and the representative length U is by the way about 3mm greater then 100km. The velocity of 100000 km/sec corresponds to a length of 103700 km.
We additionally consider two velocities of β-rays calculated in the famous experiments of Kaufmann, and to which the velocity relations 0,7202 and 0,9326 are connected. They amount ca. 216060 and 279780 km/sec and they were represented by the lengths of 272400 and 503400 km.
For v = c we have U = ∞.
In graphical illustration these relations can be easily summarized.
If we take u as abscissa and as ordinate, then (1) will be represented by curve T. The straight line G or the first term in the infinite row (2), corresponds to the ordinary definition . The straight line is the inflexion tangent of T in O, so it fits well to the curve in the very far surrounding of the coordinate origin.
Fig. 1 can provide us good services for the composition of velocities. In addition to the resultant length U we can immediately take the corresponding velocity v from the figure.
If the velocities and enclose the angle α = 0, i.e. if they lie in the same direction, then . The resultant velocity follows from the formula
or
Although the resultant is smaller as the sum of the components, it will be represented (as in ordinary mechanics) by a length equal to the sum of the lengths representing the components. It is namely in this case .
The figure in the 6th Göttingen lecture by Poincaré is not in agreement with this definition.[1]
If we compose two equal velocities to which corresponds the length , then the resultant will be represented by the length .
2. Lorentz-Einstein transformation as translation
We take this transformation in the form as it was given by me.[2]
(4) |
The variables x, y, z, l I interpret as homogeneous Weierstrass coordinates in Lobachevskian three-dimensional space. If X, Y, Z are Lobachevskian right angled coordinates, ξ, η ζ the perpendiculars that fall from point M upon the planes of the right angles coordinate system, and r is the distance of that point from the coordinate origin, then we get the following equations from the easily constructed figure
(5) |
(6) |
(7) |
(8) |
These Weierstrass coordinates satisfy, as it is easily seen, the quadratic equation[3]
(9) |
This is therefore the invariant of first kind of the transformation defined by equations (4), or by
(10) |
If we take
(11) |
then we get to distance areas[4]
- ↑ H. Poincaré, Sechs Vorträge aus der reinen Mathematik und mathematischen Physik, 1910, p. 52
- ↑ This journal. 11, 93, 1910
- ↑ For these coordinates see Liebmann, Nichteuklidische Geometrie, p. 166, and Killing, Die nichteuklidischen Raumformen.
- ↑ The general equation of the distance area was given by me in a treatise in "Rad jugoslavenske akademije" 175, 215-240, 1908. There, the middle plane was defined by the ends ξ, η and the angle α. For the determination of the plane see my work: Zur nichteuklidischen analytischen Geometrie in Attil del IV congresso dei matematici, Roma 1909, 2, 213-226.