instantaneously accelerating, but never acquires a nonzero velocity. As Jackson observes, this isn't a convenient frame to write equations of motion in, so it is convenient to define also a non-inertial reference frame where the electron is both stationary and non-accelerating. This frame is determined at each point in time by rotating the inertial rest frame by an amount based on the angular velocity of the Thomas precession and the elapsed time from an arbitrary reference time when we can have performed the initial boost. Strictly, this is the Fermi-Walker frame and is the frame that I am referring to. Other descriptions of it are found in for example Munoz, "Spin–orbit interaction and the Thomas precession: A comment on the lab frame point of view," Am. J. Phys. 69(5), May 2001, and Schild and Schlosser, "Fokker Action Principle for Particles with Charge, Spin, and Magnetic Moment," JMP 6(8), 1965, and many other sources.

If I can simply say that I am talking about Fermi-Walker frames, then what I am proposing is that if I am to write down the equations of electrodynamics is such frames, I should not have to add pseudoforces of rotation (that is, Coriolis, centrifugal, and Euler forces) as I would ordinarily in a frame that is rotating relative to an inertial frame.

I haven't a clue what you're talking about with respect to magnetic fields, or why you think this has anything whatsoever to do with them, but let's start with the above.

I might mention Larmor's theorem and the formal similarity of the magnetic force to a Coriolis force, but I came upon it in passing when I was trying to write the equations of classical motion of two interacting point charges in the Fermi-Walker frame of one of them. I was doing this as a proficiency test of my understanding of the problem of the classical hydrogen atom with an electron with intrinsic spin and intrinsic magnetic moment.

I needed to get descriptions in both the Fermi-Walker and lab frames and confirm they were consistent, because I wanted to be able to build a classical positronium atom model where the electron and positron both have intrinsic spins. Initially though I was neglecting the intrinsic spins and intrinsic magnetic moments, in order to simplify the problem. When I sat down to write the equation of motion of the positron in the Fermi-Walker frame of the electron, I initially did put in explicitly the Coriolis force I expected to be required to describe the motion of the positron. To my surprise this term was exactly the same as the magnetic force term when I wrote the equation of motion in the lab frame.

I felt like I had uncovered something important, but after a day or so I realized that something had to be wrong because a Coriolis force lives only in the rotating frame and has to vanish in the inertial frame. I kept thinking about it however because it seemed extremely unlikely to me that the magnetic force magnitude would equal the expected magnitude of the Coriolis force of the Thomas precession by mere coincidence.

That was over three years ago. I can post a link to where I tried to discuss it on phyorg. It took about a year before it dawned on me that a lack of a Coriolis force in the Thomas-precessing Fermi-Walker frame would have to manifest as an actual force in the inertial lab frame. So then having figured it out, I thought, it would be easy to show how it works. But it took another two years, working off and on, before I could get it to work out.

I am still not sure if I was just interpreting the angular velocity equation in Jackson incorrectly, or whether it is stated incorrectly, but I had a sign error that caused me to end up with a too-strong magnetic force in the inertial frame if I assumed no Coriolis force in the Fermi-Walker frame. I eventually realized I had to change the sign, and right after that I remembered that Malykin had said the same thing (that the sign in Jackson is wrong), and so I used Malykin's form of the angular velocity of the Thomas precession and got that the magnetic force could be equated with an anti-Coriolis force of the Thomas precession, as I think I show in my paper.

(Strictly speaking, what I try to show is that the lab frame equations of motion for the classical electromagnetic two-body problem without spin are equivalent to a force in the Fermi-Walker frame that is strictly radial. I realize that in the nonrelativistic treatment I use that the argument is short of airtight, and that's why I said it's only a plausibilty argument for the magnetic force being the anti-Coriolis force of the Thomas precession. But it is at least a quantitative plausibilty argument. However as I have already mention I have a relativistic version that retains all relatiivistic terms including delay-related ones to order (v/c)^2 that I hope to make public in a month or so. It was actually essentially finished when I did the short version as previously linked, but I was having trouble interpreting it. I was not expecting the additional dynamical restriction beyond the need for equal particle masses. I thought it was just not working out. So I tried the simplified version and realized that there is apparently an additional dynamical restriction that the two particle velocities be equal magnitude and opposite direction. Certainly I was not happy with this restriction but it seems that it is demanded by relativistic covariance if the magnetic part of the Lorentz force is to be as usually accepted. But since such a restriction can't be enforced generally I think the implication is that the magnetic force must be more complex than now recognized. We also need to account for the anticentrifugal and anti-Euler forces that covariance demands.

