40: file eq40.gif
or, if we square this equation, according to the equation
x2 + y2 + z2 = c^2t2 = 0 . . . (10).
It is required by the law of propagation of light, in conjunction with
the postulate of relativity, that the transmission of the signal in
question should take place -- as judged from K1 -- in accordance with
the corresponding formula
r' = ct'
or,
x'2 + y'2 + z'2 - c^2t'2 = 0 . . . (10a).
In order that equation (10a) may be a consequence of equation (10), we
must have
x'2 + y'2 + z'2 - c^2t'2 = s (x2 + y2 + z2 - c^2t2) (11).
Since equation (8a) must hold for points on the x-axis, we thus have s
= I. It is easily seen that the Lorentz transformation really
satisfies equation (11) for s = I; for (11) is a consequence of (8a)
and (9), and hence also of (8) and (9). We have thus derived the
Lorentz transformation.
The Lorentz transformation represented by (8) and (9) still requires
to be generalised. Obviously it is immaterial whether the axes of K1
be chosen so that they are spatially parallel to those of K. It is
also not essential that the velocity of translation of K1 with respect
to K should be in the direction of the x-axis.
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