I begin by saying that I'm a math student who has recently come across the study of special relativity, so I apologize in advance for my lack of physical insight. In the famous train paradox, just to briefly recall it, there is a train of length $L$ (measured at rest with respect to the tunnel reference) which travel at a costant fraction of the speed of light $0<\gamma<1=c$. There is also a tunnel, of length $h$, with $h<L$ and $h>L'$ where $L'$ is the measure of the length of the train for a frame of reference at rest with respect to the tunnel. Now, since $h>L'$ it is possible for a frame of reference at rest with respect to the tunnel to close SIMULTANEOUSLY (with respect to the tunnel reference) the two holes of the tunnel itself when the train is all inside the tunnel (this is possible since $h>L'$). It's clear to me that in the frame of reference of the train the two events of "closing" are not simultaneous and so a person inside the train would see one hole closing before the other. My question is purely physical and maybe silly, but I'm not able to figure it out properly: how is it possible for the frame of reference of the tunnel to capture the train inside if physically the "real" length of it is greater than $h$? In my head this would not be possible because when the two holes are closing simultaneously at least one of the hole is crashed by the train. What am I missing?
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$\gamma$ is the Lorentz factor given by $$\frac{1}{\sqrt{1 - \beta^2}}$$ where $\beta$ is a common variable used for the fractional velocity, that is $$\beta=\frac{v}{c}$$ – Triatticus Mar 28 '22 at 20:14
2 Answers
The "proper length" of an object (I think what you're calling the "real" length) is its length according to an observer who is stationary with respect to it. The proper length of the train is longer than the proper length of the tunnel, which results in the mundane observation that you can't park the train fully inside the tunnel - when both the train and the tunnel are at rest according to a trackside observer, the train is too long to have both ends inside simultaneously.
The mundane observation of what happens when everything is at rest does not indicate what happens when one object or the other is moving at a significant fraction of $c$. As the train moves at different speeds with respect to different observers, it has different length according to those observers. A trackside observer will see that the speeding train has a length shorter than its proper length - this isn't an optical illusion, by any measure the trackside observer chooses to use, the train actually is shorter. The trackside observer can never observe the train at rest parked fully inside the tunnel, but can witness it fit momentarily as it speeds through at near-lightspeed. Simply put, the trackside observer sees that the train is shorter when it's moving than when it is at rest.
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It is a consequence of the fact that the clocks may be syncronized for all wagons of the train, and for each correspondent distance on the station. But the clocks of the train are not syncronized with the clocks of the station.
So, Bob at the first wagon will see a shutter (like from a diafragm) closing and opening the tunnel ahead at $11:59:59$ according to his clock. At same time he sees the clock at his side on the track, that is showing $12:00:00$.
Alice at the last wagon will see another shutter closing and opening the tunnel behind at $12:00:01$ according her clock. At same time she sees the clock at her side on the track, that is showing $12:00:00$.
So, there is nothing wrong for the people in the train, because the events are not simultaneous. But they will record that the station clocks are showing the same time.
The people in the station simply has a mechanism to drive the shutters at $12:00:00$ according to synchronized clocks nearby each device, and it works with no damage in the train.
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