Translation:Application of Lobachevskian Geometry in the Theory of Relativity

by Vladimir Varićak

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]

If the velocities $v_{1}$ and $v_{2}$ enclose the angle α, and

${\frac {v_{1}}{c}}=\operatorname {th} \,\ u_{1},\ {\frac {v_{2}}{c}}=\operatorname {th} \,u_{2},$

then lay off the line $OA=u_{1}$ from point O into the direction of $v_{1}$, and apply the line $AC=u_{2}$ under the angle α. The resultant corresponds to the line $OC=u$. In the Lobachevskian triangle OAC the relation is given

$\operatorname {ch} \,u=\operatorname {ch} \,u_{1}\ \operatorname {ch} \,u_{2}+\operatorname {sh} \,u_{1}\operatorname {sh} \,u_{2}\cos \ \alpha$

If we denote herein

$\operatorname {ch} \,u_{i}={\frac {1}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}},\ \operatorname {sh} \,u_{i}={\frac {\frac {v_{i}}{c}}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}},$

then we obtain after some simple transformations the general Einsteinian addition theorem for velocities. In the case $\alpha ={\tfrac {\pi }{2}}$, we have

$\operatorname {ch} \,u=\operatorname {ch} \,u_{1}\operatorname {ch} \,u_{2}$

or

$\operatorname {th} ^{2}u=\operatorname {th} ^{2}u_{1}+\operatorname {th} ^{2}u_{2}-\operatorname {th} ^{2}u_{1}\operatorname {th} ^{2}u_{2}$

respectively

$v={\sqrt {v_{1}^{2}+v_{1}^{2}-\left({\frac {v_{2}v_{2}}{c}}\right)^{2}}}$

That this addition is not commutative can be easily seen from the first figure of Sommerfeld that, however, we now have to interpret as a figure in the Lobachevskian plane. Additionally we have to put

$OA=BD=u_{1},\ AC=OB=u_{2}$ $OC=OD=u.$

In hyperbolic geometry the angle sum in any triangle is smaller than two right angles. Hence

$\alpha _{1}+\alpha _{2}<{\frac {\pi }{2}},$

thus OD does not coincide with the direction of OC. For the direction difference $\delta =\sphericalangle COD$ we find

$\operatorname {cotg} \,\delta =\operatorname {tg} \,\left(\alpha _{1}+\alpha _{2}\right)={\frac {\operatorname {th} \,u_{1}\operatorname {sh} \,u_{1}+\operatorname {th} \,u_{2}\operatorname {sh} \,u_{2}}{\operatorname {sh} \,u_{1}\operatorname {sh} \,u_{2}-\operatorname {th} \,u_{1}\operatorname {th} \,u_{2}}}$ $={\frac {v_{1}^{2}{\sqrt {1-\left({\frac {v_{2}}{c}}\right)^{2}}}+v_{2}^{2}{\sqrt {1-\left({\frac {v_{1}}{c}}\right)^{2}}}}{v_{1}v_{2}\left(1-{\sqrt {1-\left({\frac {v_{1}}{c}}\right)^{2}}}{\sqrt {1-\left({\frac {v_{2}}{c}}\right)^{2}}}\right)}}$

It is also

$\operatorname {tg} \,{\frac {\delta }{2}}=\operatorname {th} \,{\frac {u_{1}}{2}}\operatorname {th} \,{\frac {u_{2}}{2}}$

If $v_{1}$ and $v_{2}$ are not in the xy-plane, but arbitrarily in space, then we obtain six terminal points, while above we only had points C and D.

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 $u_{x},u_{y},u_{z}$ of a point uniformly moving in relation to S'. If $u_{z'}=0$, then

${\frac {u_{y}}{u_{x}}}={\frac {u_{y'}}{u_{x'}+v}}\cdot {\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}$

If we take $c=\infty$, then the straight line upon which that point is moving, encloses the angle λ with the x-axis. However, if c remains finite and equal to the propagation velocity of light in empty space, then we find the direction coefficients of that straight line as:

$\operatorname {tg} \,\lambda '={\frac {\operatorname {tg} \,\lambda }{\operatorname {ch} \,u}}$

If u is the hypotenuse and ${\tfrac {\pi }{2}}-\lambda$ is an acute angle in the right angled Lobachevskian triangle, then $\lambda '$ is the second acute angle. It will be the smaller, the greater the translation velocity of S'. For v = c we have $\lambda '=0$.

We define s as the extension of a stationary electron in the direction of the x-axis. If it is set into motion with velocity v in the same direction, then its contracted extension is

$s'={\frac {s}{\operatorname {ch} \,u}}$

Upon the distance line y = u having the x-axis as its center line and u as its parameter, and beginning from its intersection point U with the ordinate axis, we measure the length UM = s. The abscissa of M is $s'$.

In the same way the dilation of a clock in uniform motion relative to a reference frame can be interpreted.

