Relativistic Fluids, Hydrodynamic Frames and their Galilean versus Carrollian Avatars

TL;DR

The paper develops Galilean and Carrollian hydrodynamics on arbitrary backgrounds by combining local diffeomorphism/Weyl invariance with systematic large-$c$ and small-$c$ limits in Zermelo and Papapetrou–Randers gauges. It reveals how hydrodynamic-frame invariance behaves differently under Galilean and Carrollian limits and shows that Noetherian currents tied to spacetime isometries may fail to exist in Newton–Cartan or Carroll spacetimes. The work also uncovers how extra degrees of freedom emerge in contraction limits, yielding richer but more delicate conservation structures, and relates these findings to holographic contexts and Aristotelian dynamics. Overall, it provides a unified path to derive and compare Galilean and Carrollian fluids, clarifying conserved currents, frame invariances, and the role of background geometry.

Abstract

We comprehensively study Galilean and Carrollian hydrodynamics on arbitrary backgrounds, in the presence of a matter/charge conserved current. For this purpose, we follow two distinct and complementary paths. The first is based on local invariance, be it Galilean or Carrollian diffeomorphism invariance, possibly accompanied by Weyl invariance. The second consists in analyzing the relativistic fluid equations at large or small speed of light, after choosing an adapted gauge, ADM-Zermelo for the former and Papapetrou-Randers for the latter. Unsurprisingly, the results agree, but the second approach is superior as it effortlessly captures more elaborate situations with multiple degrees of freedom. It furthermore allows to investigate the fate of hydrodynamic-frame invariance in the two limits at hand, and conclude that its breaking (in the Galilean) or its preservation (in the Carrollian) are fragile consequences of the behaviour of transport attributes at large or small $c$. Both methods do also agree on the doom of Noetherian currents generated in the relativistic theory by isometries: non-trivial currents are not always guaranteed in Newton-Cartan or Carroll spacetimes as a consequence of Galilean or Carrollian isometries. Comparison of Galilean and Carrollian fluid equations exhibits a striking but often superficial resemblance, which we comment in relation to black-hole horizon dynamics, awkwardly akin to Navier-Stokes equations. This congruity is authentic in one instance though and turns out then to describe Aristotelian dynamics, which is the last item in our agenda.