TABLE OF CONTENTS Relativistic three-velocities and their Einstein velocity addition law mesh extraordinarily well with the notion of the relativistic mass, as illustrated in Figs. 10.2 and 10.3, pp. 370 371. These figures indicate that (i) relativistic three-velocities, (ii) relativistic mass, and (iii) hyperbolic geometry, interplay in Einsteinian mechanics in a way fully analogous to the interplay of (i) Newtonian velocities, (ii) classical mass, and (iii) Euclidean geometry in Newtonian mechanics
The considerable increase of the mass of particles in particle accelerators has to be taken into account in the design of accelerators. Owing to the reality of the relativistic mass [Anderson (1967), p. 199][Tsai (1986)], Einstein addition of three-velocities is an indispensable relativistic tool. Contrasting the harmonious interplay between three-velocities and the relativistic mass, the latter is in conflict with Minkowskian four-vector spacetime approach. Explanation and rationalization of the resulting never-ending debates about the status of mass in special relativity, calling it "the messy mass", are provided by Beisbart and Jung [Beisbart and Jung (2004)]. We will find that within the frame of Einsteinian relativity, as opposed to Minkowskian relativity, the problem that Beisbart and Jung explain and rationalize in [Beisbart and Jung (2004)] does not exist.
Taking three-velocities and their Einstein addition law as a primitive concept in special relativity, one can derive Minkowski's four velocities and their Lorentz transformation law, as we show in Sec. 10.15. In contrast, taking four-velocities and their Lorentz transformation law as a primitive concept in special relativity leaves...
