The Relativistic Composite-Velocity Reciprocity Principle

Gyrogroup theory [A.A. Ungar, Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics, Found. Phys. 27 (1997), pp. 881-951] enables the study of the algebra of Einstein’s addition to be guided by analogies shared with the algebra of vector addition. The capability of gyrogroup theory to capture analogies is demonstrated in this article by exposing the Relativistic Composite-Velocity Reciprocity Principle. The breakdown of commutativity in the Einstein velocity addition ⊕ of relativistically admissible velocities seemingly gives rise to a corresponding breakdown of the relativistic composite-velocity reciprocity principle, since seemingly (i) on one hand the velocity reciprocal to the composite velocity u⊕v is −(u⊕v) and (ii) on the other hand it is (−v)⊕(−u). But, (iii) −(u⊕v) 6= (−v)⊕(−u). We remove the confusion in (i), (ii) and (iii) by employing the gyrocommutative gyrogroup structure of Einstein’s addition and, subsequently, present the relativistic composite-velocity reciprocity principle with the Thomas rotation that it involves. —