Relativity
The Special and General Theory

I PLACE a meter-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 meter-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 meter-rod is moving with the velocity v relative to K. It therefore follows that the length of a rigid meter-rod moving in the direction of its length with a velocity v is

√

1 - v2/c2

of a meter. 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