The Einstein Relativistic Velocity Model of Hyperbolic Geometry and Its Plane Separation Axiom

Abstract

The relativistically admissible velocities of Einstein’s special theory of relativity are regulated by the Beltrami–Klein ball model of the hyperbolic geometry of Bolyai and Lobachevsky. It is shown in this expository article that the Einstein velocity addition law of relativistically admissible velocities enables Cartesian coordinates to be introduced into hyperbolic geometry, resulting in the Cartesian–Beltrami-Klein ball model of hyperbolic geometry. Suggestively, the latter is increasingly becoming known as the Einstein Relativistic Velocity Model of hyperbolic geometry. Mobius addition is a transformation of the ball linked to Clifford algebra. Einstein addition and Mobius addition in the ball of the Euclidean n-space are isomorphic to each other, and they share remarkable analogies with vector addition. Thus, in particular, Einstein (Mobius) addition admits scalar multiplication, giving rise to gyrovector spaces, just as vector addition admits scalar multiplication, giving rise to vector spaces. Moreover, the resulting Einstein (Mobius) gyrovector spaces form the algebraic setting for the Beltrami-Klein (Poincare) ball model of n-dimensional hyperbolic geometry, just as vector spaces form the algebraic setting for the standard Cartesian model of n-dimensional Euclidean geometry. As an illustrative novel example special attention is paid to the study of the plane separation axiom (PSA) in Euclidean and hyperbolic geometry.