The Behaviour Of Measuring–rods And Clocks In MotionApril 8, 2022
I place a metre-rod in the x’-axis of moving
K' in a way that its beginning coincides with the point
x' = 0 while its end coincides with the point
x' = 1.
What is the length of the metre rod relative to the non-moving
This is answered by asking where the beginning and end of the rod lie with respect to non-moving
K at a time
t of non-moving
K. The first equation of the Lorentz transformation shows that the values of these two points at the time
t = 0 is=
x (beginning of rod) = 0 √ (1 − (v2/c2) x (end of rod) = 1 √ (1 − (v2/c2)
the distance between the points being=
But the metre-rod is moving with the velocity
v relative to non-moving
K. It therefore follows that the length of a rigid metre-rod moving in the direction of its length with a velocity
√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
For faster speeds, the square-root becomes imaginary. This means 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.
c follows from the equations of the Lorentz transformation. These become meaningless if
v becomes greater than
If, on the contrary, we had considered a metre-rod at rest in the x-axis with respect to non-moving
K, then the length of the rod as viewed from
K' would have been
1 − v 2 c 2. This is in accordance with my principle of relativity.
x, y, z, t, are merely the results of measurements of the measuring-rods and clocks. If we used Galilei transformation, the rod would not contract as a consequence of its motion.
Let us now consider a seconds-clock which is permanently situated at the origin
(x' = 0 ) of
t' = 0 and
t' = 1 are two successive ticks of this clock. The 1st and 4th equations of the Lorentz transformation give for these two ticks=
t = 0 and
t = 1 / √ (1-(v2/c2))
As judged from non-moving
K, the clock is moving with the velocity
v as judged from this viewpoint, the time which elapses between two strokes of the clock is not 1 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.