therefore follows that the length of a rigid metre-rod moving in the
direction of its length with a velocity _v_ is
image008
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
image009
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
image010
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.
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