Page:Grundgleichungen (Minkowski).djvu/12

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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.