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CFT dual of the AdS Dirichlet problem: fluid/gravity on cut-off surfaces

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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.

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Authors and Affiliations

  1. Centre for Particle Theory & Department of Mathematical Sciences, Science Laboratories, South Road, Durham, DH1 3LE, United Kingdom

    Daniel Brattan, Joan Camps & Mukund Rangamani

  2. Junior Fellow, Harvard Society of Fellows, Harvard University, Cambridge, MA, 02138, USA

    R. Loganayagam

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  1. Daniel Brattan
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  2. Joan Camps
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  3. R. Loganayagam
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  4. Mukund Rangamani
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Correspondence to Joan Camps.

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ArXiv ePrint: 1106.2577

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Brattan, D., Camps, J., Loganayagam, R. et al. CFT dual of the AdS Dirichlet problem: fluid/gravity on cut-off surfaces. J. High Energ. Phys. 2011, 90 (2011). https://doi.org/10.1007/JHEP12(2011)090

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