Page:VaricakRel1910a.djvu/1



By V.

For the composition of velocities in the theory of relativity, the formulas of spherical geometry with imaginary sides are valid, as it was recently shown by in this journal. Now, the non-euclidean Geometry of and  is the imaginary counter-image of the spherical geometry, and it is easily seen that an interesting field of application offers itself for the hyperbolic geometry.

As relative motion of reference frames with superluminal speed does not occur, we can always put:

$\frac{v}{c}=\operatorname{th}\, u$

The factor $$\left(1-\left(\tfrac{v}{c}\right)^{2}\right)^{-\tfrac{1}{2}}$$ that plays an important role in the transformation equations and the formulas derived from them, goes over into $$\operatorname{ch}\, u$$. If we additionally put l = ct, then the transformation equations read:

or in infinitesimal form

The inverse transformation is

The hyperbolas that are invariant with respect to these transformation

$\omega(l,x)\equiv l^{2}-x^{2}+\mathsf{const}=0$

are their orbital curves, as $$U\omega=0$$. The absolute invariant is the coordinate origin. When a point

$x=\operatorname{sh}\, u_{1},\ l=\operatorname{ch}\, u_{1}$

is subjected to transformation (1), then it goes over to the point of a hyperbola $$l^{2}-x^{2}=+1$$ corresponding to the parameter $$u_{1}-u$$. Emerging from infinity, the moving point goes from the negative side into infinity. Parameter u is the measuring unit of the double hyperbolic sector corresponding to angle ψ. It is

$\operatorname{th}\, u=\operatorname{tg}\,\psi,\ \operatorname{sh}\, u=\varkappa\ \sin\ \psi,\ \operatorname{ch}\, u=\varkappa\ \cos\ \psi$

and equations (3) go over into

by which 's coordinate transformation is defined. Here, $$\varkappa$$ denotes the radius vector of the corresponding point of the hyperbola.

If u is interpreted as length, then it can be seen from the relations

$\frac{v}{c}=\operatorname{th}\, u=\operatorname{tg}\,\ \psi=\sin\ \operatorname{gd}\ u=\cos\Pi(u)$

that the related parallel angle is complementary to the corresponding ian or so called transcendent angle.