# Page:Grundgleichungen (Minkowski).djvu/14

of x, y, z, t in x', y', z', t' with essentially real co-efficients, whereby the aggregrate ${\displaystyle -x^{2}-y^{2}-z^{2}+t^{2}}$ transforms into ${\displaystyle -x'^{2}-y'^{2}-z'^{2}+t'^{2}}$, and to every such system of values x, y, z, t with a positive t, for which this aggregate ${\displaystyle >0}$, there always corresponds a positive t'; this last is quite evident from the continuity of the aggregate x, y, z, t.

The last vertical column of co-efficients has to fulfill, the condition

 (22) ${\displaystyle \alpha _{14}^{2}+\alpha _{24}^{2}+\alpha _{34}^{2}+\alpha _{44}^{2}=1}$

If ${\displaystyle \alpha _{14}=0,\ \alpha _{24}=0,\ \alpha _{34}=0}$ then ${\displaystyle \alpha _{44}=1}$, and the Lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.

If ${\displaystyle \alpha _{14},\ \alpha _{24},\ \alpha _{34}}$ are not all zero, and if we put

${\displaystyle \alpha _{14}:\alpha _{24}:\alpha _{34}:\alpha _{44}={\mathfrak {v}}_{x}:{\mathfrak {v}}_{y}:{\mathfrak {v}}_{z}:i}$,
${\displaystyle q={\sqrt {{\mathfrak {v}}_{x}^{2}+{\mathfrak {v}}_{y}^{2}+{\mathfrak {v}}_{z}^{2}}}<1}$.

On the other hand, with every set of value of ${\displaystyle \alpha _{14},\ \alpha _{24},\ \alpha _{34},\ \alpha _{44}}$ which in this way fulfill the condition 22) with real values of ${\displaystyle {\mathfrak {v}}_{x}+{\mathfrak {v}}_{y}+{\mathfrak {v}}_{z}}$, we can construct the special Lorentz-transformation (16) with ${\displaystyle \alpha _{14},\ \alpha _{24},\ \alpha _{34},\ \alpha _{44}}$ as the last vertical column, — and then every Lorentz-transformation with the same last vertical column ${\displaystyle \alpha _{14},\ \alpha _{24},\ \alpha _{34},\ \alpha _{44}}$ supposed to be composed of the special Lorentz-transformation, and a rotation of the spatial co-ordinate system round the null-point.

The totality of all Lorentz-Transformations forms a group.

Under a space-time vector of the 1st kind shall be understood a system of four magnitudes ${\displaystyle \varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}}$ with the condition that in case of a Lorentz-transformation it is to be replaced by the set ${\displaystyle \varrho '_{1},\ \varrho '_{2},\ \varrho '_{3},\ \varrho '_{4}}$, where these are the values ${\displaystyle x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}}$ obtained by substituting ${\displaystyle \varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}}$ for ${\displaystyle x_{1},\ x_{2},\ x_{3},\ x_{4}}$ in the expression (21).

Besides the time-space vector of the 1st kind ${\displaystyle x_{1},\ x_{2},\ x_{3},\ x_{4}}$ we shall also make use of another spacetime vector of the first kind ${\displaystyle u_{1},\ u_{2},\ u_{3},\ u_{4}}$, and let us form the linear combination