Covariant Galilean versus Carrollian hydrodynamics from relativistic fluids

TL;DR

This paper develops covariant equations for non-relativistic fluids on arbitrary spaces, derived from relativistic hydrodynamics, and explores applications in atmospheric physics and gravitational spacetime solutions.

Contribution

It introduces a covariant formulation of Galilean and Carrollian hydrodynamics from relativistic limits, including dissipative effects, and constructs invariant coordinate frames for these theories.

Findings

Derived first-derivative-order Galilean fluid equations including Navier-Stokes.

Applied the framework to rotating and inflating surfaces in atmospheric physics.

Connected Carrollian fluids to solutions of Einstein's equations via the Robinson-Trautman equation.

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.