Length contraction

Key ideas

• Just as observers disagree on time intervals, they also disagree on length — this is length contraction.
• Proper length, $l_0$, is the length of an object measured by an observer at rest relative to the object (the object is not moving in their frame).
• An observer who sees the object moving past them at speed $v$ measures a shorter length, $l$:

$l = \frac{l_0}{\gamma}$

• Each variable, with sub-bullets:
  • $l$ — the contracted length, measured by an observer who sees the object move (m)
  • $l_0$ — the proper length, measured in the object's own rest frame (m)
  • $\gamma$ — the Lorentz factor, $\gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}}$, calculated exactly as for time dilation
• Since $\gamma \ge 1$, $l \le l_0$: the proper length is always the longest length any observer measures. This is the mirror image of time dilation — moving clocks run slow, moving objects measure short.
• Length contraction only happens along the direction of motion. Dimensions perpendicular to the motion (e.g. the height of a spacecraft moving horizontally) are unaffected.
• Length contraction and time dilation are two sides of the same relationship: they use the same $\gamma$, calculated from the same relative speed $v$.

Tips

• Always state whose frame gives the proper length before calculating — the object's own rest frame, never the frame of an observer who sees it moving.
• If a question gives you $\gamma$ (or lets you calculate it) from a time-dilation part, you can reuse the same $\gamma$ for a length-contraction part in the same scenario — no need to recompute it.
• A sanity check: contracted length should always come out smaller than the proper length given. If it doesn't, check you divided rather than multiplied.