Möbius Transformation and Einstein Velocity Addition in the Hyperbolic Geometry of Bolyai and Lobachevsky

Abstract

In this chapter, dedicated to the 60th Anniversary of Themistocles M. Rassias, Möbius transformation and Einstein velocity addition meet in the hyperbolic geometry of Bolyai and Lobachevsky. It turns out that Möbius addition that is extracted from Möbius transformation of the complex disc and Einstein addition from his special theory of relativity are isomorphic in the sense of gyrovector spaces.(1) Möbius addition in the ball R n c forms the algebraic setting for the Cartesian-Poincaré ball model of hyperbolic geometry, and (2) Einstein addition in the ball R n c forms the algebraic setting for the Cartesian-Beltrami-Klein ball model of hyperbolic geometry, just as the common (3) vector addition in the space R n forms the algebraic setting for the standard Cartesian model of Euclidean geometry.Remarkably, Items (1)-( 3) enable Möbius addition in R n c , Einstein addition in R n c , and the standard vector addition in R n to be studied comparatively, as in [72]. for all a, b, u, v ∈ R n c . Hence, gyr[u, v] is an isometry of R n c , keeping the norm of elements of the ball R n c invariant, (34) gyr[u, v]w = w Accordingly, gyr[u, v] represents a rotation of the ball R n c about its origin for any u, v∈R n c . The invertible self-map gyr[u, v] of R n c respects Einstein addition in R n c , (35) gyr[u, v](a⊕b) = gyr[u, v]a⊕gyr[u, v]bfor all a, b, u, v∈R n c , so that gyr [u, v] is an automorphism of the Einstein groupoid (R n c , ⊕). We recall that an automorphism of a groupoid (R n c , ⊕) is a bijective self-map of the groupoid R n c that respects its binary operation, that is, it satisfies (35). Under bijection composition the automorphisms of a groupoid (R n c , ⊕) form a group known as the automorphism group, and denoted Aut(R n c , ⊕). Being special automorphisms, gyrations gyr [u, v]∈Aut(R n c , ⊕), u, v ∈R n c , are also called gyroautomorphisms, gyr being the gyroautomorphism generator called the gyrator.

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