Non-relativistic conformal symmetries in fluid mechanics

TL;DR

This paper explores the symmetries of non-relativistic fluid mechanics, highlighting how incompressible and compressible fluids exhibit different symmetry groups, and clarifies why certain conformal symmetries are absent in the non-relativistic limit.

Contribution

It clarifies the relationship between relativistic conformal symmetries and non-relativistic fluid symmetries, explaining the absence of Conformal Galilei group as a symmetry in fluid dynamics.

Findings

Incompressible fluids have Galilei group symmetries with dilations.

Compressible fluids exhibit expanded Schroedinger group symmetries.

Conformal Galilei group is not a symmetry due to non-relativistic limit subtleties.

Abstract

The symmetries of a free incompressible fluid span the Galilei group, augmented with independent dilations of space and time. When the fluid is compressible, the symmetry is enlarged to the expanded Schroedinger group, which also involves, in addition, Schroedinger expansions. While incompressible fluid dynamics can be derived as an appropriate non-relativistic limit of a conformally-invariant relativistic theory, the recently discussed Conformal Galilei group, obtained by contraction from the relativistic conformal group, is not a symmetry. This is explained by the subtleties of the non-relativistic limit.