Well I hope that helps. I can explain a lot further and I am even willing to try your rocketship question but for the time being I am having very little free time to think about it in a new way. However another way to think about it that I have already figured out is how the anticentrifugal force looks to the children on the magic inertialess merry-go-round. They also think there is an attractive force present. They see the people standing in the park as orbiting around them, and can even calculate (if they are exceptionally smart children) what is the force law needed to hold the people in orbit, just as we can calculate the gravitational force law from the orbital motion of the moon.

 

[psionl0]

Imagine if you saw children playing on a merry-go-round, but that they weren't experiencing any centrifugal or Coriolis forces due to the rotation. If they toss a ball back and forth, the ball goes in a straight line path in a coordinate system that is rotating with the merry-go-round. As the merry-go-round spins faster and faster, the children continue to play without being thrown off by what they would normally experience as a centrifugal force. What would we be forced to conclude?

This reminds me of a "what if" that my physics teacher put to the class once:

You are floating around in space and go to sleep. When you wake up you find that the rock that was near you is now orbiting you in a near perfect circle. Curious, you grab the rock as it goes past your hands only to find the rock pulling away from you. Startled, you release the rock and see it moving away from you in ever increasing spirals.

How could that be?​

As it turns out, the answer is that while you were sleeping, somebody came around and started spinning your body. A similar answer might also apply to the merry go round example.

Centrifugal force, coriolis force and precession are all Newtonian phenomena. The only time you would need to bring relativity into it is if you were trying to explain magnetic forces.

 

[sol invictus]

Jackson makes the careful distinction that the rest frame defined this way is actually a sequence of inertial frames. In the rest frame sequence as defined this way, the electron is always instantaneously accelerating, but never acquires a nonzero velocity.

Jackson says nothing of the kind. How can something be accelerating from rest and not acquire a velocity?

Instead, Jackson observes that the frame defined by boosting to the rest frame from the lab frame is rotated with respect to the frame defined by boosting from the previous instantaneous rest frame to the current one, but it's the latter procedure that makes the equations look simplest.

If I can simply say that I am talking about Fermi-Walker frames

To be very clear - are you talking about the frames defined by Jackson 11.118?

then what I am proposing is that if I am to write down the equations of electrodynamics is such frames, I should not have to add pseudoforces of rotation (that is, Coriolis, centrifugal, and Euler forces) as I would ordinarily in a frame that is rotating relative to an inertial frame.

Nonsense. That's not an inertial frame, so of course there will be pseudoforces. Without doing a calculation to be sure, for a circular motion the origin of that frame should just orbit around the origin of the lab frame, while at the same time precessing. Obviously there are pseudo-forces in such a frame. Whatever gave you the idea there aren't??

If we are talking about non-relativistic velocities v<<c, we should be able to forget about Thomas precession entirely. In that case the origin of either of these frames (Thomas precessing or not) is orbiting around the origin on the lab frame, but the axes remain perfectly parallel. The pseudoforces are manifestly not zero and are quite different than in a rotating frame, but in any case are trivial to calculate.

When we get v~c, there's this additional relativistic effect that differentiates the frame that Thomas precesses from the frame that doesn't. My intuition is that the difference is that the frame that Thomas precesses has simpler inertial forces, I suspect without any velocity dependent ones (i.e. no Coriolis-type force). But I'd have to check to be sure (it's not hard).

 

[Eggs Ackley]

Jackson says nothing of the kind. How can something be accelerating from rest and not acquire a velocity?

Jackson says (page 543 of 2nd edition),"The electron's rest frame of coordinates is defined as a co-moving sequence of inertial frames whose successive origins move at each instant with the velocity of the electron."

So, 1) these are all inertial frames; and 2) the electron is momentarily stationary in all of them. Therefore the electron is accelerating in all of them, but our attention switches to the next frame in the sequence before it has a chance to move anywhere or a acquire a non-infinitesimal velocity. Or do you have a different interpretation of what Jackson is saying here?

Instead, Jackson observes that the frame defined by boosting to the rest frame from the lab frame is rotated with respect to the frame defined by boosting from the previous instantaneous rest frame to the current one, but it's the latter procedure that makes the equations look simplest.

Yes he does and he does also call that an electron rest frame. I was writing from memory and was thinking he did not also explicity refer to that as a rest frame. I like the way Munoz distinguishes these two frames as the boosted lab frame and the Fermi-Walker frame, respectively. This is how I think of them and in the relativistic version of my paper (not yet finished) this is the terminology I also use.

To be very clear - are you talking about the frames defined by Jackson 11.118?

