Fields and fluids on curved non-relativistic spacetimes

TL;DR

The work advances non-relativistic geometry by promoting Newton–Cartan data to an extended Galilean framework with veilbein $e^A_{\mu}$, Galilean spin connection $\omega^A{}_B$ (including a boost part $\varpi^a$), and separate mass and electromagnetic gauge fields $a_{\mu}$ and $A_{\mu}$. It provides a manifestly boost-invariant formulation of non-relativistic actions and Ward identities, and develops first-order dissipative fluid dynamics on curved backgrounds with independent mass and charge currents. Through a rigorous entropy-current analysis, the authors derive the full set of allowed transport coefficients (parity-even and parity-odd) and corresponding Kubo formulas, highlighting new constraints and magnetization contributions (e.g., $\tilde m$, $\tilde m_{\epsilon}$). The framework accommodates multicomponent fluids and yields a richer transport structure than prior treatments, with potential applications to condensed matter systems, non-relativistic holography, and QHE-like effective theories. Overall, the paper delivers a comprehensive, geometrically grounded approach to non-relativistic fluids in curved spacetimes, tying background geometry, symmetry, and transport together in a boost-invariant formalism.

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 [arXiv:1503.02682].