The Pole In The Barn Paradox

Saturday, October 27, 2018

The Pole In The Barn Paradox

The pole in the barn paradox is a famous "paradox" in special relativity. The key with "paradox's" in special relativity is that after considering all the fine details; you can see how they are resolved. The theory is self consistent and so there can be no actual paradoxes.

A picture of a similar paradox involving a ladder moving into a garage.

The Paradox

There is a pole which has a rest length greater than the length of a barn. We know that objects in motion become length contracted and so when this pole is moving very quickly relative to the barn; it's length becomes shorter than the barn. We watch the pole rush by and as it is length contracted it goes into the barn, disappears from view, and then exits the other side.
For someone moving along with the pole however they see something else. The barn is in motion in their frame of reference while they are stationary with the pole and so to the them, it is the barn that is length contracted. In their frame of reference the pole does not fit in the barn at all.
So from one person's point of view the pole fits in the barn and from the other person's point of view it doesn't?

So who's right? What's going on?

Both observers are correct. The reason why is that length invokes the idea of simultaneity which we have found changes from frame to frame. From our stationary perspective, the pole is rushing to the right towards the barn. This means clocks towards the right of the pole are behind clocks in the back. We can therefore explain why team pole (the pole's frame of reference) say that the pole does not fit. This is because they measure the position of the back of the pole first, let it move and then measure the front according to us.
Team pole can also explain our measurement. To them the barn rushes by to the left and so clocks towards the left of the barn are behind clocks towards the right. This means team poles see us measure the front of the pole first (since these clocks are ahead), wait a little while the pole moves, and then measure the back.

Lock The Doors

So we have resolved this paradox but now lets make it harder. What happens if we lock the door of the barn. What if in our frame we wait till the pole is in the barn and then we bring down steel doors. What happens in team pole's frame that means there is no paradox?
It can't be the case that in one frame a pole gets locked in a barn while in the other it doesn't?

To make this problem easier to solve, we will change the mechanism we use to stop the pole. Instead of bringing down doors we will get a line of people in the barn. At the moment the front of the pole is in line with the front of the barn, the line of people will simultaneously grab the pole causing it to stop. This has the same effect as the steel doors coming down and stopping the pole.

Team pole have a different idea of simultaneity since they are moving relative to us. The barn is rushing in the opposite direction and so what team barn call the front of the barn is towards the back of the motion. This means clocks here are ahead of clocks towards the back of the barn. This means the first person grabs the front of the pole while the rest of the pole still moves a little into the barn. The next person then grabs the next point on the pole while the rest moves again a little. This continues but because all people don't grab the pole at the same time; the back parts of the pole have extra time to move into the barn. The pole becomes compressed due to this and when all people have grabbed the pole, team pole also see the pole fitting inside the barn.

As you can see there is no actual paradox. When you remember that simultaneity is relative and that it effects the idea of length, you can logically think through what each person sees and why they see it.

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