Covariant Galilean versus Carrollian hydrodynamics from relativistic fluids

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.

$$ \mathcal{N} $$-extended Chern-Simons Carrollian supergravities in 2 + 1 spacetime dimensions

“…Finally, all these ultra-relativistic theories constructedà la CS could have some applications in the context of Carrollian fluids (and their relations with flat holography, see Refs. [31][32][33][34]).…”

“…Afterwards, in [15], the AdS Carroll CS gravity theory was discussed for the first time. 1 Non-relativistic symmetry groups play a remarkable role also in holography [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. More specifically, in the work [24], connections among Carrollian physics, holography of flat space, and the Bondi-Metzner-Sachs (BMS) algebra were discovered and followed up in [25] (see also Refs.…”

“…[26,27] and [28]). Besides, conformal extensions of the Carroll group were explored and related to the BMS group in [29,30], while in [31][32][33][34] it was shewed the way in which Carrollian structures and geometry emerge in the flat holography and fluid/gravity correspondence framework.Recently, in [35] the construction of the three-dimensional N = 1 CS supergravity theory invariant under the so-called AdS Carroll superalgebra (ultra-relativistic contraction of the N = 1 AdS superalgebra, see [14]), together with the study of its flat limit, has been presented for the first time. In the study done in [35], the method introduced in [36] was adopted.…”

“…[26,27] and [28]). Besides, conformal extensions of the Carroll group were explored and related to the BMS group in [29,30], while in [31][32][33][34] it was shewed the way in which Carrollian structures and geometry emerge in the flat holography and fluid/gravity correspondence framework.…”

Kinematical versus dynamical contractions of the de Sitter Lie algebras

“…compared to the Galilei group in the study of gravitational waves [29,30], of confined dynamical systems [31], of gravity [32] and of covariant hydrodynamics [33].…”