Effective actions for relativistic fluids from holography

  • Jan de Boer
  • Michal P. Heller
  • Natalia Pinzani-Fokeeva
Open Access
Regular Article - Theoretical Physics

Abstract

Motivated by recent progress in developing action formulations of relativistic hydrodynamics, we use holography to derive the low energy dissipationless effective action for strongly coupled conformal fluids. Our analysis is based on the study of novel double Dirichlet problems for the gravitational field, in which the boundary conditions are set on two codimension one timelike hypersurfaces (branes). We provide a geometric interpretation of the Goldstone bosons appearing in such constructions in terms of a family of spatial geodesics extending between the ultraviolet and the infrared brane. Furthermore, we discuss supplementing double Dirichlet problems with information about the near-horizon geometry. We show that upon coupling to a membrane paradigm boundary condition, our approach reproduces correctly the complex dispersion relation for both sound and shear waves. We also demonstrate that upon a Wick rotation, our formulation reproduces the equilibrium partition function formalism, provided the near- horizon geometry is properly accounted for. Finally, we define the conserved hydrodynamic entropy current as the Noether current associated with a particular transformation of the Goldstone bosons.

Keywords

AdS-CFT Correspondence Effective field theories 

References

  1. [1]
    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] [INSPIRE].CrossRefADSGoogle Scholar
  2. [2]
    G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics, JHEP 09 (2002) 043 [hep-th/0205052] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  3. [3]
    G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics. 2. Sound waves, JHEP 12 (2002) 054 [hep-th/0210220] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  4. [4]
    S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].CrossRefADSGoogle Scholar
  5. [5]
    M. Rangamani, Gravity and Hydrodynamics: Lectures on the fluid-gravity correspondence, Class. Quant. Grav. 26 (2009) 224003 [arXiv:0905.4352] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  6. [6]
    L.D. Landau and E.M. Lifshitz, Fluid Mechanics. Course of Theoretical Physics. Volume 6, Pergamon Press (1959).Google Scholar
  7. [7]
    N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma, Constraints on Fluid Dynamics from Equilibrium Partition Functions, JHEP 09 (2012) 046 [arXiv:1203.3544] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  8. [8]
    K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom, Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109 (2012) 101601 [arXiv:1203.3556] [INSPIRE].CrossRefADSGoogle Scholar
  9. [9]
    S. Bhattacharyya, Entropy Current from Partition Function: One Example, JHEP 07 (2014) 139 [arXiv:1403.7639] [INSPIRE].CrossRefADSGoogle Scholar
  10. [10]
    B. Carter, Elastic Perturbation Theory in General Relativity and a Variation Principle for a Rotating Solid Star, Commun. Math. Phys. 30 (1973) 261.CrossRefADSMATHGoogle Scholar
  11. [11]
    S. Dubovsky, T. Gregoire, A. Nicolis and R. Rattazzi, Null energy condition and superluminal propagation, JHEP 03 (2006) 025 [hep-th/0512260] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  12. [12]
    S. Dubovsky, L. Hui, A. Nicolis and D.T. Son, Effective field theory for hydrodynamics: thermodynamics and the derivative expansion, Phys. Rev. D 85 (2012) 085029 [arXiv:1107.0731] [INSPIRE].ADSGoogle Scholar
  13. [13]
    J. Bhattacharya, S. Bhattacharyya and M. Rangamani, Non-dissipative hydrodynamics: Effective actions versus entropy current, JHEP 02 (2013) 153 [arXiv:1211.1020] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  14. [14]
    G. Ballesteros, The effective theory of fluids at NLO and implications for dark energy, JCAP 03 (2015) 001 [arXiv:1410.2793] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  15. [15]
    F.M. Haehl, R. Loganayagam and M. Rangamani, The eightfold way to dissipation, Phys. Rev. Lett. 114 (2015) 201601 [arXiv:1412.1090] [INSPIRE].CrossRefADSGoogle Scholar
  16. [16]
    F.M. Haehl, R. Loganayagam and M. Rangamani, Adiabatic hydrodynamics: The eightfold way to dissipation, JHEP 05 (2015) 060 [arXiv:1502.00636] [INSPIRE].CrossRefADSGoogle Scholar
  17. [17]
    J.S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2 (1961) 407 [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  18. [18]
    L.V. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [Sov. Phys. JETP 20 (1965) 1018] [INSPIRE].
  19. [19]
    D. Nickel and D.T. Son, Deconstructing holographic liquids, New J. Phys. 13 (2011) 075010 [arXiv:1009.3094] [INSPIRE].CrossRefADSGoogle Scholar
