Relativity of simultaneity

The two postulates force an immediate and startling conclusion about a term we've been using loosely: simultaneous. In Newtonian physics, 'at the same time' is absolute — if two events happen simultaneously for one observer, they happen simultaneously for everyone, everywhere, no matter how fast they're moving. Special relativity denies this, and the cleanest way to see why is Einstein's own thought experiment: a train and a platform.

Picture a very long train moving at speed v past a station platform. Two bolts of lightning strike the front and rear of the train at the two points on the platform that the train's ends happen to be passing — and a platform observer standing exactly halfway between those two points sees both flashes arrive at her at the same instant. Since light from each strike traveled the same distance at the same speed c to reach her, she correctly concludes the two strikes were simultaneous — in the platform frame.

Now consider a passenger sitting exactly at the midpoint of the train itself. By the time the light from the two strikes reaches the platform's fixed midpoint, the train — and the passenger — have moved forward by a small amount. She is moving toward the point where the front flash occurred and away from the point where the rear flash occurred. Because light travels at the same speed c in her frame too (postulate 2 again — there is no 'her light is faster' escape hatch), the wavefront from the front strike reaches her before the wavefront from the rear strike. She has no choice but to conclude the front of the train was struck first.

This isn't a trick of light-travel-time bookkeeping that a cleverer observer could correct away — it's a genuine disagreement about which events are simultaneous, and the equation below says exactly how large it is. The cross term vΔx/c² is what separates the platform's Δt = 0 from the train's Δt′ ≠ 0: it vanishes only when the two events happen at the same place (Δx = 0) or when there's no relative motion (v = 0), which is why simultaneity feels absolute at everyday speeds and only reveals its relative nature as v approaches c.