CFT dual of the AdS Dirichlet problem: Fluid/Gravity on cut-off surfaces

TL;DR

The paper investigates the gravitational Dirichlet problem in AdS with a fixed timelike cut-off Σ_D and identifies its dual CFT interpretation within the fluid/gravity framework.A long-wavelength gradient expansion is used to map Dirichlet data on Σ_D to a boundary conformal fluid living on a dynamical background, while the induced hypersurface fluid is non-conformal but shares the same shear viscosity, revealing a dressed-fluid picture.A critical radial position r_{D,snd} signals potential acausality in the hypersurface fluid, motivating a non-relativistic (BMW) scaling that yields a well-posed incompressible Navier–Stokes dynamics on Σ_D, and enabling a near-horizon, membrane-like/ Galilean regime in AdS/CFT.In the near-horizon limit, the boundary geometry degenerates to Newton–Cartan (Galilean) structure, and a bulk/co-metric description supports a two-region interpretation consistent with the membrane paradigm.Overall, the work connects bulk Dirichlet boundary conditions to non-local boundary deformations, clarifies the Dirichlet dictionary, and situates Dirichlet dynamics within holographic RG and non-relativistic holography.

Abstract

We study the gravitational Dirichlet problem in AdS spacetimes with a view to understanding the boundary CFT interpretation. We define the problem as bulk Einstein's equations with Dirichlet boundary conditions on fixed timelike cut-off hypersurface. Using the fluid/gravity correspondence, we argue that one can determine non-linear solutions to this problem in the long wavelength regime. On the boundary we find a conformal fluid with Dirichlet constitutive relations, viz., the fluid propagates on a `dynamical' background metric which depends on the local fluid velocities and temperature. This boundary fluid can be re-expressed as an emergent hypersurface fluid which is non-conformal but has the same value of the shear viscosity as the boundary fluid. The hypersurface dynamics arises as a collective effect, wherein effects of the background are transmuted into the fluid degrees of freedom. Furthermore, we demonstrate that this collective fluid is forced to be non-relativistic below a critical cut-off radius in AdS to avoid acausal sound propagation with respect to the hypersurface metric. We further go on to show how one can use this set-up to embed the recent constructions of flat spacetime duals to non-relativistic fluid dynamics into the AdS/CFT correspondence, arguing that a version of the membrane paradigm arises naturally when the boundary fluid lives on a background Galilean manifold.

Figures (2)

• Figure 1: Schematic representation of the Dirichlet problem we consider in this paper. The Dirichlet surface is taken to be at some value $r = r_D$ where we impose boundary conditions on the fields. The solutions will further be constrained by requiring that they be regular on any putative horizon ${\mathcal{H}}^+$ (shown in the figure) or the origin. The question we are after is what is the boundary image of this Dirichlet data?
• Figure 2: Schematic representation of the gravitational Dirichlet problem in the fluid/gravity regime. The causal structure of the fluid/gravity spacetimes is illustrated emphasizing the tubewise approximation; in each tube the geometry resembles that of a uniformly boosted Schwarzschild-AdS$_{d+1}$ black hole. Suitable choices of the Dirichlet surface allow us to find the map between the boundary data ${\mathfrak X}$ and the Dirichlet hypersurface data $\hat{\mathfrak X}$ within each tube, rendering the problem tractable.