Ehrenfest Paradox

Steve Landsburg, an economist/mathematician who fancies himself an expert on relativity (he has written a book on GR) poses the following problem:.

A circular train (front of the locomotive attached to the rear of the caboose) sits on a circular track. At some point, the train accelerates and starts traveling around the track. Because the train is moving, I (an observer stationary relative to the track) should see it shrink. But the track doesn’t shrink. So the train can’t stay on the track, and gets pulled inward, ending up inside the track. On the other hand, the passengers say the track has shrunk, so they should expect to get pushed outside the track. How can everyone be right?

This is a famous puzzle known as The Ehrenfest Paradox, but I think that Landsburg's "solution" is less than adequate - not really even correct.

Any comments?

UPDATE: Here is what I came up with. It has one major flaw that I really should try to fix:

I really like this problem and had several happy ours struggling to understand it, but I don’t think your solution is ideal. The trouble is that it depends on a Procrustean stretching the train in an unphysical fashion to make it fit. If you accelerate the train in a conventional fashion, say by electric motors in each car, you find that the train really does shrink and fall off the track to the inside. To see this quantitatively, imagine not a circular track but a long, thin racetrack shape, with negligible ends and long sides of length L/2. Train and track are each initially of length L. After acceleration to speed v, Jeeter, the observer on the train, measures the track to have Lorentz contracted length L*sqrt(1-(v/c)^2). The part of the train on his side has length L/2, as before, but the part of the train on the opposite side is shortened to length (L/2)*sqrt(1-(2v/c)^2). If we do the arithmetic in the binomial approximation, we see that the total length of the track minus the length of train is (L/2)(v/c)^2. The stationary observer sees each half of the train shortened by (L/4)*(v/c)^2, again in the binomial approximation, so they agree that the train is shorter than the track by (L/2)(v/c)^2.
The train **does** fall off the track to the inside. Note, by the way, that this form is a close spatial analog of the so-called twin paradox. The key point of asymmetry is the same – one observer gets accelerated and the other doesn’t.
In addition to the liberties of using the binomial approximation, I also used Galilean addition of velocities, but that’s a higher order correction unless v is close to c.

UPDATE II: OK, let me eat some (not all) of my words. Call it a limited, modified, partial hang out) Consider a train of cars of length l moving at speed v along a track. At some time t=0 in the track frame each car begins accelerating uniformly.

Meanwhile, back on the train, observers stationed on the train notice something funny. Do to the relativity of simultaneity, the front of the car in front of him began accelerating earlier than the back by an amount of time dt = l*v*gamma, where gamma = 1/Sqrt(1-(v/c)^2). Also, the back of the car behind him started dt later. This is what Steve Landsberg said, right? Right.

If the train is circular, and you go around the train, each observer notices that he started later than the guy in front of him and earlier than the guy behind him – very odd, but a manifestation of the fact that these observers can’t synchronize their clocks.

So how do the cars feel about being stretched like this? Well they resist fiercely and then they either stretch, break, or move inward toward the center of the track. Steve focussed on the details of the acceleration, and decided that the train didn’t shrink. I focussed on the physics of railroad cars being stretched, and concluded that it must.

It turns out that this is another famous paradox in relativity called Bell’s paradox. The upshot is that uniform acceleration of a rigid body can’t occur in special relativity. An accelerated body either deforms or accelerates nonuniformly.

Links to more on Bell’s paradox in the comments.