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Length Contraction

The measurement of length involves the concept of a rod or a scale. The positions of both ends of the rod must be measured simultaneously in a frame in order to define its length. However, since space-time measurements are frame dependent, we shall see that length of the rod also is relative.

Consider two Lorentz frames S and S’; S’ moving relative to S with velocity v along X-axis. The rod therefore is a moving object with time dependent are recorded in both S and S. Hence we get

x₂’ – x₁’ = [(x₂ – x₁) – v (t₂ – t₁)]

However, the spatial distance represents length of the rod if positions of two ends are recorded simultaneously. That is, if t₂’ = t₁’, then (x₂’ – x₁’) denotes length of the rod in its rest-frame S’ (i.e. the frame in which the rod is at rest). This is called its proper length, denoted by l₀. Similarly, the length of l of the moving rod as observed from S is equal to (x₂ – x₁) only if position measurements are done in S simultaneously for both ends, i.e. if t₂ = t₁. Hence, we get

l₀ = l

Since > l, l < l₀. The length of the moving rod in the direction of its motion is observed to be less than its proper length. This is known as length contraction (Lorentz-Fitzgerald contraction) of the moving object.