One of them was expected and I posted about it previously (but see also here), but the other was completely unexpected.

Vladimir Hnizdo's paper, "Spin-orbit coupling and the conservation of angular momentum" (http://arxiv.org/abs/1103.4092) has now been published in European Journal of Physics (subscription or payment required). This paper cites two of my arxiv papers (0905.0927 and 0709.0319) and also acknowledges me as raising the issue the paper addresses.

The other citation is in a commissioned review article, "Resource Letter EM-1: Electromagnetic Momentum," (http://ajp.aapt.org/resource/1/ajpias/v80/i1/p7_s1) by David J. Griffiths, in American Journal of Physics. (There is apparently no arxiv e-print but the references may be found here. The citation of my paper is also noted here.) It appeared in January but I just found out about it last week. My paper is cited as one of three examples of ongoing research related to "hidden" momentum. My arxiv paper is in response to one of the other two articles, that was published in American Journal of Physics, that I commented on, and as I posted about previously, my comment was published in the journal, followed by a response by the original paper authors that seemed to negate my comment. Then, when I wrote a further response, it was rejected.

I don't mean to fault American Journal of Physics particularly for not publishing my paper that is now being cited by their own comissioned review article, as I understand they have many many submissions and as a pedagogical journal have to be averse to controversy. However I am very pleased that this paper is being cited and by such a distinguished author. It is especially nice because I had essentially given up on that paper and had no further plans to submit it anywhere, and yet it took a lot of time to do the work and write and I thought it obtained an interesting result. Now it seems reasonable to think I was not the only one to find the results interesting (and in fairness, two of three of the referees were positive about it).

I do hope and even expect that eventually I will be getting my work published in mainstream peer-reviewed journals, but until it is, getting cited in mainstream journals is a pretty good substitute. If it was a choice to get published but not cited, versus not published but cited by such distinguished authors, I would have to choose the latter.

## Saturday, March 10, 2012

## Saturday, March 3, 2012

### Status of my paper on the origin of the magnetic and strong forces

I posted my paper, "Does Thomas precession cause rotational pseudoforces in particle rest frames?" (http://arxiv.org/abs/1108.4343) in August 2011 and updated it in early September 2011. The update added the assessment of whether the kinematically-necessary anti-centrifugal force of the Thomas precession could plausibly correspond to the strong force that accounts for quarks combining to form nucleons. As I described in earlier posts, it has been submitted to several journals, but not peer-reviewed. Physica Scripta sent it out for review, but then withdrew it when the first referee objected because I called it a plausibilty argument. Since then I've been working on a more convincing, particularly more relativistically precise, argument that I hope the journals will consider more suitable for publication. It is taking quite a while to carry this out, but in the process I have found a number of flaws in the paper as currently posted that I want to acknowledge my awareness of.

Fixing the flaws in the paper will help make the next version more convincing, especially by removing the restriction of the current version to bound motion. The approach I'm taking is not just to fix the flaws, however. It is rather to present a completely rigorous electrodynamic analysis for general motion to order (v/c)^2 (and to (v/c)^4 in the case of circular bound motion), which is much more complicated than the minimally-relativistic treatment I posted on arxiv and previously submitted, and so is taking quite a while.

I'm still in the process of carrying out the analysis, and don't yet have a complete and convincing argument, but I have a few things I think are worth reporting prior to availibilty of a new version.

I also want to mention that I found some related prior work after posting the current arxiv versions, and there is a paper that appeared on arxiv shortly after mine, by Royer (http://arxiv.org/abs/1109.3624), that also describes the magnetic force as the anti-Coriolis force of the Thomas precession. One prior paper is by Bergstrom, "On the Origin of the Magnetic Field". Another is "New perspectives on the classical theory of motion, interaction and geometry of space-time," by A. R. Hadjesfandiari. (http://vixra.org/pdf/1011.0058v1.pdf)

My interpretation of all of these papers is that they have a different perspective than mine. They are concerned with the dynamics or kinematics of the particle experiencing a magnetic or general electrodynamic force and as mediated by a given electromagnetic field, whereas my approach is more oriented around a direct two-particle interaction, and the kinematical consequences of requiring physics to provide consistent descriptions from the point of view of various observers, in particular an observer moving with the particle that is the source of the field.

