Problem — Tunnel at 0.8c

Problem

A train of proper length $L=200$ m travels at $v=0.8\,c$ towards a tunnel at rest, itself also $L=200$ m long. There are gates at both ends. (1) How long does the train measure in the tunnel’s SR (SRB)? (2) In the train’s SR (SRA), how much time passes between the closing of the exit gate and that of the entrance gate?

Solution

Lorentz factor. $\gamma = \frac{1}{\sqrt{1-0.8^2}} = \frac{1}{\sqrt{0.36}} = \frac{5}{3} \approx 1.667$

(1) Length of the train in the tunnel’s SR. The train is moving relative to SRB, so it appears contracted along the direction of motion: $L_{(B)} = \frac{L}{\gamma} = \frac{200\ \text{m}}{5/3} = 200\cdot\frac{3}{5}\ \text{m} = \ev{L_{(B)} = 120\ \text{m}}$ The contracted train (120 m) is shorter than the tunnel (200 m): it fits comfortably inside, and in SRB the two gates can close simultaneously with the train trapped.

(2) Offset between the closures in the train’s SR. In the tunnel’s SR the events $E_1$ (entrance closes) and $E_2$ (exit closes) are simultaneous and spatially separated by the tunnel’s proper length $L=200$ m. By the relativity of simultaneity, in the train’s SR the interval is $\Delta t_{(A)} = \gamma\,\frac{v}{c^2}\,L = \gamma\,\frac{v\,L}{c^2}$ Substituting $v=0.8\,c$, i.e. $v = 0.8\cdot 3\cdot 10^8$ m/s, and $c^2 = 9\cdot 10^{16}$ m²/s²: $\Delta t_{(A)} = \frac{5}{3}\cdot\frac{0.8\cdot 3\cdot 10^8\cdot 200}{9\cdot 10^{16}} = \frac{5}{3}\cdot\frac{4.8\cdot 10^{10}}{9\cdot 10^{16}} = \frac{5}{3}\cdot 5.33\cdot 10^{-7}\ \text{s}$

$\ev{\Delta t_{(A)} \approx 8.9\cdot 10^{-7}\ \text{s}}$

Interpretation. In the train’s SR the tunnel is contracted ($120$ m) and shorter than the train ($200$ m): it seems impossible to close both gates with the train inside. The paradox is resolved precisely thanks to this $\Delta t_{(A)}$: the exit gate closes and reopens before the entrance gate does, so there is never an instant, for the train driver, in which the train is trapped between two closed gates.

Links

Topics: Relatività ristretta Concepts: Contrazione delle lunghezze · Relatività della simultaneità Skills: Cambio di sistema di riferimento