Hyperbolic trigonometry in the Einstein relativistic velocity model of hyperbolic geometry
This paper offers a new way of presenting and even generalizing the well-known law of adding relativistically admissible velocities \[ u\oplus v= {1\over 1+(u\cdot v)/c^2} \Biggl\{u+{1\over \gamma_u} v+{1\over c^2} {\gamma_u\over 1+ \gamma_u} (u\cdot v)u\Biggr\} \] considered as an operation in the open \(c\)-ball \(\mathbb{R}^3_c= \{v\in \mathbb{R}^3:\|v\|< c\}\). Different from other usual techniques (e.g. spinors or indefinite inner products), and aiming more profoundly than at the formula itself (as the Antippa's generalization, etc.), the author presents specific new algebraic structures, distinguished by the prefix gyro, namely gyrogroups and gyrovector spaces. The germs of the idea of gyrostructures are to be found in the \(K\)-loop theory (initiated by the same author since 1989), and in some older models of hyperbolic geometry. In particular, a starting point is the Möbius addition in the complex open unit disk \(D\), where \(\text{gyr}: D\times D\to \Aut(D,\oplus)\) takes the form \(\text{gyr}[a,b]= (1+ a\overline b)(1+\overline ab)^{-1}\). In an abstract gyrogroup \((G,\oplus)\), gyr acts like \(\text{gyr}[a, b]z= -(a\oplus b)\oplus (a\oplus (b\oplus z))\), and does ``correct'' associativity, commutativity, etc. in the sense that \[ a\oplus (b\oplus c)= (a\oplus b)\oplus \text{gyr}[a, b]c,\quad\text{etc.}. \] The example of a gyrogroup of order 16, generated by the MAGMA software package, shows how diversified the sphere of gyrogroups is. The general setting of gyrostructures allows extending the Einstein's addition law to arbitrary open \(c\)-balls \(\mathbb{V}_c\) in abstract real inner product spaces, but also developing the hyperbolic trigonometry in the corresponding gyrovector space (the sine and cosine rules, the Pythagorean relation, etc.). Sections 8-11 contain direct applications involving the relativistic acceleration, geodesics, and dual structures (which make use of the secondary operation, defined by \(a\boxplus b= a\oplus \text{gyr}[a,-b]b\)).
scientific article; zbMATH DE number 1528542
scientific article