# 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) $w^{2}_{1} + w^{2}_{2} + w^{2}_{3} + w^{2}_{4} = -1$.

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.