Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 4 / 17C / d / Solution

With light signals, $u=c$, the train-frame relativistic velocity-addition formula gives signal velocities $+c,-c$. The endpoints are equally distant from the source in that frame, andThe two reception events are simultaneous in $S$ but not in $S^{\prime }$: in the station frame the rear moves toward its light signal while the front moves away. Directly, $\Delta x=x_B-x_A=-L$ and $\Delta t=0$, so the Lorentz transformation gives $\Delta t^{\prime }=\gamma (\Delta t+v\Delta x/c^2)=-\gamma Lv/c^2$. This is relativity of simultaneity, concerning spatially separated events; it does not contradict the invariant light speed or the causal ordering of events on a single observer's world line.