Page:Grundgleichungen (Minkowski).djvu/11

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(14) ,

further[1]

(15) ,

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) .

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

(17) d. i.

is transformed into

d. i.

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

transformed into or in other words,

(18) ,

is an invariant in a Lorentz-transformation.

  1. The brackets shall only summarize the expressions, which are related to the index, and shall denote the vector product of and .