Holographic Lovelock gravities and black holes

Open Access


We study holographic implications of Lovelock gravities in AdS spacetimes. For a generic Lovelock gravity in arbitrary spacetime dimensions we formulate the existence condition of asymptotically AdS black holes. We consider small fluctuations around these black holes and determine the constraint on Lovelock parameters by demanding causality of the boundary theory. For the case of cubic Lovelock gravity in seven spacetime dimensions we compute the holographic Weyl anomaly and determine the three point functions of the stress energy tensor in the boundary CFT. Remarkably, these correlators happen to satisfy the same relation as the one imposed by supersymmetry. We then compute the energy flux; requiring it to be positive is shown to be completely equivalent to requiring causality of the finite temperature CFT dual to the black hole. These constraints are not stringent enough to place any positive lower bound on the value of viscosity. Finally, we conjecture an expression for the energy flux valid for any Lovelock theory in arbitrary dimensions.


AdS-CFT Correspondence Field Theories in Higher Dimensions Black Holes Anomalies in Field and String Theories 


  1. [1]
    D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  2. [2]
    B. Zwiebach, Curvature squared terms and string theories, Phys. Lett. B 156 (1985) 315 [SPIRES].ADSGoogle Scholar
  3. [3]
    B. Zumino, Gravity theories in more than four-dimensions, Phys. Rept. 137 (1986) 109 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  4. [4]
    Q. Exirifard and M.M. Sheikh-Jabbari, Lovelock gravity at the crossroads of Palatini and metric formulations, Phys. Lett. B 661 (2008) 158 [arXiv:0705.1879] [SPIRES].MathSciNetADSGoogle Scholar
  5. [5]
    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].MATHMathSciNetADSGoogle Scholar
  6. [6]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [SPIRES].MATHMathSciNetGoogle Scholar
  7. [7]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [SPIRES].MathSciNetADSGoogle Scholar
  8. [8]
    S. Nojiri and S.D. Odintsov, On the conformal anomaly from higher derivative gravity in AdS/CFT correspondence, Int. J. Mod. Phys. A 15 (2000) 413 [hep-th/9903033] [SPIRES].MathSciNetADSGoogle Scholar
  9. [9]
    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
  10. [10]
    M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, The viscosity bound and causality violation, Phys. Rev. Lett. 100 (2008) 191601 [arXiv:0802.3318] [SPIRES].CrossRefADSGoogle Scholar
  11. [11]
    D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [SPIRES].CrossRefADSGoogle Scholar
  12. [12]
    A. Buchel and R.C. Myers, Causality of holographic hydrodynamics, JHEP 08 (2009) 016 [arXiv:0906.2922] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    D.M. Hofman, Higher derivative gravity, causality and positivity of energy in a UV complete QFT, Nucl. Phys. B 823 (2009) 174 [arXiv:0907.1625] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  14. [14]
    J. de Boer, M. Kulaxizi and A. Parnachev, AdS 7/CFT 6 , Gauss-Bonnet gravity and viscosity bound, JHEP 03 (2010) 087 [arXiv:0910.5347] [SPIRES].CrossRefGoogle Scholar
  15. [15]
    X.O. Camanho and J.D. Edelstein, Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity, JHEP 04 (2010) 007 [arXiv:0911.3160] [SPIRES].CrossRefGoogle Scholar
  16. [16]
    A. Buchel et al., Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [SPIRES].CrossRefGoogle Scholar
  17. [17]
    R.R. Metsaev and A.A. Tseytlin, Curvature cubed terms in string theory effective actions, Phys. Lett. B 185 (1987) 52 [SPIRES].MathSciNetADSGoogle Scholar
  18. [18]
    R.C. Myers and J.Z. Simon, Black hole thermodynamics in Lovelock gravity, Phys. Rev. D 38 (1988) 2434 [SPIRES].MathSciNetADSGoogle Scholar
  19. [19]
    J.T. Wheeler, Symmetric solutions to the Gauss-Bonnet extended einstein equations, Nucl. Phys. B 268 (1986) 737 [SPIRES].CrossRefADSGoogle Scholar
  20. [20]
    M.H. Dehghani and R. Pourhasan, Thermodynamic instability of black holes of third order Lovelock gravity, Phys. Rev. D 79 (2009) 064015 [arXiv:0903.4260] [SPIRES].MathSciNetADSGoogle Scholar
  21. [21]
    M.H. Dehghani, N. Alinejadi and S.H. Hendi, Topological black holes in Lovelock-Born-Infeld gravity, Phys. Rev. D 77 (2008) 104025 [arXiv:0802.2637] [SPIRES].MathSciNetADSGoogle Scholar
  22. [22]
    M.H. Dehghani, N. Bostani and S.H. Hendi, Magnetic branes in third order Lovelock-Born-Infeld gravity, Phys. Rev. D 78 (2008) 064031 [arXiv:0806.1429] [SPIRES].MathSciNetADSGoogle Scholar
  23. [23]
