Page:Relativity (1931).djvu/62

XII

THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION

I PLACE a metre-rod in the $x'$-axis of $K'$ in such a manner that one end (the beginning) coincides with the point $x'=0$, whilst the other end (the end of the rod) coincides with the point $x'=1$. What is the length of the metre-rod relatively to the system $K$? In order to learn this, we need only ask where the beginning of the rod and the end of the rod lie with respect to $K$ at a particular time $t$ of the system $K$. By means of the first equation of the Lorentz transformation the values of these two points at the time $t=0$ can be shown to be

${\begin{aligned}x_{\mathrm {(beginning\ of\ rod)} }&=0\cdot {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\\x_{\mathrm {(end\ of\ rod)} }&=1\cdot {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\mathrm {,} \end{aligned}}$

the distance between the points being ${\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$.

But the metre-rod is moving with the velocity $v$ relative to $K$. It therefore follows that the length of a rigid metre-rod moving in the direction of its length with a velocity $v$ is ${\sqrt {1-{\tfrac {v^{2}}{c^{2}}}}}$ of a metre. The rigid rod is thus shorter when in motion than

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