Two mirrors. A photon bouncing between them. One bounce is one tick. The clock on the left is standing still; the one on the right is flying past you at the speed above. Both photons travel at exactly the same speed — that is the one rule — but the moving photon has further to go, because the mirrors move out from under it. So its tick takes longer. That is time dilation, and there is nothing else to it.
Unfold that diagonal into a triangle. The upright is the distance light covers in a stationary tick. The base is how far the clock slid sideways. The hypotenuse is the path the moving photon actually took. Because all three are travelled at the same speed c, the sides are just times — and Pythagoras hands you the whole of special relativity.
Time stretches by γ and length squashes by exactly the same γ. Here is a 100-metre train at rest, and the same train as you measure it going past.
This is the strange one, and it is the heart of the theory. Lightning strikes both ends of a moving train. Watch it twice: once from the platform, once from inside the train. Same events, same light, different answer to which happened first.
Up is time, across is space. A light ray is a 45° line, and because light must stay at 45° for everyone, moving to a different speed tilts both axes towards the light line like a pair of scissors. Your “now” is the flat line of dots — watch it tip as you change speed. That tipping is simultaneity coming apart.
Fire a rocket forward from a rocket. Common sense adds the speeds. Reality does not — and the result never reaches c, no matter what you put in.
Relativity is not a curiosity in a textbook. Satellite navigation is a clock comparison, and it only works because both halves of the theory are corrected for. The satellites are moving, which slows their clocks; they are also higher up the gravity well, which speeds them up by more.
Accelerate at a comfortable 1 g — ordinary Earth gravity, so the ship has a floor — flip over halfway and slow down. This is the twin paradox with an engine that at least obeys mechanics.
Newton is not wrong at everyday speeds. He is indistinguishable from right. Here is how much relativity actually changes things at speeds people have reached.
| Speed | m/s | v/c | γ − 1 | Clock drift in a year |
|---|
No changes to this tool’s own behaviour since the earliest archived release (v3.71). The full history for the site is in the changelog.
Relativity has a reputation for being difficult. It rests on a single sentence: everybody measures light at the same speed. Not that light is fast, not that nothing can beat it — that if you measure the speed of a light beam you get 299,792,458 metres per second whether you are sitting still, running towards it, or chasing it at nine tenths of that yourself.
That is not obvious and it is not intuitive, but it is measured, repeatedly, to absurd precision. And once you accept it, everything else on this page is forced. If light's speed cannot change between observers, then the things we use to measure speed — distance and time — must change instead. Relativity is not a claim about light. It is what happens to everything else in order to keep light's speed fixed.
The clock in the first panel is the whole argument. A photon bounces between two mirrors; each bounce is a tick. Set that clock moving and the photon must travel a diagonal, because the far mirror has slid sideways by the time the light arrives. The diagonal is longer than the straight-up path. The photon cannot compensate by going faster. Therefore the tick takes longer.
Unfold that into a right triangle and the sides are: light's straight-up path, the sideways slide, and the diagonal. Pythagoras gives t = t₀ / √(1 − v²/c²), which is the Lorentz factor, and no further physics is required. Every other number on this page is that factor wearing a different hat.
Time dilation is strange but swallowable. Simultaneity is the one that hurts. Two events that happen at the same moment for you happen at different moments for someone moving relative to you — and neither of you is wrong. There is no universal “now” sweeping through the universe. The panel with the train shows it directly: the same two lightning strikes, the same light, and two irreconcilable answers to which came first.
The order only becomes fixed for events close enough in space and far enough apart in time that light could travel between them — and those are exactly the pairs that could be cause and effect. Relativity protects causality and abandons simultaneity, which is a trade almost nobody expects.
GPS is a clock comparison. Each satellite broadcasts the time; your phone works out how far away each one is from how stale the signal is, and trilaterates. Get the clocks wrong and the position is wrong. The satellites lose about 7 microseconds a day to their speed and gain about 46 to being higher in Earth's gravity, for a net gain near 38 microseconds a day. Uncorrected, that is a position error growing by around 11 kilometres every day. The correction is designed into the satellites' clocks before launch.
This is special relativity in flat spacetime, plus the weak-field gravitational time dilation that GPS needs — enough for every number here, and nowhere near enough for a black hole. There is no curvature, no tensors, no frame dragging. The journey calculator uses constant proper acceleration, which is honest mechanics but wildly optimistic engineering: the fuel needed to hold 1 g for years is the reason nobody has gone anywhere. The visual effects cover Doppler shift only; a real relativistic view would also distort angles and brightness, which is a rendering problem rather than a physics one. And the diagrams are schematic — they are drawn to be understood, not to scale.