XI

to the impenetrability of solid bodies. In every such framework we imagine three surfaces perpendicular to each other marked out, and designated as "co-ordinate

planes" ("co-ordinate system"). A co-ordinate system $K$ then corresponds to the embankment, and a co-ordinate system $K^{\prime }$ to the train. An event, wherever it may have taken place, would be fixed in space with respect to $K$ by the three perpendiculars $x$, $y$, $z$ on the co-ordinate planes, and with regard to time by a time-value $t$. Relative to $K^{\prime }$, the same event would be fixed in respect of space and time by corresponding values $x^{\prime }$, $y^{\prime }$, $z^{\prime }$, $t^{\prime }$, which of course are not identical with $x$, $y$, $z$, $t$. It has already been set forth in detail how these magnitudes are to be regarded as results of physical measurements.

Obviously our problem can be exactly formulated in the following manner. What are the values $x^{\prime }$, $y^{\prime }$, $z^{\prime }$, $t^{\prime }$, of an event with respect to $K^{\prime }$, when the magnitudes $x$, $y$, $z$, $t$, of the same event with respect to $K$ are given? The relations must be so chosen that the law of the transmission of light in vacuo is satisfied for one and the same ray of light (and of course for every ray) with respect to $K$ and $K^{\prime }$. For the relative orientation in space of the co-ordinate systems indicated in the diagram ([fig:2]Fig. 2), this problem is solved by means of the equations:

$x^{\prime }\&=\frac{x-vt}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}},\mathrm{\backslash displaybreak}[1]$

$$\begin{gathered} y^{\prime }\&=y, \\ {z}^{\prime }\&=z, \\ {t}^{\prime }\&=\frac{t-{\displaystyle \frac{v}{c^2}}·x}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}. \end{gathered}$$

This system of equations is known as the "Lorentz

transformation." A simple derivation of the Lorentz transformation is given in [appendix:I]Appendix I.

If in place of the law of transmission of light we had taken as our basis the tacit assumptions of the older mechanics as to the absolute character of times and lengths, then instead of the above we should have obtained the following equations:

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 $K^{\prime }$. A light-signal is sent along the

positive $x$-axis, and this light-stimulus advances in

accordance with the equation $x=ct,$

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 $x^{\prime }$ and $t^{\prime }$. In point of fact, if we substitute for $x$ the value $ct$ in the first and fourth equations of the Lorentz transformation, we obtain: