Relativistic Fluids, Hydrodynamic Frames and their Galilean versus Carrollian Avatars

TL;DR

This paper explores the relationships between relativistic, Galilean, and Carrollian hydrodynamics on arbitrary backgrounds, analyzing their invariances, equations, and connections to black-hole horizon dynamics, revealing both similarities and fundamental differences.

Contribution

It introduces two complementary methods for deriving Galilean and Carrollian hydrodynamics from relativistic fluids, clarifies the fate of hydrodynamic-frame invariance, and compares these limits to black-hole horizon dynamics.

Findings

Both methods agree on the results, with the second method capturing more complex scenarios.

Hydrodynamic-frame invariance is fragile in Galilean and preserved in Carrollian limits.

Galilean and Carrollian fluid equations show striking similarities, linked to black-hole horizon dynamics.

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.