Q.Explain length contraction.

Length contraction is the relativistic effect whereby the length of an object, measured by an observer relative to whom it is moving, is found to be smaller than its proper length, by the factor $\sqrt{1-v^2/c^2}$.

Let a rod be at rest in frame $S'$, which moves with velocity $v$ along the $x$-axis relative to frame $S$. Let $L_0 = x_2' - x_1'$ be the proper length of the rod — its length as measured by an observer at rest relative to the rod (in $S'$), where $x_1'$ and $x_2'$ are the (time-independent) positions of its two ends in $S'$.

An observer in frame $S$, relative to whom the rod is moving, must measure the positions $x_1$ and $x_2$ of the two ends of the rod at the same instant of time $t$ (in $S$) to obtain its length $L = x_2 - x_1$ in that frame — measuring the ends at different times would give a meaningless result for a moving object.

Using the Lorentz transformation $x' = \gamma(x - vt)$, where $\gamma = 1/\sqrt{1-v^2/c^2}$, and applying it to both ends at the same time $t$:

$x_1' = \gamma(x_1 - vt), \qquad x_2' = \gamma(x_2 - vt)$

Subtracting,

$L_0 = x_2' - x_1' = \gamma(x_2 - x_1) = \gamma L$

so that

$L = \dfrac{L_0}{\gamma} = L_0\sqrt{1 - \dfrac{v^2}{c^2}}$

Since $\gamma \ge 1$ for any $v>0$, we have $L \le L_0$: the length measured by an observer relative to whom the rod moves is always less than (or equal to) its proper length. This is length contraction (also called the Lorentz–FitzGerald contraction).

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