The Pole in the Barn Paradox

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• What are the observations when both the front and the back door of the barn are closed?
•When both the doors are open.
•When the back door is closed but front door is open.
•And finally, the conclusion from this paradox.

In: Physics

The “paradox” is about length shortening when moving at a high speed. The pole would be too long to fit in the barn when stationary but if the pole is moving at a high speed its length is apparently shortened so it could fit in the barn with both doors closed. The problem is that space is actually shortened when something is moving at a high speed so the barn itself would also appear to be shortened not just the pole. This preserves the observation of the length of the pole from any frame of reference.

This is the sort of paradox you get when you only half commit.

You shorten the pole you carry though the barn by treating it as an object within the theory of relativity.

You keep the barn as is by treating it as if it was an object obeying newtonian laws.

You can’t do that. You have to treat both hypothetical object as is they existed within the same theory. Otherwise you get nonsense.

You can’t say that you for example open the 2nd door and close the 1st one as soon as the pole reaches it.

That is pretending that you have a pole flying through your barn at near relativistic speeds, but that you can see and react and send commands at faster than light speeds.

“At the same time” is not a thing that exist when you are dealing with relativity. If your barn is 100m long than anything happening at one end can’t affect the other end before 100m /c time has passed.

For 100m that is about a third of a microsecond and we normally treat is as “at the same time”, but normally we also don’t deal with relativistic poles flying through barns.

The problem is that “at the same time” doesn’t mean the same thing in both frames of reference (pole runner and observer).

The key thing with the Pole/Ladder in Barn ‘Paradox’ is that events don’t have to happen at the same time for everyone. There is no such thing as simultaneity.

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Due to Special Relativity if something is moving relative to you it is squished in the direction of relative travel from your perspective.

For example, something travelling at 3/5ths the speed of light, will only appear 4/5ths the length it “should.”

So if we run a 5m long ladder through a 4m long barn, and it is moving at 3/5ths the speed of light compared to the barn, from the barn’s perspective the ladder will only be 4m long, so there will be an instant where it is entirely in the barn.

But SR is symmetric(ish). From the ladder’s perspective it is the barn moving at 3/5ths the speed of light. So the 4m long barn will only be 3.2m long from the ladder’s point of view – at no point will the ladder be entirely within the barn.

Wikipedia has some great diagrams for this. [This shows things from the barn’s perspective](https://en.wikipedia.org/wiki/File:Ladder_Paradox_GarageScenario.svg); the ladder is squished by SR, so there is an instant where it is entirely within the barn. [This shows things from the ladder’s perspective](https://en.wikipedia.org/wiki/File:Ladder_Paradox_LadderScenario.svg); the barn is squished by SR, so there is no point where the ladder is entirely within the barn.

The key thing to realise is that *times* are also different for the ladder and the barn. We can’t deal with this just thinking about space, we need [the full Minkowski diagram](https://en.wikipedia.org/wiki/File:LadderParadox1_Minkowski.svg) (from the barn’s perspective). There is a lot going on here, so let’s break it down:

* the blue x/t-axes represent space and time from the barn’s perspective. The origin O is the entrance of the barn, when the front of the ladder reaches it. The blue shaded area is the barn.

* the red x’/t’-axes represent space and time from the ladder’s perspective (skewed by how fast it is going). The red shaded area is the ladder.

* Point A is when and where the front of the ladder hits the back of the barn. Point D is when/where the back of the ladder reaches the front of the barn.

B is where the back of the ladder is when A happens *from the barn’s perspective*, so AB is the ladder at that point in time; the ladder is entirely within the barn. But C is where the back of the ladder is when A happens *from the ladder’s perspective*, so when the front of the ladder reaches the back of the barn, the ladder is the line AC (not AB). Events A and B happen at the same time for the barn, but at different times for the ladder. Similarly A and C are the same time for the ladder but different times for the barn.

D – the point where the back of the ladder reaches the front of the barn happens *before* A for the barn, but *after* A for the ladder. We can also see (although not marked on that diagram) that when the back of the ladder reaches the front of the barn (so the bold red line starts at D) the front of the ladder will be a long way out the other side of the barn (off to the top-right of A).

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So what happens if we start closing doors?

We *could* close both the front and back doors at the same time, very briefly. *But only from the barn’s perspective.* Going back [to the original diagram](https://en.wikipedia.org/wiki/File:LadderParadox1_Minkowski.svg), we could leave the back of the barn closed until the front of the ladder reaches it at A, and then open it. And we could close the front of the barn when the back of the ladder reaches it at D. From the barn’s perspective, D happens before A (time going upwards for the barn, horizontal lines represent things happening at the same time). But from the ladder’s perspective A and C happen at the same time (lines parallel to that line represent things happening at the same time, time goes up-and-right). So D happens some time after A. From the ladder’s point of view, the back of the barn opens before the front of the barn closes and all is good with the world.

There is plenty of time where both doors are open for the ladder, even if there is no time both doors are open for the barn.

If we close the back door, we’ll need to slow down the ladder so it comes to a stop. If we do this gently, from the barn’s perspective the ladder will return to its normal length (so we’d ‘see’ the back of the ladder start slowing down before the front does – ‘stretching’ the ladder, and it won’t fit in the barn).

If we leave the back door shut and let the ladder smash into it, then our idea of solid objects breaks down. The “knowledge” that the barn is closed can’t travel faster than the speed of light. So the back of the ladder won’t learn that the front has stopped until some time later (marked as point F [on this diagram](https://en.wikipedia.org/wiki/File:Junk1.png)). After the front of the ladder hits the back of the barn (event A) the front moves from A to E. From the barn’s perspective, during that time, the back of the ladder moves from B to F (where the back of the ladder comes crashing to a halt). But from the ladder’s perspective the back of the ladder moves from C, through D and B, to F. The ladder is going to get crushed as the back slams into the front, collapsing it down.

Also, we probably blow out the back of the barn. The energies involved are going to be massive. Don’t try this at home.