Skip to content

Length contraction

Space and time · level 7, lesson 4 of 6

A question to start

Seen from the ground, a muon survives because its clock runs slow. But to the muon, its own clock is perfectly normal. So from the muon's point of view, how does it reach the ground in time?

The idea

From the muon's point of view, the muon is still and the Earth's air rushes up towards it at nearly \(c\). Moving lengths shrink along the direction of motion:

\[L = \frac{L_0}{\gamma}.\]

\(L_0\) is the length measured by someone at rest beside it, and \(L\) is the length measured by someone it moves past. This is length contraction. It affects only lengths along the motion, not across it.

For a muon with \(\gamma = 10\), the 15 km of air is only 1.5 km deep. At nearly \(c\) that takes about 5 µs to pass, a couple of lifetimes. The ground and the muon tell different stories, slow clocks or short air, but they agree on what happens: the muon arrives.

15 km of airseen from the ground1.5 km of airseen by the muonmuon

Worked examples

1. The muon's air. \(L = 15 \div 10 = 1.5\) km.

2. Time to cross it. \(1{,}500 \div 300{,}000{,}000 = 5\) µs, about 2.3 lifetimes of 2.2 µs. The same answer as the ground's view in the last lesson.

3. A fast ship. A ship 100 m long passes at \(0.6c\), where \(\gamma = 1.25\). You measure it as \(100 \div 1.25 = 80\) m long.

Try it

  1. Which lengths shrink: along the motion or across it?
  2. A rod 2 m long moves at \(0.8c\) (\(\gamma = 1.667\)). How long do you measure it?
  3. Does the crew of the ship in example 3 notice their ship is shorter?
  4. Why do the ground and the muon agree on whether it arrives?
Answers
  1. Along the motion only.
  2. \(2 \div 1.667 = 1.2\) m.
  3. No. To them it is 100 m; it is you who are moving, as they see it.
  4. Time dilation in one view and length contraction in the other give the same number of lifetimes.

Common mistake

"It only looks shorter." It is not an illusion of light or eyesight. Measured carefully, with any method, the moving length is shorter.

Where it leads