Basically, but in my posted paper I am believing what Malykin says about Jackson's Eq. 11.119 (2nd Ed) which is that it is not correct for that frame. Malykin says that is the equation for the observed precession of the rest of the world from the Fermi-Walker frame, which I don't think is how Jackson intends it. This whole issue is obviated when I treat the problem more relativistically, because it isn't necessary to put in the Thomas precession angular velocity explicitly in that case, to still obtain the result that the magnetic part of the Lorentz force corresponds to the lack of Coriolis force in the Fermi-Walker frame.

Nonsense. That's not an inertial frame, so of course there will be pseudoforces.

I am careful to say, at least most of the time, that I am only talking about the rotational pseudoforces, and not all of the inertial forces. The inertial forces due to linear acceleration must still be taken account of, as they are in my paper.

Without doing a calculation to be sure, for a circular motion the origin of that frame should just orbit around the origin of the lab frame, while at the same time precessing. Obviously there are pseudo-forces in such a frame. Whatever gave you the idea there aren't??

If we are talking about non-relativistic velocities v<<c, we should be able to forget about Thomas precession entirely. In that case the origin of either of these frames (Thomas precessing or not) is orbiting around the origin on the lab frame, but the axes remain perfectly parallel. The pseudoforces are manifestly not zero and are quite different than in a rotating frame, but in any case are trivial to calculate.

When we get v~c, there's this additional relativistic effect that differentiates the frame that Thomas precesses from the frame that doesn't. My intuition is that the difference is that the frame that Thomas precesses has simpler inertial forces, I suspect without any velocity dependent ones (i.e. no Coriolis-type force). But I'd have to check to be sure (it's not hard).

So you are almost agreeing with me anyhow. In which case it is only necessary to work out what are the consequences. If you accept that there are not Coriolis-type of forces, then do you not think there will also be an absence of a centrifugal force? An Euler force absence too? How would these absences be viewed from the lab frame?

I do disagree about Thomas precession being of very limited consequence at the atomic scale, as well. I think it is much more important than is currently thought. If you start doing these quite simple calculations, I think you may well get a clue before long that there is a lot of interest here that has been previously overlooked. Then perhaps you will share my burden.

 

[Eggs Ackley]

If we are talking about non-relativistic velocities v<<c, we should be able to forget about Thomas precession entirely. In that case the origin of either of these frames (Thomas precessing or not) is orbiting around the origin on the lab frame, but the axes remain perfectly parallel.

This is not consistent with my understanding. At the atomic scale, it's true, the Thomas precession frequency is much smaller than the orbital frequency of, say, the Bohr model ground state in hydrogen. So they don't remain perfectly parallel. It's a small change over an orbit, I agree, but after a lot of orbits, the rest frame will eventually make a whole revolution.

 

[sol invictus]

So, 1) these are all inertial frames; and 2) the electron is momentarily stationary in all of them. Therefore the electron is accelerating in all of them, but our attention switches to the next frame in the sequence before it has a chance to move anywhere or a acquire a non-infinitesimal velocity. Or do you have a different interpretation of what Jackson is saying here?

That's fine.

Yes he does and he does also call that an electron rest frame. I was writing from memory and was thinking he did not also explicity refer to that as a rest frame. I like the way Munoz distinguishes these two frames as the boosted lab frame and the Fermi-Walker frame, respectively. This is how I think of them and in the relativistic version of my paper (not yet finished) this is the terminology I also use.

They're both rest frames. The difference is whether or not they rotate around the electron.

I am careful to say, at least most of the time, that I am only talking about the rotational pseudoforces, and not all of the inertial forces. The inertial forces due to linear acceleration must still be taken account of, as they are in my paper.

I don't know what you mean. The acceleration is not linear in any case. My hunch is that in the non-relativistic case, there is a pseudoforce that's of constant magnitude and going around in a circle. Once relativistic effects are included, there either is or isn't an extra Coriolis-type piece.

So you are almost agreeing with me anyhow. In which case it is only necessary to work out what are the consequences. If you accept that there are not Coriolis-type of forces, then do you not think there will also be an absence of a centrifugal force? An Euler force absence too? How would these absences be viewed from the lab frame?

What are you talking about? This is all totally trivial. There are whatever forces there are to keep everything consistent, and it's very simple to work out. How can you possibly have written a paper about this and not know the answer? Again, in the NR case, I think it's just a force that spins around in a circle, with constant magnitude and independent of position. That's not much like the forces in a rotating frame, but so what?

