Conformal and projective symmetries in Newtonian cosmology

Understand

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.