Place a metre-rod in the -axis of
in such a manner that one end
(the beginning) coincides with the point
whilst the other end
(the end of the rod) coincides with the point
. What is the length
of the metre-rod relatively to the system
? 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
at a particular time
of the system
. By means
of the first equation of The Lorentz transformation
the values of
these two points at the time
can be shown to be
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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 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 we should have
,
and for stiII greater velocities the square-root becomes imaginary.
From this we conclude that in the theory of relativity the velocity
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 as a limiting velocity also
clearly follows from the equations of the Lorentz transformation, for
these became meaningless if we choose values of
greater than
.
If, on the contrary, we had considered a metre-rod at rest in the
-axis with respect to
, then we should have found that the length of
the rod as judged from
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
, 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 () of
.
and
are two successive ticks of
this clock. The first and fourth equations of the Lorentz
transformation give for these two ticks:
and
As judged from , the clock is moving with the velocity
; as judged
from this reference-body, the time which elapses between two strokes
of the clock is not one second, but
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
plays the part of an unattainable limiting velocity.
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