Relativistic fluids, hydrodynamic frames and their Galilean versus Carrollian avatars

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, Arnowitt-Deser-Misner-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: conserved 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.

Automated summary

This article comprehensively studies Galilean and Carrollian hydrodynamics on arbitrary backgrounds, taking into account a matter/charge conserved current. Through two distinct and complementary methods, the results agree that the second approach is superior and can capture more complex situations with multiple degrees of freedom. Additionally, the study investigates the fate of hydrodynamic-frame invariance in the two limits, concluding that its breaking or preservation depends on the behavior of transport attributes at large or small speed of light. The research also reveals that conserved currents are not always guaranteed in Newton-Cartan or Carroll spacetimes due to Galilean or Carrollian isometries. Comparing the fluid equations of Galilean and Carrollian hydrodynamics, a superficial resemblance is found and commented on in relation to black-hole horizon dynamics, resembling Navier-Stokes equations. However, in one instance, the congruity turns out to describe Aristotelian dynamics.