Page:Grundgleichungen (Minkowski).djvu/11



further

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

Now we shall make a very important observation about the vectors $$\mathfrak{w}$$ and $$\mathfrak{w}'$$. We can again introduce the indices 1, 2, 3, 4, so that we write $$x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}$$ instead of x,' y,' z,' it' , and $$\varrho'_{1},\ \varrho'_{2},\ \varrho'_{3},\ \varrho'_{4}$$ instead of $$\varrho'\mathfrak{w}'_{x'}\ \varrho'\mathfrak{w}'_{y'}\ \varrho'\mathfrak{w}'_{z'}\ i\varrho'$$. 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

is transformed into

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

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

is an invariant in a -transformation.