On Nonlinear Inertial Transformations

TL;DR

This work determines the most general inertial-frame transformations between observers, showing they are nonlinear yet projective-linear, and expresses them in a fractional form that pairs temporal and spatial parts. The authors derive the inertiality constraints as coupled nonlinear PDEs and solve them by isolating a Schwarzian condition for the temporal function, leading to a projective-linear solution $f(t,\vec{x}) = \frac{b_1(\vec{x}) + b_2 t}{b_3(\vec{x}) + b_4 t}$ and a corresponding spatial solution for $\vec{g}$. They demonstrate that, when demanding invariance of the speed of light in all directions, the nonlinear transformations reduce to affine linear ones, effectively reducing the number of postulates needed for special relativity and revealing a natural higher-dimensional vector-space structure for spacetime. Additionally, they introduce a higher-dimensional Schwarzian operator $\mathcal{S}_{\mu\nu\rho\lambda}(f)$ whose kernel comprises projective-linear transformations, thus tying the kinematic constraints to a robust mathematical framework and connecting relativity with projective geometry and Schwarzian theory.

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.