The train and tunnel paradox

A train of proper length $L$ travels at speed $v$ close to $c$ towards a tunnel of proper length $L$ (the same length as the train “at rest”). At the tunnel’s mouths there are two automatic gates, able to close and reopen instantaneously. The question is simple: can the train, for an instant, be trapped inside the tunnel with both gates closed?

Tunnel reference frame (SRB). The train is contracted: we see it as $L/\gamma < L$ long, so it fits comfortably. We can close simultaneously the two gates at the instant the last carriage has just crossed the entrance: for a brief instant the train is trapped inside the tunnel. Immediately after we reopen the gates and the train continues. No problem.

Train reference frame (SRA). From the driver’s point of view it is the tunnel that is contracted, at $L/\gamma < L$. Then the train, which remains $L$ long (its proper length), is longer than the tunnel. It seems impossible for both gates to close with the train inside: at least part of the train should stick out of one gate or the other. A contradiction?

Where is the catch? In simultaneity. Call $E_1$ the event “the entrance gate closes” and $E_2$ “the exit gate closes”. In SRB they are simultaneous. But they are spatially separated by an amount equal to $L$ (the proper length of the tunnel), so they cannot be simultaneous in SRA. By the formula for simultaneity:

$\Delta t_{(A)} = \gamma\,\frac{v}{c^2}\,L \neq 0$

In fact, since $E_2$ is downstream of the train’s motion, in SRA the event $E_2$ occurs before $E_1$: the exit gate closes and reopens before the entrance gate closes. From the train’s viewpoint, the complete sequence of the four events is:

1. the exit gate closes ($E_2$): the locomotive has not yet arrived;
2. the exit gate reopens ($E_4$): the locomotive passes through freely;
3. the entrance gate closes ($E_1$): the last carriage is already inside and has by now crossed the entrance;
4. the entrance gate reopens ($E_3$): the train continues.

There is never an instant, in SRA, in which both gates are closed at the same time — which is precisely what would be needed to “imprison” the train. The two descriptions are different but perfectly consistent: no contradiction.

In the tunnel’s frame the two gates close simultaneously: $E_1$ and $E_2$ lie on the same vertical, i.e. at the same instant $t_{(B)}$.

In the train’s frame the closing of the exit gate $E_2$ happens before that of the entrance gate $E_1$: the train is never “entirely inside” with both gates closed.

Links

Topics: Special relativity Concepts: Relativity of simultaneity Skills: Constructing Minkowski diagrams Methods: Minkowski diagram

Related exercises: Space-type or time-type · Problem — Tunnel at 0.8c · Problem — Simultaneity (true or false)