Page:The Meaning of Relativity - Albert Einstein (1922).djvu/49

Rh $$K'$$ relatively to $$K$$, instead of the imaginary angle $$\Psi$$. We have, first,

Since for the origin of $$K'$$, i.e., for $$x_1 = 0$$, we must have $$x_1 = vl$$, it follows from the first of these equations that

and also {{MathForm2|(28)| $$ \left. \begin{align} \sin\psi & = \frac{-iv}{\sqrt{1-v^2}} \\ \cos\psi & = \frac{1}{\sqrt{1-v^2}} \end{align} \right\}, $$}} so that we obtain {{MathForm2|(29)| $$ \left. \begin{align} x_1' & = \frac{x_1 - vl}{\sqrt{1 - v^2}} \\ l' & = \frac{l - vx_1}{\sqrt{1 - v^2}} \\ x_2' & = x_2 \\ x_3' & = x_3 \end{align} \right\} $$}}

These equations form the well-known special Lorentz transformation, which in the general theory represents a rotation, through an imaginary angle, of the four-dimensional system of co-ordinates. If we introduce the ordinary time $$t$$, in place of the light-time $$l$$, then in (29) we must replace $$l$$ by $$ct$$ and $$v$$ by $$\frac{v}{c}$$.

We must now fill in a gap. From the principle of the constancy of the velocity of light it follows that the equation