Page:The Meaning of Relativity - Albert Einstein (1922).djvu/48

can also conclude that the coefficients $b_{\mu \alpha }$ must satisfy the conditions

$b_{\mu \alpha }b_{\nu \alpha }=\delta _{\mu \nu }=b_{\alpha \mu }b_{\alpha \nu }$

(25)

Since the ratios of the $x_{\nu }$ are real, it follows that all the $\alpha _{\mu }$ and the $b_{\mu \alpha }$ are real, except $a_{4},b_{41},b_{42},b_{43},b_{14},b_{24},$ and $b_{34}$, which are purely imaginary.

Special Lorentz Transformation. We obtain the simplest transformations of the type of (24) and (25) if only two of the co-ordinates are to be transformed, and if all the $a_{\mu }$, which determine the new origin, vanish. We obtain then for the indices I and 2, on account of the three independent conditions which the relations (25) furnish,

$\left.{\begin{aligned}x_{1}'&=x_{1}\cos \phi -x_{2}\sin \phi \\x_{2}'&=x_{1}\sin \phi +x_{2}\cos \phi \\x_{3}'&=x_{3}\\x_{4}'&=x_{4}\end{aligned}}\right\}$

(26)

This is a simple rotation in space of the (space) co-ordinate system about $x_{3}$-axis. We see that the rotational transformation in space (without the time transformation) which we studied before is contained in the Lorentz transformation as a special case. For the indices 1 and 4 we obtain, in an analogous manner,

$\left.{\begin{aligned}x_{1}'&=x_{1}\cos \psi -x_{4}\sin \psi \\x_{4}'&=x_{1}\sin \psi +x_{4}\cos \psi \\x_{2}'&=x_{2}\\x_{3}'&=x_{3}\end{aligned}}\right\}$

(26a)

On account of the relations of reality $\psi$ must be taken as imaginary. To interpret these equations physically, we introduce the real light-time $l$ and the velocity $v$ of