There are a bazillion alleged “paradoxes” in special relativity, all based on exactly the same fallacy, but I might have just invented a brand-new one—-where “invented” is shorthand for “confused the hell out of myself for a while”. When I finally got up and drew a picture (as opposed to lying in bed with my eyes closed doing something that felt like thinking), it became clear that, sure enough, it was the same old fallacy again (how could it not have been?), but in a new enough guise that someone reading this might find it amusing.
So: 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?
Yes, the train is moving in a circle and is therefore not in an inertial frame. No, that has nothing to do with resolving this. I won’t spoil the fun by posting the answer just yet. I had a very “d’oh!” moment when I got it.
I’m not sure whether I’m a geek for figuring this out or a dork for not seeing it instantly.