Wednesday, November 13, 2013

A new version of my magnetic force paper on Arxiv

It's here.  It isn't the final version, but it has significant improvements compared to previous.  Section IIb is improved in the sense that there are no leftover terms in the magnetic force derived as a Coriolis effect of the relative rotation of the lab frame relative to the field source particle rest frame as seen by the test particle co-moving observer (TPCMO).  This is a result of having the correct sign on the Thomas precession as observed by the TPCMO, which is opposite of that seen by an inertial observer of an accelerated frame, as usually is provided in textbooks.  The explanation of how this happens is at the end of new Appendix A.

The new Appendix A also has a complete derivation of the Thomas precession using very elementary analysis that I hope is more transparent than other derivations, and may be unique in its own right.  I needed such a derivation because unlike other derivations that focus on the precession of a spinning particle, this one is focused on kinematics more generally, I'd say, and so obtains directly standard kinematical effects of rotation, such as that the velocity of a particle in a rotating frame is the velocity in the non-rotating plus an angular velocity of the rotation crossed with the radius vector to the particle from the center of rotation.  This is particularly important because it has been argued previously (by Bergstrom) that even though the magnetic force is clearly a Coriolis effect of the Thomas precession, it cannot give rise to an anticentrifugal forces because it applies only at a point and not more globally.  Bergstrom invents an interpretation that there is a "mosaic" of transformations between non-inertial and inertial reference frames such that the rotation applies only at the center of rotation, but I believe this interpretation is without real basis, and furthermore is disproved by the analysis in my Appendix A of version 7.  It seems pretty clear that the sole purpose of Bergstrom's interpretation is to avoid the otherwise obvious conclusion that if the Thomas precession causes a Coriolis effect as the magnetic force, then it must also cause a centrifugal-like force.  So, I believe this clears the way for a convincing relativistic argument that there need to be anti-centrifugal and anti-Euler forces.

I also used this update as an opportunity to introduce for the first time on arxiv the hypothesis that the anti-centrifugal force is the ultra-strong force that binds preons to from quarks.

The improvements to section IIb make it fully consistent with that part of the talk I gave at the PIERS conference last August.  Unfortunately due to confusion related to finding a sign error at the last minute and the deadline for the paper, they didn't get into the paper published in the conference proceedings. I discussed that sign error in at least one previous post.  Later on perhaps I will make a corrected version of that and post it on Reasearchgate.  The charts I gave as the talk for the PIERS conference are already posted there.  The talk also has an overview of the analysis that is now in Appendix A, but Appendix A is more advanced and more rigorous, in particular in how the partial derivative of time in the TPCMO's frame with respect to source particle rest frame time should be obtained.  The version in the talk gets the right result but the reasoning behind it is not quite right.  Getting it through a defensible derivation is a very significant improvement, I feel.

The path should now be clear to complete the analysis and obtain a relativistically exact (to order v^2/c^2) derivation of the magnetic force as a Coriolis effect of the Thomas precession.  This should also bring along an anti-Euler force of the Thomas precession, if one exists as I think necessary.  The anti-centrifugal force with be strongly implied, but can't be proven until the analysis is extended to order v^4/c^4.  But of course, as mentioned previously, it can already be found in Maxwell-Lorentz electrodynamics, if one knows where to look.

Saturday, October 12, 2013

The Magnetic Force as the Ultra-Strong Force that Binds Preons to form Quarks and Leptons

I want to make the point in this post, that although one could easily dismiss my contention that existence of the Thomas precession along with electrostatic forces implies existence of an anti-centrifugal force that can overcome electrostatic repulsion as speculative and unproven (and you'd be right), it is a different matter so far as the existence of strong magnetic force that can do the same is concerned.  Anyone with an undergraduate physics student's understanding of electrodynamics can see this for their self with a half-hour's worth of derivation.

To derive the interparticle separation, between two like charges, where the magnetic force will overcome electrostatic repulsion, while not leading to a mass for the bound composite that exceeds the proton mass, one can simply evaluate the magnetic part of the Lorentz force for a first relativistic charge moving in the magnetic field of a second relativistically-moving charge, that is also accelerating due to electrostatic forces due to the presence nearby of the first charge.  Everything needed is in a standard electrodynamics textbook such as Jackson or Griffiths, and on just a couple of pages (or on wikipedia, alternatively).

