A clock made of light
To reason cleanly, strip a clock down to its essence: two parallel mirrors a distance L apart with a light pulse bouncing between them. One 'tick' is one round trip. In the clock's own rest frame the pulse goes straight up and back, so one tick takes \Delta t_0 = 2L/c. Because this clock is at rest where the ticks happen, it measures the proper time — the time between two events read by a single clock present at both.
Watch it tick slower
Now let the same clock glide past you at speed v, its mirrors still L apart, moving sideways. In your frame the pulse cannot go straight up — while it climbs, the top mirror slides forward, so the pulse must travel a longer diagonal to catch it. Postulate 2 fixes the pulse's speed at exactly c, no faster. A longer path at the same speed takes more time: the moving clock's tick is stretched. Slide the speed toward c below and watch the diagonal — and the tick — grow without bound.
Pythagoras for half a tick in your frame: the light's diagonal (hypotenuse) has vertical leg L and horizontal leg the distance the clock slides. Solve for Δt.
The Lorentz factor
Solving that triangle for \Delta t and using \Delta t_0 = 2L/c gives the central result of the whole subject. The stretch factor has a name and a symbol, \gamma, the Lorentz factor. This is time dilation: the time you measure for a moving clock's tick is always longer than the clock's own proper time, by the factor \gamma.
Time dilation. Δt₀ is the proper time (the clock's own tick); Δt is what a frame that sees the clock moving measures. Since γ ≥ 1, the moving clock always runs slow.
Feel the numbers. At v = 0.1c, \gamma \approx 1.005 — half a percent, still tiny. At v = 0.6c, \gamma = 1.25. At v = 0.8c, \gamma = 1.667. At v = 0.99c, \gamma \approx 7.09. The curve is gentle until you get close to c, then it rockets toward infinity — which is the first hint that reaching c is impossible for anything with mass.
For everyday speeds γ is a hair above 1, so Δt ≈ Δt₀ and Newton's shared time re-emerges. This is why we never noticed relativity before clocks and particles got fast enough.
It is real, and it is measured
This is not a thought experiment. A muon — a heavy cousin of the electron — has an average lifetime of only 2.2\ \mu\text{s} at rest. Cosmic rays create muons high in the atmosphere moving at about $0.98c$ (\gamma \approx 5). Classically they should decay after travelling only \sim\!650 m and almost none should reach the ground. In fact large numbers do, because in our frame their internal clock is dilated to about 11\ \mu\text{s}, letting them cover several kilometres before decaying. Real muons span a range of energies, but the effect is exactly as \gamma predicts.
Atomic clocks flown around the world (Hafele–Keating, 1971) came back reading slightly different from clocks left at home, matching the combined predictions of special and general relativity to within experimental error. And the GPS satellites in your phone must correct their clocks for relativistic effects every day, or positions would drift by kilometres. Time dilation is engineering, not philosophy.