04. The pole and the barn

A 20-meter pole is carried at $\gamma = 2$ toward a 10-meter barn with doors at both ends. The farmer says: contracted to 10 meters, the pole fits — slam both doors simultaneously, and for an instant it is entirely inside. The runner says: the barn is contracted to 5 meters, and the pole never fit at any moment. Both cannot be right about the doors — can they?

Solution

Both are right, and the doors never contradict either of them.

“The pole fits” secretly means “both ends are inside at the same time” — it is a simultaneity claim, not an invariant statement. The two door-slams are simultaneous in the barn frame, and in no frame moving along the barn’s axis. Which one comes first for the runner is fixed by $t' = \gamma\,(t - vx/c^2)$: of two events simultaneous in the barn frame, the one farther along the direction of motion is the earlier one. So in the runner’s frame the far door slams and reopens first — just as the pole’s front end arrives at it, with fifteen meters of pole still trailing outside the entrance of the five-meter barn — and only $\gamma v L/c^2 \approx 58\ \mathrm{ns}$ later does the near door slam behind the rear end, by which time the front end is already fifteen meters past the far doorway. The slams happen in sequence, and at no moment is the pole enclosed. The two events are spacelike separated, so their time-order is frame-dependent; give the barn a hair of clearance so the fit is strict rather than grazing, and no door ever touches the pole in either account.

The paradox only acquires teeth if you demand the pole stop inside the barn — and then Problema I collapses it: the pole cannot stop rigidly. The front stops first, a compression wave runs back, and what ends up enclosed is a genuinely shortened, crumpled pole in every frame. The shortening is violent: even with the stress front running back at the absolute limit $c$, the pole comes to rest with proper length $L_0\sqrt{(1-\beta)/(1+\beta)} \approx 5.4\ \mathrm{m}$ — better than fifty percent strain, which no elastic limit survives. That is why it stays crumpled rather than springing back out.

Deeper in the notebook: 02. The Ladder (Pole-and-Barn) Paradox · 01. Timelike, Null and Spacelike Separation; the Causal Order