This is not consistent with my understanding. At the atomic scale, it's true, the Thomas precession frequency is much smaller than the orbital frequency of, say, the Bohr model ground state in hydrogen. So they don't remain perfectly parallel. It's a small change over an orbit, I agree, but after a lot of orbits, the rest frame will eventually make a whole revolution.

Huh? Thomas precession is a relativistic effect. Take v/c->0, and it goes away. That means for NR velocities these two frames are essentially identical, so just use the non-rotating version. Then it's just a frame that orbits the origin of the lab frame, with the obvious pseudoforces.

 

[sol invictus]

There would have to be a sort of centripetal force present on the merry-go-round preventing the children from being flung off.

It turns out, there is a relativistic effect that has been known to exist for almost 100 years, and is a certain consequence of the mathematics of relativity, that does exactly what our magic merry-go-round does.

This is totally wrong. Physics is independent of coordinate choice. If the children have to hang on in one frame (for instance, the lab frame) they obviously have to hang on in every other frame. You never need to change frames at all, you can simply use the lab frame.

This effect is called the Thomas precession. It happens to reference frames that are simultaneously translating and cross-wise accelerating relative to an inertial reference frame. These frames must rotate, but if the idea of a reference frame rotating is to have any non-trivial meaning, it seems to me it must mean an absense of rotational pseudoforces due to the rotation. It is a most fantastic thing that is apparently being almost completely overlooked. I think, the missing Coriolis force is what we see as the magnetic force.

Complete nonsense. There is a magnetic force in every frame. Not only that, magnetic forces only act on charged particles, with a strength proportional to their charge. They cannot have anything at all to do with pseudoforces, which act on everything with a strength proportional to mass.

Doesn't this idea seem attractive at all? Doesn't any plausible explanation for so important a thing as the strong force deserve at least a review?

Not in the slightest - it's patent nonsense.

 

[Eggs Ackley]

Sorry for the long gap between replies. At about the time of the last posting, I realized there were some substantive errors in my argument, and so felt the need to fix it if possible prior to further discussion. I was successful at fixing the things that were incorrect about the last version linked (which was actually arxiv version 2) and posted a revised version in April. It has an errata section that describes the problems. These were things that had been bothering me but I couldn't quite put my finger on them in the initial versions from last fall. Then I realized there were two different major issues that needed to be addressed. One was, anytime there is a change in the rate of rotation of the merry-go-round, there is the presence of the Euler pseudoforce, which I was neglecting in the initial versions, although I mentioned it. There is only one case where it is properly neglected in the classical electrodynamic two-body problem, and that is for bound circular motion, where the angular velocity of the Thomas precession is unvarying. That was the case I did originally, anyhow, but I was expecting the anti-Coriolis to magnetic force correspendence to be more general, and it was reflected in the point of view. When I realized the anti-Euler force is essential in all other cases, it made the whole problem make more sense.

The second problem was my failure to recognize that the magnetic force is only part of the anti-Coriolis force. In fact the anti-Coriolis force contains the complete magnetic force (which was how I recognized the correspondence), but it contains other terms as well. Basically, the problem is that the expected Coriolis force in the electron rest frame is the cross product of the angular velocity (of the Thomas precession) with the velocity of the proton in the electron rest frame, but the proton velocity in the electron rest frame is not the same as the proton velocity in the center-of-mass frame. It gets an added component due to the electron velocity. So there is an extra vector triple product term with the electron velocity appearing twice, compared to the vector triple product that corresponds directly to the magnetic force. This extra vector triple product needs either to vanish somehow or be absorbed somehow into the exact relativistic dynamics. I was partially successful with this in the version I posted in April, but today I have tentatively resolved the whole issue.

A vector triple product can be written as the sum of two terms that each involve a dot product of two of the vectors, times one of the vectors in the inner cross product. I realized that one of these can create an inverse Lorentz factor (i.e., a relativistic "gamma") squared factor that is already found in the exact relativistic equations. The other half of the vector triple product vanishes for bound circular motion, which allowed me to handle that case completely, if somewhat notionally, in the April version, to which I now link: http://arxiv.org/abs/1108.4343v3 . (The un-magnetic-like triple product is the right-most term of Eq. 19 and its factorization and reduction occur in Eqs 20 and 21).

