Page:Relativity (1931).djvu/163

Thus we have obtained the Lorentz transformation for events on the $x$-axis. It satisfies the condition

$x'^{2}-c^{2}t'^{2}=x^{2}-c^{2}t^{2}$

(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

$\left.{\begin{aligned}y'=y\\z'=z\end{aligned}}\right\}$

(9).

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

$r={\sqrt {x^{2}+y^{2}+z^{2}}}=ct$,

or, if we square this equation, according to the equation

$x^{2}+y^{2}+z^{2}-c^{2}t^{2}=0$

(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,

$x'^{2}+y'^{2}+z'^{2}-c^{2}t'^{2}=0$

(10a)