Journal of High Energy Physics

, 2011:141

Wilsonian approach to fluid/gravity duality

  • Irene Bredberg
  • Cynthia Keeler
  • Vyacheslav Lysov
  • Andrew Strominger


The problem of gravitational fluctuations confined inside a finite cutoff at radius r = rc outside the horizon in a general class of black hole geometries is considered. Consistent boundary conditions at both the cutoff surface and the horizon are found and the resulting modes analyzed. For general cutoff rc the dispersion relation is shown at long wavelengths to be that of a linearized Navier-Stokes fluid living on the cutoff surface. A cutoff-dependent line-integral formula for the diffusion constant D (rc) is derived. The dependence on rc is interpreted as renormalization group (RG) flow in the fluid. Taking the cutoff to infinity in an asymptotically AdS context, the formula for D(∞) reproduces as a special case well-known results derived using AdS/CFT. Taking the cutoff to the horizon, the effective speed of sound goes to infinity, the fluid becomes incompressible and the Navier-Stokes dispersion relation becomes exact. The resulting universal formula for the diffusion constant D(horizon) reproduces old results from the membrane paradigm. Hence the old membrane paradigm results and new AdS/CFT results are related by RG flow. RG flow-invariance of the viscosity to entropy ratio \( \frac{\eta }{s} \) is shown to follow from the first law of thermodynamics together with isentropy of radial evolution in classical gravity. The ratio is expected to run when quantum gravitational corrections are included.


Classical Theories of Gravity Black Holes Holography and condensed matter physics (AdS/CMT) 


