An extension of the principle of relativity for one-dimensional space

TL;DR

This paper extends the principle of relativity to include accelerated frames in one-dimensional space, deriving new transformations and algebraic structures, revealing an infinite-dimensional but trivial extension of the Poincaré algebra.

Contribution

It introduces an extended algebraic framework for accelerated reference frames in 1D space, generalizing the Poincaré algebra with explicit coordinate transformations and generators.

Findings

Derived explicit coordinate transformations for uniform acceleration

Obtained infinitesimal generators and their commutation relations

Found the extended algebra to be infinite-dimensional and Abelian

Abstract

The class of accelerated reference frames has been studied, on the basis of Fermi-Walker coordinates; both in the cases of uniform and arbitrary accelerations. In the first case, explicit formulae for the transformation of coordinates have been derived and, in both cases, we have also obtained the infinitesimal generators and their commutation relations. The outcome has been an extension of the Poincar\'e algebra (in 1+1 dimensions), which is infinite dimensional in the case of general acceleration. This extension turns out to be trivial, in the sense that it is Abelian.