Page:Grundgleichungen (Minkowski).djvu/12

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

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

so that

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

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

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

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

The differentials denoting the displacements of matter occupying the spacetime point $x_{1},\ x_{2},\ x_{3},\ x_{4}$ 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 ${\mathfrak {w}}'$ with the ratios ${\frac {dx'}{dt'}},\ {\frac {dy'}{dt'}},\ {\frac {dz'}{dt'}}$ as components.

Now it is quite apparent that the system of values

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

is transformed into the values

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

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

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