On Nonlinear Inertial Transformations

TL;DR

This paper derives the most general form of inertial transformations, revealing their potential nonlinearity and the conditions under which they reduce to affine linear transformations that preserve the speed of light, thus refining the foundations of special relativity.

Contribution

It provides a complete derivation of nonlinear inertial transformations and shows that those preserving the speed of light are necessarily affine linear, reducing the postulates needed for special relativity.

Findings

Inertial transformations are generally nonlinear and governed by Schwarzian differential equations.

Transformations preserving the speed of light are constrained to be affine linear.

Spacetime acquires a vector space structure in higher dimensions due to these 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.