Run towards a beam of light and you would expect to meet it faster. You do not. Every measurement ever made says light arrives at the same speed however you are moving, and once you take that seriously something else has to bend instead. What bends is time. Everything on this page follows from that one refusal, and it has been checked to more decimal places than almost anything else in physics.
On the left, a photon bouncing between two mirrors β one tick per round trip. On the right, the same clock moving past you. Its photon has to travel diagonally, because it must also keep up with the mirrors, and a diagonal is longer than a straight up-and-down. The photon cannot go faster to compensate, so the tick simply takes longer. That is the entire argument, and the factor it produces is the hypotenuse of a right-angled triangle.
Ξ³ is 1.005 at a tenth of light speed and 1.15 at half β which is why nobody noticed for two hundred years. It only bites near the end: the curve is almost flat until about 0.8c and then goes vertical. Distances contract by the same factor in the direction of travel, so from the traveller's point of view nothing slowed down at all β the journey simply got shorter.
| speed | Ξ³ | a year on board is | a metre looks |
|---|
Cosmic rays make muons about fifteen kilometres up. A muon lasts 2.2 microseconds on average, and even at nearly light speed that is only about 650 metres' worth of travel. Almost none should survive the trip β and yet detectors at sea level are full of them. Their internal clock runs slow by exactly the factor above, and the discrepancy is not small: it is a factor of a hundred million.
One twin flies out and returns; the other stays. Both would say the other one's clock ran slow while they were moving apart, which sounds like a contradiction. It is not, because the situations are not the same: only the traveller turns around, and only the traveller feels it. The worldlines below make the asymmetry visible β the bent path is genuinely shorter in the way that counts.
Every satellite in the GPS constellation carries an atomic clock, and every one of them is deliberately set to tick at the wrong rate before launch. Two corrections, in opposite directions: the satellite's speed makes its clock run slow by about 7.2 microseconds a day, and its height in weaker gravity makes it run fast by about 45.7. The net 38.5 microseconds a day sounds trivial until you multiply by the speed of light β which is where this page's arithmetic gives about eleven kilometres of position error per day if you ignore it.
One honest split, though: only the first of those two is special relativity, which is what this page is about. The larger correction comes from general relativity β gravity, not motion β which is a different and much harder theory. The light clock argument here cannot produce it. Both are computed on this page because GPS needs both, but they are not the same idea, and anyone telling you relativity is one theory is compressing twenty years of Einstein's work into a sentence.