The Bergstrom paper, from the early 1970s, attempts to explain the magnetic force as a Coriolis force. The magnetic force however is properly an anti-Coriolis force (in that it accounts for the

Royer neatly avoids the problem of the what is the proper Thomas precession angular velocity, by doing the analysis directly with successive Lorentz transforms. That leads to the correct interpretation that the magnetic force is an anti-Coriolis force. However, Royer mentions that there is no anti-centrifugal force expected, as I agree for his approach. That is, there is no anti-centrifugal force in the electromagnetic field. The need for the anti-centrifugal force doesn't become obvious until one tries to describe an electrodynamic interaction in the rest reference frame of the source particle.

I found the Hadjesfandiari paper shortly after finishing the second (and currently-posted) version of my paper. When I realized that the Lorentz force must be incomplete, omitting as it does both anti-centrifugal and anti-Euler forces, I googled "Lorentz force incomplete" and turned it up. (It is from late 2010 and so is prior to mine being posted, although there is evidence on the web of me having the basic idea in the summer of 2008. That was when I noticed that the expected Coriolis force on a moving charge in the rest frame of a charged particle (of equal mass) has the same form and magnitude as the magnetic force.)

Now about my paper, the first thing I want to mention is that I now believe that the restriction I made to bound motion was based on an error of understanding and is unnecessary. It seemed at the time though to solve a problem I was having with getting the magnetic force to equate exactly to the anti-Coriolis force of the Thomas precession. The problem is that in the laboratory frame where (say) the center of mass of two interacting charged particles is stationary, the magnetic force on either particle depends only on the velocity of that particle relative to the local magnetic field. The source-particle velocity enters through the magnetic field, but not in the interaction of the other particle with the magnetic field. But, in the source-particle rest frame the expected (but absent) coriolis force is based on the relative velocity of the non-source particle to the source particle, which is equal to the velocity difference between the particles in the lab frame. This led to some stray terms that made the analysis seem incorrect, but I could see they would vanish for bound motion, so I decided to make that assumption in order to get a working paper. Now however I am fairly certain that those extra terms are just part of the electrodynamics in the lab frame. Showing this will be something that makes my whole thesis much more convincing, but I am still working out the details. Since there is of course no restriction to bound motion in the laws of electrodynamics, it isn't very good to have to make it for my argument that the magnetic force is an anti-Coriolis force of the Thomas precession.

Another flaw with the current version of my paper is that it fails to be sufficiently cognizant of the general necessity of including the anti-Euler force. I mention that there must be an anti-Euler force, and that there isn't one currently in electrodynamics (although it may correspond to the weak force), but I failed to take to heart that it needed to be included in the analysis unless the motion is restricted to zero radial motion between the particles. The next version will be at least cognizant of this fact, and possibly may include the anti-Euler force explictly to order (v/c)^2. I have tentatively obtained a description of it to order (v/c)^2, which I would like to publish as soon as possible. It's quite simple to describe to this order, although it has a very complex description when higher order terms are included. These high-order terms will become significant at nuclear scales, which opens up the possibility of the anti-Euler force accounting for the weak force, which also has a complicated description and is significant only at nuclear scale. However the anti-Euler force if I have things right also has a effect at order (v/c)^2 that I'm very eager to incorporate in my positronium atom model, as I have mentioned in at least one other post here. At order (v/c)^2 the anti-Euler force can be relevant at the atomic scale.

Still another problem with the current version is that it isn't sufficiently careful about keeping the order of the analysis consistent. This leads to a confusing and unconvincing handling of the centrifugal force. The centrifugal force is an order (v/c)^4 effect, while the magnetic force is a (v/c)^2 effect, so it would have been better to throw the former away sooner than to carry it along as long as I did. On the other hand, working the analysis at order (v/c)^4 is complicated but possible in the case of circular motion (where the delay equation can be solved exactly), and it becomes clear that the Lorentz force as we know it today does not account for the needed anti-centrifugal force. This is an analysis I hope to include in a future version, but it may not be the next. Getting everything to work out exactly at order (v/c)^4 has eluded me so far and I don't want to hold up the next version just for that.

