Page:Grundgleichungen (Minkowski).djvu/10

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can easily be transferred to any other axis when the system of axes are subjected to a transformation about this last axis. So we came to a more general law: —

Let \mathfrak{v} be a vector with the components \mathfrak{v}_{x},\ \mathfrak{v}_{y},\ \mathfrak{v}_{z}, and let \left|\mathfrak{v}\right|=q<1. By \mathfrak{\bar{v}} we shall denote any vector which is perpendicular to \mathfrak{v}, and by \mathfrak{r_{v}}, \mathfrak{r_{\bar{v}}} we shall denote components of \mathfrak{r} in direction of \mathfrak{\bar{v}} and \left|\mathfrak{v}\right|.

Instead of x, y, z, t, new magnetudes x,' y,' z,' t' will be introduced in the following way. If for the sake of shortness, \mathfrak{r} is written for the vector with the components x, y, z in the first system of reference, \mathfrak{r}' for the same vector with the components x', y', z' in the second system of reference, then for the direction of \mathfrak{v} we have

(10) \mathfrak{r'_{v}}=\frac{r_{v}-qt}{\sqrt{1-q^{2}}},

and for every perpendicular direction \mathfrak{\bar{v}}

(11) \mathfrak{r'_{\bar{v}}}=\mathfrak{r_{\bar{v}}},

and further

(12) t'=\frac{-q\mathfrak{r_{v}}+t}{\sqrt{1-q^{2}}}

The notations \mathfrak{r'_{v}} and \mathfrak{r'_{\bar{v}}} are to be understood in the sense that with the directions \mathfrak{v}, and every direction \mathfrak{v} perpendicular to \mathfrak{\bar{v}} in the system x, y, z are always associated the directions with the same direction cosines in the system x', y', z' ,

A transformation which is accomplished by means of (10), (11), (12) with the condition 0< q < 1 will be called a special Lorentz-transformation. We shall call \mathfrak{v} the vector, the direction of \mathfrak{v} the axis, and the magnitude of \mathfrak{v} the moment of this transformation.

If further \varrho' and the vectors \mathfrak{w}',\ \mathfrak{e}',\ \mathfrak{m}', in the system x', y', z' are so defined that,

(13) \varrho'=\frac{\varrho(-q\mathfrak{w_{v}}+1)}{\sqrt{1-q^{2}}},