Relativity Challenge

[kamil9876]

This one was inspired by dcc's spam in the relativity thread:

A proper length of a train is equal to the proper length of a tunnel. The train goes through the tunnel at a very fast speed. According to length contraction, the train is shorter than the tunnel and hence is covered completely by the tunnel. However, from the train's reference frame the tunell is moving very fast and so it is length contracted and so the train is never completely covered by the tunnel.

SOlve this apparent paradox

[kamil9876]

Okay maybe as an extension/hint:

Imagine that in the tunnel's reference frame, once the train's front is on the verge of leaving the tunnel, a boulder falls at the entrance that it came in. Because the train is shorter than the tunnel the boulder doesn't hit the train. However in the train's reference frame once the front of the train is on the verge of leaving, the back is still sticking out the tunnel and so the train would get struck by the boulder. SO does the train really get struck or not?

I think this is what dcc meant by 'does the car get wet'

[Mao]

Imagine that in the tunnel's reference frame, once the train's front is on the verge of leaving the tunnel, a boulder falls at the entrance that it came in. Because the train is shorter than the tunnel the boulder doesn't hit the train. However in the train's reference frame once the front of the train is on the verge of leaving, the back is still sticking out the tunnel and so the train would get struck by the boulder. SO does the train really get struck or not?

what frame is that? cannot assume simultaneity.

[kamil9876]

Mao, I see your understanding of relativity is good, but I'm worried about your reading skills

Okay maybe as an extension/hint:

Imagine that in the tunnel's reference frame, once the train's front is on the verge of leaving the tunnel, a boulder falls at the entrance that it came in.

[evaporade]

Relative to the train:
Event A (front leaving tunnel) and event B (rear entering tunnel) are not simultaneous. In fact event A occurs later than event B. At the time that event B occurs, the front of the train is still inside the tunnel.

[/0]

Relativity makes my head hurt -.-

[kamil9876]

Yep, well done evaporade. So the train does not get hit by the boulder in both frames. Assuming it's simultaenous in tunnel frame(train leaving end and boulder falling) then the train misses the boulder by length contraction. However the event of the boulder falling and the train's front being at the tunnel exit are not simultaenous in train frame. Specifically, the boulder falling happens at a later time and at this time the train has completely passed the rear.
THis can be quanititavely justified with lorentz transformations

[Mao]

Mao, I see your understanding of relativity is good, but I'm worried about your reading skills

Okay maybe as an extension/hint:

Imagine that in the tunnel's reference frame, once the train's front is on the verge of leaving the tunnel, a boulder falls at the entrance that it came in.

* Mao facepalms

Despite the fact that I do realise the importance of that statement, and I highlighted the segment immediately after it, I still managed to not read that bit... I really do wonder what my head does these days...

[kamil9876]

Well there were 2 commas in that sentence so it was pretty long. Plus it was past your bedtime.