Holographic stress tensor for non-relativistic theories

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 Schrödinger 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 Schrödinger 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. * S.F. Ross@durham.ac.uk

What do non-relativistic CFTs tell us about Lifshitz spacetimes?

“…This different scaling in fact means the notion of a boundary itself is altered; using the Penrose definition of a conformal boundary leads to a degenerate boundary metric. However, more careful treatments have shown that there is still a reasonable notion of a boundary as the location where metric components go to infinity, and holographic calculations can be performed using suitable prescriptions [5][6][7][8][9][10][11].…”

“…We will consider an ansatz which allows for analysis of all these cases. Much recent progress has been made in creating a complete bulk/boundary dictionary for nonrelativistic systems [5,6,8,10]. In the well studied case of Lorentzian AdS/CFT, an important part of this dictionary is the correspondence between normalizable modes, which scale as r ∆ + near the boundary, and states in the Hilbert space of the dual field theory.…”

“…4 Note that this is in contrast with the computation of AdS/CFT correlators, where B is interpreted as the response to turning on a source A. 5 For simplicity, we have assumed 1 < k < 2. For 0 < k ≤ 1, the horizon falloff ∼ 1/ζ k is insufficiently fast, and the potential becomes long-ranged.…”

“…Thus, to create an arbitrary state in the bulk we can use either bulk operators or boundary operators that are 'smeared' over x and t in an appropriate way. For example, for a single-particle state we have 5) so the state |E, p AdS can be built entirely out of local boundary operators, and so on.…”

Analytic Lifshitz black holes in higher dimensions

“…[12,13,14]. The holographic description of asymptotically Lifshitz spacetimes was studied in [15]. More recent investigations related to Lifshitz black holes can be found in Refs.…”