Time dilation (A-level only)

AQA·Physics·3.12.3.3

Time dilation

Two observers moving relative to each other can measure different time intervals between the same two events. In special relativity, the interval measured in the frame where the events happen at the same position is shorter than the interval measured in a frame moving relative to it.

The interval measured in the frame where both events occur at the same position is called the proper time, $$t_0$$. A single clock at rest in that frame can be present at both events and record the interval between them.

In an inertial frame moving at speed $$v$$ relative to that clock, the two events occur at different positions. The interval $$t$$ measured in that frame is:

t=t01−v2c2t = \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}

Here $$c$$ is the speed of light in a vacuum. For any relative speed greater than zero and less than $$c$$, the denominator is less than one, so $$t$$ is greater than $$t_0$$. This difference between the measured intervals is time dilation: the moving clock records less time between the events than the interval measured in the other frame.

As $$v$$ approaches $$c$$, the denominator approaches zero and the factor relating $$t$$ to $$t_0$$ increases. At speeds much smaller than $$c$$, the denominator is close to one, so the difference between the two intervals is small.

Muon decay evidence for time dilation

Muons are particles produced high in Earth’s atmosphere. Many travel towards the ground at speeds close to the speed of light, but they can decay before they arrive.

Decay is statistical: individual muons do not all decay after the same time. Instead, a population of muons is described by a mean lifetime. Some muons decay sooner than this mean and some survive longer.

Muon lifetimes in different frames

The mean lifetime measured in the muons’ own rest frame is their proper lifetime, $$\tau_0$$. In the ground frame, those same muons are moving, and their mean lifetime is longer because of time dilation. Special relativity predicts:

τ=τ01−v2/c2\tau = \frac{\tau_0}{\sqrt{1-v^2/c^2}}

Here, $$\tau$$ is the mean lifetime in the ground frame, $$v$$ is the muons’ speed relative to the ground, and $$c$$ is the speed of light. At speeds close to $$c$$, the predicted increase in lifetime is substantial.

Muon survival at ground level

In the ground frame, muons need time to travel from their point of production to the ground. If their mean lifetime in that frame were only their proper lifetime, many would decay during the journey and far fewer would be expected to reach the ground. The longer mean lifetime predicted by time dilation means that a larger fraction survives.

The observed survival of atmospheric muons is therefore evidence for time dilation. Direct measurements of moving muons’ mean lifetimes also show the increase predicted by special relativity.