Covariant Galilean versus Carrollian hydrodynamics from relativistic fluids

TL;DR

The work provides a comprehensive, covariant framework for non-relativistic hydrodynamics by deriving Galilean and Carrollian fluid equations from relativistic fluids in suitably chosen backgrounds. It introduces the Zermelo frame for Galilean limits and Randers–Papapetrou backgrounds for Carrollian limits, enabling first-order dissipative hydrodynamics on time-dependent geometries. Key results include a covariant, generalized Navier–Stokes form for incompressible Galilean fluids on curved spaces and a Carrollian fluid theory where dynamics are anchored to geometry, notably yielding Robinson–Trautman/Calabi-flow dynamics in 2D. The study also highlights a self-dual, duality-inspired connection between the two contractions and discusses holographic contexts and potential physical applications in atmospheric physics and flat-space holography.

Abstract

We provide the set of equations for non-relativistic fluid dynamics on arbitrary, possibly time-dependent spaces, in general coordinates. These equations are fully covariant under either local Galilean or local Carrollian transformations, and are obtained from standard relativistic hydrodynamics in the limit of infinite or vanishing velocity of light. All dissipative phenomena such as friction and heat conduction are included in our description. Part of our work consists in designing the appropriate coordinate frames for relativistic spacetimes, invariant under Galilean or Carrollian diffeomorphisms. The guide for the former is the dynamics of relativistic point particles, and leads to the Zermelo frame. For the latter, the relevant objects are relativistic instantonic space-filling branes in Randers-Papapetrou backgrounds. We apply our results for obtaining the general first-derivative-order Galilean fluid equations, in particular for incompressible fluids (Navier-Stokes equations) and further illustrate our findings with two applications: Galilean fluids in rotating frames or inflating surfaces and Carrollian conformal fluids on two-dimensional time-dependent geometries. The first is useful in atmospheric physics, while the dynamics emerging in the second is governed by the Robinson-Trautman equation, describing a Calabi flow on the surface, and known to appear when solving Einstein's equations for algebraically special Ricci-flat or Einstein spacetimes.