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Time dilation

Space and time · level 7, lesson 2 of 6

A question to start

A clock is made of a pulse of light bouncing between two mirrors. What happens to its ticking when it moves past you?

The idea

In the light clock, the mirrors are a distance \(L\) apart. At rest, one tick, up and down, takes \(2L/c\).

Now the clock moves sideways at speed \(v\). From outside, the light has to go along a slanted path, which is longer. But light still goes at \(c\). So each tick takes longer. Pythagoras on the slanted path gives

\[t = \gamma\,t_0, \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.\]

Here \(t_0\) is the time on the moving clock, and \(t\) is the time you measure. The Lorentz factor \(\gamma\) is always at least 1. Moving clocks run slow. This is time dilation, and it is true of every kind of clock, including hearts and atoms.

Speed \(\gamma\)
\(0.1c\) 1.005
\(0.5c\) 1.155
\(0.8c\) 1.667
\(0.9c\) 2.29
\(0.995c\) 10.0
at restmoving: longer pathv →

Worked examples

1. At \(0.8c\). \(v^2/c^2 = 0.64\). \(\sqrt{1 - 0.64} = 0.6\). \(\gamma = 1/0.6 \approx 1.667\).

2. A moving hour. A clock moving at \(0.8c\) shows 1 hour passing. You measure \(1.667\) hours, about 1 hour 40 minutes.

3. Where Pythagoras comes in. In time \(t\) the clock moves \(vt/2\) sideways while the light goes \(ct/2\) along a slant from one mirror to the other. So \((ct/2)^2 = L^2 + (vt/2)^2\). Solve for \(t\): \(t = \frac{2L/c}{\sqrt{1 - v^2/c^2}}\).

Try it

  1. Why does a moving light clock tick slower?
  2. Find \(\gamma\) at \(0.6c\).
  3. A clock moving at \(0.6c\) shows 10 minutes. How long do you measure?
  4. Is the moving clock "broken"?
Answers
  1. Its light takes a longer path, but still at speed \(c\), so each tick takes longer.
  2. \(\sqrt{1 - 0.36} = 0.8\), so \(\gamma = 1.25\).
  3. \(1.25 \times 10 = 12.5\) minutes.
  4. No. To anyone moving with it, it ticks normally. It is time itself that is measured differently.

Common mistake

"Only light clocks slow down." If a moving wristwatch kept a different time from a moving light clock beside it, the crew could tell they were moving. Einstein's first idea says they cannot. So all clocks slow together.

Where it leads