HKB
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Thus we have obtained the Lorentz transformation for events on the x-axis. It satisfies the condition
x′2 – c2t′2 = x2 – c2t2 . . . . . . (8a).
The extension of this result, to include events which take place outside the x-axis, is obtained by retaining equations (8) and supplementing them by the relations
image048
In this way we satisfy the postulate of the constancy of the velocity of light in vacuo for rays of light of arbitrary direction, both for the system K and for the system K′. This may be shown in the following manner.
We suppose a light-signal sent out from the origin of K at the time t = 0. It will be propagated according to the equation
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or, if we square this equation, according to the equation
x2 + y2 + z2 – c2t2 = 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 K′—in accordance with the corresponding formula
r′ = ct′
or,
*x′*2 + *y′*2 + z′2 – c2t′2 = 0 . . . . . . (10a).
In order that equation (10a) may be a consequence of equation (10), we must have
*x′*2 + *y′*2 + z′2 – c2t′2 = σ (x2 + y2 + z2 – c2t2) (11).
Since equation (8a) must hold for points on the x-axis, we thus have σ = 1. It is easily seen that the Lorentz transformation really satisfies equation (11) for σ = 1; 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 K′ be chosen so that they are spatially parallel to those of K. It is also not essential that the velocity of translation of K′ with respect to K should be in the direction of the x-axis. A simple consideration shows that we are able to construct the Lorentz transformation in this general sense from two kinds of transformations, viz. from Lorentz transformations in the special sense and from purely spatial transformations. which corresponds to the replacement of the rectangular co-ordinate system by a new system with its axes pointing in other directions.
Mathematically, we can characterise the generalised Lorentz transformation thus:
It expresses x′, y′, x′, t′, in terms of linear homogeneous functions of x, y, x, t, of such a kind that the relation
*x′*2 + y′2 + z′2 – c2t′2 = x2 + y2 + z2 – c2t2 (11a).
is satisficd identically. That is to say: If we substitute their expressions in x, y, x, t, in place of x′, y′, x′, t′, on the left-hand side, then the left-hand side of (11a) agrees with the right-hand side.
APPENDIX II
MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”)
(SUPPLEMENTARY TO SECTION XVII)
We can characterise the Lorentz transformation still more simply if we introduce the imaginary
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in place of t, as time-variable. If, in accordance with this, we insert
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and similarly for the accented system K′, then the condition which is identically satisfied by the transformation can be expressed thus:
x1′2 + x2′2 + x3′2 + x4′2 = x12 + x22 + x32 + x42 (12).
That is, by the afore-mentioned choice of “coordinates,” (11a) [see the end of Appendix II] is transformed into this equation.
We see from (12) that the imaginary time co-ordinate x4, enters into the condition of transformation in exactly the same way as the space co-ordinates x1, x2, x3. It is due to this fact that, according to the theory of relativity, the “time” x4, enters into natural laws in the same form as the space co ordinates x1, x2, x3.