Page:VaricakRel1910a.djvu/1

Application of Lobachevskian geometry in the theory of relativity.

By V. Varićak

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 Sommerfeld in this journal.^[1] Now, the non-euclidean Geometry of Lobachevsky and Bolyai 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:^[2]

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

The factor $\left(1-\left({\tfrac {v}{c}}\right)^{2}\right)^{-{\tfrac {1}{2}}}$ that plays an important role in the Lorentz-Einstein 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:^[3]

$l'=-x\ \operatorname {sh} \,u+l\ \operatorname {ch} \,u$ $x'=x\ \operatorname {ch} \,u-l\ \operatorname {sh} \,u$ $y'=y,\ z'=z.$

(1)

or in infinitesimal form

$Uf=-l{\frac {\partial f}{\partial x}}-x{\frac {\partial f}{\partial l}}$

(2)

The inverse transformation is

$l=x'\ \operatorname {sh} \,u+l'\ \operatorname {ch} \,u$ $x=x'\ \operatorname {ch} \,u+l'\ \operatorname {sh} \,u$ $y=y',\ z=z'.$

(3)

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.^[4] 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

$l=\varkappa \left(x'\sin \ \psi +l'\cos \ \psi \right)$ $x=\varkappa \left(x'\cos \ \psi +l'\sin \ \psi \right)$ $y=y',\ z=z'.$

(4)

by which Minkowski's coordinate transformation is defined.^[5] 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 Gudermannian or so called transcendent angle.^[6]

1. This journal 10, 828, 1909.
2. $\operatorname {th} \,u$ means tangens hyperbolicus of u, as well as ch u and sh u cosinus and sinus hyperbolicus.
3. Einstein, Jahrbuch der Redioaktivität 4, 420, 1908
4. This reminds on the hyperbolic motion by Born, Ann. d. Phys. 11, 25, 1909.
5. In connection with Minkowski's views on space and time, one will probably give its due interest the booklet of M. Palágy, Neue Theorie des Raumes und der Zeit, Leipzig 1901.
6. See Engel-Lobatschefskij, Zwei geometrische Abhandlungen. Leipzig 1898, p. 246