    T. Takahashi and J. Soda, Stability of Lovelock black holes under tensor perturbations, Phys. Rev. D 79 (2009) 104025 [arXiv:0902.2921] [SPIRES].MathSciNetADSGoogle Scholar
  24. [24]
    T. Takahashi and J. Soda, Instability of small Lovelock black holes in even-dimensions, Phys. Rev. D 80 (2009) 104021 [arXiv:0907.0556] [SPIRES].MathSciNetADSGoogle Scholar
  25. [25]
    R.-G. Cai, Gauss-Bonnet black holes in AdS spaces, Phys. Rev. D 65 (2002) 084014 [hep-th/0109133] [SPIRES].ADSGoogle Scholar
  26. [26]
    R.-G. Cai, L.-M. Cao, Y.-P. Hu and S.P. Kim, Generalized Vaidya spacetime in Lovelock gravity and thermodynamics on apparent horizon, Phys. Rev. D 78 (2008) 124012 [arXiv:0810.2610] [SPIRES].MathSciNetADSGoogle Scholar
  27. [27]
    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  28. [28]
    M. Henningson and K. Skenderis, Holography and the Weyl anomaly, Fortsch. Phys. 48 (2000) 125 [hep-th/9812032] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  29. [29]
    F. Bastianelli, S. Frolov and A.A. Tseytlin, Conformal anomaly of (2, 0) tensor multiplet in six dimensions and AdS/CFT correspondence, JHEP 02 (2000) 013 [hep-th/0001041] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  30. [30]
    E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [SPIRES].MATHMathSciNetGoogle Scholar
  31. [31]
    S.W. Hawking and D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  32. [32]
    O. Aharony, A. Fayyazuddin and J.M. Maldacena, The large-N limit of N = 2, 1 field theories from three-branes in F-theory, JHEP 07 (1998) 013 [hep-th/9806159] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  33. [33]
    A. Fayyazuddin and M. Spalinski, Large-N superconformal gauge theories and supergravity orientifolds, Nucl. Phys. B 535 (1998) 219 [hep-th/9805096] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  34. [34]
    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].CrossRefMathSciNetADSGoogle Scholar
  35. [35]
    D.G. Boulware and S. Deser, String generated gravity models, Phys. Rev. Lett. 55 (1985) 2656 [SPIRES].CrossRefADSGoogle Scholar
  36. [36]
    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] [SPIRES].CrossRefADSGoogle Scholar
  37. [37]
    Holography and hydrodynamics: diffusion on stretched horizons, JHEP10 (2003) 064 [hep-th/0309213] [SPIRES].
  38. [38]
    R. Brustein and A.J.M. Medved, The ratio of shear viscosity to entropy density in generalized theories of gravity, Phys. Rev. D 79 (2009) 021901 [arXiv:0808.3498] [SPIRES].ADSGoogle Scholar
  39. [39]
    F.-W. Shu, The quantum viscosity bound in Lovelock gravity, Phys. Lett. B 685 (2010) 325 [arXiv:0910.0607] [SPIRES].ADSGoogle Scholar
  40. [40]
    X.O. Camanho and J.D. Edelstein, Causality in AdS/CFT and Lovelock theory, to appear.Google Scholar
  41. [41]
    D.T. Son and A.O. Starinets, Minkowski-space correlators in AdS/CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  42. [42]
    C.P. Herzog and D.T. Son, Schwinger-Keldysh propagators from AdS/CFT correspondence, JHEP 03 (2003) 046 [hep-th/0212072] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  43. [43]
    D. Marolf, States and boundary terms: subtleties of lorentzian AdS/CFT, JHEP 05 (2005) 042 [hep-th/0412032] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  44. [44]
    K. Skenderis and B.C. van Rees, Real-time gauge/gravity duality, Phys. Rev. Lett. 101 (2008) 081601 [arXiv:0805.0150] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  45. [45]
    G. Kofinas and R. Olea, Vacuum energy in Einstein-Gauss-Bonnet AdS gravity, Phys. Rev. D 74 (2006) 084035 [hep-th/0606253] [SPIRES].MathSciNetADSGoogle Scholar
  46. [46]
    O. Mišković and R. Olea, Counterterms in dimensionally continued AdS gravity, JHEP 10 (2007) 028 [arXiv:0706.4460] [SPIRES].ADSGoogle Scholar
  47. [47]
    G. Kofinas and R. Olea, Universal regularization prescription for Lovelock AdS gravity, JHEP 11 (2007) 069 [arXiv:0708.0782] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  48. [48]
    G. Kofinas and R. Olea, Universal kounterterms in Lovelock AdS gravity, Fortsch. Phys. 56 (2008) 957 [arXiv:0806.1197] [SPIRES].MATHCrossRefMathSciNetGoogle Scholar

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© The Author(s) 2010

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Jan de Boer
    • 1
  • Manuela Kulaxizi
    • 1
  • Andrei Parnachev
    • 2
  1. 1.Department of PhysicsUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.C.N.Yang Institute for Theoretical PhysicsStony Brook UniversityStony BrookU.S.A.

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