Geometries for Possible Kinematics

Abstract

The algebras for all possible Lorentzian and Euclidean kinematics with $\mathfrak{so}(3)$ isotropy except static ones are re-classified. The geometries for algebras are presented by contraction approach. The relations among the geometries are revealed. Almost all geometries fall into pairs. There exists $t\leftrightarrow 1/({\nu }^{2}t)$ correspondence in each pair. In the viewpoint of differential geometry, there are only 9 geometries, which have right signature and geometrical spatial isotropy. They are 3 relativistic geometries, 3 absolute-time geometries, and 3 absolute-space geometries.

40 pages, 7 figures

References in corpus (2)

• Snyder's Model -- de Sitter Special Relativity Duality and de Sitter Gravity
• The Principle of Relativity and Special Relativity Triple

Cited by in corpus (3)

• Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time
• Deformed Carroll particle from 2+1 gravity
• Possible Supersymmetric Kinematics