This came up in our special relativity assignment, and I hadn't heard of it before. I'm still not quite satisfied by the explanations I've seen online. Apparently, there is a recent disagreement with the (accepted) solution. So one statement of the problem is as follows:
A rod is sliding on a table with a constant velocity (in a direction along its length). There is a hole in the table. The width of the hole (in the rest frame of the table) equals the rest length of the rod. From the table frame, the rod will be length contracted, and hence will fall into the hole. But, from the rod's rest frame, the width of the hole will be length contracted and the rod will not fall through.
So, what happens?
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Let ends of rod be A and A' and the ends of the hole be B and B'. Let event 1 be the event of A and B occupying the same space co-ordinates and event 2 be the event of A' and B' occupying the same space co-ordinates, say both in the table frame.
The space separation between the two events (in the table frame) is l_0 (rest length of the rod) and the time separation is l_0(1-1/\gamma)/v. A simple calculation shows that the events are separated by a spacelike interval and hence, the order of occurrence of the events may be different in different frames.
I had read this problem earlier in Kleppner & Kolenkow's Mechanics and also in Griffiths' Electrodynamics (one of my favourite physics texts). But only now did I work it out. :)
What happens, you ask? My guess is that the rod falls because if you are bringing gravity into consideration, then the calculations in the frame of the table are not incorrect atleast, with a fixed earth. In the frame of reference of the rod, which has a moving planet below it, you must call for general relativistic calculations, I think.
If you try to run this experiment, the rod will of course, begin to tilt the moment its centre of mass crosses one end of the hole, and so it is doomed into the abyss.
Ah, one case where Newtonian mechanics yields the same result as General Relativity in all its glory! Albeit for different reasons. :P
All right. Forget about gravity. The original problem we were given had a ring and a rod. The diameter of the ring equals the rest length of the rod. The ring and rod are moving at right angles to each other with constant velocities. Will the rod pass through the ring? From the rod's reference frame, what happens when end A meets edge B? Will the rod deform ( as some sites seems to suggest)?
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