Linear Form of 3-scale Relativity Algebra and the Relevance of Stability

TL;DR

This paper demonstrates that the algebra underlying Triply Special Relativity can be reformulated as a stable Lie algebra, aligning it with historical algebraic structures and confirming the absence of a Quadruply Special Relativity within this framework.

Contribution

It shows how to linearize the algebra of Triply Special Relativity and connects it to a known stable Lie algebra, clarifying the algebraic structure and limitations of extended relativity theories.

Findings

The algebra of Triply Special Relativity can be expressed as a stable Lie algebra.

The stable Lie algebra aligns with Vilela Mendes' proposal and Yang's algebra.

There is no Quadruply Special Relativity within the Lie algebra framework.

Abstract

We show that the algebra of the recently proposed Triply Special Relativity can be brought to a linear (ie, Lie) form by a correct identification of its generators. The resulting Lie algebra is the stable form proposed by Vilela Mendes a decade ago, itself a reapparition of Yang's algebra, dating from 1947. As a corollary we assure that, within the Lie algebra framework, there is no Quadruply Special Relativity.