Length Contraction: Moving Rulers Shrink

The rule: shorter along the motion

Imagine a spaceship 100 metres long resting in dock. Now watch it scream past you at 0.99c. You will measure it to be only about 14 metres long — squashed nose-to-tail, even though it has not been crushed and the crew inside notice nothing wrong. This is length contraction: a moving object is shorter along its direction of motion than it is at rest.

The same gamma, now dividing

The amount of shrinking is set by the exact same stretch factor from the last lesson — the Lorentz factor gamma. But where time dilation multiplied time by gamma, length contraction divides length by gamma. The longest length, measured by someone at rest with the object (its proper length), gets divided down for anyone watching it fly by:

    gamma = 1 / sqrt(1 - v^2/c^2)

    measured length = proper length / gamma

    e.g. v = 0.99c -> gamma = 7.09
         100 m / 7.09  =  14.1 m

Time stretches (multiply by gamma); length shrinks (divide by gamma). One factor governs both.

The muon's own story

Remember the muons raining down from the sky? From the ground's point of view they survive the 15 km trip because their internal clocks run slow — time dilation. But ride along with a muon and its own clock ticks perfectly normally; it still only lives about 2 microseconds. So how does it possibly reach the ground? Because from the muon's seat, it is the atmosphere that is rushing up at it at 0.99c, and that 15 km column of air is length-contracted.

  Ground frame:                 Muon's frame:
  --------------                ------------
  thick 15 km of air            air squashed to 15 / 7.09
  muon lives LONG               = about 2.1 km
  (clock dilated x7)            muon lives its normal ~2 us
                                but only ~2 km to cross

        BOTH frames agree:  the muon hits the ground.

Two descriptions, one outcome. Ground says 'slow clock'; muon says 'short distance'. Both predict it lands.

Shrink 15 km by gamma ≈ 7 and you get only about 2 km of air to cross — a distance the muon clears easily within its ordinary 2-microsecond lifetime. Same muon, same landing, but each observer explains it with a different effect.

Two views of one truth

This is the heart of it: length contraction and time dilation are not two separate laws but two views of the single fact that space and time mix together. Whether an observer accounts for a result using stretched time or squeezed distance depends only on which frame they sit in — and crucially, they always agree on what actually happens (the muon lands; the clocks read what they read).