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The following problem is taken from Exercise 1.7 in David Morin's "Special Relativity: For the Enthusiastic Beginner":

A train and a tunnel both have proper length L. The train moves toward the tunnel at speed $v$. A bomb is located at the front of the train. The bomb is designed to explode when the front of the train passes the far end of the tunnel. A deactivation sensor is located at the back of the train. When the back of the train passes the near end of the tunnel, the sensor sends a signal to the bomb, telling it to disarm itself. Does the bomb explode?

The solution given in the book is in the affirmative. It is certainly obvious from the train frame. Since the tunnel is length-contracted, the front of the train passes the far end before the back of the train passes near end of the tunnel. However, as seen from the tunnel frame, it seems we have a contradiction as the order of the two events reverses.

The resolution to this paradox given in the book says the deactivation device cannot instantaneously tell the bomb to disarm itself. A signal takes time to travel to the front of train, and it is calculated that, even if the signal has speed $c$, the transmission time is still longer than the time it takes for the front of the train to pass the far end of the tunnel. Thus, the bomb explodes in the train frame as well.

Nevertheless, the solution doesn't seem quite satisfactory to me. What if we change the question to "Does the deactivation sensor send a signal to disarm the bomb?" This certainly looks like a frame-independent statement and all observers must agree on whether or not the deactivation sensor initiate a signal.

Question:

Is the statement "The deactivation sensor sends a signal to disarm the bomb" frame-independent in the train-tunnel paradox? If yes, what is the correct answer to the above statement?

Edit: I don't think the proposed duplicate pinpoints the loophole behind the apparent "contradiction". The below two answers did a much better job clarifying my doubt.

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Yes. It is frame-independent. From the tunnel frame it is obvious that the signal is sent, as this happens before the bomb explodes in that frame. From the train's frame, the bomb explodes before the signal is sent, but that doesn't mean the signal is not sent. This is because the bomb is at the front of the train, therefore not all the train explodes at the same time. When the bomb explodes, the front of the train explodes, but the end of the train (and therefore the sensor) is yet perfectly still, because whatever destruction the sensor may suffer due to the shock wave produced by the explosion takes at least a time $L/c$ to reach the end of the train. On the other hand, the time that it takes the back of the train to enter the tunnel since the activation of the bomb is $$\displaystyle\frac{l_{\text{tunnel}}-L}{v}=\frac{\displaystyle\frac{L}{\gamma}-L}{v}=L\ \frac{1-\gamma}{v\gamma}$$ and you can check that this time is less than $L/c$, and this means that the signal is sent before the sensor is destroyed by the wave caused by the bomb.

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  • $\begingroup$ What if the bombs are evenly distributed along the chain and explode simultaneously as this comment suggests? Is this impossible? $\endgroup$
    – YuiTo Cheng
    Commented Jul 21, 2020 at 13:59
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    $\begingroup$ If the event that causes them to explode is the train arriving at the end of the tunnel, then that is not possible, because it would violate causality. In that scenario, a bomb would explode at the end of the train just because the front of the train crossed the tunnel, without time for information to have reached from one end of the train to the other. $\endgroup$
    – Urb
    Commented Jul 21, 2020 at 14:16
  • $\begingroup$ @YuiToCheng Simultaneity is frame-dependent. If they're timed to go off simultaneously in the train frame, then in the tunnel frame the bombs at the back of the train go off first (and before the trigger is tripped, which means they have to have been pre-set to do this). On the other hand, if they're timed to be simultaneous in the tunnel frame, then in the train frame the ones at the front of the train go off first. $\endgroup$
    – Gordon Davisson
    Commented Jul 21, 2020 at 23:15
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Is the statement "The deactivation sensor sends a signal to disarm the bomb" frame-independent in the train-funnel paradox? If yes, what is the correct answer to the above statement?

Yes, it is a frame independent statement. You can think of it as a measurement. The sensor measures something and reports the value of that measurement. The outcome of any measurement is frame independent, meaning for example that I may observe your watch to be running slow, but both you and I will agree on what your watch actually says. You will believe that it is running correctly and I will believe it is running slow, but whatever it reads we both agree.

As described the sensor sends the signal. The signal is sent whenever the back of the train passes near the end of the tunnel. The back of the train passes near the end of the tunnel in all frames so the signal is sent in all frames. The bomb explodes before the signal is sent in the train frame, but the explosion travels (much) slower than c and therefore the deactivation sensor in the rear is still intact as the rear reaches the end of the tunnel.

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  • $\begingroup$ Thanks for your response. But as observed from the train frame, the front of the train will pass the far end of the tunnel first, and the train will explode instantaneously so that the the deactivation sensor located at the back of the train doesn't function any more. How do you account for that scenario? $\endgroup$
    – YuiTo Cheng
    Commented Jul 21, 2020 at 12:12
  • $\begingroup$ @YuiToCheng Nothing is instantaneous. I would highly recommend working with Lorentz Transformations instead of length contractions/time dilations. Then you can easily determine the space-time coordinates for each event of interest and see how they all line up in each frame. It might be beneficial for you to do this, although Dale's answer would benefit greatly from showing this as well. $\endgroup$
    – BioPhysicist
    Commented Jul 21, 2020 at 12:55
  • $\begingroup$ The problem has the bomb at the front of the train, so the answer is a simple "yes". Of course, since things are "pre programmed", the bombs could very well be distributed and simultaneous across the train, and then the problem has some subtleties. $\endgroup$
    – JEB
    Commented Jul 21, 2020 at 13:20
  • $\begingroup$ @YuiTo Cheng I added some additional explanation in the answer. $\endgroup$
    – Dale
    Commented Jul 21, 2020 at 15:14

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