PlanetPhysics/Behaviour of Measuring Rods and Clocks in Motion

The Behaviour of Measuring-Rods and Clocks in Motion
From Relativity: The Special and General Theory by Albert Einstein 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 $$x'=I$$. 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

$$\begin{matrix} x_{\mbox{(begining of rod)}} &=& 0 \overline{\sqrt{I-\frac{v^2}{c^2}}} \\ x_{\mbox{(end of rod)}} &=& 1 \overline{\sqrt{I-\frac{v^2}{c^2}}} \end{matrix}$$ ~

\noindent the distance between the points being $$\sqrt{I-v^2/c^2}$$.

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 $$\sqrt{I-v^2/c^2}$$ 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 $$\sqrt{I-v^2/c^2} = 0$$, and for stiII 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 became 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 $$\sqrt{I-v^2/c^2}$$; 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 $$z, y, x, t$$, 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 Galileian 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 $$t'=I$$ are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks:

$$t = 0$$

\noindent and

$$t' = \frac{I}{\sqrt{I-\frac{v^2}{c^2}}}$$ ~

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

$$\frac{I}{\sqrt{I-\frac{v^2}{c^2}}}$$ ~

\noindent seconds, {\it 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.