Non-relativistic conformal symmetries in fluid mechanics

TL;DR

This work systematically analyzes non-relativistic conformal symmetries in fluid dynamics, distinguishing Schrödinger-type and Conformal Galilean symmetries. It shows incompressible fluids admit Galilean invariance plus independent space/time dilations but not CGA or CG expansions, while compressible fluids can exhibit expanded Schrödinger symmetry in the absence of dissipation. Dissipation reduces the symmetry to Galilei × arbitrary dilations, breaking expansions; the study also clarifies the fate of relativistic conformal symmetry under NR limits, arguing that CGA does not survive as a NR symmetry. The findings resolve conflicting claims and illuminate how central extensions (mass) constrain possible NR symmetries in fluids.

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.