Time dilation: moving clocks run slower

Turning Points in Physics | AQA A-Level Physics

Key Definitions

Time dilation: The phenomenon where a clock moving relative to an observer ticks more slowly than an identical clock at rest in the observer's frame.

Proper time ($\Delta t_0$): The time interval measured by a clock that is present at both events (i.e. the clock is in the same frame as the events). It is always the shortest measured time interval.

The light clock derivation

Imagine a "light clock" made of two mirrors separated by a distance $d$, with a pulse of light bouncing between them.
In the clock's own rest frame, the time for one tick is $\Delta t_0 = 2d/c$ (light travels straight up and down).
For an observer watching the clock move sideways at speed $v$, the light follows a longer, diagonal path. By Pythagoras, the total path length is $2\sqrt{d^2 + (v\Delta t/2)^2}$.
Since the speed of light is the same in both frames (Einstein's second postulate), the longer path takes a longer time. Solving gives:

$$\Delta t = \frac{\Delta t_0}{\sqrt{1 - v^2/c^2}} = \gamma \Delta t_0$$

where $\gamma = 1/\sqrt{1 - v^2/c^2}$ is the Lorentz factorThe factor $\gamma = 1/\sqrt{1 - v^2/c^2}$ that appears in all special relativity equations. It equals 1 at rest and increases without limit as $v$ approaches $c$.. Since $\gamma \geq 1$, the dilated time $\Delta t$ is always greater than or equal to the proper time $\Delta t_0$.

The muon lifetime experiment

Muons are subatomic particles created when cosmic rays hit the upper atmosphere at an altitude of roughly 10 km. They travel towards the Earth's surface at about 0.98$c$.
The muon's half-life in its own rest frame is about 1.6 microseconds. In that time, even at nearly the speed of light, it should only travel about 0.5 km before decaying. Very few should reach sea level.
In practice, many muons are detected at sea level. From the Earth's frame, the muon's internal clock runs slowly (time dilation). With $\gamma \approx 5$ at $v = 0.98c$, the muon's observed lifetime is about 8 microseconds, giving it time to travel roughly 2.4 km per half-life, enough to survive the journey.
This is direct experimental evidence for time dilation.

Common Mistake

Students often get proper time backwards. The proper time $\Delta t_0$ is measured in the frame where both events happen at the same place (e.g. the muon's own rest frame for its creation and decay). The dilated time $\Delta t$ is always larger. If you are asked "which observer measures the proper time," it is the one travelling with the clock or particle.