If time dilates, lengths contract. It is the other side of the same coin: an object in motion turns out to be shorter, measured along the direction of its motion, than it is in the frame where it is at rest. But by how much, exactly, and for whom?
Law — Length contraction
If an object (or a racetrack) has rest length , i.e. measured in the frame where it is at rest, then in the frame in which the object moves at speed its measured length is The contraction occurs only along the direction of motion, not perpendicular to it.
Two points deserve emphasis. First: the rest length is the largest possible; any observer who sees the object in motion measures it as shorter, by a factor . Second: only the dimensions parallel to the motion contract. A metre stick flying by horizontally appears shortened horizontally but keeps its usual height; a sphere in motion becomes a “squashed” ellipsoid, flattened only in the direction of the velocity.
Let us return to the rabbit race. The observer at rest says the track is ls long and that the rabbits run from its two ends. But Matilda, in her own reference frame, sees the track as shorter: it is she who is at rest (with respect to herself), while the finish line comes towards her. With we have , so for Matilda the track is
The result is consistent with proper time: Matilda travels at for s, covering ls — exactly the contracted length of the track. Time dilation and length contraction are thus two descriptions of the same phenomenon, seen by different observers: what for the spectator is “a long track crossed slowly” is for Matilda “a short track crossed in little proper time”.
Links
Topics: Special relativity Concepts: Length contraction · Time dilation Skills: Changing reference frame Methods: Minkowski diagram
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