# Page:Grundgleichungen (Minkowski).djvu/14

of x, y, z, t in x', y', z', t' with essentially real co-efficients, whereby the aggregrate $-x^{2} - y^{2} - z^{2} + t^{2}$ transforms into $-x'^{2} - y'^{2} - z'^{2} + t'^{2}$, and to every such system of values x, y, z, t with a positive t, for which this aggregate $>0$, 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) $\alpha^{2}_{14} + \alpha^{2}_{24} + \alpha^{2}_{34} + \alpha^{2}_{44} =1$

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

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

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

On the other hand, with every set of value of $\alpha_{14},\ \alpha_{24},\ \alpha_{34},\ \alpha_{44}$ which in this way fulfill the condition 22) with real values of $\mathfrak{v}_{x}+\mathfrak{v}_{y}+\mathfrak{v}_{z}$, we can construct the special Lorentz-transformation (16) with $\alpha_{14},\ \alpha_{24},\ \alpha_{34},\ \alpha_{44}$ as the last vertical column, — and then every Lorentz-transformation with the same last vertical column $\alpha_{14},\ \alpha_{24},\ \alpha_{34},\ \alpha_{44}$ 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 $\varrho_{1},\ \varrho_{2},\ \varrho_{3},\ \varrho_{4}$ with the condition that in case of a Lorentz-transformation it is to be replaced by the set $\varrho'_{1},\ \varrho'_{2},\ \varrho'_{3},\ \varrho'_{4}$, where these are the values $x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}$ obtained by substituting $\varrho_{1},\ \varrho_{2},\ \varrho_{3},\ \varrho_{4}$ for $x_{1},\ x_{2},\ x_{3},\ x_{4}$ in the expression (21).

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