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Consider a particle travelling at 0.9c through a 9-light-second long tube which erects a wall at exactly 10 seconds to block off the particle. For a stationary (relative to the tube) observer, he sees that it takes the particle exactly 10 seconds to traverse through the tube, so he sees the particle getting blocked by the wall.

However, from the particle's perspective, its journey through the tube is shortened by a factor of $\gamma = 1/\sqrt{1-0.9^2}$, so it will take less than 10 seconds to go through it. Further, the wall's erection will be dilated according to the particle, so it's erected after 10 seconds. Thus from the particle's perspective it is able to travel through the tube unblocked, which contradicts what the observer sees.

This is an obvious paradox, which part of my reasoning was flawed?

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As Dale said, you forgot the relativity of simultaneity, which caused you to misapply the concept of time dilation.

If we stick with the coordinates suggested by JEB, the particle enters the tube at t=t'=0. In the frame of the tube, t=0 is also the time when the construction of the wall started at the other end. But in your frame, the time at the other end of the tube is t'=0, which owing to the relativity of simultaneity is not t=0 but some much later time in t when the wall is already well under way to being finished.

The concept of time dilation is that the elapsed time between two events that occur at the same place in one frame is less than the elapsed time between them in another moving frame. So you can apply that in one of two ways in your scenario. One is that you can compare the 10s it took to build the wall in the tube frame with the corresponding time it took in your frame. For that you would have to work out what time the construction started in your frame (which certainly wasn't t'=0). The other is that you could consider just the time taken to complete the remaining part of the wall that had still to be built at t'=0 when you entered the tube. Either way, you will find all the numbers are consistent with the fact that the particle hits the wall.

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  • $\begingroup$ ah perfectly explained $\endgroup$ Commented Jan 22, 2022 at 8:38
  • $\begingroup$ Why, thank you, Joshua- you are very kind! $\endgroup$ Commented Jan 22, 2022 at 9:08
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This is an obvious paradox, which part of my reasoning was flawed?

You forgot the relativity of simultaneity. This is the resolution of most of the SR paradoxes.

Although the wall construction is indeed time dilated in the particle’s frame, the construction started much earlier. Sufficiently early that all frames agree that the particle hits the wall.

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  • $\begingroup$ can you back this up with calculations? thank you $\endgroup$ Commented Jan 22, 2022 at 1:07
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For calculations, set $c=1$, and use coordinates $(t, x)$. Place the tube entrance/entry at both origins, where (un)primed frame is (tube) particle:

$$ E_0=(t,x)=(0,0)$$ $$E_0=(t',x')=(0,0)$$

and the wall erection in the tube frame:

$$ E_1=(t_1,x_1)=(10,9)$$

Now transform $E_1$ to the particle frame:

$$ t'_1=\gamma(t_1-\beta x_1)=\gamma(10-0.9\cdot 9)\approx 4.46$$

$$ x'_1=\gamma(x_1-\beta t_1)=\gamma(9-0.9\cdot 10)=0$$

So the particle doesn't move in the primed frame, and after 4.36 seconds, it hits the wall.

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  • $\begingroup$ Ah interesting, this makes sense (mostly). From the particle frame its $x_1'=0$, why does that prove it hits the wall? Isn't the particle frame always at rest anyways? ------------ Also, it seems that my fallacy is in applying the time dilation and length contraction formulae i.e., $t'=\gamma t$ and $x'=x/\gamma$, because in that case I get that (a) the particle has to travel less distance, thus time is shorter and (b) the wall takes longer time to erect in the particle's frame, and overall the particle won't hit the wall. Could you explain why I cannot use time dilation formula here? $\endgroup$ Commented Jan 22, 2022 at 3:57
  • $\begingroup$ Do not worry I have figured it out $\endgroup$ Commented Jan 22, 2022 at 5:58

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