First, calculate the acceleration of a charge with arbitrary rest mass in the non-radiative electric field of a second nearby charge, as a function of the separation between the charges and their velocities.  This must be done using the proper relativistic forms for both the electric field, using the electric field derived from the Lienard-Wiechert potentials, and for the resulting acceleration due to the electric force, which must be based on the relativistic equivalent of Newton's law of inertia.  

Next, get the magnetic part of the radiative field due to the accelerated (second) charge, again using the Lienard-Wiechert field expressions.  Assume the second charge is moving at approximately (i.e., asymptotically close to) the speed of light perpendicularly to the direction of its acceleration.  This is consistent with

Sunday, October 6, 2013

The Preon Model as a Possible Application for Relativistic Kinematical Forces

A few days ago I read an article in the November 2012 Scientific American, by Don Lincoln of Fermilab, "The Inner Life of Quarks," that describes arguments that quarks and leptons are not themselves fundamental, but rather are made up of more fundamental objects named "preons."  The fact that leptons and quarks come in three known "families", with a hierarchy of increasing mass across them, and where the members of  heavier families decay rapidly into the equivalent members of the lightest family, suggests the more-massive families' members are just excited forms of the lightest. Since it is difficult to see how a truly fundamental object can have excited states, more fundamental constituents seem likely.

In the preon model described in the article, which is said to be only one of several that have been put forward, there are two different fundamental preons, termed the "plus (+)" and the "zero (0)", along with their antiparticles.  The plus and its antiparticle have electrical charge, while the zero and its antiparticle do not.
The gluons that are the carriers of the strong force are also composite objects made of the same set of preons.

Lincoln writes and illustrates that the preon model does a good job of representing the known hierarchy of the various families of quarks and leptons and their associated bosonic force particles.  But, according to the article, there is a fundamental problem with the preon model: because the masses of the leptons and quarks are already established, there is no room in it to accommodate the masses of force carrying particles needed to bind like-charged preons into the various charged quarks and leptons.  For example, gluons in the Standard Model account for most of the proton mass, while the up and down quarks are quite light comparatively.  Particles to bind preons with similar charges as quarks would need to be at least as massive as the gluons, and so would lead to larger quark masses than observed. This is where the anti-centrifugal force of the Thomas precession can help.  It provides a mechanism for overcoming electrostatic repulsion that does not require mediation by massive force-carrying particles.  It is simply a kinematical necessity of

Saturday, September 7, 2013

Slides from my talk at PIERS 2013 Conference

I have posted a pdf file of the slides from my PIERS 2013 talk, on Researchgate, here.

The material in the file is quite a bit beyond the version of the paper in the conference proceedings, and the slightly newer version (v6) currently posted on arxiv.  I'm working now towards completing a new version for arxiv that will provide the material outlined in the talk, and perhaps complete the relativistic proof to order v^2/c^2 that the magnetic force can be interpreted as a Coriolis effect of the Thomas precession.  In the meantime, though, the slides from the talk are to the best of my knowledge more correct than the arxiv version or the PIERS paper, where they differ.

Saturday, July 13, 2013

A New Version of My Magnetic Force Paper on Arxiv

The new version of my paper explaining the origin of the magnetic force as being a kinematical consequence of Thomas precession has been up for over a week now here. It's similar to the conference version I put on Researchgate and linked to previously, but it has two new appendices, an Errata section, and a change to the explanation of how the magnetic force is related to Thomas precession. 

The conference version of the paper was improved in various ways compared to the previous arxiv version (v5), and so the new posting on arxiv is much better than the last version, I think.  First, it fixes the glaring sign error I mentioned previously.  More importantly, it has a much better description of what is the expected anti-centrifugal force of the Thomas precession, that shows explicitly the inverse-cube dependence on interparticle separation (as necessary to overcome the inverse-square character of electrostatic repulsion), and then it shows how this prediction agrees with existing Maxwell-Lorentz electrodynamics in the case of bound circular motion, getting much better agreement than previously.  Previously I had an extra Lorentz (gamma) factor squared, which would certainly not be negligible given that it is an ultrarelativistic case where the anti-centrifugal or strong magnetic force becomes significant.

