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

If $$\alpha_{14}=0,\ \alpha_{24}=0,\ \alpha_{34}=0$$ then $$\alpha_{44}=1$$, and the 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 -transformation (16) with $$\alpha_{14},\ \alpha_{24},\ \alpha_{34},\ \alpha_{44}$$ as the last vertical column, — and then every -transformation with the same last vertical column $$\alpha_{14},\ \alpha_{24},\ \alpha_{34},\ \alpha_{44}$$ supposed to be composed of the special -transformation, and a rotation of the spatial co-ordinate system round the null-point.

The totality of all -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 -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