A Derivation of Three-Dimensional Inertial Transformations

TL;DR

This paper generalizes inertial frame transformations to three dimensions, deriving a set consistent with special relativity and exploring implications for accelerated systems and absolute simultaneity.

Contribution

It extends the Mansouri-Sexl transformation framework to three dimensions and analyzes the logical consistency of inertial transformations with absolute simultaneity.

Findings

Transformations are kinematically equivalent to special relativity.

A theory maintaining absolute simultaneity is uniquely consistent with accelerated motion.

Algebraic properties of the derived transformations are discussed.

Abstract

The derivation of the transformations between inertial frames made by Mansouri and Sexl is generalised to three dimensions for an arbitrary direction of the velocity. Assuming lenght contraction and time dilation to have their relativistic values, a set of transformations kinematically equivalent to special relativity is obtained. The ``clock hypothesis'' allows the derivation to be extended to accelerated systems. A theory of inertial transformations maintaining an absolute simultaneity is shown to be the only one logically consistent with accelerated movements. Algebraic properties of these transformations are discussed. Keywords: special relativity, synchronization, one-way velocity of light, ether, clock hypothesis.