Galilei covariance and (4,1) de Sitter space

TL;DR

This paper introduces a vector space framework for Galilei transformations, explores their covariant structure, and demonstrates the embedding of Euclidean space into (4,1) de Sitter space, facilitating Lie algebra analysis.

Contribution

It develops a covariant tensor analysis of Galilei transformations within a new vector space and embeds Euclidean space into (4,1) de Sitter space for advanced Lie algebra analysis.

Findings

Galilei transformations are linear in a new vector space G

Euclidean space is embedded into (4,1) de Sitter space

Covariant tensor analysis of Galilei group is established

Abstract

A vector space G is introduced such that the Galilei transformations are considered linear mappings in this manifold. The covariant structure of the Galilei Group (Y. Takahashi, Fortschr. Phys. 36 (1988) 63; 36 (1988) 83) is derived and the tensor analysis is developed. It is shown that the Euclidean space is embedded the (4,1) de Sitter space through in G. This is an interesting and useful aspect, in particular, for the analysis carried out for the Lie algebra of the generators of linear transformations in G.