Classical and quantum temperature fluctuations via holography

  • Alexander Balatsky
  • Sven Bjarke GudnasonEmail author
  • Yaron Kedem
  • Alexander Krikun
  • Lárus Thorlacius
  • Konstantin Zarembo
Open Access
Regular Article - Theoretical Physics


We study local temperature fluctuations in a 2+1 dimensional CFT on the sphere, dual to a black hole in asymptotically AdS spacetime. The fluctuation spectrum is governed by the lowest-lying hydrodynamic modes of the system whose frequency and damping rate determine whether temperature fluctuations are thermal or quantum. We calculate numerically the corresponding quasinormal frequencies and match the result with the hydrodynamics of the dual CFT at high temperature. As a by-product of our analysis we determine the appropriate boundary conditions for calculating low-lying quasinormal modes for a four-dimensional Reissner-Nordström black hole in global AdS.


AdS-CFT Correspondence Holography and condensed matter physics (AdS/CMT) Black Holes 


Open Access

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  1. [1]
    J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].ADSMathSciNetGoogle Scholar
  2. [2]
    S. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
  3. [3]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    S. Hawking and D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577.ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].zbMATHMathSciNetGoogle Scholar
  6. [6]
    E.J. Brynjolfsson and L. Thorlacius, Taking the temperature of a black hole, JHEP 09 (2008) 066 [arXiv:0805.1876] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    A. Balatsky and J.-X. Zhu, Quantum Nyquist temperature fluctuations, Physica E 18 (2003) 341 [cond-mat/0202521].ADSCrossRefGoogle Scholar
  8. [8]
    E. Lifshitz and L. Landau, Statistical physics, Course of Theoretical Physics volume 5, Butterworth-Heinemann (1980).Google Scholar
  9. [9]
    H. Nyquist, Thermal agitation of electric charge in conductors, Phys. Rev. 32 (1928) 110 [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    C.S. Peca and P.S. Lemos, Jose, Thermodynamics of Reissner-Nordstrom Anti-de Sitter black holes in the grand canonical ensemble, Phys. Rev. D 59 (1999) 124007 [gr-qc/9805004] [INSPIRE].
  11. [11]
    G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics, JHEP 09 (2002) 043 [hep-th/0205052] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    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].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    J.J. Friess, S.S. Gubser, G. Michalogiorgakis and S.S. Pufu, Expanding plasmas and quasinormal modes of anti-de Sitter black holes, JHEP 04 (2007) 080 [hep-th/0611005] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    G. Michalogiorgakis and S.S. Pufu, Low-lying gravitational modes in the scalar sector of the global AdS 4 black hole, JHEP 02 (2007) 023 [hep-th/0612065] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  15. [15]
    P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].ADSMathSciNetGoogle Scholar
  17. [17]
    P. Kovtun and A. Ritz, Universal conductivity and central charges, Phys. Rev. D 78 (2008) 066009 [arXiv:0806.0110] [INSPIRE].ADSGoogle Scholar
  18. [18]
    J.S.F. Chan and R.B. Mann, Scalar wave falloff in asymptotically Anti-de Sitter backgrounds, Phys. Rev. D 55 (1997) 7546 [gr-qc/9612026] [INSPIRE].ADSGoogle Scholar
  19. [19]
    G.T. Horowitz and V.E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027 [hep-th/9909056] [INSPIRE].ADSMathSciNetGoogle Scholar
  20. [20]
    V. Cardoso and J.P.S. Lemos, Quasinormal modes of Schwarzschild anti-de Sitter black holes: Electromagnetic and gravitational perturbations, Phys. Rev. D 64 (2001) 084017 [gr-qc/0105103] [INSPIRE].ADSMathSciNetGoogle Scholar
  21. [21]
    I.G. Moss and J.P. Norman, Gravitational quasinormal modes for Anti-de Sitter black holes, Class. Quant. Grav. 19 (2002) 2323 [gr-qc/0201016] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    E. Berti and K.D. Kokkotas, Quasinormal modes of Reissner-Nordstrom-Anti-de Sitter black holes: scalar, electromagnetic and gravitational perturbations, Phys. Rev. D 67 (2003) 064020 [gr-qc/0301052] [INSPIRE].ADSMathSciNetGoogle Scholar
  23. [23]
    E. Berti, V. Cardoso and A.O. Starinets, Quasinormal modes of black holes and black branes, Class. Quant. Grav. 26 (2009) 163001 [arXiv:0905.2975] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    F. Mellor and I. Moss, Stability of black holes in de Sitter space, Phys. Rev. D 41 (1990) 403 [INSPIRE].ADSMathSciNetGoogle Scholar
  25. [25]
    G. Siopsis, Low frequency quasi-normal modes of AdS black holes, JHEP 05 (2007) 042 [hep-th/0702079] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    V. Cardoso, O.J.C. Dias, G.S. Hartnett, L. Lehner and J.E. Santos, Holographic thermalization, quasinormal modes and superradiance in Kerr-AdS, JHEP 04 (2014) 183 [arXiv:1312.5323] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  27. [27]
    O.J.C. Dias and J.E. Santos, Boundary conditions for Kerr-AdS perturbations, JHEP 10 (2013) 156 [arXiv:1302.1580] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  28. [28]
    R.A. Konoplya and A. Zhidenko, Stability of higher dimensional Reissner-Nordstrom-Anti-de Sitter black holes, Phys. Rev. D 78 (2008) 104017 [arXiv:0809.2048] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    S. Chandrasekhar, The mathematical theory of black holes, Oxford University Press, Oxford U.K. (1998).zbMATHGoogle Scholar
  30. [30]
    A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Charged AdS black holes and catastrophic holography, Phys. Rev. D 60 (1999) 064018 [hep-th/9902170] [INSPIRE].ADSMathSciNetGoogle Scholar
  31. [31]
    J. Sonner and A.G. Green, Hawking radiation and non-equilibrium quantum critical current noise, Phys. Rev. Lett. 109 (2012) 091601 [arXiv:1203.4908] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    A. Kundu and S. Kundu, Steady-state physics, effective temperature dynamics in holography, arXiv:1307.6607 [INSPIRE].
  33. [33]
    S. Nakamura and H. Ooguri, Out of equilibrium temperature from holography, Phys. Rev. D 88 (2013) 126003 [arXiv:1309.4089] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Alexander Balatsky
    • 1
    • 2
  • Sven Bjarke Gudnason
    • 1
    Email author
  • Yaron Kedem
    • 1
  • Alexander Krikun
    • 1
    • 3
  • Lárus Thorlacius
    • 1
    • 4
    • 5
  • Konstantin Zarembo
    • 1
    • 3
    • 6
  1. 1.Nordita, KTH Royal Institute of Technology and Stockholm UniversityStockholmSweden
  2. 2.Institute for Materials ScienceLos Alamos National LaboratoryLos AlamosU.S.A.
  3. 3.Institute of Theoretical and Experimental PhysicsMoscowRussia
  4. 4.University of Iceland, Science InstituteReykjavikIceland
  5. 5.The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm UniversityAlbaNova University CentreStockholmSweden
  6. 6.Department of Physics and AstronomyUppsala UniversityUppsalaSweden

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