x1 = x — vt
                                y1 = y
                                z1 = z
                                t1 = 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 K1.  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 x1 and t1.  In point of fact, if we substitute for x the value ct in the first and fourth equations of the Lorentz transformation, we obtain: 

eq. 3:  file eq03.gif

eq. 4:  file eq04.gif

from which, by division, the expression

x1 = ct1

immediately follows.  If referred to the system K1, the propagation of light takes place according to this equation.  We thus see that the velocity of transmission relative to the reference-body K1 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.

  Notes

*) A simple derivation of the Lorentz transformation is given in Appendix I.

THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION

Place a metre-rod in the x1-axis of K1 in such a manner that one end (the beginning) coincides with the point x1=0 whilst the other end (the end of the rod) coincides with the point x1=I.  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

eq. 05a:  file eq05a.gif

eq. 05b:  file eq05b.gif

the distance between the points being eq. 06 .

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 eq. 06 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 eq. 06a ,