The barn-pole paradox

The setup

A pole longer than a barn moves toward the barn at relativistic speed. In the barn frame, the moving pole is length-contracted and may fit entirely inside the barn at one instant. The barn doors could briefly close simultaneously.

In the pole's rest frame, the pole has its proper length, while the barn is length-contracted and even shorter. So how can the pole fit?

This is the barn-pole paradox.

Length contraction in the barn frame

Let the pole's proper length be $L_0$. In the barn frame, the pole moves at speed $v$, so its measured length is

L=rac{L_0}{gamma}

If $L$ is less than the barn length $B$, then there is an instant in the barn frame when the entire pole is inside the barn.

The front and back doors closing simultaneously in the barn frame means

$Delta t=0$

for the two door-closing events in that frame.

The pole frame

In the pole frame, the barn moves and is length-contracted:

B'=rac{B}{gamma}

The pole definitely does not fit inside the contracted barn at a single pole-frame instant. The resolution is that the door closings are not simultaneous in the pole frame.

Using the Lorentz time transformation for separated events,

ight)$$ If $Delta t=0$ in the barn frame and the doors are separated by $Delta x=B$, then $$Delta t'=-gammarac{vB}{c^2}$$ So the door events occur at different times in the pole frame. ## Physical interpretation In the pole frame, the front door closes and opens before the back end of the pole reaches it, while the back door closes at a different time. There is no single moment in the pole frame when both doors enclose the pole. In the barn frame, both doors can close simultaneously while the contracted pole is inside. Both descriptions are consistent because simultaneity differs between frames. ## What if the doors trap the pole? If the doors physically stop the pole, the problem becomes more complex because forces and signals propagate through the pole at finite speeds, not instantly. The pole cannot know everywhere at once that its front has hit a door. Stress waves travel through the material at speeds less than or equal to $c$. Rigid bodies in the classical sense do not exist in relativity. Perfect rigidity would require instantaneous communication across an object, violating causality. ## Lessons from the paradox The paradox teaches that length contraction cannot be separated from relativity of simultaneity. To say an object fits inside a space means its endpoints are inside at the same time in a chosen frame. Another frame uses a different definition of same time. ## The big idea The barn-pole paradox is resolved by relativity of simultaneity. In the barn frame, the moving pole is contracted and can fit between simultaneously closed doors. In the pole frame, the barn is contracted, but the door closings are not simultaneous. No contradiction occurs because simultaneity for separated events is frame-dependent.