Special Relativity for the Full Speed Range -- speed slower than $C_R$ also equal to and faster than $C_R$

Abstract

In this paper, we establish a theory of Special Relativity valid for the entire speed range without the assumption of constant speed of light. Two particles species are defined, one species of particles have rest frames with rest mass, and another species of particles do not have rest frame and can not define rest mass. We prove that for the particles which have rest frames, the Galilean transformation is the only linear transformation of space-time that allows infinite speed of particle motion. Hence without any assumption, an upper bound of speed is required for all non-Galilean linear transformations. We then present a novel derivation of the mass-velocity and the mass-energy relations in the framework of relativistic dynamics, which is solely based on the principle of relativity and basic definitions of relativistic momentum and energy. The generalized Lorentz transformation is then determined. The new relativistic formulas are not related directly to the speed of light $c$, but are replaced by a Relativity Constant $C_R$ which is an universal speed constant of the Nature introduced in relativistic dynamics. Particles having rest mass are called tardyons moving slower than $C_R$. Particles having neither rest frames nor rest mass are called $tachyons$ moving faster than $C_R$, and with the real mass-velocity relation $m=|\vec{p}_\infty |(v^2-C_R^2)^{-1/2}$ where $\vec{p}_\infty $ is the finite momentum of tachyon at infinite speed. Moreover, the particles with constant-speed $C_R$, also having neither rest frames nor rest mass, are called $constons$. For all particles, $p^2=\vec{p}^2-({E^2}/{C_R^2})$ remains invariant under transformation between inertia frames. The invariant reads $p^2=-m_0^2C_R^2 <0$ for tardyons, $p^2 =0$ for constons and $p^2=|\vec{p}_\infty|^2 >0$ for tachyons, respectively.