# Page:Grundgleichungen (Minkowski).djvu/11

 (14) ${\displaystyle \varrho '{\mathfrak {w'_{v}}}={\frac {\varrho {\mathfrak {w_{v}}}-\varrho q}{\sqrt {1-q^{2}}}},\ \varrho '{\mathfrak {w'_{\bar {v}}}}=\varrho {\mathfrak {w_{\bar {v}}}}}$,

further[1]

 (15) ${\displaystyle {\begin{array}{c}({\mathfrak {e}}'+i{\mathfrak {m}}')_{\mathfrak {\bar {v}}}={\frac {({\mathfrak {e}}+i{\mathfrak {m}}-i[{\mathfrak {w}},\ {\mathfrak {e}}+i{\mathfrak {m}}])_{\bar {v}}}{\sqrt {1-q^{2}}}}\\({\mathfrak {e}}'+i{\mathfrak {m}}')_{\mathfrak {v}}=({\mathfrak {e}}+i{\mathfrak {m}}-i[{\mathfrak {w}},\ {\mathfrak {e}}+i{\mathfrak {m}}])_{\mathfrak {v}}\end{array}}}$,

Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes.

The solution of the equations (10), (11), (12) leads to

 (16) ${\displaystyle {\mathfrak {r_{v}}}={\frac {{\mathfrak {r'_{v}}}+qt'}{\sqrt {1-q^{2}}}},\ {\mathfrak {r_{\bar {v}}}}={\mathfrak {r'_{\bar {v}}}},\ t={\frac {q{\mathfrak {r'_{v}}}+t'}{\sqrt {1-q^{2}}}}}$.

Now we shall make a very important observation about the vectors ${\displaystyle {\mathfrak {w}}}$ and ${\displaystyle {\mathfrak {w}}'}$. We can again introduce the indices 1, 2, 3, 4, so that we write ${\displaystyle x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}}$ instead of x,' y,' z,' it' , and ${\displaystyle \varrho '_{1},\ \varrho '_{2},\ \varrho '_{3},\ \varrho '_{4}}$ instead of ${\displaystyle \varrho '{\mathfrak {w}}'_{x'}\ \varrho '{\mathfrak {w}}'_{y'}\ \varrho '{\mathfrak {w}}'_{z'}\ i\varrho '}$. Like the rotation round the z-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also linear transformations with the determinant +1, so that

 (17) ${\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2},}$ d. i. ${\displaystyle x^{2}+y^{2}+z^{2}-t^{2}}$

is transformed into

${\displaystyle x_{1}^{'2}+x_{2}^{'2}+x_{3}^{'2}+x_{4}^{'2},}$ d. i. ${\displaystyle x'^{2}+y'^{2}+z'^{2}-t'^{2}.}$

On the basis of the equations (13), (14), we shall have

${\displaystyle -(\varrho _{1}^{2}+\varrho _{2}^{2}+\varrho _{3}^{2}+\varrho _{4}^{2})=\varrho ^{2}(1-{\mathfrak {w}}_{x}^{2}-{\mathfrak {w}}_{y}^{2}-{\mathfrak {w}}_{z}^{2})=\varrho ^{2}(1-{\mathfrak {w}}^{2})}$

transformed into ${\displaystyle \varrho '(1-{\mathfrak {w}}'^{2})}$ or in other words,

 (18) ${\displaystyle \varrho {\sqrt {1-{\mathfrak {w}}^{2}}}}$,

is an invariant in a Lorentz-transformation.

1. The brackets shall only summarize the expressions, which are related to the index, and ${\displaystyle [{\mathfrak {w}},{\mathfrak {e}}+i{\mathfrak {m}}]}$ shall denote the vector product of ${\displaystyle {\mathfrak {w}}}$ and ${\displaystyle +i{\mathfrak {m}}}$.