Time Dilation

The proper time $\Delta t_0$ is the interval measured by a clock that is present at both events — the traveller's own watch, carried along for the whole journey. Every other observer measures more. The reason is not mysterious once you build a clock out of light. Put two mirrors a fixed distance apart and let a pulse bounce between them; one round trip is one tick. Now watch that clock go past you. The pulse must travel a longer, diagonal path to get from one mirror to the other, and because light has the same speed for you as for the clock, it cannot compensate by going faster. So the tick takes longer. Nothing in that argument mentions mirrors, and that is the point — a caesium standard, a muon's decay and a heartbeat all slow by the same factor, because it is time that is being stretched and not the mechanism.

The classic demonstration falls out of the sky. Cosmic rays create muons about 15 km up, and a muon's mean lifetime at rest is 2.2 µs — enough for light itself to cover only 660 m. Classically almost none should reach sea level. At $\beta = 0.995$, though, $\gamma = 1/\sqrt{1 - 0.990} = 10$, so in the ground's frame the mean lifetime is 22 µs and the mean distance about 6.6 km. Enough survive that they arrive in quantity, and Bruno Rossi and David Hall measured exactly this in 1941 by counting muons on the summit of Mount Washington and again at sea level. A more everyday case: GPS satellites orbit at 3.87 km/s, so $\beta = 1.29 \times 10^{-5}$ and their clocks lose about 7 µs per day to motion. They also gain about 45 µs per day from sitting higher in Earth's gravity, which is a general-relativistic effect and not on this page; the net +38 µs per day is built into the system, and without it positions would drift by roughly ten kilometres a day.

This page and the length-contraction page describe one fact from two chairs. Ask the muon and it will tell you its clock is perfectly normal and the atmosphere ahead of it was only 1.5 km thick. Both accounts predict the same muon arriving at the same detector, which is the sense in which relativity is not about appearances: each observer's story is internally consistent and they agree on every observable event. Hafele and Keating settled it directly in 1971 by flying caesium clocks around the world eastward and westward on commercial airliners and comparing them against the clocks left at the Naval Observatory.

The twin paradox is not a paradox, and the resolution is not subtle. The setup looks symmetric — each twin measures the other's clock as slow — but it is not symmetric, because only one twin turns around. Turning around means accelerating, which means switching from one inertial frame to another, and the travelling twin's account of what "now" means back on Earth jumps discontinuously during the turn. The stay-at-home twin never changes frames; the traveller does, and can feel it. That asymmetry is physical, and it is why the traveller really is the younger one when they meet again. Two smaller points. "Proper" in proper time does not mean correct — it descends from the French propre, meaning one's own, and $\Delta t_0$ is the clock's own time. And $\Delta t_0$ can never exceed $\Delta t$; the solver rejects that combination, and if you have hit the guard you have almost certainly entered the two times the wrong way round.