Resolving the Pole & Barn Paradox with Relativity

[espen180]

Homework Statement

The Pole and Barn Paradox
Suppose a very fast runner (v=0.600c) holding a long horizontal pole runs through a barn that is open at both ends. The length of the pole (in its rest frame) is 6.00m, and the length of the barn (in its rest frame) is 5.00m. In the barn's reference frame, the pole will undergo length contraction and the entire pole can fit inside the barn at the same time. But in the runner's reference frame, the barn will undergo length contraction, and the entire pole can never be entirely within the barn at any time! Explain the resolution of this paradox. (A spacetime diagram may be useful.)

Homework Equations

u = speed of reference frame S'
v = speed of particle

Spacetime diagram slopes:
Slpoe of worldline of particle: c/v

S' reference frame axis slopes:
Slope of x'-axis: u/c
Slope of ct'-axis: c/u

The Attempt at a Solution

Let S be the reference frame of that barn, where the barn is at rest.
Let S' be the reference frame of the runner, where the pole is at rest.
S' is moving inertially in relation to S at a speed of 0.600c in the positive x-direction.
Let event A be the time and position in S when the forward facing end of the pole reaches x=5.

Due to length contraction, the pole is 4.8 meters long in S, and the barn is 4 meters long in S'.

Since the pole is at rest in S', the worldline of the front end of the pole will be parallel to the ct'-axis and intersect the x-axis at x=5. It will also intersect the x'-axis at x'=6.25.
Event A is NOT simultaneous in S and S'. By the time the runner measures that the front end of the pole has passed reached the end of the barn, an observer at rest inside the barn will measure a fraction of the pole have already passed through the barn.
In short, the pole passes through the barn later in S' then in S.

That is as far as i have gotten with this problem. is it a suffiecient explanation? If not, am I on the right track?

Thanks for any help.

 

[CompuChip]

I think you need to be a bit more careful with expressions like "the pole passes through the barn".
I suggest considering the following two events in S and S':
A) the front of the pole reaches the front end of the barn
B) the back of the pole reaches the back end of the barn

You can imagine for example, closing the front and back doors of the barn when A and B occur, respectively.

 

[sylas]

...
Since the pole is at rest in S', the worldline of the front end of the pole will be parallel to the ct'-axis and intersect the x-axis at x=5. It will also intersect the x'-axis at x'=6.25.
Event A is NOT simultaneous in S and S'. By the time the runner measures that the front end of the pole has passed reached the end of the barn, an observer at rest inside the barn will measure a fraction of the pole have already passed through the barn.
In short, the pole passes through the barn later in S' then in S.

That is as far as i have gotten with this problem. is it a suffiecient explanation? If not, am I on the right track?

Looks very good to me. Worst problem is incorrect spelling of parallel.

You can also improve this sentence: By the time the runner measures that the front end of the pole has passed reached the end of the barn, an observer at rest inside the barn will measure a fraction of the pole have already passed through the barn.

I know what you mean... but a phrase like "by the time the runner ..., an observer at rest ..." appears to speak of a time which is common to the runner and the observer.

It might be better to say: "for the runner, the front end of the pole reaches the end of the barn in the same instant that the back end is..., whereas for an observer at rest in the barn, the front end of the pole reaches the end of the barn in the same instant than the back end is..."

Cheers -- sylas

 

[BAnders1]

Relativity of Simultaneity. If you call event A the event in which the front of the pole exits the front of the barn and event B the event in which the pole exits the back of the barn, these two events will occur at different times in different frames.

If, for example, event A and B occur at the same time in the barn's frame, they will occur at different times in the runner's frame, and event A will occur before event B.

 

[fleem]

Super-simplified explanation: its the same as fitting a 6-foot pole in a square box with 5-foot sides. You simply rotate the pole so it fits in the box at an angle. Velocity rotates the pole so it can fit in the barn. Whether the barn or the pole is rotated depends on the ref frame of the observer.

 

[Doc Al]

Super-simplified explanation: its the same as fitting a 6-foot pole in a square box with 5-foot sides. You simply rotate the pole so it fits in the box at an angle. Velocity rotates the pole so it can fit in the barn. Whether the barn or the pole is rotated depends on the ref frame of the observer.

Oh really?

So if the barn were really narrow, things would work differently? I don't think so.

 

[fleem]

Oh really?

So if the barn were really narrow, things would work differently? I don't think so.

By "small box" I assume you mean a very young box (in a Lorentz diagram). Yes, if the age of the box is less than the time it takes for the pole to move through it, then the pole wouldn't fit. Very astute observation. However, I suspect the the OP was not contemplating a box that existed only a couple of femtoseconds, because he probably would have mentioned that, if he were.

 

[Doc Al]

By "small box" I assume you mean a very young box (in a Lorentz diagram).

I never mentioned anything about a "small box". I did mention narrow, since you spoke about the pole rotating. (Narrow in the usual spatial sense.)

Yes, if the age of the box is less than the time it takes for the pole to move through it, then the pole wouldn't fit. Very astute observation. However, I suspect the the OP was not contemplating a box that existed only a couple of femtoseconds, because he probably would have mentioned that, if he were.

What has this to do with your statement that "velocity rotates the pole"? I presume you meant to say something about the pole being rotated in spacetime, not space. The pole is certainly not rotated in space, and the situation is not made clearer by mentioning the width of the barn.

 

[espen180]

Thanks for all replies! :)

So my general statement above was correct, but I should change my events to A and B as described above?