Conformal and projective symmetries in Newtonian cosmology

TL;DR

This paper explores non-relativistic conformal symmetries in Newtonian cosmology using geometric frameworks, revealing the structure of symmetry groups in cosmological models and highlighting unique dilation features.

Contribution

It characterizes the maximal symmetry group of Newtonian cosmology and explicitly constructs its conformal-Bargmann extension, contrasting it with other known symmetry groups.

Findings

Maximal symmetry group is 13-dimensional and isomorphic to a conformal-Newton-Cartan group.

Explicit construction of the conformal-Bargmann extension.

Independent space and time dilations are present, unlike in Schrödinger or Conformal Galilei groups.

Abstract

Definitions of non-relativistic conformal transformations are considered both in the Newton-Cartan and in the Kaluza-Klein-type Eisenhart/Bargmann geometrical frameworks. The symmetry groups that come into play are exemplified by the cosmological, and also the Newton-Hooke solutions of Newton's gravitational field equations. It is shown, in particular, that the maximal symmetry group of the standard cosmological model is isomorphic to the 13-dimensional conformal-Newton-Cartan group whose conformal-Bargmann extension is explicitly worked out. Attention is drawn to the appearance of independent space and time dilations, in contrast with the Schr{\"o}dinger group or the Conformal Galilei Algebra.