Advertisement

From Navier-Stokes to Einstein

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

Abstract

We show by explicit construction that for every solution of the incompressible Navier-Stokes equation in p + 1 dimensions, there is a uniquely associated “dual” solution of the vacuum Einstein equations in p + 2 dimensions. The dual geometry has an intrinsically flat timelike boundary segment Σc whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a “near-horizon” limit in which Σc becomes highly accelerated. The near-horizon expansion in gravity is shown to be mathematically equivalent to the hydrodynamic expansion in fluid dynamics, and the Einstein equation reduces to the incompressible Navier-Stokes equation. For p = 2, we show that the full dual geometry is algebraically special Petrov type II. The construction is a mathematically precise realization of suggestions of a holographic duality relating fluids and horizons which began with the membrane paradigm in the 70’s and resurfaced recently in studies of the AdS/CFT correspondence.

Keywords

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

References

  1. [1]
    S. Bhattacharyya, S. Minwalla and S.R. Wadia, The incompressible non-relativistic Navier-Stokes equation from gravity, JHEP 08 (2009) 059 [arXiv:0810.1545] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    Y. Oz and M. Rabinovich, The Penrose inequality and the fluid/gravity correspondence, JHEP 02 (2011) 070 [arXiv:1011.5895] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    T. Damour, Quelques propriétés mécaniques, électromagnétiques, thermodynamiques et quantiques des trous noirs, Ph.D. thesis, Université Pierre et Marie Curie, Paris VI, Paris, France (1979).Google Scholar
  4. [4]
    T. Damour, Surface effects in black hole physics, in Proceedings of the second Marcel Grossmann meeting on general relativity, R. Ruffini ed., North Holland, The Netherlands (1982).Google Scholar
  5. [5]
    R. Price and K. Thorne, Membrane viewpoint on black holes: properties and evolution of the stretched horizon, Phys. Rev. D 33 (1986) 915 [INSPIRE].MathSciNetADSGoogle Scholar
  6. [6]
    T. Jacobson, Thermodynamics of space-time: the Einstein equation of state, Phys. Rev. Lett. 75 (1995)1260 [gr-qc/9504004] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. [7]
    R. Bousso, A covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    G. Policastro, D. Son and A. Starinets, The shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 87 (2001) 081601 [hep-th/0104066] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics, JHEP 09 (2002) 043 [hep-th/0205052] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    P. Kovtun, D.T. Son and A.O. Starinets, Holography and hydrodynamics: diffusion on stretched horizons, JHEP 10 (2003) 064 [hep-th/0309213] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    A. Karch, Experimental tests of the holographic entropy bound, hep-th/0311116 [INSPIRE].
  12. [12]
    P. Kovtun, D. Son and A. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    E. Gourgoulhon and J.L. Jaramillo, A 3 + 1 perspective on null hypersurfaces and isolated horizons, Phys. Rept. 423 (2006) 159 [gr-qc/0503113] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72 (2005)086009 [hep-th/0506184] [INSPIRE].ADSGoogle Scholar
  15. [15]
    V.E. Hubeny, M. Rangamani, S. Minwalla and M. Van Raamsdonk, The fluid-gravity correspondence: the membrane at the end of the universe, Int. J. Mod. Phys. D 17 (2009) 2571 [INSPIRE].ADSGoogle Scholar
  16. [16]
    S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    S. Bhattacharyya et al., Forced fluid dynamics from gravity, JHEP 02 (2009) 018 [arXiv:0806.0006] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    C. Eling, Hydrodynamics of spacetime and vacuum viscosity, JHEP 11 (2008) 048 [arXiv:0806.3165] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    S. Bhattacharyya, R. Loganayagam, I. Mandal, S. Minwalla and A. Sharma, Conformal nonlinear fluid dynamics from gravity in arbitrary dimensions, JHEP 12 (2008) 116 [arXiv:0809.4272] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    I. Fouxon and Y. Oz, Conformal field theory as microscopic dynamics of incompressible euler and Navier-Stokes equations, Phys. Rev. Lett. 101 (2008) 261602 [arXiv:0809.4512] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    R.K. Gupta and A. Mukhopadhyay, On the universal hydrodynamics of strongly coupled CFTs with gravity duals, JHEP 03 (2009) 067 [arXiv:0810.4851] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    I. Fouxon and Y. Oz, CFT hydrodynamics: symmetries, exact solutions and gravity, JHEP 03 (2009) 120 [arXiv:0812.1266] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    C. Eling, I. Fouxon and Y. Oz, The incompressible Navier-Stokes equations from membrane dynamics, Phys. Lett. B 680 (2009) 496 [arXiv:0905.3638] [INSPIRE].MathSciNetADSGoogle Scholar
