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Galilean relativity and its invariant bilinear forms

Herintsitohaina Ratsimbarison

Abstract

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

Galilean relativity and its invariant bilinear forms

Abstract

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

Paper Structure

This paper contains 10 sections, 5 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Galilean bilinear forms
  3. Remark:
  4. Problem:
  5. Remarks:
  6. Questions:
  7. Remarks:
  8. Poisson structure from Galilean bilinear forms
  9. Conclusion
  10. Acknowledgements

Key Result

Proposition 2.1

For symmetric g, A is an isometry of (M,g) iff g(Ax,Ax) = g(x,x) $\forall x\in M$.

Theorems & Definitions (10)

  • Definition 2.1
  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.3
  • Proposition 3.1
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.2