Take the usual ladder paradox in special relativity, but this time the ladder and the barn have the same length, at rest. Since each one of them contracts when viewed in the rest frame of the other, I would guess that:
- an observer on the ladder would see that the barn is too small and the ladder doesn't fit inside,
- an observer at the barn would see that the ladder is contracted so now it can fit in the barn with room to spare.
My problem is, I tried solving it with the diagrams in Minkowski spacetime (drawing the barn as two parallel lines and intersecting with the two lines of the ladder, etc.). Basically the diagram turns out to be like this one (from https://en.wikipedia.org/wiki/Ladder_paradox)
except that the red segment has a length equal to the horizontal width of the blue band, that is two "big squares" of the grid. Wikipedia says (referring to the diagram in the image):
We see that such line segments never lie fully inside the blue band; that is, the ladder never lies fully inside the garage.
But in my case, the red segment can very well fit inside the blue band, and I think this tells me that the ladder would fit in the barn even in the ladder rest frame.
I think I did something wrong with this reasoning, but I can't find what it is...