XXXII

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 $K^{\prime }$–-in accordance with the corresponding formula $r^{\prime }=c{t}^{\prime },$ or, $x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-c^2{t}^{\prime 2}=0.(10a)$ In order that equation eqn:(10a) may be a consequence of equation eqn:(10), we must have $x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-c^2{t}^{\prime 2}=\sigma (x^2+y^2+z^2-c^2{t}^2).(11)$

Since equation eqn:(8a) must hold for points on the $x$-axis, we thus have $\sigma =1$. It is easily seen that the Lorentz transformation really satisfies equation eqn:(11)

for $\sigma =1$; for eqn:(11) is a consequence of eqn:(8a) and eqn:(9), and hence also of eqn:(8) and eqn:(9). We have thus derived the Lorentz transformation.

The Lorentz transformation represented by eqn:(8) and eqn:(9) still requires to be generalised. Obviously it is immaterial whether the axes of $K^{\prime }$ be chosen so that they are spatially parallel to those of $K$. It is also not essential that the velocity of translation of $K^{\prime }$ 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 $x^{\prime }$, $y^{\prime }$, $z^{\prime }$, $t^{\prime }$, in terms of linear homogeneous functions of $x$, $y$, $z$, $t$, of such a kind that the relation $x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-c^2{t}^{\prime 2}=x^2+y^2+z^2-c^2{t}^2(11a)$ is satisfied identically. That is to say: If we substitute their expressions in $x$, $y$, $z$, $t$, in place of $x^{\prime }$, $y^{\prime }$, $z^{\prime }$, $t^{\prime }$, on the left-hand side, then the left-hand side of eqn:(11a) agrees with the right-hand side.

IIMinkowski's Four-dimensional Space ("World")[Supplementary to [chapter:XVII]Section XVII]

We can characterise the Lorentz transformation

still more simply if we introduce the imaginary $\sqrt{-1}·ct$ in place of $t$, as time-variable. If, in accordance with this, we insert

$$\begin{gathered} x_1\&=x, \\ {x}_2\&=y, \\ {x}_3\&=z, \\ {x}_4\&=\sqrt{-1}·ct, \end{gathered}$$

and similarly for the accented system $K^{\prime }$, then the condition which is identically satisfied by the transformation can be expressed thus: ${x_1^{\prime }}^2+{x_2^{\prime }}^2+{x_3^{\prime }}^2+{x_4^{\prime }}^2={x_1}^2+{x_2}^2+{x_3}^2+{x_4}^2.(12)$

That is, by the afore-mentioned choice of "co-ordinates," eqn:(11a) is transformed into this equation.

We see from eqn:(12) that the imaginary time co-ordinate $x_4$

enters into the condition of transformation in exactly the same way as the space co-ordinates $x_1$, $x_2$, $x_3$. It is due to this fact that, according to the theory of

relativity, the "time" $x_4$ enters into natural laws in the same form as the space co-ordinates $x_1$, $x_2$, $x_3$.