Page:Relativity (1931).djvu/166

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 ${\sqrt {-1}}.ct$ in place of $t$, as time-variable. If, in accordance with this, we insert

${\begin{aligned}x_{1}&=x\\x_{2}&=y\\x_{3}&=z\\x_{4}&={\sqrt {-1}}.ct\end{aligned}}$

and similarly for the accented system $K'$, then the condition which is identically satisfied by the transformation can be expressed thus:

$x_{1}'^{2}+x_{2}'^{2}+x_{3}'^{2}+x_{4}'^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$

(12).

That is, by the afore-mentioned choice of “coordinates” (11a) is transformed into this equation. We see from (12) that the imaginary time coordinate $x_{4}$, 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”

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