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
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