Lifshitz Hydrodynamics from Lifshitz Black Branes with Linear Momentum

TL;DR

This work constructs a novel class of 4D $z=2$ Lifshitz black branes with nonzero linear momentum, realized in an Einstein–Proca–dilaton theory that descends from a Scherk–Schwarz reduction of AdS$_5$ with a free scalar. The boundary theory inherits a Newton–Cartan geometry, and the dual fluid exhibits Lifshitz hydrodynamics on NC space where the fluid velocity magnitude acts as a chemical potential conjugate to mass density, signaling explicit breaking of particle number. Thermodynamics of the branes obeys $\mathcal{E}+P=Ts+\frac12\rho V^2$ with $P=\mathcal{E}$ in $d=2$ and Lifshitz scaling, and the first law is supported by Noether/bulk charges, including a horizon–boundary link $V^y=-\frac{N(R_h)}{R_h}$. Lifshitz perfect fluids thus emerge from a twisted null reduction of a Schrödinger fluid with broken particle number, providing a concrete holographic realization of nonrelativistic hydrodynamics on Newton–Cartan backgrounds with novel thermodynamic structure and momentum control. The results offer a framework to study Lifshitz hydrodynamics holographically and suggest directions to generalize to broader EPD models and to include additional bulk charges or connections to alternative nonrelativistic gravities.

Abstract

We construct a new class of 4-dimensional z=2 Lifshitz black branes that have a nonzero linear momentum. These are solutions of an Einstein-Proca-dilaton model that can be obtained by Scherk-Schwarz circle reduction of AdS_5 gravity coupled to a free real scalar field. The boundary of a bulk Lifshitz space-time is a Newton-Cartan geometry. We show that the fluid dual to the moving Lifshitz black brane leads to a novel form of Lifshitz hydrodynamics on a Newton-Cartan space-time. Since the linear momentum of the black brane cannot be obtained by a boost transformation the velocity of the fluid or rather, by boundary rotational invariance, its magnitude plays the role of a chemical potential. The conjugate dual variable is mass density. The Lifshitz perfect fluid can be thought of as arising from a Schroedinger perfect fluid with broken particle number symmetry.