10.7 Gyrogeodesics, Gyromidpoints and Gyrocentroids

A point v in the Einstein gyrovector space ( , ?, ?), or the Einstein gyrovector plane ( , ?, ?) shown in Fig. 10.1, represents all the inertial frames ? _v with relativistically admissible velocity v relative to a rest frame ? ₀. Accordingly, the relativistic velocity of frame ? _v relative to frame ? _u is ? u ? v, and the relativistic velocity of frame ? _u relative to frame ? _v is ? v ? u. Remarkably, these two Einsteinian reciprocal velocities are not reciprocal in the classical sense since they are related by the identity

(10.22)

that involves a Thomas precession. Identity (10.22) is obtained by employing the gyroautomorphic inverse property, Def. 3.1 and Theorem 3.2, of Einstein addition, according to which ?( a ? b) = ? a? b, and the gyro-commutative law of Einstein addition. The classical counterpart of (10.22), - u + v = -(- v + u), is known as the principle of reciprocity.

Figure 10.1: The Einstein Gyrovector Plane. Points of the disc = { v ? ? ²: ? v ? < c} represent two-dimensional relativistically admissible velocities v relative to some inertial rest frame with velocity 0. The latter, accordingly, is represented by the origin of the disc. While the disc origin is distinguished...