A new relativity paradox

 Recently, there was an article in PRL called "Trouble with the Lorentz Law of Force: Incompatibility with Special Relativity and Momentum Conservation".  I admit I didn't read the article, because I felt that the summary in Science was sufficient.

Figure from Science, credit P. Huey

The basic paradox is as follows.  You have a charge and a magnetic dipole, motionless with respect to each other.  In the reference frame where they are motionless, they exert no forces on each other.  In the reference frame where they are both moving, there will be an electric dipole in the magnet.  If you think of the magnetic dipole as a loop of current, the electric dipole can be understood as the result of Lorentz contraction (as discussed in an old post).

The positive end of the electric dipole will be repelled by the charge, while the negative end will be attracted.  Therefore, the charge applies a torque on the magnet, causing it to spin.  (The moving charge also applies a magnetic force, but this does not counteract the torque.)  But how can the magnet spin in one frame, but not spin in another?  Therein lies the paradox.

The author of the paper concludes that the Lorentz force law needs to be revised in the presence of polarized or magnetic materials.  However, I do not think this is the correct resolution for the following two reasons:

1. It is easy to construct the same paradox without any polarized or magnetic material.  Replace the magnet with a loop of current, which functions as a magnet.
2. It is easy (but tedious) to prove that the Lorentz force law is the same in all reference frames.  Showing that it behaves differently in two frames is akin to proving 1 = 2; the correct conclusion is that you made a mistake somewhere.

But I must admit that I could not figure out where the mistake was.  It's a good paradox!  Instead, I solved the problem using the research method.  I read two responses by Daniel Vanzella and Kirk McDonald (both of which said the same thing).  The resolution to the paradox is tricky, and I will attempt to summarize it for a lay audience.

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We need to talk about the momentum of the electromagnetic field itself.  It's well-known that light, a fluctuation in the electromagnetic field, can carry momentum.  But even constant fields can have momentum.  The momentum is proportional to the cross product of the electric and magnetic fields.  When you have an electric charge and magnetic dipole near each other, the momentum of the electromagnetic field is not zero.  But the total momentum of the magnetic dipole must be zero, since it is motionless.  Therefore, there must be some hidden mechanical momentum in the opposite direction.

In this figure, the magnetic dipole is represented by a loop of current.  The magnetic field produced by the current goes through the loop out of the screen.  The electromagnetic momentum is downwards.  The hidden mechanical momentum is upwards.

I claim that there's hidden mechanical momentum, but where does that come from?  I believe that it depends on the precise nature of the magnetic dipole, but we can at least consider a simple example.  In the above figure, we imagine that the magnetic dipole is actually a square loop of counter-clockwise current (with no dissipation).  Charged particles in section A are being slowed down by repulsion from the charge on the left.  Similarly, charged particles in section C are being sped up.  Therefore, the particles in section D are moving faster than the particles in section B.  And recall that in relativity, faster particles are heavier.  Therefore, there is more momentum in D than in B, and the total momentum is upwards

And that's how a loop that is motionless can still have momentum.

It turns out that even when we use a moving reference frame, there is some mechanical momentum upwards.  Torque, which normally causes an object to spin, is really the change in angular momentum around a single pivot point.  The angular momentum is proportional to the cross product of the momentum and the distance from the pivot point.  In the moving reference frame, there is torque on the loop, but this does not cause the momentum to change, or cause the object to spin.  It causes the distance from the pivot point to change (because the loop is moving, while the pivot point is motionless).  Thus the paradox is resolved.

Upon reading this resolution, I was puzzled, because there must be an equal and opposite torque applied somewhere else in the system.  I figured out that an opposite torque is applied to the momentum of the electromagnetic field.

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The responses from Vanzella and McDonald appeared before the paper was even officially published.  That might be kind of embarrassing for PRL.  Or maybe not?  Presumably, PRL felt that even if the paper was shot down, it would still generate productive discussion.

Note that there is precedence for revising old textbook equations.  I'm thinking of the Aharanov-Bohm effect (which shows that magnetic field has an effect on particles which go around the field even if they don't go through the field) and Berry's curvature (which modifies the laws of motion for particles within a crystal structure).  In both cases, the fundamental equations were known, but physicists overlooked a few of their strange consequences.