# Page:Grundgleichungen (Minkowski).djvu/10

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

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, ${\displaystyle {\mathfrak {r}}}$ is written for the vector with the components x, y, z in the first system of reference, ${\displaystyle {\mathfrak {r}}'}$ for the same vector with the components x', y', z' in the second system of reference, then for the direction of ${\displaystyle {\mathfrak {v}}}$ we have

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

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

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

and further

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

The notations ${\displaystyle {\mathfrak {r'_{v}}}}$ and ${\displaystyle {\mathfrak {r'_{\bar {v}}}}}$ are to be understood in the sense that with the directions ${\displaystyle {\mathfrak {v}}}$, and every direction ${\displaystyle {\mathfrak {v}}}$ perpendicular to ${\displaystyle {\mathfrak {\bar {v}}}}$ 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 ${\displaystyle 0 will be called a special Lorentz-transformation. We shall call ${\displaystyle {\mathfrak {v}}}$ the vector, the direction of ${\displaystyle {\mathfrak {v}}}$ the axis, and the magnitude of ${\displaystyle {\mathfrak {v}}}$ the moment of this transformation.

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

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