  24. [24]
    C. Eling and Y. Oz, Relativistic CFT hydrodynamics from the membrane paradigm, JHEP 02 (2010) 069 [arXiv:0906.4999] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    M.F. Paulos, Transport coefficients, membrane couplings and universality at extremality, JHEP 02 (2010) 067 [arXiv:0910.4602] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    T. Padmanabhan, Thermodynamical aspects of gravity: new insights, Rept. Prog. Phys. 73 (2010)046901 [arXiv:0911.5004] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    C. Eling, I. Fouxon and Y. Oz, Gravity and a geometrization of turbulence: an intriguing correspondence, arXiv:1004.2632.
  28. [28]
    I. Bredberg, C. Keeler, V. Lysov and A. Strominger, Wilsonian approach to fluid/gravity duality, JHEP 03 (2011) 141 [arXiv:1006.1902] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    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] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    T. Damour and M. Lilley, String theory, gravity and experiment, arXiv:0802.4169 [INSPIRE].
  31. [31]
    M. Rangamani, Gravity and hydrodynamics: lectures on the fluid-gravity correspondence, Class. Quant. Grav. 26 (2009) 224003 [arXiv:0905.4352] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    V.E. Hubeny, The fluid/gravity correspondence: a new perspective on the membrane paradigm, Class. Quant. Grav. 28 (2011) 114007 [arXiv:1011.4948] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    S.W. Hawking and J. B. Hartle, Energy and angular momentum flow into a black hole, Commun. Math. Phys. 27 (1972) 283.MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    J.B. Hartle, Tidal friction in slowly rotating black holes, Phys. Rev. D 8 (1973) 1010 [INSPIRE].ADSGoogle Scholar
  35. [35]
    J.B. Hartle, Tidal shapes and shifts on rotating black holes, Phys. Rev. D 9 (1974) 2749 [INSPIRE].ADSGoogle Scholar
  36. [36]
    A. Petrov, Einstein spaces, Pergamon Press, U.K. (1969).zbMATHGoogle Scholar
  37. [37]
    E. Hertl, C. Hoenselaers, D. Kramer, M. Maccallum and H. Stephani, Exact solutions of Einsteins field equations, Cambridge University Press, Cambridge U.K. (2003).Google Scholar
  38. [38]
    R. Milson, A. Coley, V. Pravda and A. Pravdova, Alignment and algebraically special tensors in Lorentzian geometry, Int. J. Geom. Meth. Mod. Phys. 2 (2005) 41 [gr-qc/0401010] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    E. Gourgoulhon, Generalized Damour-Navier-Stokes equation applied to trapping horizons, Phys. Rev. D 72 (2005) 104007 [gr-qc/0508003].MathSciNetADSGoogle Scholar
  40. [40]
    K.S. Thorne, R.H. Price and D.A. Macdonald, Black holes: the membrane paradigm, Yale University Press, New Haven U.S.A. (1986).Google Scholar
  41. [41]
    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] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    O. Saremi, Shear waves, sound waves on a shimmering horizon, hep-th/0703170 [INSPIRE].
  43. [43]
    R. Baier, P. Romatschke, D.T. Son, A.O. Starinets and M.A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance and holography, JHEP 04 (2008) 100 [arXiv:0712.2451] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    R. Brustein and A. Medved, The ratio of shear viscosity to entropy density in generalized theories of gravity, Phys. Rev. D 79 (2009) 021901 [arXiv:0808.3498] [INSPIRE].ADSGoogle Scholar
  45. [45]
    M. Rangamani, S.F. Ross, D. Son and E.G. Thompson, Conformal non-relativistic hydrodynamics from gravity, JHEP 01 (2009) 075 [arXiv:0811.2049] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    M. Mia, K. Dasgupta, C. Gale and S. Jeon, Five easy pieces: the dynamics of quarks in strongly coupled plasmas, Nucl. Phys. B 839 (2010) 187 [arXiv:0902.1540] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    J. McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys. 2010 (2010) 723105 [arXiv:0909.0518] [INSPIRE].Google Scholar
  49. [49]
    S. Sachdev, Condensed matter and AdS/CFT, arXiv:1002.2947 [INSPIRE].
  50. [50]
    V.E. Hubeny and M. Rangamani, A holographic view on physics out of equilibrium, arXiv:1006.3675 [INSPIRE].
  51. [51]
    T. Faulkner, H. Liu and M. Rangamani, Integrating out geometry: holographic Wilsonian RG and the membrane paradigm, JHEP 08 (2011) 051 [arXiv:1010.4036] [INSPIRE].MathSciNetADSGoogle Scholar
  52. [52]
    A.O. Starinets, Quasinormal spectrum and the black hole membrane paradigm, Phys. Lett. B 670 (2009)442 [arXiv:0806.3797] [INSPIRE].MathSciNetADSGoogle Scholar
  53. [53]
    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] [INSPIRE].ADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

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

Personalised recommendations