Page:The Meaning of Relativity - Albert Einstein (1922).djvu/50

has a significance which is independent of the choice of the inertial system; but the invariance of the quantity $\sum \Delta x_{\nu }^{2}$ does not at all follow from this. This quantity might be transformed with a factor. This depends upon the fact that the right-hand side of (29) might be multiplied by a factor $\lambda$, independent of $v$. But the principle of relativity does not permit this factor to be different from 1, as we shall now show. Let us assume that we have a rigid circular cylinder moving in the direction of its axis. If its radius, measured at rest with a unit measuring rod is equal to $R_{0}$, its radius $R$ in motion, might be different from $R_{0}$ since the theory of relativity does not make the assumption that the shape of bodies with respect to a space of reference is independent of their motion relatively to this space of reference. But all directions in space must be equivalent to each other. $R$ may therefore depend upon the magnitude $q$ of the velocity, but not upon its direction; $R$ must therefore be an even function of $q$. If the cylinder is at rest relatively to $K'$ the equation of its lateral surface is

$x'^{2}+y'^{2}=R_{0}^{2}$

If we write the last two equations of (29) more generally

${\begin{aligned}x_{2}'&=\lambda x_{2}\\x_{3}'&=\lambda x_{3}\end{aligned}}$

then the lateral surface of the cylinder referred to $K$ satisfies the equation

$x^{2}+y^{2}={\frac {R_{0}^{2}}{\lambda ^{2}}}$