A train on a track--relativity paradox

While answering a question over on Worldbuilding.SE I found myself looking at a situation that I can't figure out:

You have a train track of length L that makes a very large circle. You have a train that fills the whole track and is moving down that track at such a speed that the Lorentz factor is 2.

The track is length L. When the train was sitting still it was also of length L. Now that it's up to speed it's length is L/2--yet it's still riding on a track of L. Since the train goes all the way around it has no way to change length, nor can any issue of the location of observers explain this.

How can this be resolved?

Edit: While I agree that an observer at one spot on the train won't see a circular track I don't see how this avoids the paradox. The observer will see the track shrunk in the direction the observer is currently moving but it will be just as far across. The whole track has a length of 2 pi * r, Lorentz isn't going to change the track in the direction that the observer isn't currently moving in. Since we are at Lorentz factor 2 the train length is only pi * r but even if you shrink it to zero in the direction the observer is moving you have a track of 4 r length. The train still doesn't fit.

Answer

I believe you'll find exactly this question posed and answered here.

In brief: Let's look at one car on the train, which takes up a small enough part of the track that we can treat is as roughly straight.

If you are stationary relative to the track, and if the entire car accelerates "all together" according to you (as opposed to, say, the front accelerating before the back does) then you cannot see the car shrink. After all, the front and back of the car have the same velocity at every instant, so the distance between them can't change.

An observer on the car, though, does see the size of the car change, because he sees the front of the car accelerate before the back does, causing the car to stretch. Therefore the car looks smaller to you than it does to the observer on the car, which is just what SR requires.

So in summary: You don't see the cars shrink, and (for the same reasons) you don't see the spaces between the cars shrink, which is why you don't see the train having any problem staying on the track.