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

TL;DR

This paper explores the gravitational Dirichlet problem in AdS spacetimes, revealing how boundary and hypersurface fluids emerge and behave, including non-relativistic limits, within the fluid/gravity correspondence framework.

Contribution

It introduces a novel boundary and hypersurface fluid interpretation of the Dirichlet problem in AdS, including non-conformal and non-relativistic regimes, linking to flat spacetime duals.

Findings

Boundary fluid propagates on a dynamical background metric.

Hypersurface fluid is non-conformal but shares shear viscosity with boundary fluid.

Non-relativistic behavior emerges below a critical cut-off radius.

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.