Relativity - The Special and General Theory, page 89 by Albert Einstein
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90
hen x and t are given.
A light-signal, which is proceeding along the positive axis of x, is transmitted according to the equation
x = ct
or
x - ct = 0 . . . (1).
Since the same light-signal has to be transmitted relative to K1 with the velocity c, the propagation relative to the system K1 will be represented by the analogous formula
x' - ct' = O . . . (2)
Those space-time points (events) which satisfy (x) must also satisfy (2). Obviously this will be the case when the relation
(x' - ct') = l (x - ct) . . . (3).
is fulfilled in general, where l indicates a constant ; for, according to (3), the disappearance of (x - ct) involves the disappearance of (x' - ct').
If we apply quite similar considerations to light rays which are being transmitted along the negative x-axis, we obtain the condition
(x' + ct') = µ(x + ct) . . . (4).
By adding (or subtracting) equations (3) and (4), and introducing for convenience the constants a and b in place of the constants l and µ, where
eq. 29: file eq29.gif
and
eq. 30: file eq30.gif
we obtain the equations
eq. 31: file eq31.gif
We should thus have the solution of our problem, if the constants a and b were known. These result from the following discussion.
For the origin of K1 we have permanently x' = 0, and hence according to the first of the equations (5)
eq. 32: file eq32.gif
If we call v the velocity with which the origin of K1 is moving relative to K, we then have
eq. 33: file eq33.gif
The same value v can be obtained from equations (5), if we calculate the velocity of another point of K1 relative to K, or the velocity (directed towards the negative x-axis) of a point of K with respect to K'. In short, we can designate v as the relative velocity of the two systems.
Furthermore, the principle of relativity teaches us that, as judged from K, the length of a unit measuring-rod which is at rest with refere