Vladimir Varićak

“ The analogies that exist between relativity theory and Lobachevskian geometry are in any case interesting. The formulas of recent mechanics for are reduced to the formulas of Newtonian mechanics. Similarly also the Lobachevskian geometry, if we take the so called radius of curvature as infinite, goes over into the euclidean geometry. ”

“ 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 occurrence of Ehrenfest's paradox is understandable, when one clings to the standpoint taken by Lorentz in the formulation of his contraction hypothesis, i.e., when one sees the contraction of moving rigid bodies in the direction of motion as a change which takes place in an objective way. Every element of the periphery will be changed independently of the observer according to Lorentz, while the elements of the radius remain non-contracted. ”

“ Here, it would not be hard to introduce the reduced velocity v, as well as to calculate the deviation of the resultant in the general case, where the components enclose an arbitrary angle with each other. If we compose three to each other perpendicular velocities in space, then we arrive at six resultants of different direction.
In Lobachevskian geometry there are no similar figures. On the other hand, in relativity theory there is no kinematic similarity. ”

“ It's known that one has to take as being negative for angle , since it is supplementary to . In which relation the magnitude of angle is with respect to angle , depends on the direction of motion of the mirror relative to the light source. In the case considered, angle is larger than , since it is related to the smaller perpendicular as parallel angle. ”

“ It seems convenient to introduce a name for length U. In my cited Serbian paper I have called it pseudo velocity. The easiest way would be, if we simply denote U and u as (physical) velocity and denote v as reduced velocity. As long as they are small, they practically cannot be distinguished. The pseudo velocity of light is infinitely great. By this definition it seems natural to us, that the velocity of light constitutes the upper limit for velocities. [8] ”

“ During uniform motion the relation between the unit of distance and the unit of time is equal to the unit of velocity. Thus it is
or
The image of uniform motion is the perpendicular to the X-axis (Fig. 5) . By the coordinate transformation as indicated in Fig. 4, also the uniformity of motion is canceled. ”

“ The speed of light plays in physics the role of an infinitely great velocity. It cannot be reached by an accumulation of velocities less than the speed of light; it also cannot be changed by addition or subtraction of a velocity less than the speed of light. Now, by a suitable definition we can simply make that the speed of light is represented by an infinite quantity. As unit length we take , i.e. ”

“ We denote l as the new time that emerges from the old time by multiplication with the speed of light c as measured in the resting medium. We don't want to think of the considered clock as adjusted in the usual way, but as a simple clockwork that indicates, how often a certain (always repeating) process under the same circumstances, has happened since a certain event indicating the beginning of time counting. ”

“ For the ratio of amplitudes and frequencies, Einstein gives the following equation:
Due to relation (2) we can write it in the form
or also
If equation (7) is considered, then it becomes
However, for the reflected ray it is
according to formula (28) on p. ”

“ Here it was presupposed that this body is not subjected to external forces. However, if external forces act on that body, which are in equilibrium with each other, i.e. they don't accelerate the body, then according to the investigations of Einstein [10] its kinetic energy is (oddly enough) increased by
According to our definition this expression goes over into
We take a Lobachevskian right angled triangle. ”

“ This real way together with the time ray constitutes an angle φ, that I call the time angle of uniform motion."
The velocity was expressed by him by the tangent of time angle φ.
The image of accelerated motion is the distance line. To its intersection with the ordinate axis we attribute the velocity 0; in its additional points we have
The velocity increases together with the distance up to the velocity of light. ”

“ In addition, I want to show by some examples, how Einstein's formulas of can be interpreted as real in the Lobachevskian geometry.
Equations (3) in § 5 of the mentioned paper of Einstein define (in respect to the stationary system S) the velocity components of a point uniformly moving in relation to S'. ”

“ According to Minkowski the world line illustrating uniform motion, is the straight line in the xt-plane trough the coordinate origin.
In this connection the following thought of Palagyi [14] is interesting: "What we consider as the path traveled by the material point within a space imagined as stationary, is only the apparent path of the point ”

“ In Lobachevskian geometry no parallelograms exist; the resultant of two velocities cannot be represented by a diagonal of a parallelogram. As a consequence the components are noncommutative. Because of simplicity we take two velocities under an angle . From formula (5) we obtain
from which we can easily derive
or
In figure 3 we have
In hyperbolic geometry the sum of two angles in any triangle is less than two right angles. ”

“ For a normally incident light ray we have , also , the formula (66) goes over to
The relation of the frequencies and the amplitudes can in this case be illustrated as the relation of two coaxial limiting arcs. The corresponding figure would be the same as in Fig. 11, we only have to take . ”