thus
and eventually
(24) |
This relation is valid between the Weierstrass coordinates of every single point. It is known which role this invariant plays in Minkowski's four-dimensional interpretation of relativity theory.
5. The Lorentz-Einstein transformation.
The Galilei-Newton transformation
(25) |
represents the translation along the X-axis in euclidean space. The Lorentz-Einstein transformation
(26) |
similarly can be interpreted as a translation along the X-axis in Lobachevskian space.
If we remain in the plane then we can say: The Lorentz-Einstein transformation defines a motion along the distance line with the X-axis as its center line.[1]
This distance line Y = b is the location of the points having a constant distance b from the X-axis. The length of its arc between two points M and is (Fig. 6).
(27) |
The displacement by the distance s along that equidistant line is defined by the equations
(28) |
For the passage from to we have
or
(29) |
- ↑ On the transformations of the Lobachevskian plane see my relevant papers in Rad jugoslavenske akademije 165, 50-80, 236-244, 1906, or the short excerpt therefrom in Jahresber. d. Deutsch. Mathematiker-Ver. 17, 80-83, 1908.