# Does Mansuripur's Paradox involve fictitious charges?

Mansuripur's Paradox involves a magnet moving at relativistic speeds in an external electric field.

Additional: thanks to Retarded Potential, who found the original paper.

If I understand correctly, the fictitious charges on either side of the magnet should cause a torque, which of course does not actually exist. Apparently this is conventionally resolved by adding an (equally fictitious) internal angular momentum. Mansuripur appears to be arguing that a better resolution would be to add a (fictitious) additional term to the Lorentz force law.

Am I right in thinking fictitious charges are involved? If so, why are they introduced, i.e., what is the advantage in doing so? If not, where does the torque really come from?

• Mansuripur maintains that there is a torque in one inertial frame and none in another, which is true for his gedanken experiment. He advocates replacing the Lorentz law of force with the Einstein-Laub law of force (which contains a correction) while relativists maintain that the covariant 4-vector formulation properly describes the difference. Interestingly, when Griffiths and Hnizdo use a Gilbert dipole, they get the same answer as Mansuripur. The problem is, a Gilbert dipole isn't currently allowed in Maxwell's equations (because magnetic monopoles haven't been observed yet!) Feb 4 '13 at 22:10
• Feb 4 '13 at 22:14
• @daaxix: I have to admit haven't seen a copy of the original paper, my understanding is based on the magazine article and a brief scan of one of the papers written in response so may well be incorrect. In Mansuripur's formulation of his paradox, what is the source of the torque, if it isn't fictitious charges on either side of the magnet? Feb 4 '13 at 22:38
• As far as I understand, he uses point charge and a really tiny loop of current (or magnetic dipole) no fictitious charges... Then when you use the Lorentz law of force you get the torque in the moving frame. When the Lorentz law of force is Lorentz transformed via the 3-vector formalism to the moving frame, you get the torque. If you had instead used the 4-vector formalism to do the Lorentz (tensor) transformation instead, you would get a so called "hidden" momentum...but it appears that to get the same equation as Mansuripur you need to assume monopoles prior to the 4-vector transformation Feb 4 '13 at 22:52
• Never heard of this one, interesting. MIT has a resource page on it, as well as on other quirky SR problems. That page includes this link to a quite recent paper -- updated less than two weeks ago! -- by Kirk T. McDonald at Princeton, who has a knack for finding and collecting such problems. Finally, arXiv has a May 2012 paper by David Cross that claims resolution. A final note: Sometimes these are simpler than they might seem. Feb 5 '13 at 0:54