  1. [1]
    S.W. Hawking and J.B. Hartle, Energy and angular momentum flow into a black hole, Commun. Math. Phys. 27 (1972) 283 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    J.B. Hartle, Tidal friction in slowly rotating black holes, Phys. Rev. D 8 (1973) 1010 [SPIRES].ADSGoogle Scholar
  3. [3]
    J.B. Hartle, Tidal shapes and shifts on rotating black holes, Phys. Rev. D 9 (1974) 2749 [SPIRES].ADSGoogle Scholar
  4. [4]
    T. Damour, Quelques propriétés mécaniques, électromagnétiques, thermodynamiques et quantiques des trous noirs, Thèse de Doctorat d’Etat, Université Pierre et Marie Curie, Paris VI, Fance (1979).Google Scholar
  5. [5]
    T. Damour, Black hole Eddy currents, Phys. Rev. D 18 (1978) 3598 [SPIRES].ADSGoogle Scholar
  6. [6]
    R.L. Znajek, The electric and magnetic conductivity of a Kerr hole, Mon. Not. Roy. Astron. Soc. 185 (1978) 833.ADSGoogle Scholar
  7. [7]
    T. Damour, Surface effects in black hole physics, in the proceedings of the 2nd Marcel Grossmann Meeting on general Relativity, R. Ruffini ed., North-Holland, The Netherlands (1982).Google Scholar
  8. [8]
    R.H. Price and K.S. Thorne, Membrane viewpoint on black holes: properties and evolution of the stretched horizon, Phys. Rev. D 33 (1986) 915 [SPIRES].MathSciNetADSGoogle Scholar
  9. [9]
    K.S. Thorne, R.H. . Price and D.A. Macdonald, Black holes: the membrane paradigm, Yale University Press, U.S.A. (1986), p. 367.Google Scholar
  10. [10]
    T. Damour and M. Lilley, String theory, gravity and experiment, arXiv:0802.4169 [SPIRES].
  11. [11]
    G. Policastro, D.T. Son and A.O. Starinets, The shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 87 (2001) 081601 [hep-th/0104066] [SPIRES].ADSCrossRefGoogle Scholar
  12. [12]
    G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics, JHEP 09 (2002) 043 [hep-th/0205052] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    P. Kovtun, D.T. Son and A.O. Starinets, Holography and hydrodynamics: diffusion on stretched horizons, JHEP 10 (2003) 064 [hep-th/0309213] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    H. Kodama, A. Ishibashi and O. Seto, Brane world cosmology: gauge-invariant formalism for perturbation, Phys. Rev. D 62 (2000) 064022 [hep-th/0004160] [SPIRES].MathSciNetADSGoogle Scholar
  15. [15]
    C.P. Herzog, The hydrodynamics of M-theory, JHEP 12 (2002) 026 [hep-th/0210126] [SPIRES].ADSCrossRefGoogle Scholar
  16. [16]
    G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics. II: sound waves, JHEP 12 (2002) 054 [hep-th/0210220] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    A. Buchel, N = 2* hydrodynamics, Nucl. Phys. B 708 (2005) 451 [hep-th/0406200] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72 (2005) 086009 [hep-th/0506184] [SPIRES].ADSGoogle Scholar
  19. [19]
    P. Benincasa, A. Buchel and R. Naryshkin, The shear viscosity of gauge theory plasma with chemical potentials, Phys. Lett. B 645 (2007) 309 [hep-th/0610145] [SPIRES].ADSGoogle Scholar
  20. [20]
    Y. Kats and P. Petrov, Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory, JHEP 01 (2009) 044 [arXiv:0712.0743] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, Viscosity bound violation in higher derivative gravity, Phys. Rev. D 77 (2008) 126006 [arXiv:0712.0805] [SPIRES].ADSGoogle Scholar
  22. [22]
    P. Kovtun and A. Ritz, Universal conductivity and central charges, Phys. Rev. D 78 (2008) 066009 [arXiv:0806.0110] [SPIRES].ADSGoogle Scholar
  23. [23]
    A.O. Starinets, Quasinormal spectrum and the black hole membrane paradigm, Phys. Lett. B 670 (2009) 442 [arXiv:0806.3797] [SPIRES].MathSciNetADSGoogle Scholar
  24. [24]
    S. Bhattacharyya, S. Minwalla and S.R. Wadia, The incompressible non-relativistic Navier-Stokes equation from gravity, JHEP 08 (2009) 059 [arXiv:0810.1545] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    A. Buchel, R.C. Myers and A. Sinha, Beyond η/s = 1/4π, JHEP 03 (2009) 084 [arXiv:0812.2521] [SPIRES].ADSCrossRefGoogle Scholar
  26. [26]
    A. Buchel et al., Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [SPIRES].ADSCrossRefGoogle Scholar
  27. [27]
    S. Cremonini, K. Hanaki, J.T. Liu and P. Szepietowski, Black holes in five-dimensional gauged supergravity with higher derivatives, JHEP 12 (2009) 045 [arXiv:0812.3572] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    R.C. Myers, M.F. Paulos and A. Sinha, Holographic hydrodynamics with a chemical potential, JHEP 06 (2009) 006 [arXiv:0903.2834] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    S. Cremonini, K. Hanaki, J.T. Liu and P. Szepietowski, Higher derivative effects on η/s at finite chemical potential, Phys. Rev. D 80 (2009) 025002 [arXiv:0903.3244] [SPIRES].ADSGoogle Scholar
  30. [30]
    D.T. Son and A.O. Starinets, Viscosity, black holes and quantum field theory, Ann. Rev. Nucl. Part. Sci. 57 (2007) 95 [arXiv:0704.0240] [SPIRES].ADSCrossRefGoogle Scholar
  31. [31]
    A. Buchel and J.T. Liu, Universality of the shear viscosity in supergravity, Phys. Rev. Lett. 93 (2004) 090602 [hep-th/0311175] [SPIRES].ADSCrossRefGoogle Scholar
  32. [32]
    N. Iqbal and H. Liu, Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm, Phys. Rev. D 79 (2009) 025023 [arXiv:0809.3808] [SPIRES].ADSGoogle Scholar
  33. [33]
    C. Eling, I. Fouxon and Y. Oz, The incompressible Navier-Stokes equations from membrane dynamics, Phys. Lett. B 680 (2009) 496 [arXiv:0905.3638] [SPIRES].MathSciNetADSGoogle Scholar
  34. [34]
    C. Eling and Y. Oz, Relativistic CFT hydrodynamics from the membrane paradigm, JHEP 02 (2010) 069 [arXiv:0906.4999] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    M.F. Paulos, Transport coefficients, membrane couplings and universality at extremality, JHEP 02 (2010) 067 [arXiv:0910.4602] [SPIRES].ADSCrossRefGoogle Scholar
  36. [36]
    N. Banerjee and S. Dutta, Nonlinear hydrodynamics from flow of retarded Green’s function, JHEP 08 (2010) 041 [arXiv:1005.2367] [SPIRES].ADSCrossRefGoogle Scholar
  37. [37]
    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    J. McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys. 2010 (2010) 723105 [arXiv:0909.0518] [SPIRES].Google Scholar
  39. [39]
    S. Sachdev, Condensed matter and AdS/CFT, arXiv:1002.2947 [SPIRES].
  40. [40]
    J. Polchinski, Semi-holographic Fermi liquids, talk at Strings 2010, March 15–19, College Station, U.S.A. (2010).Google Scholar
  41. [41]
    S. Bhattacharyya et al., Forced fluid dynamics from gravity, JHEP 02 (2009) 018 [arXiv:0806.0006] [SPIRES].ADSCrossRefGoogle Scholar
  42. [42]
    A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [SPIRES].MathSciNetADSGoogle Scholar
  43. [43]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [SPIRES].MathSciNetADSMATHGoogle Scholar
  44. [44]
    T. Jacobson, Thermodynamics of space-time: the Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  45. [45]
    E.P. Verlinde, On the origin of gravity and the laws of Newton, arXiv:1001.0785 [SPIRES].
  46. [46]
    M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [SPIRES].
  47. [47]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [SPIRES].ADSMATHCrossRefGoogle Scholar
  48. [48]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    I.R. Klebanov and M.J. Strassler, Supergravity and a confining gauge theory: duality cascades and χ SB -resolution of naked singularities, JHEP 08 (2000) 052 [hep-th/0007191] [SPIRES].MathSciNetADSCrossRefMATHGoogle Scholar
  50. [50]
    R. Gopakumar, S. Minwalla, N. Seiberg and A. Strominger, OM theory in diverse dimensions, JHEP 08 (2000) 008 [hep-th/0006062] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c-theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [SPIRES].MathSciNetMATHGoogle Scholar
  52. [52]
    T. Padmanabhan, Thermodynamical aspects of gravity: new insights, Rept. Prog. Phys. 73 (2010) 046901 [arXiv:0911.5004] [SPIRES].ADSCrossRefGoogle Scholar
  53. [53]
    R.C. Myers, M.F. Paulos and A. Sinha, Quantum corrections to η/s, Phys. Rev. D 79 (2009) 041901 [arXiv:0806.2156] [SPIRES].ADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  • Irene Bredberg
    • 1
  • Cynthia Keeler
    • 1
  • Vyacheslav Lysov
    • 1
  • Andrew Strominger
    • 1
  1. 1.Center for the Fundamental Laws of NatureHarvard UniversityCambridgeU.S.A.

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