Page:Relativity (1931).djvu/129

validity of the law of transmission of light for all Galileian systems of reference.

Minkowski found that the Lorentz transformations satisfy the following simple conditions. Let us consider two neighbouring events, the relative position of which in the four-dimensional continuum is given with respect to a Galileian reference-body $K$ by the space co-ordinate differences $dx$, $dy$, $dz$ and the time-difference $dt$. With reference to a second Galileian system we shall suppose that the corresponding differences for these two events are $dx'$, $dy'$, $dz'$, $dt'$. Then these magnitudes always fulfil the condition.^[1]

$dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}=dx'^{2}+dy'^{2}+dz'^{2}-c^{2}dt'^{2}$.

The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude

$ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}$,

which belongs to two adjacent points of the four-dimensional space-time continuum, has the same value for all selected (Galileian) reference-bodies. If we replace $x$, $y$, $z$, ${\sqrt {-1}}ct$, by $x_{1}$, $x_{2}$, $x_{3}$$x_{4}$, we also obtain the result that

$ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}$

is independent of the choice of the body of refer-

1. Cf. Appendices I and II. The relations which are derived there for the co-ordinates themselves are valid also for co-ordinate differences, and thus also for co-ordinate differentials (indefinitely small differences).