On Nonlinear Inertial Transformations

Abstract: It is often assumed that the most general transformation between two inertial reference frames is affine linear in their Cartesian coordinates, an assumption which is however not true. We provide a complete derivation of the most general inertial frame transformation, which is indeed nonlinear; along the way, we shall find that the conditions of preserving the Law of Inertia take the form of Schwarzian differential equations, providing perhaps the simplest possible physics setting in which the Schwarzian derivative appears. We then demonstrate that the most general such inertial transformation which further preserves the speed of light in all directions is, however, still affine linear. Physically, this paper may be viewed as a reduction of the number of postulates needed to uniquely specify special relativity by one, as well as a proof that inertial transformations automatically imbue spacetime with a vector space structure, albeit in one higher dimension than might be expected. Mathematically, this paper may be viewed as a derivation of the higher-dimensional analog of the Schwarzian differential equation and its most general solution.