Relativity: The Special and General Theory · Albert Einstein

Chapter XII: The Behaviour of Measuring-Rods and Clocks in Motion

13 of 37 · 3 min of reading

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 x Subscript ( beginning of rod ) equals 0 dot StartRoot 1 minus (v squared)/(c squared) EndRoot x Subscript ( end of rod ) equals 1 dot StartRoot 1 minus (v squared)/(c squared) EndRoot 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, y, x, 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 (1)/(StartRoot 1 minus StartFraction v squared Over c squared) EndRoot EndFraction.

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)/(StartRoot 1 minus StartFraction v squared Over c squared) 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.

· · ·

End of part 13 of 37