the behaviour of measuring-rods and clocks in motion
place a metre-rod in the x′-axis of k′ in such a manner that one end (the beginning) coincides with the point x prime equals 0 whilst the other end (the end of the rod) coincides with the point x prime equals 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 equals 0 can be shown to be
startlayout 1st row 1st column x subscript left-parenthesis beginning of rod right-parenthesis 2nd column equals 3rd column 0 dot startroot 1 minus startfraction v squared over c squared endfraction endroot 2nd row 1st column x subscript left-parenthesis end of rod right-parenthesis 2nd column equals 3rd column 1 dot startroot 1 minus startfraction v squared over c squared endfraction endroot endlayout
the distance between the points being startroot 1 minus v squared slash c squared endroot.
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 startroot 1 minus v squared slash c squared endroot 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 equals c we should have startroot 1 minus v squared slash c squared endroot equals 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 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 startroot 1 minus v squared slash c squared endroot; 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 comma y comma x comma 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 prime equals 0) of k′. t prime equals 0 and t prime equals 1 are two successive ticks of this clock. the first and fourth equations of the lorentz transformation give for these two ticks:
t equals 0
and
t prime equals startfraction 1 over startroot 1 minus startfraction v squared over c squared endfraction endroot endfraction period
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 startfraction 1 over startroot 1 minus startfraction v squared over c squared endfraction endroot endfraction 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.