The new version, like the conference version, assumes that the correct form for the angular velocity of the Thomas precession, as observed from the laboratory frame, is as given by Jackson and most other authors, as opposed to the formula according Malykin (derived by Ritus, originally, references are in my paper on arxiv).  This was very nice in getting agreement in the ultrarelativistic case on the Lorentz factors, and for a brief while I thought it might also be working better in obtaining the magnetic force in the low velocity limit, but now I am having severe doubts.  Finding the sign error flustered and confused me into thinking I could solve problems with that part too using the Jackson formula, but as I think about it further I'm suspecting that Malykin is nonetheless correct.  It may be possible to reinterpret my anticentrifugal force to be consistent with Malykin, since Malykin does not say that the Jackson formula is wrong, merely that it applies to observations made from the accelerating reference frame rather than from the lab frame.

The Errata section retracts the conjecture of the last two versions that the expected magnetic-like force due to kinematical consequences of acceleration of the field-source charge, that I call a quasi-magnetic force, might account for the electron gyromagnetic ratio being (about) twice the classically-expected value.  The doubling of the strength of the spin-orbit coupling it predicts would only happen in positronium atoms, not in hydrogen atoms.  The Errata section also mentions my doubts about the correctness of using the Jackson Thomas precession angular velocity formula.  I hope to get it resolved pretty soon.  The fully relativistic derivation should answer the question.  I already have quite a lot of it done, and it seems to strongly support Malykin, still, to me.  I'm eager to return to working on it in earnest, but next I have to make the slides for my conference talk, and then give the talk.  When I have the slides I plan to make them generally accessible.

The new appendices together are an explicit demonstration of the relativistic-kinematical character of the magnetic force, and how it can always be related to a Coulomb force between two charges in at least one inertial reference frame.  This should not be controversial or any kind of surprise, but I have never seen this derivation in any textbook, so I thought it worth putting in.  It doesn't involve the Thomas precession explicitly.

Another thing perhaps worth mentioning: the conference version of the paper has some different expository content than the arxiv version.  The abstract and introduction in the conference version were written from scratch, and I didn't copy that content into the arxiv version.  I did shorten the introduction of the arxiv version, though, compared to previous, since it was saying a lot of things would be done in the paper that haven't been done yet.  I'll bring that part back when they are.

Thursday, June 20, 2013

My submittal to the PIERS Conference

PIERS is the Progress in Electromagnetics Research Symposium in Stockholm this August, where I am accepted to give a talk.  Acceptance for the talk is separate from publication of a paper in the proceedings, though, and I haven't yet received a decision about my submitted paper.  But, as I mentioned previously, I discovered a sign error that enabled a major re-write since the paper was submitted. These changes make the whole argument much more convincing, I believe.   Also, the current arxiv version (v5) is obsolete and it will be some time before I can get a revision up of that.  So, I have posted the revised conference paper as a dataset on Researchgate here

This version is the first where I could make the argument work using the equation for the angular velocity of the Thomas precession per the Jackson Classical Electrodynamics textbook, which is the more widely-accepted form compared to that proposed by Malykin, that I was forced to use previously.  

Thursday, June 13, 2013

Another sign error and more

There is a sign error in my arxiv paper, "The Magnetic Force as a Kinematical Consequence of the Thomas Precession," versions v4 and v5,  The sign error was apparently introduced when I changed notation slightly between versions 3 and 4, from referring to an electron and positron (or proton) using subscripts e and p, to a field source particle and a test particle using subscripts s and t.  Somehow the sign got switched so that now  Eq. (9) obviously does not follow from Eq. (8), and furthermore the Coulomb force per Eq. (9)  is repulsive for opposite charges. 

The sign was correct in previous versions and had been checked carefully, and the change of notation was not a big deal, so I apparently didn't feel a need to recheck it, unfortunately, because it is pretty obvious under even just a casual perusal.

The reason I'm finding it now, though, is because I got a different clue there might be a sign error in that part, due to the fact that I recently found that if I use the Jackson form for the angular velocity of the Thomas precession, rather than the Malykin form, I can successfully predict the form of what I'm calling the strong magnetic force, including all of the gamma (Lorentz) factors, as the anti-centrifugal force of the Thomas precession.  These gamma factors are highly significant in the highly relativistic case where the strong magnetic force manifests, and so the fact that I was missing them using the Malykin form but getting them all correct using the Jackson form is very significant, I believe.  Up until a few weeks ago, I had thought that the form of the strong magnetic force was supporting that Malykin is correct, so I was quite surprised to see it's apparently the other way around.

