Page:Grundgleichungen (Minkowski).djvu/14

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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