I posted that version because it improves over the previous but I didn't resubmit for publication for two reasons. One was that the argument can be dismissed as not relativistic enough, in that an exact relativistic treatment has various terms of order (v/c)^2 that could potentially ruin or obviate the correspondence. The second reason is that I felt I needed to show a more general correspondence than just for bound circular motion in order to have a convincing argument. So I have been working on both of these tasks since I posted the April version. I got the exact (to order (v/c)^2 relativistic version of the circular motion case to work about a month ago. I had been working on the more general motion case concurrently but unable to get it to work out, until today, when I found a mistake that had introduced extra unwanted force terms. Now I am in a position to explain provisionally how the magnetic force can arise generally, as part of the anti-Coriolis force of the Thomas precession. In the case of non-circular motion so that the last term on my Eq 20 cannot be assumed to vanish, it will be be cancelled by the then-non-vanishing anti-Euler force. Also, the other terms at order (v/c)^2 introduced by the anti-Euler force are subsumed into the standard relativistic description derived from the Lienard-Wiechert potentials and properly accounting for delay.

Sol Invictus, I am aware of the conceptual difficulty of equating a pseudo-force, that depends on the mass of body being acted on, to an actual force, that does not. My equations only work when the "proton" mass is the same as the electron's, for this reason. I've been aware of the problem since I first noticed that the Coriolis force in the electron rest frame equates, apart from a sign, to the magnetic force. It's a strange thing, but I believe it has to have significance, because it is fair to demand that our physical description be consistent from the point of view of all observers. On the one hand, if there are no pseudoforces of rotation in Thomas precessing frames, then simple kinematics requires a compensating force in inertial frames. On the other hand, I don't believe that an assumption of present pseudoforces in Thomas-precessing frames can lead to a consistent picture, for reasons that are perhaps even more involved than what I have tried probably unsuccessfully to explain here, but I would be interested if you can do it. You did say this is all trivial.

 

[quarky]

Here's a link to a you-tube i made that demonstrates a more complex merry-go-round:

http://www.youtube.com/watch?v=EVyuQchUdiE&feature=related

Its my invention; my farm; my old lady; my hillbilly neighbors.
Hopefully, it explains nothing.

 

[Eggs Ackley]

Cute.

Did you ever have a Marx-a-copter?

 

[Brian-M]

Imagine if you saw children playing on a merry-go-round, but that they weren't experiencing any centrifugal or Coriolis forces due to the rotation.

Sounds like a very boring ride.

If they toss a ball back and forth, the ball goes in a straight line path in a coordinate system that is rotating with the merry-go-round.

If they can throw a ball back and forth, that means inertia is still present.

As the merry-go-round spins faster and faster, the children continue to play without being thrown off by what they would normally experience as a centrifugal force. What would we be forced to conclude?

The simplest solution would be that the merry-go-round is stationary and we're the ones that are rotating around it.

Are we experiencing centrifugal forces as we watch the merry-go-round?

If not, there must be hidden forces at work precisely balancing the rotational forces on the merry-go-round to create a rotating environment that does not seem to be rotating to those within it

It turns out, there is a relativistic effect that has been known to exist for almost 100 years, and is a certain consequence of the mathematics of relativity, that does exactly what our magic merry-go-round does. This effect is called the Thomas precession. It happens to reference frames that are simultaneously translating and cross-wise accelerating relative to an inertial reference frame. These frames must rotate, but if the idea of a reference frame rotating is to have any non-trivial meaning, it seems to me it must mean an absense of rotational pseudoforces due to the rotation. It is a most fantastic thing that is apparently being almost completely overlooked. I think, the missing Coriolis force is what we see as the magnetic force.

Why are you looking for a real-life physical theory to explain a hypothetical scenario that does not occur in real-life?

 

[Eggs Ackley]

Why are you looking for a real-life physical theory to explain a hypothetical scenario that does not occur in real-life?

Because it seems to explain the origin of the magnetic force, and I suspect, the strong and electro-weak forces as well.

 

[Brian-M]

Because it seems to explain the origin of the magnetic force, and I suspect, the strong and electro-weak forces as well.

How so?

(You're not suggesting that the children circling the hub of the merry-go-round represent electrons circling a nucleus are you?)

 

[Eggs Ackley]

How so?

(You're not suggesting that the children circling the hub of the merry-go-round represent electrons circling a nucleus are you?)

The merry-go-round represents a reference frame that is moving with an elecron in an atom. This is called the electron rest frame (ERF). If the electron has orbital angular momentum, then the ERF rotates due to Thomas precession ( http://en.wikipedia.org/wiki/Thomas_precession ).

The children then represent other particles nearby the electron. There are inertial forces due to the tremendous acceleration but I believe not due to the rotation of the Thomas precession. When the lack of the Coriolis force in the ERF is transformed to inertial frames, the equation of motion of a particle experiencing magnetic force due to the electron motion results.