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

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

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, {{MathForm2|(26)| $$ \left. \begin{align} x_1' & = x_1 \cos\phi - x_2 \sin\phi \\ x_2' & = x_1 \sin\phi + x_2 \cos\phi \\ x_3' & = x_3 \\ x_4' & = x_4 \end{align} \right\} $$}}

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, {{MathForm2|(26a)| $$ \left. \begin{align} x_1' & = x_1 \cos\psi - x_4 \sin\psi \\ x_4' & = x_1 \sin\psi + x_4 \cos\psi \\ x_2' & = x_2 \\ x_3' & = x_3 \end{align} \right\} $$}}

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