  20. [20]
    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
  21. [21]
    I. Heemskerk and J. Polchinski, Holographic and Wilsonian Renormalization Groups, JHEP 06 (2011) 031 [arXiv:1010.1264] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  22. [22]
    T. Sakai and S. Sugimoto, Low energy hadron physics in holographic QCD, Prog. Theor. Phys. 113 (2005) 843 [hep-th/0412141] [INSPIRE].CrossRefADSMATHGoogle Scholar
  23. [23]
    N. Arkani-Hamed, H. Georgi and M.D. Schwartz, Effective field theory for massive gravitons and gravity in theory space, Annals Phys. 305 (2003) 96 [hep-th/0210184] [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  24. [24]
    S.F. Hassan and R.A. Rosen, Bimetric Gravity from Ghost-free Massive Gravity, JHEP 02 (2012) 126 [arXiv:1109.3515] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  25. [25]
    T. Faulkner and J. Polchinski, Semi-Holographic Fermi Liquids, JHEP 06 (2011) 012 [arXiv:1001.5049] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  26. [26]
    C.P. Herzog and D.T. Son, Schwinger-Keldysh propagators from AdS/CFT correspondence, JHEP 03 (2003) 046 [hep-th/0212072] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  27. [27]
    T. Damour, Black Hole Eddy Currents, Phys. Rev. D 18 (1978) 3598 [INSPIRE].ADSGoogle Scholar
  28. [28]
    K.S. Thorne, R. Price and D. MacDonald, Black Holes: The Membrane Paradigm, The Silliman Memorial Lectures Series, Yale University Press (1986) [ISBN: 978-0300037708].Google Scholar
  29. [29]
    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
  30. [30]
    M. Parikh and F. Wilczek, An Action for black hole membranes, Phys. Rev. D 58 (1998) 064011 [gr-qc/9712077] [INSPIRE].MathSciNetADSGoogle Scholar
  31. [31]
    J. de Boer, M.P. Heller and N. Pinzani-Fokeeva, Testing the membrane paradigm with holography, Phys. Rev. D 91 (2015) 026006 [arXiv:1405.4243] [INSPIRE].ADSGoogle Scholar
  32. [32]
    J.B. Hartle and S.W. Hawking, Wave Function of the Universe, Phys. Rev. D 28 (1983) 2960 [INSPIRE].MathSciNetADSGoogle Scholar
  33. [33]
    V. Balasubramanian, B. Czech, B.D. Chowdhury and J. de Boer, The entropy of a hole in spacetime, JHEP 10 (2013) 220 [arXiv:1305.0856] [INSPIRE].CrossRefADSGoogle Scholar
  34. [34]
    V. Balasubramanian, B.D. Chowdhury, B. Czech, J. de Boer and M.P. Heller, Bulk curves from boundary data in holography, Phys. Rev. D 89 (2014) 086004 [arXiv:1310.4204] [INSPIRE].ADSGoogle Scholar
  35. [35]
    T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS 2, Phys. Rev. D 83 (2011) 125002 [arXiv:0907.2694] [INSPIRE].ADSGoogle Scholar
  36. [36]
    R. Emparan, R. Suzuki and K. Tanabe, The large D limit of General Relativity, JHEP 06 (2013) 009 [arXiv:1302.6382] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  37. [37]
    R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].MathSciNetADSGoogle Scholar
  38. [38]
    A. Nicolis, R. Penco and R.A. Rosen, Relativistic Fluids, Superfluids, Solids and Supersolids from a Coset Construction, Phys. Rev. D 89 (2014) 045002 [arXiv:1307.0517] [INSPIRE].ADSGoogle Scholar
  39. [39]
    C. Adam, T. Klähn, C. Naya, J. Sanchez-Guillen, R. Vazquez and A. Wereszczynski, Baryon chemical potential and in-medium properties of BPS skyrmions, Phys. Rev. D 91 (2015) 125037 [arXiv:1504.05185] [INSPIRE].ADSGoogle Scholar
  40. [40]
    M. Crossley, P. Glorioso, H. Liu and Y. Wang, Off-shell hydrodynamics from holography, arXiv:1504.07611 [INSPIRE].
  41. [41]
    S. Endlich, A. Nicolis and J. Wang, Solid Inflation, JCAP 10 (2013) 011 [arXiv:1210.0569] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  42. [42]
    I. Low and A.V. Manohar, Spontaneously broken space-time symmetries and Goldstones theorem, Phys. Rev. Lett. 88 (2002) 101602 [hep-th/0110285] [INSPIRE].CrossRefADSGoogle Scholar
  43. [43]
    P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72 (2005) 086009 [hep-th/0506184] [INSPIRE].ADSGoogle Scholar
  44. [44]
    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].MathSciNetCrossRefADSGoogle Scholar
  45. [45]
    S. Bhattacharyya et al., Local Fluid Dynamical Entropy from Gravity, JHEP 06 (2008) 055 [arXiv:0803.2526] [INSPIRE].CrossRefADSGoogle Scholar
  46. [46]
    I. Booth, M.P. Heller and M. Spalinski, Black Brane Entropy and Hydrodynamics, Phys. Rev. D 83 (2011) 061901 [arXiv:1010.6301] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Jan de Boer
    • 1
  • Michal P. Heller
    • 2
  • Natalia Pinzani-Fokeeva
    • 1
  1. 1.Institute of PhysicsUniversiteit van AmsterdamAmsterdamThe Netherlands
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada

Personalised recommendations