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Particle dynamics on torsional galilean spacetimes

by José Figueroa-O'Farrill, Can Görmez, Dieter Van den Bleeken

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Submission summary

Authors (as Contributors): José Figueroa-O'Farrill · Can Görmez · Dieter Van den Bleeken
Submission information
Preprint link: scipost_202209_00005v1
Date accepted: 2022-12-23
Date submitted: 2022-09-04 19:03
Submitted by: Figueroa-O'Farrill, José
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
Approach: Theoretical


We study free particle motion on homogeneous kinematical spacetimes of galilean type. The three well-known cases of Galilei and (A)dS-Galilei spacetimes are included in our analysis, but our focus will be on the previously unexplored torsional galilean spacetimes. We show how in well-chosen coordinates free particle motion becomes equivalent to the dynamics of a damped harmonic oscillator, with the damping set by the torsion. The realization of the kinematical symmetry algebra in terms of conserved charges is subtle and comes with some interesting surprises, such as a homothetic version of hamiltonian vector fields and a corresponding generalization of the Poisson bracket. We show that the Bargmann extension is universal to all galilean kinematical symmetries, but also that it is no longer central for nonzero torsion. We also present a geometric interpretation of this fact through the Eisenhart lift of the dynamics.

Published as SciPost Phys. 14, 059 (2023)

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2022-12-2 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202209_00005v1, delivered 2022-12-02, doi: 10.21468/SciPost.Report.6246


1- Contains novel and interesting results.
2- Use powerful mathematical methods.
2- It is very rigorous and clearly written.


- I do not see any weak point.


The dynamics of free particles moving in geodesics on a torsional galilean spacetime was analyzed. One of the main results of the article is that the dynamics of free particles, using a specific coordinate system, is described by a damped harmonic oscillator. Damping is then related to the presence of torsion, generalizing previous results found in the literature.

The action principle, used to describe the dynamics of the particles, is not invariant under time translations, but it rescales by a constant. As a consequence, the equations of motion are invariant under this symmetry, and an associated conserved charge can be constructed. The kinematical algebra is then realized using a suitable modification of the Poisson brackets. Surprisingly, if the standard Poisson brackets are used, the charges also close in an algebra.

Finally, in section 4 the dynamics is described in covariant way using Newton-Cartan geometry, without specifying a particular coordinate system.

The article is clear, well-written and rigorous. The results are novel and very interesting. In my opinion this is really an excellent paper.

I recommend the article for publication in its current form.

Requested changes

1- No changes

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: perfect

Anonymous Report 1 on 2022-11-28 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202209_00005v1, delivered 2022-11-28, doi: 10.21468/SciPost.Report.6203


1- Extremely well structured and easy to follow
2- Self-contained
3- Very interesting physics supported by solid mathematics


None that were apparent to me.


This work focuses on torsional Galilean spacetimes and free particle motion on these backgrounds. The authors show with great care that in suitable coordinates, the equations of motion reduce to those of a damped harmonic oscillator. A good portion of this work is dedicated to setting the stage for this main result. The authors first provide a review of torsional Galilean spacetimes and introduce the necessary mathematics to describe these kinematical spacetimes before they derive their main result. The remainder of this work is dedicated to a detailed discussion on how to realize the underlying kinematical Lie algebra as symmetries of the damped harmonic oscillator (including a covariant description) and in terms of conserved charges.

This paper is exceptionally well written and a joy to read. All necessary concepts and underlying mathematical structures are appropriately introduced, and the relevant literature is adequately referred to. The authors also take great care to provide a physical interpretation of the (maybe for some readers abstract) mathematical notions that underlie their analysis.

Maybe the only "weakness" (though this depends on personal preferences) is that this work derives a fascinating result -- the appearance of the damped harmonic oscillator as the equations of motion -- and does not follow up on it besides some hints on future directions. Something as fundamental as the (damped) harmonic oscillator might hint at a lot of interesting physics and relations to other fields to be explored.

In summary, this is excellent work with interesting physical interpretations written clearly and concisely, and I recommend publication in its current form.

Requested changes


  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: perfect

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