Hyperscaling-Violating Lifshitz hydrodynamics from black-holes: Part II

TL;DR

This work extends holographic hydrodynamics to hyperscaling-violating Lifshitz theories with a U(1) symmetry, deriving non-relativistic fluid dynamics from HV black holes via a derivative expansion. The analysis shows that the resulting hydrodynamics is governed by Newton–Cartan geometry with Milne invariance, featuring a conserved mass current and a pressure redefinition that reduces the dynamics to standard non-relativistic hydrodynamics with a chemical potential for the mass current; remarkably, the bulk viscosity vanishes and the shear-viscosity–entropy ratio saturates the universal bound $rac{ ilde{ ext{η}}}{ ilde{ ext{s}}}=rac{1}{4\pi}$. A thorough holographic renormalization yields explicit expressions for the stress tensor, current, and entropy current, and reveals a consistent Ward identity $(z-rac{ heta}{d-1})oldsymbol{ ext{E}}=(d-1- heta)P$ reflecting Lifshitz scaling with hyperscaling violation. The paper also clarifies the role of dimensional reduction in HV settings, showing how bulk viscosity can be tuned by the internal volume and how the HV hydrodynamics corresponds to a reduced theory with an extra scalar source; conformally flat boundary backgrounds are analyzed to illustrate simplifications and to connect with a broader class of HV fluids. Overall, the results provide a universal framework for HV Lifshitz hydrodynamics in holography and illuminate how dimensional reduction and NC geometry shape transport in these non-relativistic quantum critical systems.

Abstract

The derivation of Lifshitz-invariant hydrodynamics from holography, presented in [arXiv:1508.02494] is generalized to arbitrary hyperscaling violating Lifshitz scaling theories with an unbroken U(1) symmetry. The hydrodynamics emerging is non-relativistic with scalar "forcing". By a redefinition of the pressure it becomes standard non-relativistic hydrodynamics in the presence of specific chemical potential for the mass current. The hydrodynamics is compatible with the scaling theory of Lifshitz invariance with hyperscaling violation. The bulk viscosity vanishes while the shear viscosity to entropy ratio is the same as in the relativistic case. We also consider the dimensional reduction ansatz for the hydrodynamics and clarify the difference with previous results suggesting a non-vanishing bulk viscosity.