I 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
x_((beginning of rod)) = 0·√(1 - v²/c²), x_((end of rod)) = 1·√(1 - v²/c²),
the distance between the points being
√(1 - v²/c²)
. 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
√(1 - v²/c²)
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
√(1 - v²/c²) = 0
, 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 become 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
√(1 - v²/c²)
; 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 x , y , z , 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 Galilei 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 t = 1/(√(1 - v²/c²)).
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
1/(√(1 - v²/c²))
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.