Page:The Meaning of Relativity - Albert Einstein (1922).djvu/44

settled by means of the principle of the constancy of the velocity of light and the principle of special relativity.

To this end we think of space and time physically defined with respect to two inertial systems, $K$ and $K'$, in the way that has been shown. Further, let a ray of light pass from one point $P_{1}$ to another point $P_{2}$ of $K$ through a vacuum. If $r$ is the measured distance between the two points, then the propagation of light must satisfy the equation

$r=c.\Delta t.$

If we square this equation, and express $r^{2}$ by the differences of the co-ordinates, $\Delta x_{\nu }$, in place of this equation we can write

$\sum (\Delta x_{\nu })^{2}-c^{2}\Delta t^{2}=0$

(22)

This equation formulates the principle of the constancy of the velocity of light relatively to $K$. It must hold whatever may be the motion of the source which emits the ray of light.

The same propagation of light may also be considered relatively to $K'$ in which case also the principle of the constancy of the velocity of light must be satisfied. Therefore, with respect to $K'$, we have the equation

$\sum (\Delta x_{\nu }')^{2}-c^{2}\Delta t'^{2}=0$

(22a)

Equations (22a) and (22) must be mutually consistent with each other with respect to the transformation which transforms from $K$ to $K'$. A transformation which effects this we shall call a "Lorentz transformation."

Before considering these transformations in detail we