Relativity (1931)/Section 12

by Albert Einstein

Layout 2

 

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 when at rest, and the more quickly it is moving, the shorter is the rod. For the velocity $v=c$ we should have ${\sqrt {1-{\tfrac {v^{2}}{c^{2}}}}}=0$, 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 $K'$ would have been ${\sqrt {1-{\tfrac {v^{2}}{c^{2}}}}}$; 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 $(x'=0)$ of $K'$. $t'=0$ and $t'=1$ are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks:

$t=0$

and

$t={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

As judged from $K$, the clock is moving with the velocity $v$; as judged from this reference-body, the time which elapses between two strokes of the clock is not one second, but ${\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$ seconds, i.e. a somewhat larger time. As a consequence of its motion the clock goes more slowly than when at rest. Here also the velocity $c$ plays the part of an unattainable limiting velocity.