Page:Relativity (1931).djvu/87

$t'={\frac {t-{\tfrac {v}{c^{2}}}x}{\sqrt {1-{\tfrac {v^{2}}{c^{2}}}}}}$

Moreover, according to this equation the time difference $\Delta t'$ of two events with respect to $K'$ does not in general vanish, even when the time difference $\Delta t$ of the same events with reference to $K$ vanishes. Pure “space-distance” of two events with respect to $K$ results in “time-distance” of the same events with respect to $K'$. But the discovery of Minkowski, which was of importance for the formal development of the theory of relativity, does not lie here. It is to be found rather in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space.^[1] In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate $t$ by an imaginary magnitude ${\sqrt {-1}}$. $ct$ proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same rôle as the three space co-ordinates. Formally, these four co-ordinates

1. Cf. the somewhat more detailed discussion in Appendix II.