Einstein’s velocity addition law and its hyperbolic geometry

Einstein's velocity addition law and its hyperbolic geometry

Following a brief review of the history of the link between Einstein’s velocity addition law of special relativity and the hyperbolic geometry of Bolyai and Lobachevski, we employ the binary operation of Einstein’s velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry. In full analogy with Euclidean geometry, we show in this article that the in...

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) th...

The relativistic velocity addition law optimizes a forecast gambler’s profit

We extend the projective covariant bookmaker’s bets model to the forecasting gamblers case. The probability of correctness of forecasts shifts probabilities of branching. The formula for the shift of probabilities leads to the velocity addition rule of the special theory of relativity. In the absence of information about bookmaker’s wagers the stochastic logarithmic rates completely determines ...

The Relativistic Velocity Addition Law through Special Relativity

It is shown that if we can define a physical quantity with proper character in a given inertial reference frame (kinematic, dynamic, electromagnetic in its nature) which transforms when detected from a reference frame relative to which it moves with velocity ux as

Hyperbolic Geometry