Galilea relativity and its invariant bilinear forms

TL;DR

This paper explores the mathematical structure of bilinear forms invariant under Galilean transformations, revealing a family parametrized by a Galilean invariant vector and linking it to Poisson brackets.

Contribution

It introduces a new family of invariant bilinear forms on R3+1 and analyzes their properties, including the associated Poisson brackets.

Findings

Identified a family of bilinear forms invariant under Galilean boosts and rotations.

Connected the antisymmetric part of these forms to Poisson brackets.

Provided insights into the physical interpretation of the invariant vector.

Abstract

We construct the family of bilinear forms gG on R3+1 for which Galilea boosts and spatial rotations are isometries. The key feature of these bilinear forms is that they are parametrized by a Galilea invariant vector whose physical interpretation is rather unclear. At the end of the paper, we construct the Poisson bracket associated to the (nondegenerate) antisymmetric part of gG.