HKB
[9] A simple derivation of the Lorentz transformation is given in Appendix I.
If in place of the law of transmission of light we had taken as our basis the tacit assumptions of the older mechanics as to the absolute character of times and lengths, then instead of the above we should have obtained the following equations:
x′ = x – vt
y′ = y
z′ = z
t′ = t
This system of equations is often termed the “Galilei transformation.” The Galilei transformation can be obtained from the Lorentz transformation by substituting an infinitely large value for the velocity of light c in the latter transformation.
Aided by the following illustration, we can readily see that, in accordance with the Lorentz transformation, the law of the transmission of light in vacuo is satisfied both for the reference-body K and for the reference-body K′. A light-signal is sent along the positive x-axis, and this light-stimulus advances in accordance with the equation
x = ct,
i.e. with the velocity c. According to the equations of the Lorentz transformation, this simple relation between x and t involves a relation between x′ and t′. In point of fact, if we substitute for x the value ct in the first and fourth equations of the Lorentz transformation, we obtain:
image005
from which, by division, the expression
x′ = ct′
immediately follows. If referred to the system K′, the propagation of light takes place according to this equation. We thus see that the velocity of transmission relative to the reference-body K′ is also equal to c. The same result is obtained for rays of light advancing in any other direction whatsoever. Of cause this is not surprising, since the equations of the Lorentz transformation were derived conformably to this point of view.
XII.
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′ = 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
image006
the distance between the points being
image007
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
image008
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
image009
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
image010
this is quite in accordance with the principle of relativity which forms the basis of our considerations.