## Tuesday, March 25, 2014

### Pole in the Barn Paradox

According to the special theory of relativity an object that is in motion relative to a givenframe of reference contracts – becomes shorter in length – along the direction in which it is moving.  But this seems to lead to a paradox.  Consider a pole that is 15 meters long when at rest, and consider a barn that is 10 meters long when at rest.  Suppose that the barn has a door at each end.  Clearly the pole should not fit inside the barn with both doors closed.

Now suppose you can get a friend to run through the barn carrying the pole at (4/5)c – i.e. at 4/5 the speed of light.  Then, from the barns rest frame, the barn is still 10 meters long, but the pole is only 9 meters long (i.e. the length of the pole in the barns rest frame = 10m x sqrt(1-(v/c)^2) = 15 x (3/5) = 9m). So, for a brief moment, as the pole passes through the barn, the pole fits completely inside the barn with both doors closed.
However, from your friend’s frame of reference, where the pole is at rest and barn is moving toward him at (4/5)c, the pole remains 15m long. But the barn becomes shorter.  Its length becomes 6m  (i.e. the length of the barn in the poles rest frame = 10m x sqrt(1-(v/c)^2) = 10m x (3/5) = 6m). So, as it passes through there is no way that the pole should fit completely inside the barn with both doors closed.
But the pole either fits completely inside the barn (for a brief moment as it passes through), or it doesn’t.  This is the paradox.
So from the point of view of the pole, its length never changes, it is always 15 meters long, and the length of the barn is ever decreasing so that from the poles frame the pole could never fit within the barn all at once.  And, from the point of view of the barn, the pole is ever decreasing so at some point it should be able to fit within both doors of the barn.  So the diagram shows that in the barn frame the pole will fit and in the pole frame it is quite to long to fit.  Which one of these is the correct assumption?

The solution to the paradox is in the relativity of simultaneity.  That is that what one observer sees as simultaneous is not the same as what another observer sees as simultaneous.  “At the same time” does not apply to those events which are separated by space in special relativity theory.  This thought experiment shows that there are circumstances when there is no correct answer.  Neither observer has privileged status or rightness.  And even that all included observers can claim to be correct, at least in their own independent reference frames, even if their ordering of events differs from another’s.

This article originally written February 25th, 2008 for OU PHIL 3623 - Physics and Cosmology.