The Intrinsic Beauty, Harmony and Interdisciplinarity in Einstein Velocity Addition Law: Gyrogroups and Gyrovector Spaces

The only justification for the Einstein velocity addition law ‎appeared to be its empirical adequacy‎, ‎so that the ‎intrinsic beauty and harmony in Einstein addition remained for a long time ‎a mystery to be conquered‎. ‎Accordingly‎, ‎the aim of this expository article is to present ‎(i) the Einstein relativistic vector addition‎, ‎(ii) the resulting Einstein scalar multiplication‎, ‎(iii) the Einstein relativistic mass‎, ‎and ‎(iv) the Einstein relativistic kinetic energy‎, ‎along with remarkable analogies with classical results in ‎groups and vector spaces that these ‎Einstein concepts capture in gyrogroups and gyrovector spaces‎. ‎Making the unfamiliar familiar‎, ‎these analogies uncover the ‎intrinsic beauty and harmony in the underlying ‎Einstein velocity addition law of relativistically admissible velocities‎, ‎as well as its interdisciplinarity‎.