Page:VaricakRel1910b.djvu/2

Sciences and Arts at Agram,^[1] I also mentioned the investigations concerning the admissible curvature measure of space or the length of the absolute unit distance of hyperbolic space.

All lengths with which we are concerned vanish against the unit distance and therefore the formulas of Lobachevskian geometry reduce themselves in regions of our empirical space to expressions of ordinary euclidean geometry. To make clear the relation of those geometries by an analogy from physics, I alluded to the relation of the mechanics of electrons to Newtonian mechanics.^[2]

The Lorentz-Fitzgerald hypothesis of electron contraction led me to the assumption, whether this contraction could be interpreted as a consequence of geometrical anisotropy of space. It seemed to me that this contraction is analogues to the deformation of lengths in a very familiar interpretation of Lobachevskian geometry.^[3] Now it seems that my assumption of the connection of non-euclidean geometry with relativity theory can be realized. The appearance of a work by Herglotz^[4] urged me to immediately publish the following remarks, as his work in one point partially goes into the same direction as the related investigations of mine. The actual content of my report is of course completely different.

1. The substitution ${\tfrac {v}{c}}=\operatorname {th} \,u$

This substitution paved me the way to the non-euclidean interpretation of relativity theory. Now, subsequently I noticed that Minkowski^[5] once put

$-i\ \operatorname {tg} \,i\ \psi ={\frac {e^{\psi }-e^{-\psi }}{e^{\psi }+e^{-\psi }}}=q$

that is the expression of the velocity relation as tangens hyperbolicus, but he didn't pay further attention to the middle term of this relation.

J. J. Thomson^[6] has expressed the relation of the electron's velocity to the velocity of light as the sinus of a certain angle θ. If I define this angle as a complement to the parallel angle belonging to length U, then I again arrive at that substitution. The various ways upon which we arrive at this substitution, are increasing the confidence in it.

During the composition of velocities $v_{1}$ and $v_{2}$ enclosing the angle α, we have to construct the Lobachevskian triangle with sides $u_{1}$ and $u_{2}$ with the enclosed angle π-α. The length U of measure unit u is attributed to the velocity v by the relation

$u=\operatorname {th} \,^{-1}{\frac {v}{c}}$

(1)

Following the English style of writing $\operatorname {th} \,^{-1}{\tfrac {v}{c}}$ denotes the inverse function of the hyperbolic tangent. Now we want to investigate whether this definition is not in sharp contrast to the ordinary illustration of velocities. The distances proportional to the relevant velocities are used in ordinary mechanics as representing the velocities of uniform motions. Formula (1) leads to the same result at the limits of our ordinary experience. Only at velocities nearly comparable to the velocity of light, a notable difference occurs that quickly leads to infinite distortion. As unit distance we use the path of light in one second. Then

${\frac {U}{c}}=\operatorname {th} \,^{-1}{\frac {v}{c}}={\frac {v}{c}}+{\frac {1}{3}}\left({\frac {v}{c}}\right)^{3}+{\frac {1}{5}}\left({\frac {v}{c}}\right)^{5}+\dots$

(2)

If we take v = 1 km/sec at first, then

${\frac {U}{c}}={\frac {1}{3\cdot 10^{5}}}+{\frac {1}{3^{4}\cdot 10^{15}}}+{\frac {1}{5\cdot 3^{5}\cdot 10^{25}}}+\dots$

(3)

If we neglect everything after the first term on the right-hand side, then we commit an error that not even exerts an influence upon the 10th decimal. So by our definition, we have for a velocity of 1 km/sec a length of 1 km as the representative.

If we take the velocity of 100 km/sec which is in any case an extremely high velocity in ordinary mechanics, then it is given

1. Published in "Rad jugoslavenske akademije" 169, 110-194, 1907.
2. l.c., p. 172.
3. My remarks on that interpretation were published in "Rad jugoslavenske akademije" 154, 81-131, 1903. With this interpretation is also conerned A. Schwarz: Zur Theorie der reellen linearen Transformationen und der Lobatschefskijschen Geometrie, Wien, Sitzungsb., II. Abt., 99, 1890. Bonolas, Index operum ad geometriam absolutam spectantium, in Klausenburger Festschrift, is not mentioned in that treatise.
4. Herglotz, Über den vom Standpunkt des Relativitätsprinzips aus als "starr" zu bezeichnenden Körper. Ann. der Phys. 31, 383, 1910
5. Minkowski, Die Grundgleichungen usw., Göttinger Nachrichten, p. 59, 1908
6. Lodge, Elektronen, p. 120