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Relativity: The Special and General Theory

The Behaviour of Measuring-Rods and Clocks in
Motion
Place a metre-rod in the x1-axis of K1 in such a manner that one end (the beginning)
coincides with the point x1=0 whilst the other end (the end of the rod) coincides with the
point x1=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 the distance between the points being
.
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
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
,
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 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 K1 would have been;
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 (x1=0) of
K1. t1=0 and t1=I are two successive ticks of this clock. The first and fourth equations of
the Lorentz transformation give for these two ticks :
t = 0
and 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
 
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