Trouble with the Lorentz law of force: Incompatibility with special relativity and momentum conservation?

In Physical Review Letters, there was a paper recently published:

Masud Mansuripur, Trouble with the Lorentz Law of Force: Incompatibility with Special Relativity and Momentum Conservation, Phys. Rev. Lett. 108 (2012) 193901, arXiv:1205.0096.

The paper considers the case of a stationary point charge at a fixed distance from a stationary magnetic dipole in some frame, stating that the magnetic dipole neither experiences a torque nor force from the stationary charge. Upon transforming to another frame, they argue that the magnetic dipole is transformed into a magnetic dipole and electrical dipole. This combination will therefore experience both an electrical force, and magnetic torque from the field of the moving charge. The authors therefore claim that there is a contradiction and therefore a problem with the consistency of the Lorentz force and special relativity.

Is the paper correct, or has the author overlooked something?

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If it is that easy to tell on a website like this, it would not have been published in PRL I suppose. It is interesting to see that a correction first proposed but then disputed by Einstein solves this incompatibility. I know to little of this subject to say anything useful. – Bernhard May 17 '12 at 18:36
– Bernhard May 17 '12 at 18:37
@Bernhard there are some highly competent people here, well capable of answering this question - which is why I posed it. – Larry Harson May 17 '12 at 18:43
Larry, could you edit the question to include a brief description of the main point of the paper and the reason why you think it might not be correct? – David Z May 17 '12 at 20:19
– Ben Crowell May 26 '13 at 16:04

Whoever the PRL referee(s) was/were, they should have sent it back to the author to put the argument into a manifestly covariant formalism. The editors should have done the same before the paper got to a referee. As it is, everybody has to waste time unpicking the 3-d vector mess. 3-d vectors have a perfectly legitimate place in Physics, but not if one is constructing arguments concerning Lorentz invariance/covariance or otherwise.

The title is misleading on the face of it, because the problem is reported as a failure of the system to conserve momentum, which is associated with translation invariance, not with Lorentz invariance.

There is only a "problem" if one is using the macroscopic form of Maxwell's equations. If one is using the macroscopic equations, the system will not be translation invariant if the material background is not homogeneous, similarly for rotation invariance and isotropy. If the background material is not homogeneous (and isotropic), momentum (and angular momentum) will not be a conserved quantities. Introducing the Einstein-Laub formula as a way to jury rig a non-covariant formalism is significantly too ad-hoc.

In any case, if a manifestly Lorentz and translation invariant Lagrangian can be constructed for a model, the forces that act in that model can be presented in a manifestly covariant way. The force equations could be arbitrarily complex, depending on what Lagrangian we introduce. The Einstein-Laub force law can only be applicable in a restricted setting, just as the Lorentz law is. One comment on the Science article linked to in comments points to a more-or-less intuitive resolution, "So what is wrong with polarization and magnetization being fundamental, given that point particles carry angular momentum and the quantum vacuum can be polarized?" Ultimately, this would have to be cashed out with a Lorentz and translation invariant Lagrangian (and then it would have to be quantized, etc.), but this seems the most positive thing to take from the paper. A panoply of Lorentz and translation invariant equations could be written down that include the displacement and the magnetic induction as well as the electric field and the magnetic field as dynamical degrees of freedom, though proving anything about any given system might be prohibitively difficult.

A lot more could be said, and I have a feeling more will be said because the paper has been linked to across the web, by ZapperZ, for example, on May 3rd. Now that the paper has been published, it's fair game. It's different enough from what most people are doing in Mathematical Physics, however, that relatively few people will care much. Anyone who is busy with their own research is unlikely to comment, unless, like me, they're cross enough. I've now commented on this paper twice (at ZapperZ long ago), however, so it's time to join the ranks of people who are ignoring it.

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The covariant transformation gives a similar, but not the same, expression for his Einstein-Laub correction. What is interesting is that the different "corrections" apparently lead to different polarization of materials formulations, which can be experimentally tested! – daaxix Feb 4 '13 at 22:35

There is a comment by Daniel A. T. Vanzella with a counter argument that essentially removes the paradox. He uses the natural covariant formulation of the problem. In this, you can see that the lorentz force has no spatial component in the charge/dipole rest frame, but the four force is not null. The dipole develops a time dependent angular momentum consistently across frames.

Here is the comment: http://arxiv.org/abs/1205.1502 and abstract:

It has been recently argued that the Lorentz force is incompatible with Special Relativity and should be amended in the presence of magnetization and polarization in order to avoid a paradox involving a magnet in the presence of an electric ﬁeld. Here we show that the appearance of such a “paradox” has nothing to do with the form of the Lorentz force but rather is a consequence of an incorrect use of relativistic mechanics. In fact, this pretense paradox is very similar to the “Trouton-Noble paradox” which has been solved more than a hundred years ago.

I suspect the problem in the original paper has to do with the author not carefully transforming the delta function charge/dipoles, but that's just a hunch.

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Magnetic dipoles are created by moving charge and will therefore experience electric forces from external electric fields. The author assumes that the magnetic dipole doesn't experience an electric force from the stationary charge, which implies the existence of an equal and opposite internal electric force to counter the effect of the external one. By forgetting this and just transforming the external one, an unbalanced effect and inconsistency is erroneously concluded.

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protected by Qmechanic♦Mar 4 at 22:11

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