Page:The Meaning of Relativity - Albert Einstein (1922).djvu/47

a co-variant equation, that is, an equation which is satisfied with respect to every inertial system if it is satisfied in the inertial system to which we refer the two given events (emission and reception of the ray of light). Finally, with Minkowski, we introduce in place of the real time co-ordinate $l=ct$, the imaginary time co-ordinate

$x_{4}=il=ict({\sqrt {-1}}=1).$

Then the equation defining the propagation of light, which must be co-variant with respect to the Lorentz transformation, becomes

$\sum \limits _{(4)}\Delta x_{\nu }^{2}=\Delta x_{!}^{2}+\Delta x_{2}^{2}+\Delta x_{3}^{2}+\Delta x_{4}^{2}=0$

(22c)

This condition is always satisfied ^[1] if we satisfy the more general condition that

$s^{2}=\Delta x_{1}^{2}+\Delta x_{2}^{2}+\Delta x_{3}^{2}+\Delta x_{4}^{2}$

(23)

shall be an invariant with respect to the transformation. This condition is satisfied only by linear transformations, that is, transformations of the type

$x_{\mu }'=a_{\mu }+b_{\mu \alpha }x_{\alpha }$

(24)

in which the summation over the $\alpha$ is to be extended from $\alpha =1$ to $\alpha =4$. A glance at equations (23) and (24) shows that the Lorentz transformation so defined is identical with the translational and rotational transformations of the Euclidean geometry, if we disregard the number of dimensions and the relations of reality. We

1. That this specialization lies in the nature of the case will be evident later.