Variation of the pole-barn or train-tunnel paradox ( does it break causality )

Question

I know that in the standard train tunnel paradox, the train shrinks in the tunnel frame and the tunnel shrinks in the train frame. Paradox is resolved by the fact that even though both doors of tunnel close simultaneously in tunnel frame, the further door closes first in the train frame.

Now, suppose we consider a scenario where , in the tunnel frame , once the rear of the train has entered the tunnel, this sliding door shuts down. Now, a sensor detects this shutting down and sends a signal to the other end of the tunnel to shut down its door, so that the train is now inside the tunnel with both doors closed. ( I know that signal travels at finite light speed, but since train travels at sub-light speed, the signal should reach the other door before train reaches it )

From the train frame,the tunnel has shrank down. But we cant use the usual relativity of simultaneity explanation because that would violate causality, since the door further along is triggered by the signal from the other door.

So, can anyone tell what each frame would observe so that causality is not violated ?

" ( I know that signal travels at finite light speed, but since train travels at sub-light speed, the signal should reach the other door before train reaches it )". .... this obviously not possible. The signal can never reach the exit door before the front of the train does. — JEB, Jul 09 '20 at 17:09
Related : (1) Special Relativity - Regarding the Simultaneity of Events During the Train Paradox.(2) Are my intuitions about special relativity right?. — Frobenius, Jul 09 '20 at 18:31
@JEB: You're a person of few words :) I know that faster than light signals would violate causality (in appropriately chosen reference frames), but why would the light signal reaching the exit door before the front of the train violate causality? In words please, as it is supposed to be obvious. — Not_Einstein, Jul 10 '20 at 01:32
@Not_Einstein I'm just saying the OP is right: if you could do this, it would violate relativity, so my conclusion is: therefore you can't do it. I think I also posted and answer, but maybe it's shadow banned? — JEB, Jul 10 '20 at 13:29

score 1 · Accepted Answer · answered Jul 09 '20 at 17:47

"( I know that signal travels at finite light speed, but since train travels at sub-light speed, the signal should reach the other door before train reaches it )"

Before throwing out relativity, it's best to check your assumptions, and that one is suspect.

The most extreme case is for a tunnel of length $L$, and a train of length $L+\epsilon$, where $\epsilon \rightarrow 0$. We can choose length units such that $L=1$ and $c=1$.

With that, the time it takes the "close" signal to reach the exit door is:

$$ t_1 = \frac 1 1 = 1 $$

The train is Lorentz contracted to $(1+\epsilon)/\gamma$, so the time it takes the front of the train to reach the exit is:

$$ t_2 = \frac{1 - (1+\epsilon)/\gamma}{\beta} = (1 - (1+\epsilon)\sqrt{1-\beta^2})/\beta $$

In the limit that $\epsilon \rightarrow 0$:

$$ t_2 = \frac 1 {\beta}(1-\sqrt{1-\beta^2}) < 1 $$

as shown in the figure:

So in the limit that $\beta \rightarrow c$, the train length $\rightarrow 0$, and it rides with the light beam to exit, but as long as $\beta < c$, the front of the train reaches the exit before the signal, preventing your causality violating condition from occurring.

Variation of the pole-barn or train-tunnel paradox ( does it break causality )

1 Answers1

Linked