Simultaneity of the lamps on the train

Problem

Qualitative: simultaneity. Two lamps switch on together in the station’s frame, one at the front and one at the back of a passing train. For the passenger on the train, which one switches on first? Argue using the relativity of simultaneity.

Solution

In the station’s frame. The two flashes leave simultaneously from the front and back of the train. In the station’s frame, by definition of the problem, the two events are contemporaneous.

Change of frame. For the passenger the train is at rest and it is the station that moves backwards. The relativity of simultaneity says that two events that are simultaneous and spatially separated in one frame are not simultaneous in another frame moving relative to it. We need to establish which of the two precedes.

Reasoning with the light fronts. Imagine the flashes meet at the middle of the train in the station’s frame. The passenger at the centre of the train moves towards the front (in the direction of $v$) and away from the back. So they move towards the light front coming from the front and “flee” from the one from the back: they receive the light from the front first. Since for them the two lamps are equidistant and light travels at the same speed $c$, they must conclude that

$\ev{\text{for the passenger, the lamp at the front of the train switches on FIRST}}$

Confirmation with the Lorentz transformation. With the front at the greater coordinate ($\Delta x > 0$ from back to front) and the train moving at $+v$, the transformation gives $\Delta t' = -\gamma\,v\,\Delta x/c^2 < 0$ for the front event: it occurs before in the train’s frame. There is no absolute “before” or “after”: the order depends on the observer.

Links

Topics: Special relativity Concepts: Relativity of simultaneity Skills: Changing reference frame