Page:Relativity (1931).djvu/60

 of times and lengths, then instead of the above we should have obtained the following equations:

$$\begin{align} x' & = x - vt \\ y' & = y \\ z' & = z \\ t' & = t \end{align}$$

This system of equations is often termed the “Galilei transformation.” The Galilei transformation can be obtained from the Lorentz transformation by substituting an infinitely large value for the velocity of light $$c$$ in the latter transformation.

Aided by the following illustration, we can readily see that, in accordance with the Lorentz transformation, the law of the transmission of light in vacuo is satisfied both for the reference-body $$K$$ and for the reference-body $$K'$$. A light-signal is sent along the positive x-axis, and this light-stimulus advances in accordance with the equation

i.e. with the velocity $$c$$. According to the equations of the Lorentz transformation, this simple relation between $$x$$ and $$t$$ involves a relation between $$x'$$ and $$t'$$. In point of fact, if we substitute for $$x$$ the value $$ct$$ in the first and fourth equations of the Lorentz transformation, we obtain:

$$\begin{align} x' & = \frac{(c - v)t}{\sqrt{1 - \frac{v^2}{c^2}}} \\ t' & = \frac{\left ( 1 - \frac{v}{c} \right ) t}{\sqrt{1 - \frac{v^2}{c^2}}} \end{align}$$