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 and . We can again introduce the indices 1, 2, 3, 4, so that we write instead of x,' y,' z,' it' , and instead of . 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
|| d. i.
is transformed into
On the basis of the equations (13), (14), we shall have
transformed into or in other words,
is an invariant in a Lorentz-transformation.
- ↑ The brackets shall only summarize the expressions, which are related to the index, and shall denote the vector product of and .