The Relativity of Simultaneity — Question 9

Question 9

Simultaneous gate closures in a tunnel are not simultaneous on a train

The entrance and exit gates of a tunnel, separated by $L=90\,\mathrm m$ in the tunnel frame, close momentarily at the same tunnel time. A train frame moves at $v=0.80c$. No gate remains closed. Use $\Delta t\prime=\gamma(\Delta t-v\Delta x/c^2)$, $\gamma=(1-v^2/c^2)^{-1/2}$; $S\prime$ moves at +v along x. Take $c=3.00\times10^8\,\mathrm{m/s}$.

1. Find the exit-closure time minus entrance-closure time in the train frame.

2. Find the separation of these two closure events using $\Delta x\prime=\gamma(\Delta x-v\Delta t)$ and explain why it is not the train-frame tunnel length.

Question 9 - Solution

Simultaneous gate closures in a tunnel are not simultaneous on a train

(a) Method and calculation

$$\gamma=5/3,\qquad\Delta t=0,\quad\Delta x=L.$$$$\boxed{\Delta t\prime=-\gamma vL/c^2=-0.400\,\mu\mathrm s}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta x\prime=\gamma L=150\,\mathrm m}.$$$$\Delta t\prime\ne0\quad\Rightarrow\quad\Delta x\prime\text{ is not a simultaneous length measurement}.$$

Physical check

The exit closes first in the train frame. A length requires endpoint positions taken at the same time in the measuring frame.