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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?

Harry Johnston
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  • 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!) – daaxix Feb 04 '13 at 22:10
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    Related: http://physics.stackexchange.com/q/28514/2451 – Qmechanic Feb 04 '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? – Harry Johnston Feb 04 '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 – daaxix Feb 04 '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. – Terry Bollinger Feb 05 '13 at 00:54
  • Having looked briefly at some of the references, I can't even figure out for sure what the paradox is. Some use coils, some magnets, some low speed, some high speeds, some motion parallel to $E$, some perpendicular to $E$... argh! Too many of the writers seem to have an aversion to adding in a simple diagram showing what they think the issue is, but no aversion at all to diving into abstruse approaches to solving... whatever it is. I flatly do not think that for this kind of problem you can skip the diagram -- that is where you start. – Terry Bollinger Feb 05 '13 at 01:40
  • @TerryBollinger: I found Mansuripur's original paper on the arXiv, the diagram is figure 1 on page 3, and the para starting section 2 describes what he thinks the paradox is. – Retarded Potential Feb 06 '13 at 06:31
  • @daaxix: looking at the original paper, I suspect that a fictitious electric dipole is being unintentionally introduced. He calculates the electric field in the moving frame, observes that it looks like the field around an electric dipole, and concludes that there is in fact an electric dipole. That reasoning is invalid, but several of the papers in response seem to accept the existence of the dipole, so I'm uncertain. I'll need to chase down some of the references. – Harry Johnston Feb 06 '13 at 20:06
  • @HarryJohnston, can you point me to which paragraph you think results in a fictitious dipole? He starts with a real electric monopole and a real magnetic "dipole" (really a tiny current loop), then transforms them... – daaxix Feb 07 '13 at 18:59
  • I was misreading his paper, sorry, so my last comment described his reasoning improperly. What I find doubtful is the transformation from equation 9 to equation 10; I suspect that the electric dipole being introduced by that transformation isn't real. However, several of the responding papers seemed to accept it. I think at least one of them has a reference, and that's what I need to look up. It seems unlikely to me that a point electric dipole can be physically meaningful, but I could be wrong. – Harry Johnston Feb 08 '13 at 03:20
  • Retarded Potential, thanks for finding that, I'll take a look. Alas, I've developed a bit of an aversion to paradoxes of this type after probing the Feynman Disk Paradox. Many papers were written over decades about that one, yet Feynman did answer it in his Lectures. He just took his sweet time doing it, making the answer hard to find. (My thanks again to @JohnMcVirgo for helping me find the original Feynman answer at the end of a later chapter.) – Terry Bollinger Feb 09 '13 at 20:38
  • Related: http://news.sciencemag.org/2013/01/purported-relativity-paradox-resolved – BlueRaja - Danny Pflughoeft Aug 20 '13 at 15:22
  • @BlueRaja-DannyPflughoeft: that article is wrong in several important ways - in particular the claim that "Thanks to the weird effects of relativity, the magnet appears to have more positive charge on one side and more negative charge on the other." is clearly wrong since charge is a scalar. (I suspect this is also the underlying problem in Mansuripur's paper, but it's not clear since he doesn't explain where his transformation of a point magnetic dipole to a point electric dipole comes from.) – Harry Johnston Aug 20 '13 at 22:00
  • @daaxix Magnetic monopoles do not exist according to Maxwell's equations. Even if ever they are observed, this will still be a fact. – my2cts Apr 03 '21 at 10:07

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