On the linearity of the generalized Lorentz transformation

Abstract

Lorentz transformations between inertial observers, along with Einstein's theory of special relativity, remedied discrepancies between Newtonian physics and Maxwell's electromagnetism caused by the use of the same time in all inertial frames. In view of the fundamental importance of the relativity between inertial observers, there have been several papers deriving generalized Lorentz transformations without using light. Proving that general transformations are linear in space and time can be done in several ways, most commonly relying on a four-dimensional Minkowski spacetime, but other approaches are possible. A method is presented here that establishes the linearity of the transformation by considering velocity transformations in the light of Einstein's first relativity postulate of 1905. Once linearity is obtained, the remainder is fairly straightforward and parallels results and methods found in the literature.