Fields and fluids on curved non-relativistic spacetimes

Abstract

We consider non-relativistic curved geometries and argue that the background structure should be generalized from that considered in previous works. In this approach the derivative operator is defined by a Galilean spin connection valued in the Lie algebra of the Galilean group. This includes the usual spin connection plus an additional "boost connection" which parameterizes the freedom in the derivative operator not fixed by torsion or metric compatibility. As an example we write down the most general theory of dissipative fluids consistent with the second law in curved non-relativistic geometries and find significant differences in the allowed transport coefficients from those found previously. Kubo formulas for all response coefficients are presented. Our approach also immediately generalizes to systems with independent mass and charge currents as would arise in multicomponent fluids. Along the way we also discuss how to write general locally Galilean invariant non-relativistic actions for multiple particle species at any order in derivatives. A detailed review of the geometry and its relation to non-relativistic limits may be found in a companion paper.

Null fluids: A new viewpoint of Galilean fluids

“…The 3 dimensional Galilean fluid was also studied by [2], however we find certain discrepancies in their and our results. A detailed comparison has been provided in Appendix C. Before closing this discussion we would like to note that [6] also constructed an equilibrium partition function and entropy current for an uncharged 3 and 4 dimensional Galilean fluid, and used it to constraint the respective constitutive relations.…”

“…We deduce that we can see null fluid as Galilean fluid written in extended space representation. Many aspects of it are already hinted by extended space construction of [2]. In the next section we extend this approach to study effect of Uð1Þ anomaly on fluid transport.…”

“…Our null fluid construction is computationally similar to [2], but has a different essence to it. Authors in [2] considered an extended (d þ 2)-dim representation of the Galilean group, and realized it with the help of a (d þ 2)-dim flat space.…”

“…We mentioned in the beginning that [2] constructed a Milne boost invariant formalism of Galilean hydrodynamics using an "extended space representation." On taking a closer look we realize that their construction is just a "bottom-top" viewpoint of our "null fluid" where authors started with a Galilean fluid and worked out the corresponding null fluid.…”

“…In this paper, apart from rediscovering the known results of [2,6] for Galilean fluids in a Milne boost covariant manner, we have also used the null fluid approach to write down the most generic constitutive relations of a "charged Galilean fluid" in arbitrary (odd and even) number of dimensions, up to first order in derivatives. Further, we have discussed the potential Uð1Þ anomalies in Galilean theories (without dwelling into their field theoretic interpretation).…”