Finally, I might mention that thanks to a friend I'm now aware of a pedagogical write-up on the Thomas precession (in use at UC Berkeley) that explicitly states there is no centrifugal force experienced by an observer in a Thomas-precessing frame, due to the Thomas precession (http://bohr.physics.berkeley.edu/classes/221/0708/notes/thomprec.pdf.) This answers the titular question of my paper in the negative. The question was really only meant to be rhetorical, in any case. I think it should be rather obvious that if the Thomas precession is to be a non-trivial effect, it must not give rise to rotational pseudoforces in the particle rest frame. The point of asking the question was that if there are indeed no rotational pseudoforces in Thomas-precessing particle rest frames, kinematics requires the presence of compensatory forces in the inertial laboratory frame. Seeing the statement explicitly in teaching materials should be justification for somebody to ask what are the kinematical implications of the absence of centrifugal forces. I believe they are profound and warrant further attention.

Fixing the flaws in the paper will help make the next version more convincing, especially by removing the restriction of the current version to bound motion. The approach I'm taking is not just to fix the flaws, however. It is rather to present a completely rigorous electrodynamic analysis for general motion to order (v/c)^2 (and to (v/c)^4 in the case of circular bound motion), which is much more complicated than the minimally-relativistic treatment I posted on arxiv and previously submitted, and so is taking quite a while.

I'm still in the process of carrying out the analysis, and don't yet have a complete and convincing argument, but I have a few things I think are worth reporting prior to availibilty of a new version.

I also want to mention that I found some related prior work after posting the current arxiv versions, and there is a paper that appeared on arxiv shortly after mine, by Royer (http://arxiv.org/abs/1109.3624), that also describes the magnetic force as the anti-Coriolis force of the Thomas precession. One prior paper is by Bergstrom, "On the Origin of the Magnetic Field". Another is "New perspectives on the classical theory of motion, interaction and geometry of space-time," by A. R. Hadjesfandiari. (http://vixra.org/pdf/1011.0058v1.pdf)

My interpretation of all of these papers is that they have a different perspective than mine. They are concerned with the dynamics or kinematics of the particle experiencing a magnetic or general electrodynamic force and as mediated by a given electromagnetic field, whereas my approach is more oriented around a direct two-particle interaction, and the kinematical consequences of requiring physics to provide consistent descriptions from the point of view of various observers, in particular an observer moving with the particle that is the source of the field.

The Bergstrom paper, from the early 1970s, attempts to explain the magnetic force as a Coriolis force. The magnetic force however is properly an anti-Coriolis force (in that it accounts for the

*absense*of a Coriolois force in the rest frame of the particle that is the source of the magnetic field). Bergstrom can't get to this result however because he is using the incorrect formula for the angular velocity of the Thomas precession, as in Moller and many subsequent textbooks, which leads to a sign error. This also hampered me for about three years, until I realized the sign had to be wrong, and then immediately after remembered that Malykin had said exactly that in his review paper (cited in my paper).Royer neatly avoids the problem of the what is the proper Thomas precession angular velocity, by doing the analysis directly with successive Lorentz transforms. That leads to the correct interpretation that the magnetic force is an anti-Coriolis force. However, Royer mentions that there is no anti-centrifugal force expected, as I agree for his approach. That is, there is no anti-centrifugal force in the electromagnetic field. The need for the anti-centrifugal force doesn't become obvious until one tries to describe an electrodynamic interaction in the rest reference frame of the source particle.

I found the Hadjesfandiari paper shortly after finishing the second (and currently-posted) version of my paper. When I realized that the Lorentz force must be incomplete, omitting as it does both anti-centrifugal and anti-Euler forces, I googled "Lorentz force incomplete" and turned it up. (It is from late 2010 and so is prior to mine being posted, although there is evidence on the web of me having the basic idea in the summer of 2008. That was when I noticed that the expected Coriolis force on a moving charge in the rest frame of a charged particle (of equal mass) has the same form and magnitude as the magnetic force.)

