Postulates and the Relativity of Simultaneity

Ask two people whether two distant events — say, a flash of lightning at the front of a train and another at the back — happened at the same time, and you expect a single honest answer: yes or no. It feels as solid as arithmetic. Yet special relativity says that answer depends on who you ask. A passenger speeding through the station and a guard standing on the platform can each measure carefully, make no mistake, and still disagree about whether the two flashes were simultaneous — and both are right.

This is not a puzzle about slow signals or sloppy clocks. It is a genuine feature of the universe, and it flows from just two assumptions Einstein made in 1905. Everything strange in relativity — slowed clocks, shrunken rulers, the cosmic speed limit — grows from these two seeds. This page plants them, and then follows the first surprising sprout: simultaneity is not absolute.

The two postulates

Einstein began not with a mountain of equations but with two short statements he refused to compromise. Read them slowly — the whole edifice balances on them.

The first postulate sounds tame — Galileo knew it, and it is why coffee pours normally on a steady aeroplane. The second is the troublemaker. In everyday life speeds add: throw a ball at 20\ \text{km/h} from a train doing 100\ \text{km/h} and the ground sees 120\ \text{km/h}. Light flatly refuses to play this game. Its speed is a fixed law of nature, and by the first postulate that law must read the same for everyone. Something else has to give — and what gives is our comfortable belief that everyone shares a single universal "now".

Why constant c breaks the universal "now"

How do you even decide that two events far apart happened "at the same time"? You cannot be in two places at once, so you need a rule. The natural one: stand exactly halfway between them, and if the light (or sound, or any signal at fixed speed) from both events reaches you together, then they were simultaneous. This is a perfectly good definition — as long as everyone agrees on the speed of the signal and on who is standing still.

But the second postulate fixes the signal speed at c for everyone, and the first postulate says nobody is truly standing still. So imagine you are midway between two events and the two flashes reach you at once — you declare them simultaneous. Now someone glides past you. In their frame they are the one at rest and you are moving; during the time the light was in flight, one event's flash had to chase after them while the other rushed to meet them. Same light speed, different distances covered relative to the moving observer — so the flashes cannot reach them together. They must conclude the events were not simultaneous. Neither of you has erred. Simultaneity has simply come apart.

The train-and-lightning thought experiment

Einstein made this concrete with a picture worth a thousand equations. A train car races along a platform at speed v. Two bolts of lightning strike, one at the back end A and one at the front end B, leaving scorch marks on both the train and the track. A guard stands on the platform exactly midway between the two scorch marks; a rider sits exactly at the middle of the train car. Reveal the figure step by step.

The guard's story. The two flashes were emitted at the same moment and travel equal distances at the same speed c, so they arrive at the midpoint together. The guard declares: the strikes were simultaneous.

The rider's story. While the light was crossing the car, the rider was carried toward the front. So the light from B (front) had a shorter distance to travel to reach her, and the light from A (back) had to catch up. The front flash arrives first. Since she sits at the exact middle of her own car and knows light travels at c for her too, she can only conclude that the front strike happened before the back one. The two observers disagree — not about what their eyes saw, but about the ordering of events in time itself.

How big is the disagreement?

The relativity of simultaneity is not just qualitative — it has a size. If two events are separated by a distance \Delta x along the direction of motion and are simultaneous in one frame, then in a frame moving at speed v they are separated in time by

\Delta t' = \frac{v\,\Delta x}{c^2}\,\gamma, \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.

The key piece is the v\,\Delta x / c^2 at the front: the further apart the events (\Delta x) and the faster the frame (v), the larger the clash. Notice the c^2 in the denominator — it is enormous, which is exactly why you never notice this at human speeds and distances. But it is never zero (as long as v \ne 0 and \Delta x \ne 0), so the universal "now" is an approximation, not a law.

Worked example. Suppose the train car is \Delta x = 300\ \text{m} long and races at a (fictional but tidy) v = 0.6c, so \gamma = 1.25. In the platform frame the strikes are simultaneous. In the train's own frame the leading-order time gap between them is

\Delta t' \approx \frac{v\,\Delta x}{c^2} = \frac{(0.6)(3\times10^8)(300)}{(3\times10^8)^2} \approx 6\times 10^{-7}\ \text{s} = 0.6\ \mu\text{s}.

Not simultaneous by about six-tenths of a microsecond — small, but a real, measurable ordering of events that the guard and rider genuinely disagree on.

Watch out — this is the single most common misreading, so pin it down now. The relativity of simultaneity is not an optical delay you could subtract off. Both observers are careful physicists: each already corrects for the travel time of light, placing themselves exactly midway between the events and reasoning about when the light was emitted, not merely when it hit their retina. After all corrections, they still disagree about which event happened first.

The disagreement survives the correction because the two observers, moving relative to each other, do not even agree on what "midway" means or on how far apart the events are — space and time are entangled. If it were a mere signal delay, a clever accountant could recover a single true ordering. No accountant can. The ordering of two spacelike-separated events is genuinely frame-dependent. (The universe protects cause and effect: events that could influence one another are never reordered, because reaching one from the other would require a signal faster than light.)

By agreeing, in advance, on a single frame and a synchronization rule. Einstein's own recipe: two clocks at rest relative to each other are synchronized if a light pulse sent from one at its time t_1, reflected at the other, and returned at t_3 marks the far clock reading exactly (t_1 + t_3)/2 at the moment of reflection. This works beautifully — within one frame. The catch is that a set of clocks synchronized this way in the platform frame will look out of sync to the train rider, marching progressively out of step down the direction of motion (the "leading clocks lag" rule). Every GPS receiver on Earth lives with exactly this: the satellite constellation is synchronized in the Earth-centered frame, and the software carries the relativistic bookkeeping so that "now" is well-defined for that agreed frame. There is no universal now, but there is a perfectly good agreed-upon now.