If we further put

$\varphi =\Pi \left(u_{1}\right),\ \varphi '=\Pi \left(u'_{1}\right),\ {\frac {v}{c}}=\operatorname {th} \,u,$

then we obtain from the first formula, the principle of Doppler:

$\nu '=\nu \left(\operatorname {ch} \,u-\operatorname {th} \,u_{1}\operatorname {sh} \,u\right)={\frac {\nu \,\operatorname {ch} \,\left(u-u_{1}\right)}{\operatorname {ch} \,u_{1}}}$

The parallel angle $\varphi =0$ corresponds to the length $u_{1}=\infty$, and so we have in this case

$\nu '=\nu e^{-u}\,$

It is also

$\nu '=\nu \left(1-u+{\frac {u^{2}}{2!}}-{\frac {u^{3}}{3!}}+\dots \right)$

Higher powers of u can be neglected for small values, and we can replace $\operatorname {th} \,{\tfrac {v}{c}}$ by ${\tfrac {v}{c}}$. The preceding formula goes over into the expression of Doppler's principle of ordinary mechanics:

$\nu '=\nu \left(1-{\frac {v}{c}}\right)$

Note, that in the primed reference frame $v'=-v$.

The ratio of the frequencies $\nu$ and $\nu '$ in the formula $\nu '=\nu e^{-u}$ can be represented as the ratio of two limiting circular arcs between two common axes.

The expression of aberration is transformed into

$\operatorname {th} \,u'_{1}=\operatorname {th} \,\left(u_{1}-u\right)$

The aberration equation is thus

$u'_{1}=u_{1}-u\,$

A light ray T coming from an infinitely distant light source, is striking the x-axis at point M under the acute angle φ. We lay off the line $MN=u$ toward the increasing abscissa, and from N apply the Lobachevskian parallel to T. This parallel $T'$ encloses the angle $\varphi '$ with the x-axis.

If $\varphi ={\tfrac {\pi }{2}}$, also $u_{1}=0$, hence $u'_{1}=-u$, and the angle $\varphi '$ goes over into its supplement $\varphi '_{1}$.

The formulas of relativity theory are very simplified in this interpretation. For example, for a moving electron of mass μ we have

longitudinal mass	$=\mu \ \operatorname {ch} ^{3}u$
transverse mass	$=\mu \ \operatorname {ch} ^{2}u$

instead of^[7]

${\frac {\mu }{\left({\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}\right)^{3}}}$ and ${\frac {\mu }{1-\left({\frac {v}{c}}\right)^{2}}}$

If UM (Fig. 1) represents the longitudinal mass, then ON is the transverse mass.

The radius of curvature R of the orbit, when a magnetic force acting normal to the velocity of the electron is present, will be

$R={\frac {c^{2}\mu }{\epsilon N}}\operatorname {sh} \,u$

instead of^[8]

$R=c^{2}{\frac {\mu }{\epsilon }}\cdot {\frac {\frac {v}{c}}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\cdot {\frac {1}{N}}$

A body in uniform translational motion in the direction of the increasing x-coordinate, has according to relativity theory the kinetic energy

$K_{0}=\mu c^{2}\left\{{\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}-1\right\}$

where μ denotes its mass in the ordinary sense.^[9] We can write this in a simpler way

$K_{0}=2\mu c^{2}\operatorname {sh} ^{2}{\frac {u}{2}}$

Instead of $\operatorname {sh} ^{2}{\tfrac {u}{2}}$ we can write $\operatorname {tg} \,p$, where p is the area of a Saccheri isosceles quadrilateral with two right angles. Its three sides enclosing two right angles, have a length of u.

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

$\Delta E=-{\frac {\left({\frac {v}{c}}\right)^{2}}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\sum \left(\xi K_{\xi }\right)$

According to our definition this expression goes over into

$\Delta E=-\operatorname {th} \,u\,\operatorname {sh} \,u\sum \left(\xi K_{\xi }\right)$

We take a Lobachevskian right angled triangle. If the legs a and b are the lines that correspond to the parallel angles ${\tfrac {1}{2}}\pi -\Pi (u)$ and $\Pi (u)$, and A and B are its acute angles, then we simply have

$\Delta E=-\operatorname {tg} \,B\sum \left(\xi K_{\xi }\right)$

From these few examples we can see, what advantage (even by mathematical evaluation) the non-euclidean interpretation of the relativity formulas could give to us. We have excellent tables for hyperbolic functions, which were published by the Simithsonian Institution in 1909.

The analogies that exist between relativity theory and Lobachevskian geometry are in any case interesting. The formulas of recent mechanics for $c=\infty$ 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. For ordinary velocities, the results calculated according to the relativity formulas, practically do not differ from these calculated according to the ordinary mechanical expressions. Also for distances of ordinary lengths, the calculations according to the Lobachevskian geometry do not differ from the euclidean calculations. In relativity theory there exists an absolute speed, in Lobachevskian geometry there exists an absolute length.

In relativity theory all bodies in motion are subjected to a certain deformation. In Poincaré's interpretation of Lobachevskian geometry, we can take the line element $d\sigma ={\tfrac {ds}{y}}$ which cannot be moved without deformation.

(Received January 19, 1910.)

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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
7. Einstein, Ann. d. Phys. 17, 919, 1905.
8. l. c., p. 921
9. Einstein, Ann. d. Phys. 23, 374, 1907.
10. l.c. p. 376