Now about my paper, the first thing I want to mention is that I now believe that the restriction I made to bound motion was based on an error of understanding and is unnecessary. It seemed at the time though to solve a problem I was having with getting the magnetic force to equate exactly to the anti-Coriolis force of the Thomas precession. The problem is that in the laboratory frame where (say) the center of mass of two interacting charged particles is stationary, the magnetic force on either particle depends only on the velocity of that particle relative to the local magnetic field. The source-particle velocity enters through the magnetic field, but not in the interaction of the other particle with the magnetic field. But, in the source-particle rest frame the expected (but absent) coriolis force is based on the relative velocity of the non-source particle to the source particle, which is equal to the velocity difference between the particles in the lab frame. This led to some stray terms that made the analysis seem incorrect, but I could see they would vanish for bound motion, so I decided to make that assumption in order to get a working paper. Now however I am fairly certain that those extra terms are just part of the electrodynamics in the lab frame. Showing this will be something that makes my whole thesis much more convincing, but I am still working out the details. Since there is of course no restriction to bound motion in the laws of electrodynamics, it isn't very good to have to make it for my argument that the magnetic force is an anti-Coriolis force of the Thomas precession.

Another flaw with the current version of my paper is that it fails to be sufficiently cognizant of the general necessity of including the anti-Euler force. I mention that there must be an anti-Euler force, and that there isn't one currently in electrodynamics (although it may correspond to the weak force), but I failed to take to heart that it needed to be included in the analysis unless the motion is restricted to zero radial motion between the particles. The next version will be at least cognizant of this fact, and possibly may include the anti-Euler force explictly to order (v/c)^2. I have tentatively obtained a description of it to order (v/c)^2, which I would like to publish as soon as possible. It's quite simple to describe to this order, although it has a very complex description when higher order terms are included. These high-order terms will become significant at nuclear scales, which opens up the possibility of the anti-Euler force accounting for the weak force, which also has a complicated description and is significant only at nuclear scale. However the anti-Euler force if I have things right also has a effect at order (v/c)^2 that I'm very eager to incorporate in my positronium atom model, as I have mentioned in at least one other post here. At order (v/c)^2 the anti-Euler force can be relevant at the atomic scale.

Still another problem with the current version is that it isn't sufficiently careful about keeping the order of the analysis consistent. This leads to a confusing and unconvincing handling of the centrifugal force. The centrifugal force is an order (v/c)^4 effect, while the magnetic force is a (v/c)^2 effect, so it would have been better to throw the former away sooner than to carry it along as long as I did. On the other hand, working the analysis at order (v/c)^4 is complicated but possible in the case of circular motion (where the delay equation can be solved exactly), and it becomes clear that the Lorentz force as we know it today does not account for the needed anti-centrifugal force. This is an analysis I hope to include in a future version, but it may not be the next. Getting everything to work out exactly at order (v/c)^4 has eluded me so far and I don't want to hold up the next version just for that.

Finally, I might mention that thanks to a friend I'm now aware of a pedagogical write-up on the Thomas precession (in use at UC Berkeley) that explicitly states there is no centrifugal force experienced by an observer in a Thomas-precessing frame, due to the Thomas precession (http://bohr.physics.berkeley.edu/classes/221/0708/notes/thomprec.pdf.) This answers the titular question of my paper in the negative. The question was really only meant to be rhetorical, in any case. I think it should be rather obvious that if the Thomas precession is to be a non-trivial effect, it must not give rise to rotational pseudoforces in the particle rest frame. The point of asking the question was that if there are indeed no rotational pseudoforces in Thomas-precessing particle rest frames, kinematics requires the presence of compensatory forces in the inertial laboratory frame. Seeing the statement explicitly in teaching materials should be justification for somebody to ask what are the kinematical implications of the absence of centrifugal forces. I believe they are profound and warrant further attention.

### Why the links no longer work

Many of the links to the literature on this blog's homepage no longer function, unfortunately, because of an apparent clamp-down by the original journals, or at least some of them, which forced the people operating the sites where they resided to take them down or behind a firewall.

I think it is especially unfortunate that the journal publishers are so touchy in the case of the sort of physics literature relevant to my work. It is often decades old and probably not of very wide interest.

I think it is especially unfortunate that the journal publishers are so touchy in the case of the sort of physics literature relevant to my work. It is often decades old and probably not of very wide interest.

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