You can rest easier now: A purported conflict between the century-old theory of classical electrodynamics and Einstein's theory of special relativity doesn't exist, a chorus of physicists says.
Last April, Masud Mansuripur, an electrical engineer at the University of Arizona in Tucson, claimed that the equation that determines the force exerted on an electrically charged particle by electric and magnetic fields—the Lorentz force law—clashes with relativity, the theory that centers on how observers moving at a constant speed relative to one another will view the same events. To prove it, he concocted a simple "thought experiment" in which the Lorentz force law seemed to lead to a paradox, which he described in Physical Review Letters (PRL). Now, four physicists independently say that they have resolved the paradox in comments in press at PRL.
"Masud is completely convinced that he's right, but he's not," says Stephen Barnett, a commenter from the University of Strathclyde in Glasgow, U.K.
To understand the Lorentz force, suppose a charged particle moves through electric and magnetic fields. The Lorentz force law states that the electric field will push the particle in the same direction as the field while the magnetic field will push it in a direction perpendicular to both the magnetic field and the particle's velocity. The force law is often used to illustrate how in relativity, for example, a force that appears to be purely electric to one observer will appear to be both electric and magnetic to an observer moving at a different speed.
But Mansuripur thought up an example in which the law seemed to lead to a logical contradiction. Consider a pointlike electric charge sitting a fixed distance from a tiny magnet (see figure). The magnet has no charge, so it experiences no force from the charge's electric field. Similarly, the unmagnetized charge does not interact with the magnet's magnetic field. So nothing happens.
Now imagine how things look from a "moving frame of reference" in which the charge and magnet both glide by at a steady speed. 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. So the point charge will pull on one side of the magnet and push on the other, creating a twisting torque—or so Mansuripur claimed.
The details go like this. The magnet can be thought of as a tiny loop of wire in which negatively charged electrons run through stationary positive ions. In the frame in which the ring is stationary, the electrons and ions are equally spaced and the ring appears uncharged. But in the moving frame, the electrons on one side of the ring move faster than those on the other relative to the observer. So thanks to the weird "Lorentz contraction" of relativity, the electrons on one side appear more tightly spaced and those on the other side more loosely spaced, creating the charge imbalance.
However, according to relativity, the magnet cannot twist in one frame and not in another, Mansuripur notes, so the results are paradoxical. To avoid the problem, he advocates replacing the Lorentz law with one that treats magnetism differently.
But Mansuripur forgot something, all four commenters argue. Thanks to the bizarreness of special relativity, the magnet also possesses an odd "hidden angular momentum" that in the moving frame constantly increases. By its very definition, a torque equals a change in angular momentum. So instead of twisting the magnet, the torque in the moving frame simply feeds the increase in hidden angular momentum. Problem solved.
Here's how the hidden angular momentum comes about. If the magnet is thought of as a current loop, then on one side of the loop the electric field from the point charge pushes the electrons in the direction in which they're already moving and boosts their energy. On the other side of the loop, the electric field opposes the electrons' motion and saps their energy. So there is a net flow of energy from one side of the loop to the other. Thanks to Einstein's equation E=mc2, that energy flow is equivalent to the movement of mass, which itself is equivalent to momentum. So the energy flow gives the magnet a sideways hidden momentum, even though it's not moving sideways.
In the moving frame, this hidden momentum also gives rise to an increasing angular momentum. To see how this works, suppose you twirl a ball on the end of a string over your head. At any moment, the ball has a momentum that points perpendicular to the string and gives it an angular momentum around your hand, and that angular momentum increases as you let out string. In the same way, in the moving frame, the receding sideways hidden momentum of the magnet leads to a steadily increasing angular momentum. And pumping up that angular momentum absolutely requires the torque that Mansuripur identifies, the commenters say.
Mansuripur is sticking to his guns. He argues that hidden momentum, which was identified in the 1960s, is an ill-defined concept that merely papers over the problem. "That's always been the problem with hidden momentum," Mansuripur says. "You know that something is missing, so you just postulate its existence." He says his approach eliminates the need for hidden momentum.
Others say that hidden momentum is part and parcel of relativity. "If you have a system with internal motion that is subject to an external force, then hidden momentum is a general property," says Daniel Vanzella, a commenter at the University of São Paulo in São Carlos, Brazil. "It's not an ad hoc invention put in to reconcile things." Vanzella also notes that mathematically, the Lorentz force law can be written in a form that guarantees it will jibe with relativity, so it's "simply impossible" for it to contradict the theory.
Some physicists argue that Mansuripur's paper should never have been published. "I couldn't disagree more," Barnett says. "Is it not better to publish things that are interesting and not obviously wrong than to kick the next theory of general relativity to the curb because it looks like rubbish?" He adds that the argument has been extremely civil: "Masud is very passionate about what he's done, but he's a complete gentleman."
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