XII. The Behaviour of Measuring-Rods and Clocks in Motion

Relativity: The Special and General TheoryAlbert Einstein

Place a metre-rod in the xx'-axis of KK' in such a manner that one end (the beginning) coincides with the point x=0x'=0 whilst the other end (the end of the rod) coincides with the point x=Ix'=I. What is the length of the metre-rod relatively to the system KK? 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 KK at a particular time tt of the system KK. By means of the first equation of the Lorentz transformation the values of these two points at the time t=0t = 0 can be shown to be

x(begining  of  rod)=0Iv2c2x_{(begining\; of\; rod)} = 0 \overline{\sqrt{I-\frac{v^2}{c^2}}} \\ x(end  of  rod)=1Iv2c2x_{(end\; of\; rod)} = 1 \overline{\sqrt{I-\frac{v^2}{c^2}}}

the distance between the points being Iv2/c2\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 vv is Iv2/c2\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=cv=c we should have Iv2/c2=0\sqrt{I-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 cc 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 cc as a limiting velocity also clearly follows from the equations of the Lorentz transformation, for these became meaningless if we choose values of vv greater than cc.

If, on the contrary, we had considered a metre-rod at rest in the xx-axis with respect to KK, then we should have found that the length of the rod as judged from KK' would have been Iv2/c2\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,tz, 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=0x'=0) of KK'. t=0t'=0 and t=It'=I are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks:

t=0t = 0


t=IIv2c2t' = \frac{I}{\sqrt{I-\frac{v^2}{c^2}}}

As judged from KK, the clock is moving with the velocity vv; as judged from this reference-body, the time which elapses between two strokes of the clock is not one second, but IIv2c2\frac{I}{\sqrt{I-\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 cc plays the part of an unattainable limiting velocity.

XIII. Theorem of the Addition of Velocities. The Experiment of Fizeau

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