TABLE OF CONTENTS Unlike the restricted Einstein velocity addition of parallel velocities, (3.150), the general Einstein velocity, (10.3), addition is unheard of in most modern relativity books. Why?
Harmony is the notion that motivates and justifies our desire to impose mathematical order on natural phenomena. The discovery of Vladimir Vari?ak in 1908 1910 [Vari?ak (1908); Vari?ak (1910a)] that Einstein's addition of relativistically admissible three-velocities has natural interpretation in the hyperbolic geometry of Bolyai and Lobachevsky was therefore a great triumph to Riemann and to the principle of harmony between mathematics and physics. For his chagrin, Vari?ak had to admit in 1924 that the adaption of vector algebra for use in hyperbolic space was just not possible [Vari?ak (1924), p. 80], as Scott Walter notes in [Walter (1999b), p. 121]. However, following Chap. 5, the introduction of vectors into hyperbolic geometry, where they are called gyrovectors, is now possible.
Riemann was aware of the possible application of his geometry to physics. In his inaugural address in 1854 on the occasion of joining the University Faculty of G ttingen he said that the value of his non-Euclidean geometry can possibly be to liberate us from preconceived ideas, should ever the time come that in the exploration of the laws of physics the concepts of Euclidean geometry may have to be abandoned. These prophetic words were literally fulfilled fifty years later by the special theory of relativity [Lanczos (1970), p. 9l], uncovered by Einstein in 1905 [Einstein (1905); Einstein (1998)].
However,...
