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
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the distance between the points being
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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
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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
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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 _K′_ would have been
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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′_ = 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
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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
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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.