I've been using the Malykin form since the first version of this paper, because in my original derivation of the magnetic force, that was the only way it would work.  In the low velocity case it's just a sign change on the angular velocity of the Thomas precession, so finding the sign error after suspecting one appears to confirm that I should be using the Jackson equation (which is also the Moller form, and I believe is the only one consistent with the Bargmann-Michel-Telegdi equation that is experimtally verified by "g minus 2" experiments.  Malykin doesn't seem to see a contradiction there though so maybe I'm missing something about that.) 

My understanding has advanced tremendously (seems like to me anyhow) since my initial derivation, and so having the sign flipping there is only making everything make better sense, not causing new problems, I don't think.  I already knew that the original derivation was incorrect, which was what prompted the creation of version 4 in January of this year.  I'm now working on a new revision (6) that I'll post as soon as I can. 

There was one thing in particular that was troubling me with the derivation of versions 4 and 5, that the opposite sign resolves.  I'm hoping I'll feel good enough about the new version when it's done to want to re-submit this paper to a journal, finally.

Perhaps I should mention that in any case I'm planning to give a talk on it at the PIERS conference in Stockholm in August.  My abstract has been accepted.  I submitted a short version of the paper, with enhancements beyond the v5 posting, similar to what I posted here a few weeks ago, for possible publication in the conference preceedings, that's in peer review.  Unfortunately, that version has the glaring sign error.  I'm working frantically to try to get a revision done that I can send with apologies to the conference people.  I hope I'll have one after this weekend.  I've been very busy with my engineering job recently, including having to travel internationally, which has made it difficult to give my paper the time it deserves.

My attitude about this is that having the right answer is the only thing that matters in the long run, so I'm not letting it get me down.  Things are making better sense all the time.

Tuesday, May 14, 2013

A correction, and resolution of a problem

In the current version of my paper on arxiv,, as well as version 4, I argue that the strength of atomic (at least) spin-orbit coupling should be doubled compared to what is predicted according to Maxwellian electromagnetism, and apart from the electron g-factor being about twice the classically expected value.  This caused me to suspect that the doubling of the g-factor could be a mistaken interpretation of the increased strength of the magnetic interaction expected when both interacting particles are free to accelerate. However, as I observe in version 5, the g-factor being closer to one than two is directly contradicted by highly precise "g minus two" experiments that measure the electron (and also muon) g-factor to sufficient precision to measure the g-factor anomaly (that is, the small deviation predicted by quantum electrodynamics of the  g-factor from the Dirac value of exactly 2).   Because these experiments utilize strong magnetic fields generated by electron currents in neutral wires, there is no doubing of the magnetic field strength expected according to the mechanism of my paper.

The resolution to this problem is to simply pay attention to what my own theory is saying.  In hydrogen or other atoms, the nucleus is much heavier than the electron and so the acceleration of the nucleus is much smaller than the acceleration of electron, and the additional magnetic interaction strength is reduced accordingly.  I had been thinking of the situation as seen from the electron rest frame, where the proton (in hydrogen, say) is relatively accelerating with the  same acceleration as that of the electron seen from an inertial frame, and thinking that this would cause a doubling, but on further reflection it is now clear (and should have been obvious) to me that the proton acceleration seen from the electron rest frame is only just how the usual magnetic field arises and can't cause a doubling.  In order for an actual doubling to occur, it would be necessary that the proton acceleration as seen from inertial frames be of the same magnitude as the electron acceleration seen from inertial frames.

I don't know why it took so long for this to become obvious to me but a couple of days ago it did and now I feel foolish.

I'll be updating my paper on arxiv to cover this and some other items including what I have already posted about regarding what is the expected form of the anti-centrifugal force of the Thomas precession.  It will probably be within a couple of weeks.

In the meantime, I can mention that based on this proper understanding, it is possible to say what should be the expected effect on the spin-orbit coupling strength due to the effect of both interaction particles being free to accelerate. In positronium, the spin-orbit coupling strength should indeed be doubled compared to that expected according to pure Maxwellian electromagnetics.  I haven't done any research yet into whether anyone has ever tried to measure the spin-orbit coupling strength in positronium, but I suspect it would be difficult. In hydrogen, on the other hand, there will be an additional magnetic interaction strength equal to the ratio of the electron to the proton masses.  This is about one part in 1836 (if memory serves) and so it might be within the realm of possibility for measurement. 

