= Tunnel photon timing in two inertial frames
{title2=$f=\beta/(1+\beta)$}
A runner enters a stationary tunnel of <length> $L$ at the tunnel-frame instant a photon is emitted towards them from the far end. The interception fraction is $f=\beta/(1+\beta)$, where $\beta=v/c$. In the runner frame, with entry at the origin, emission is at $(x',t')=(\gamma L,-\gamma vL/c^2)$ and the entrance clock simultaneously reads $-vL/c^2$. The entrance is then ahead by $\gamma\beta^2L$, so the photon travels $\gamma L$, not the contracted tunnel <length> $L/\gamma$. Subtracting that initial entrance gap from the entrance's motion during the photon flight gives the same interception fraction. This is a concrete application of <relativity of simultaneity>, <length contraction> and the <Lorentz transformation>.
Back to article page
