10.6 The Relativistic Gyrovector Space

If integer scalar multiplication n ? v is defined in the Einstein gyrogroup ( , ?) by the equation

(10.15)

for any positive integer n and then it follows from Einstein addition law, (10.3), that

(10.16)

for any positive integer n and . Suggestively, the scalar multiplication that Einstein addition admits in the relativistic velocity gyrogroup ( , ?) is defined by the equation, (6.234),

(10.17)

where r is any real number, r ? ?, , v ? 0, and r ? 0 = 0, and with which we use the notation v ? r = r ? v.

Einstein scalar multiplication turns the Einstein gyrogroup ( , ?) of relativistically admissible velocities into a gyrovector space ( , ?, ?) that possesses the following properties.

for all real numbers r, r ₁, r ₂ ? ? and admissible velocities .

Unlike vector spaces, the Einstein gyrovector space ( , ?, ?) does not possess a distributive law since, in general,

(10.18)

for r ? ? and u, .

Remarkably, the Einstein gyrovector space ( , ?, ?) of Einsteinian velocities with its gyrodistance function given by the equation

(10.19)

u, , forms the setting for the Beltrami ball model of 3-dimensional hyperbolic geometry just as the vector space ( ? ³, +, ) of Newtonian velocities with its...