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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.

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    $\begingroup$ To the train the track doesn't look like a circle. $\endgroup$ – CuriousOne Mar 15 '16 at 5:33
  • $\begingroup$ @CuriousOne Well, true, but what about an external observer? $\endgroup$ – barrycarter Mar 15 '16 at 14:30
  • $\begingroup$ To an external observer the train couplings are experiencing enormous forces (that would be tearing it apart), if I remember correctly. $\endgroup$ – CuriousOne Mar 15 '16 at 20:11
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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.

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  • $\begingroup$ OK, but if the back and front of a car have the same velocity, won't the distance between them still Lorentz contract as compared to when the car was stationary? A meter stick will contract once it speeds up and the front and back of a moving meter stick have the same velocity. $\endgroup$ – barrycarter Mar 16 '16 at 13:34
  • $\begingroup$ @barrycarter: At time 0, the left end of your (stationary) meter stick is at point 0 and the right end is at point 1. At that time (according to you, the stationary observer), the entire stick instantly starts moving rightward at $.9c$. At time 1, the left end is at point $.9c$ and the right end is at $.9c+1$. The length of the stick (as measured by you, the stationary observer) is still $1$. $\endgroup$ – WillO Mar 16 '16 at 14:08
  • $\begingroup$ I'm pretty sure you're wrong about that. $\endgroup$ – barrycarter Mar 16 '16 at 14:48
  • $\begingroup$ @barrycarter: To which end of the meter stick are you pretty sure that the formula "distance equals rate times time" does not apply? $\endgroup$ – WillO Mar 16 '16 at 15:05
  • $\begingroup$ If the car is traveling at velocity .99c, the two ends of the meter stick are never one m apart to a "stationary" observer. They are always about 43.6cm apart by Lorentz contraction. $\endgroup$ – barrycarter Mar 16 '16 at 16:31
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This is, in essence, the Ehrenfest paradox. The problem is that you are assuming that the train is perfectly rigid. Because the track is circular, the train is always accelerating, and since the reference frame of the train is accelerating the rules of special relativity are not globally valid over the entire track.

Over small regions of the track the train is not accelerating too much so things are okay and you will find that a small segment of the train will contract as normal. But if you try to look at what's happening over the entire track, you will find that the relative accelerations will induce stresses on the train. If the train is traveling at relativistic speeds these stresses will be so strong that they will cause the train to break up into small pieces, each of which will be contracted.

I should note that for realistic materials the stresses induced by relativistic length contraction will actually be minimal compared to the stresses induced by centripetal acceleration. These stresses will cause the train to fragment as soon as the transverse velocity of the train becomes comparable to the sound speed within the material of the train.

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There are a few possibilities.

One possibility, you could replace your train cars with automobiles. You could have two circular tracks each full of cars and they could all set up in different places and park and then they could all accelerate in a straight line so that they all get to gamma 2 for one instant where the person in the center of one of those circles sees all two circles worth if cars fitting on the one track. Then they keep going on their straight line, or they have to all try to turn to keep going in a circle, but they wouldn't be able to turn that hard, because steering a car going at gamma 2 is too hard.

Another option is to have a trillion cars filling the circle and have just one extra car. Position them, accelerate them up to speed gamma 1.000000000001 and you can fit all trillion and one cars in there. And now they can turn fine.

Another option is to have just one track with your trillion cars filling it at rest and as they speed up to gamma 1.000000000001 then there is room for one more length contracted car but it's not there, the cars are not touching each other.

Another option is like the previous but you attach a spring to connect each car. When the springs get pulled the spring pulls the car, but the car is less stretchy so doesn't get longer anywhere near the amount the springs do.

So now if there was a trillion cars with a trillion springs connecting them, you see the cars get smaller and the springs get stretched so that the cars take up a tiny bit more than 1/gamma of the space and the springs take up the rest. It's tiny because the cars aren't very stretchy compared to the springs, and it's more because the springs do pull and stretch the cars a tiny bit.

If you started making the springs stiffer, you'd find that more the additional space is taken up by the stretched cars, compared to the natural length if the cars (which is their rest length divided by gamma) in the limit where the springs are as stretchy as the cars then the cars basically have the spring as a component of the car and now the cars either have to break or they have to stretch (like the springs did) to keep connected as the high speed contracts the cars.

So to fit perfectly the cars have to stretch each other by a factor of gamma so that that the contracted length of the stretched car fits on the track.

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