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We consider non-relativistic curved geometries and argue that the backgroundstructure should be generalized from that considered in previous works. In thisapproach the derivative operator is defined by a Galilean spin connectionvalued in the Lie algebra of the Galilean group. This includes the usual spinconnection plus an additional "boost connection" which parameterizes thefreedom in the derivative operator not fixed by torsion or metriccompatibility. As an example we write down the most general theory ofdissipative fluids consistent with the second law in curved non-relativisticgeometries and find significant differences in the allowed transportcoefficients from those found previously. Kubo formulas for all responsecoefficients are presented. Our approach also immediately generalizes tosystems with independent mass and charge currents as would arise inmulticomponent fluids. Along the way we also discuss how to write generallocally Galilean invariant non-relativistic actions for multiple particlespecies at any order in derivatives. A detailed review of the geometry and itsrelation to non-relativistic limits may be found in a companion paper[arXiv:1503.02682].

Fields and fluids on curved non-relativistic spacetimes