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We compute constitutive relations for a charged $(2+1)$ dimensionalSchr\"odinger fluid up to first order in derivative expansion, usingholographic techniques. Starting with a locally boosted, asymptotically $AdS$,$4+1$ dimensional charged black brane geometry, we uplift that to tendimensions and perform $TsT$ transformations to obtain an effective fivedimensional local black brane solution with asymptotically Schr\"odingerisometries. By suitably implementing the holographic techniques, we compute theconstitutive relations for the effective fluid living on the boundary of thisspace-time and extract first order transport coefficients from these relations.Schr\"odinger fluid can also be obtained by reducing a charged relativisticconformal fluid over light-cone. It turns out that both the approaches resultthe same system at the end. Fluid obtained by light-cone reduction satisfies arestricted class of thermodynamics. Here, we see that the charged fluidobtained holographically also belongs to the same restricted class.

Light-Cone Reduction vs. TsT transformations : A Fluid Dynamics Perspective