Saturday, May 4, 2013

An online discussion I'm having about my model of the magnetic force

It's here, but I re-opened it recently after a hiatus, since I recently figured a lot of things out I didn't previously understand, starting here.

I am "Eggs Ackley" there.  (Eggs Ackley is a cartoon character by R.  Crumb.)

Friday, April 12, 2013

A better demonstration of the similarity of the anticentrifugal force to the strong magnetic force

The current version of my paper ( 1108.4343v5 ), like previous versions 2-4, has a section that attempts to characterize the anticentrifugal force of the Thomas precession in the highly relativistic limit, and to show that it can, like the strong force, overcome Coulomb repulsion as needed to bind quarks into nucleons.  At the time it was written, over a year ago, and until just a couple of weeks ago, I was thinking that the anticentrifugal force was not present in Maxwell-Lorentz electrodynamics, and said or implied as much in the earlier versions.  I didn't elaborate on this much, but I was thinking that the Maxwell fields didn't contain the strong force, and so although I stated or implied the Lorentz force was incomplete, I didn't expect that strong force could be added to electrodynamics by a modification of the Lorentz force law without an accompanying modification of the electromagnetic field.  Now of course I think this was a mistaken belief, and that the strong force is already apparently present in Maxwell-Lorentz electrodynamics as the magnetic force between highly relativistic mutually-Coulomb-accelerating charges.  This has resulted in an explicit form (if only approximate so far, due to my neglect so far of retardation effects, which cannot be considered insignificant here) of the strong magnetic force, which can be compared with my earlier characterization.  This comparison has forced me to realize the previous characterization is at best confusing and less than clear.

The problem of my initial characterization of the anticentrifugal force, which is in section V of the version at the link above, is that it obtains a force law that is inversely proportional to only the first power of the interparticle separation.  The Coulomb repulsion of course is inversely proportional to the square of the separation, so in order to overcome it, the anticentrifugal force should be inverse to a higher power of separation than two.  The strong magnetic force obtained in section VI is inverse to the third power of separtion, and so meets this expectation.  On the other hand, when I equated the anticentrifugal force magnitude with that of Coulomb repulsion, I got essentially the same formula that I got by equating the Coulomb repulsion with the strong magnetic force (and immediately declared success).  Naturally when I examined this situation further I was perplexed and at least a little disturbed by it.  I'm still in the process of sorting this out, but I think there's probably a straightforward explanation, that there's a hidden dependence on separation in the assumption of near light-like particle velocities, that can contribute additional inverse dependence on separation.  However, while looking into this, I realized there's an easier and I think more straightforward way to see the direct correspondence between the anticentrifugal force and the strong magnetic force.  I put this into a new draft version of my paper, but I don't want to do another update on arxiv just yet, pending addressing the issue of retardation, so I think I will copy it in here instead, for now.

The above equation  (303), derived as the anticentrifugal force, is essentially the same as Eq. (34) of my version 5 at the link above, that is derived from the Lienard-Wiechert potentials, if the test and source particles are of equal mass, apart from some gamma factors that still need to be sorted out carefully.   This shows more explicitly than the current arxiv version how the strong magnetic force is the embodiment of the anticentrifugal force of the Thomas precession.

Saturday, April 6, 2013

A. O. Barut

I want to mention that A. O. Barut argued that the strong force was plausibly related to or derivable from the magnetic force.  I have had this report for some time: Stable Particles as Building Blocks of Matter

Abstract:  Only absolutely stable indestructible particles can be truly elementary. A simple theory of matter based on the three constituents, proton, electron and neutrino (and their antiparticles), bound together by the ordinary magnetic forces is presented, which allows us to give an intuitive picture of all processes of high-energy physics, including strong and weak interactions, and make quantitative predictions.

Here is another I haven't downloaded exceptt for the free preview:

Derivation of strong and weak forces from magnetic interactions in quantum electrodynamics (QED)

Abstract: The principles of magnetic interactions between stable particles are outlined and a simple theory of matter is discussed based on absolutely stable particles proton, electron and neutrino as constituents. Experimental tests are proposed.

