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

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

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 -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 -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.