Einstein “Special Relativity” (ESR)

von Brzeski “Special Relativity” (BSR)

Lobachevskian Real 3-dim Space

$SL\left(2C\right)/SU\left(2\right)$ Quotient of Lorentz Group

$SL\left(2C\right)/SU\left(2\right)$

Quotient of Lorentz Group

Type of Lorentz Group Action on $SL\left(2C\right)/SU\left(2\right)$

Single sided (left) $x=g{x}^{\prime }$

Double sided

$x=g^{\mathrm{<mo>\; *\; </mo>}}{x}^{\prime }g$

Type of Projection (Type of Map)

Gnomonic (central), non-conformal

Stereographic,

conformal

Type of Coordinates

Projective Weierstrass

Projective rational

Resulting Model of Lobachevskian Geometry

Beltrami-Klein, non-conformal

Poincare, conformal

Relation between BSR and ESR

Isomorphic as models, Homotopic as maps

Isomorphic as models, Homotopic as maps