Tuesday, April 2, 2013

The Magnetic Force as the Strong Force

It's been over a year since I proposed that the anticentrifugal force of the Thomas precession might be identified with the strong force.  But, I thought it was something that would have to be added to electrodynamics, not already part of it.  Last week though I started thinking seriously that it needed to be in electrodynamics already, if the latter is truly Lorentz covariant, so over the weekend I looked for it and tentatively I seem to have found it.  At least, neglecting propagation delay effects (which cannot be considered insignificant so addressing them is a next step) I can show how the magnetic force between two charged particles can become attractive independent of the relative polarties of the particles  and so potentially overcome electrostatic repulsion between like charges.  When I solved for the particle separation where this would happen, I got the same result as for the anticentrifugal force.  This is in section VI of the new version of my paper, which is now publicly viewable here: .

This has developed quite abruptly and somewhat unexpectedly.  If it's meaningful, I should be able to find the anti-Euler force and another magnetic-like anti-Coriolis force for an accelerating field-source particle, so I will be looking for those.  I've looked previously for physically significant atomic-scale effects of the acceleration fields, though, with no success.  This time I'll be trying more persistently.

Wednesday, February 20, 2013

A little more about the electron g factor

I spent a little bit of time reading about the electron "g minus 2" experiments that are referred to in Jackson's Chapter 11, that are based on the BMT or Thomas's equation of motion for the spin.  These do appear to contradict my hypothesis that the electron g-factor being (approximately) 2 is a misunderstanding due to a failure to recognize that the magnetic interaction strength doubles when it is between two equal-mass free particles, compared to that given by Maxwell-Lorentz electrodynamics.  When I make my next revision to my paper I will at least mention this fact.  I may decide to de-emphasize the hypothesis that the doublling of the magnetic interaction strength can explain the electron non-unity g-factor, by removing mention of it from the abstract.

When I finish obtaining the main thrust of the paper, I will look into the matter further.  The problem here is (and I already mention this in the revision I posted on arxiv in January), that if it doesn't account for doubling the g-factor compared to classical expectations, the doubling I found would be an additional and unneeded factor of two. 

If this had been known in 1926 then there would have been no motivation for postulating the electron must have a non-unity g-factor, but now that the g-factor has been measured as being close to 2, it must be reconciled with the doubling of the magnetic interaction strength by other considerations.  There is almost room for such a reconcilation, since Bucher has shown that we should not expect the classical analog of the atomic L=1 quantum state to be a circular orbit.  This means that the Sommerfeld explanation of the anomalous fine structure could be accounting for the extra factor of two.  However, this would also require invoking the Thomas explanation for the (spin-orbit coupling) anomaly, which I believe is not correct.

I just want to register some awareness of the issue.  Right now I am focused on getting the fully relativistic version of my paper finished and reposted on arxiv, and then submitted to a journal.  It is coming along pretty well and perhaps I will be able to repost it within a month or so, or even less.

Monday, January 21, 2013

New Version of My Paper is Posted

It's uploaded but won't be publicly viewable until midnight Wednesday (23 Jan), at the earliest.  I'll try to either resist any immediate revisions or get them in before the point when it will delay the process.   When the link goes to version 4, with a 2013 date, that will be it.

I had hoped to have more relativity in this version, but I'm having to hold off on the more rigorous argument, for the moment.  I hope it will come along fairly quickly, and then I will do an arxiv update again and submit to a journal.  I don't plan to submit this one.  Apart from it probably needing a lot of writing improvements, I have at least one major section in work to add, that I hope will make the entire thing much more convincing.

The specific argument that the magnetic force is the anti-Coriolis force predicted for the lab frame by the test-particle rest frame obserever, who sees the lab frame Thomas precessing relative to the source particle rest frame (where the interaction is purely Coulombic) , is only a couple of days old at this point.  Previous arguments have always resulted in additional cross-product terms that I tried to subsume into the relativistic electrodynamics, with limited if any success.  This new argument yields explicitly just the magnetic force.  I also used to think (and say) that the anti-Euler force had to be part of the magnetic force, and was involved in the hypothesized subsumation of the extra terms.  That now seems unrealistic, and instead I now think the anti-Euler force part at order v^2/c^2 is a new force.  I guess that generally the anti-Euler force at higher order in v/c corresponds to the weak force, but I think that the piece at order v^2/c^2 may be overlooked until now.  The form of it is explicitly provided in this new version.