HKB
(SUPPLEMENTARY TO SECTION XI)
For the relative orientation of the co-ordinate systems indicated in Fig. 2, the x-axes of both systems permanently coincide. In the present case we can divide the problem into parts by considering first only events which are localised on the x-axis. Any such event is represented with respect to the co-ordinate system K by the abscissa x and the time t, and with respect to the system K′ by the abscissa x′ and the time t′. We require to find x′ and t′ when 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 K′ with the velocity c, the propagation relative to the system K′ will be represented by the analogous formula
x′ – ct′ = 0 . . . . . (2)
Those space-time points (events) which satisfy (1) must also satisfy (2). Obviously this will be the case when the relation
(x′ – ct′) = λ(x – ct) . . . (3).
is fulfilled in general, where λ 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′) = (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 λ and μ where
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and
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we obtain the equations
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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 K′ we have permanently x′ = 0, and hence according to the first of the equations (5)
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If we call v the velocity with which the origin of K′ is moving relative to K, we then have
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The same value v can be obtained from equations (5), if we calculate the velocity of another point of K′ 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 reference to K′ must be exactly the same as the length, as judged from K′, of a unit measuring-rod which is at rest relative to K. In order to see how the points of the x′-axis appear as viewed from K, we only require to take a “snapshot” of K′ from K; this means that we have to insert a particular value of t (time of K), e.g. t = 0. For this value of t we then obtain from the first of the equations (5)
x′ = ax
Two points of the x′-axis which are separated by the distance Δx′ = 1 when measured in the K′ system are thus separated in our instantaneous photograph by the distance
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But if the snapshot be taken from K′(t′ = 0), and if we eliminate t from the equations (5), taking into account the expression (6), we obtain
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From this we conclude that two points on the x-axis separated by the distance 1 (relative to K) will be represented on our snapshot by the distance
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But from what has been said, the two snapshots must be identical; hence Δx in (7) must be equal to Δx′ in (7a), so that we obtain
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The equations (6) and (7b) determine the constants a and b. By inserting the values of these constants in (5), we obtain the first and the fourth of the equations given in Section XI.
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