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