Albert Einstein (1879–1955). Relativity: The Special and General Theory. 1920.

**XII**

**Part I: The Special Theory of Relativity**

# XII. The Behaviour of Measuring-Rods and Clocks in Motion

I *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

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

*v*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

*v*= 0 we should have

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

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*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

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.