special relativity - Feynman on transverse length contraction - SR

This is what Feynman had to say about transverse length contraction in special relativity.

How do we know that perpendicular lengths do not change? The men can agree to make marks on each other’s y-meter stick as they pass each other. By symmetry, the two marks must come at the same y- and y′-coordinates, since otherwise, when they get together to compare results, one mark will be above or below the other, and so we could tell who was really moving.

I understand other arguments used to explain why we can't have transverse length contraction (e.g. an event in spacetime not being the same in all reference frames when there is transversal contraction). However, I don't understand this one.

1. If they're moving relative to each other, the marks will be the same height on each one's meter stick, so when they come together they both "decontract" so there is no contradiction and one could not tell "who was really moving".


2. Even if one could tell who was really moving, feynman's own explanation of the "twin paradox" could be applied here. When they come together one of them accelerates back to the other and he felt this so this would mean that he "absolutely" could tell he was moving away from the other person staying on earth. The one who felt the acceleration has property X when he comes back to compare; X in the twin paradox being difference in age and here being difference in position of the mark.

Maybe I totally misunderstand this argument so any help would be appreciated. Also, I would like for someone to clarify what he means by "by symmetry", symmetry in the sense that nothing should change when switching (inertial) reference frames?

Answer

I’m stationary with my stick. If there was transverse contraction/dilation, the mark on my stick will be in a different place than the moving one: there’s no need to stop the moving stick to see that. For simplicity, say the moving stick is shorter.

But you’re holding that stick. You don’t think you’re moving. But at the crossing, you see your stick marked as contracted! (You can’t see it as not; that’s not what would have happened in this hypothetical-transverse case)

So there’s the break in symmetry: you both should be able to think of the other as moving. And you can’t, because only one of the two “moving” sticks can be shorter.

Note: this is different from measuring length because that involves simultaneity along the motion. This involves simultaneity at a point, the marking-the-stick point, which is (almost) automatic.