XII

The Behaviour of Measuring-Rods and Clocks in Motion

I place a metre-rod in the $x^{\prime }$-axis of $K^{\prime }$ in such a manner that one end (the beginning) coincides with the point $x^{\prime }=0$, whilst the other end (the end of the rod) coincides with the point $x^{\prime }=1$. 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

x_{\text{(beginning of rod)}} &= 0·\sqrt{1 - \frac{v^{2}}{c^{2}}}, \ x_{\text{(end of rod)}} &= 1·\sqrt{1 - \frac{v^{2}}{c^{2}}},

the distance between the points being $\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}$. 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 $\sqrt{1-v^2/c^2}$ of a metre. The rigid rod is thus shorter when in motion than when at rest, and the more quickly it is moving, the shorter is the rod. For the velocity $v=c$ we should have $\sqrt{1-v^2/c^2}=0$, and for still greater velocities the square-root becomes

imaginary. From this we conclude that in the theory of relativity the velocity $c$ plays the part of a limiting

velocity, which can neither be reached nor exceeded by any real body.

Of course this feature of the velocity $c$ as a limiting velocity also clearly follows from the equations of the Lorentz transformation, for these become meaningless if we choose values of $v$ greater than $c$.

If, on the contrary, we had considered a metre-rod at rest in the $x$-axis with respect to $K$, then we should have found that the length of the rod as judged from $K^{\prime }$ would have been $\sqrt{1-v^2/c^2}$; this is quite in accordance with the principle of relativity which forms the basis of our considerations.

A priori it is quite clear that we must be able to learn something about the physical behaviour of measuring-rods and clocks from the equations of transformation, for the magnitudes $x$, $y$, $z$, $t$, are nothing more nor less than the results of measurements obtainable by means of measuring-rods and clocks. If we had based our considerations on the Galilei transformation we

should not have obtained a contraction of the rod as a consequence of its motion.

Let us now consider a seconds-clock which is permanently

situated at the origin ($x^{\prime }=0$) of $K^{\prime }$. $t^{\prime }=0$ and $t^{\prime }=1$ are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks:

$$\begin{gathered} t\&=0 \\ \mathrm{\backslash intertext}andt\&=\frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}. \end{gathered}$$