Problem in the Ladder paradox in relativity?

Question

The ladder paradox consists of a ladder rushing towards a garage with two open doors. In the actual paradox, proper length of the ladder is greater than that of the garage, but in this case lets consider them to be of the same proper length.

In the frame of the garage, the ladder is Lorentz contracted, and hence it is possible to "trap" the ladder inside the garage. By "trap" I mean that both doors of the garage must be closed while the ladder is completely inside the garage.

But in the frame of the ladder, the garage is Lorentz contracted, and hence it is never possible to "trap" the ladder inside the garage.

If I am able to trap the ladder in one frame, I should be able to do it in the other frame too right? Why then is there this paradox?

Answer 1

It is because of the relativity of simultaneity.

In the garage frame, the two doors are closed at the same time when the ladder is completely inside.

In the ladder frame, the front door is closed first, the front end crashes with the front door. The rear end keep moving into the garage and the rear door is closed after it enters.

So in both frames, the garage can "trap" the ladder.

Answer 2

The ladder won't be trapped in both reference frames. If an observer with the garage closes the doors at the same time, so as to "trap" the ladder inside the garage, the events corresponding to the two doors closing will be simultaneous in the garage frame, but not in the ladder frame. An observer with the ladder will see the front door closing first and the back door closing later so that the ladder won't be trapped into the garage. Look at this video for more details.