# Page:Grundgleichungen (Minkowski).djvu/12

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If we divide ${\displaystyle \varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}}$ by this magnitude, we obtain the four values

${\displaystyle w_{1}={\frac {{\mathfrak {w}}_{x}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{2}={\frac {{\mathfrak {w}}_{y}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{3}={\frac {{\mathfrak {w}}_{z}}{\sqrt {1-{\mathfrak {w}}^{2}}}},\ w_{4}={\frac {i}{\sqrt {1-{\mathfrak {w}}^{2}}}}}$,

so that

 (19) ${\displaystyle w_{1}^{2}+w_{2}^{2}+w_{3}^{2}+w_{4}^{2}=-1}$.

It is apparent that these four values, are determined by the vector ${\displaystyle {\mathfrak {w}}}$ and inversely the vector ${\displaystyle {\mathfrak {w}}}$ of magnitude ${\displaystyle <1}$ follows from the 4 values ${\displaystyle w_{1},\ w_{2},\ w_{3},\ w_{4}}$, where ${\displaystyle w_{1},\ w_{2},\ w_{3}}$ are real, ${\displaystyle -iw_{4}}$ real and positive and condition (19) is fulfilled.

The meaning of ${\displaystyle w_{1},\ w_{2},\ w_{3},\ w_{4}}$ here is, that they are the ratios of ${\displaystyle dx_{1},\ dx_{2},\ dx_{3},\ dx_{4}}$ to

 (20) ${\displaystyle {\sqrt {-(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2})}}=dt{\sqrt {1-{\mathfrak {w}}^{2}}}}$

The differentials denoting the displacements of matter occupying the spacetime point ${\displaystyle x_{1},\ x_{2},\ x_{3},\ x_{4}}$ to the adjacent space-time point.

After the Lorentz-transfornation is accomplished the velocity of matter in the new system of reference for the same space-time point x', y', z', t' is the vector ${\displaystyle {\mathfrak {w}}'}$ with the ratios ${\displaystyle {\frac {dx'}{dt'}},\ {\frac {dy'}{dt'}},\ {\frac {dz'}{dt'}}}$ as components.

Now it is quite apparent that the system of values

${\displaystyle x_{1}=w_{1},\ x_{2}=w_{2},\ x_{3}=w_{3},\ x_{4}=w_{4}}$

is transformed into the values

${\displaystyle x'_{1}=w'_{1},\ x'_{2}=w'_{2},\ x'_{3}=w'_{3},\ x'_{4}=w'_{4}}$

in virtue of the Lorentz-transformation (10), (11), (12).

The dashed system has got the same meaning for the velocity ${\displaystyle {\mathfrak {w}}'}$ after the transformation as the first system of values has got for ${\displaystyle {\mathfrak {w}}}$ before transformation.