I've watched video Einstein's Relativistic Train in a Tunnel Paradox: Special Relativity and have read wiki - Ladder_paradox. My question for the case where train is trapped in the tunnel. Unfortunately (in my opinion) in the video from train view it is still train moving, though longer than the tunnel. However, from train view it is tunnel moving which will engulf the train partly. As I understand the solution to the paradox assumes tunnel/barn are born rigid, even pointing out for the solution is that no body is perfectly rigid.

However, properly let's realize tunnel is also non-rigid and upon hitting the train with unbreakable back door front of the tunnel will continue to move further engulfing the train. As I understand from time differences/Lorentz transforms how much train should shrink/tunnel elongate does not depend on relative rigidity, but on speed of light in a vacuum - how fast info that collision occurred would travel to back of the train/front of the tunnel.

If logic of train shrinking in original explanation is correct, looks to me tunnel will elongate and train will shrink "to full extent", that is train would shrink to fit in original tunnel and tunnel would elongate to fit original train (we'll get short train in much longer for it tunnel). But that is paradox comparing to the fact that train should perfectly fit in the tunnel. Or not?

P.S. Wiki page does not consider elongating the barn and I could not find answer via web search.


Some of this depends on how the train applies its brakes. Let's suppose the brake signal is sent from the track, simultaneously in the track frame, from a position on the track to the wheel directly above it. The brakes are perfect and go from $v$ to zero instantly.

In this case the short train enters the tunnel, and knowing it will hit the exit door, the track signal is pre-computed and sent simultaneously to the train. The train is 100% in the tunnel and finds itself at rest, crunched in its Lorentz contracted state by relativistic stresses.

From the train frame, the braking is not simultaneous. The front of the train stops 1st, with the rear well outside the tunnel. As the train slams into the tunnel, the braking signal works its way backwards and finally stops the caboose as it just enters the tunnel. The train is crunched by inertia forces.

Another option is that the train controls the brakes, and the conductor knows before hand, when to apply them to all the wheels, at the same time. (As in the track case, this has to be precomputed so that the braking events at different locations can happen simultaneously).

In the train frame, the front of the train hits the exit door and the conductor stops the entire train at once. Of course, the caboose is still well outside the tunnel when it stops, and the train finds itself stretched by relativistic stresses.

From the tunnel point of view, the caboose stops 1st and as the train is stretched by inertial forces, the braking 'wave' moves up the train, reaching the cab as it hits the exit door. The train has been stretched by inertial forces.

Those are the 2 extremes. You can also work out cases in which the cab or caboose control the braking, and the signal propagates along the train at the speed of light.

If the caboose sends the signal, and applies the brakes when it enters the tunnel, the signal cannot reach the front of the train before it has exited the tunnel.

If the cab applies the brakes, then it cannot stop the caboose before it enters the tunnel.

So the point here is if the train doesn't have brakes, it just slams into the exit door. The rigidity of the tunnel door doesn't matter because it has no longitudinal extent, nor does the rigidity of the tunnel: it is passive.

If you want the train to stop, you need to define how the brakes are applied, and since this is a thought experiment, engineering factors are not a concern: you can make the entire length of the train stop simultaneously, but it matters which frame you choose to define "simultaneous".

As we know from the Bell's Spaceship paradox, an object cannot accelerate uniformly with respect to an inertial frame without experiencing relativistic stress, and that is the case here.

If the brakes are simultaneous in the train frame, the train is stretched by $\gamma$ when it stops. If they are simultaneous in the tunnel frame, then it is crunched by a factor $\gamma$ when it stops, in which case, it fits in the tunnel.

  • $\begingroup$ Thanx, but: "From the train frame, the braking is not simultaneous. The front of the train stops 1st," - no, no, no. The train cannot stop as it is already at rest in its frame, tunnel goes to it. It is my understanding now I want to "see" correct way. $\endgroup$ – Martian2020 Nov 10 '20 at 22:20
  • $\begingroup$ Interesting twist, you added trails. There are not rigid, right? As they move toward the train and it hits the brakes in the front, trails under the train will continue to go and trail should elongate. Or not? $\endgroup$ – Martian2020 Nov 10 '20 at 22:29
  • $\begingroup$ @Martian2020 while the train is not moving in its coordinates, the conductor is certain that it is moving down the track, which as for all trains: defines an absolute frame of rest. For trains. $\endgroup$ – JEB Nov 11 '20 at 1:35
  • $\begingroup$ @Martian2020 What is a trail? You mean the train cars? If so, I'll edit the answer. $\endgroup$ – JEB Nov 11 '20 at 1:38
  • $\begingroup$ Sorry, not trail, "track". "the conductor is certain that it is moving down the track" - we are in a thought experiment, why are you so sure? "Absolute frame of rest"... - as of now AFAIK SR does not have absolute frame, in non-flat spacetime we sometime cannot go w/out it... $\endgroup$ – Martian2020 Nov 11 '20 at 1:47

Thought about it for a while.

The door stops the front of the train not because tunnel is rigid, but because door is fixed to "Earth" - object with much more mass than the train. At least the solution I write here assumed that, if tunnel is not fixed to other mass then I guess relative movements would be different. Therefore from tunnel point of view door just stays in the place and tunnel length remains the same.

From train point of view, it got hit by object of large mass and train shrinks 50%. As tunnel stops, Lorentz length contraction goes away, and tunnel elongates from 1/4 to 1/2 of original train length. Therefore train fits exactly to the tunnel.


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