Time Dilation

A clock is any repeating process: a pendulum, a heartbeat, a pulse of light between two mirrors. Use the light version, insist that light goes at c for everyone, and you are forced to conclude that moving clocks tick slowly.

Sideways lengths don’t change#

First, a thought experiment. Make two identical rods, check their ends match, and paint the ends of one red, the other blue. Now slide one past the other sideways (perpendicular to their length), close enough for the paint to rub off.

If the moving rod shrank, it would leave red marks inside the blue rod’s ends. But viewed from the red rod, it is the blue rod that moves, and by the principle of relativity the moving rod should then be the one that shrinks: blue marks inside the red ends. Both can’t happen. So neither does: lengths perpendicular to the motion are unchanged.

The light clock#

Fix two mirrors a distance L apart on a rod, and bounce a light pulse between them. Each return to the bottom mirror M₁ is a “tick”. In the frame S where the mirrors are at rest, the pulse covers 2L per tick, so

Now watch the same clock from a frame S′ in which it moves sideways at speed v. Between two ticks taking Δt′, the mirrors move v Δt′ along, so the pulse travels along two slanted lines, each of length √(L² + (vΔt′/2)²). The rod length L is unchanged (it is perpendicular to the motion), and light still travels at c:

Proper time

So a moving clock runs slow by the factor γ. This holds for every kind of clock (atomic, mechanical, biological) because by postulate 1 they must all agree with the light clock in their own frame. And it is symmetric: each of two observers in relative motion finds the other’s clock running slow.

Any two events can be treated this way. If event E₁ happens at x = 0, t = 0 and event E₂ at x = L, time t, a clock moving at L/t is present at both. In its frame the events are at one place, so its reading t/γ is the proper time.