Holographic stress tensor for non-relativistic theories

TL;DR

The paper develops a holographic dictionary for non-relativistic theories dual to asymptotically Lifshitz and Schrödinger spacetimes, focusing on a finite action principle and a gauge-invariant NR stress tensor complex. It introduces a Lifshitz-specific boundary term with a boundary function $f(A_\alpha A^\alpha)$ chosen to cancel divergences, ensuring a finite on-shell action and a well-defined variational principle. The NR stress tensor complex is defined via variations of the boundary frame fields with tangent-space fixed matter fields, yielding finite, conserved charges ${\mathcal E}$, ${\mathcal E}^i$, ${\mathcal P}_i$, and $\Pi_i^{\ j}$ that reproduce expected hydrodynamic relations; in the Schrödinger case, the results agree with the AdS-based stress tensor under TsT reductions. A linearized perturbation analysis around Lifshitz confirms finiteness of the stress tensor and clarifies how bulk falloffs map to boundary data, though non-linear issues and vector-field dual operators warrant further study. Overall, the work provides a concrete, finite holographic framework for non-relativistic theories with Lifshitz and Schrödinger asymptotics and lays out directions for extensions and embeddings.

Abstract

We discuss the calculation of the field theory stress tensor from the dual geometry for two recent proposals for gravity duals of non-relativistic conformal field theories. The first of these has a Schrodinger symmetry including Galilean boosts, while the second has just an anisotropic scale invariance (the Lifshitz case). For the Lifshitz case, we construct an appropriate action principle. We propose a definition of the non-relativistic stress tensor complex for the field theory as an appropriate variation of the action in both cases. In the Schrodinger case, we show that this gives physically reasonable results for a simple black hole solution and agrees with an earlier proposal to determine the stress tensor from the familiar AdS prescription. In the Lifshitz case, we solve the linearised equations of motion for a general perturbation around the background, showing that our stress tensor is finite on-shell.