A scalar field instability of rotating and charged black holes in (4+1)-dimensional Anti-de Sitter space-time



We study the stability of static as well as of rotating and charged black holes in (4+1)-dimensional Anti-de Sitter space-time which possess spherical horizon topology. We observe an instability related to the condensation of a charged, tachyonic scalar field and construct “hairy” black hole solutions of the full non-linear system of coupled Einstein, Maxwell and scalar field equations. We observe that the limiting solution for small horizon radius is either a hairy soliton solution or a singular solution that is not a regular extremal solution. Within the context of the gauge/gravity duality the condensation of the scalar field describes a holographic conductor/superconductor phase transition on the surface of a sphere.


Black Holes Classical Theories of Gravity 


  1. [1]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS/CFT correspondence, hep-th/0201253 [INSPIRE].
  3. [3]
    M.K. Benna and I.R. Klebanov, Gauge-string dualities and some applications, arXiv:0803.1315 [INSPIRE].
  4. [4]
    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) 1133] [hep-th/9711200] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  5. [5]
    S.S. Gubser, Breaking an abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].ADSGoogle Scholar
  6. [6]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a holographic superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    G.T. Horowitz and M.M. Roberts, Holographic superconductors with various condensates, Phys. Rev. D 78 (2008) 126008 [arXiv:0810.1077] [INSPIRE].ADSGoogle Scholar
  9. [9]
    C.P. Herzog, Lectures on holographic superfluidity and superconductivity, J. Phys. A 42 (2009) 343001 [arXiv:0904.1975] [INSPIRE].Google Scholar
  10. [10]
    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Classical and Quantum Gravity 26 (2009), no. 22 224002 [arXiv:0903.3246].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    G.T. Horowitz, Introduction to holographic superconductors, arXiv:1002.1722.
  12. [12]
    P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Annals Phys. 144 (1982) 249.MathSciNetADSMATHCrossRefGoogle Scholar
  13. [13]
    C. Herzog, P. Kovtun and D. Son, Holographic model of superfluidity, Phys. Rev. D 79 (2009) 066002 [arXiv:0809.4870] [INSPIRE].MathSciNetADSGoogle Scholar
  14. [14]
    P. Basu, A. Mukherjee and H.-H. Shieh, Supercurrent: vector hair for an AdS black hole, Phys. Rev. D 79 (2009) 045010 [arXiv:0809.4494] [INSPIRE].ADSGoogle Scholar
  15. [15]
    D. Arean, P. Basu and C. Krishnan, The many phases of holographic superfluids, JHEP 10 (2010) 006 [arXiv:1006.5165] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    Y. Brihaye and B. Hartmann, Holographic superfluids as duals of rotating black strings, JHEP 09 (2010) 002 [arXiv:1006.1562] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    Y. Brihaye and B. Hartmann, Holographic superfluid/fluid/insulator phase transitions in 2 + 1 dimensions, Phys. Rev. D 83(2011) 126008 [arXiv:1101.5708] [INSPIRE].ADSGoogle Scholar
  18. [18]
    J. Sonner and B. Withers, A gravity derivation of the Tisza-Landau model in AdS/CFT, Phys. Rev. D 82 (2010) 026001 [arXiv:1004.2707] [INSPIRE].ADSGoogle Scholar
  19. [19]
    J. Sonner, A rotating holographic superconductor, Phys. Rev. D 80 (2009) 084031 [arXiv:0903.0627] [INSPIRE].ADSGoogle Scholar
  20. [20]
    B. Carter, Hamilton-Jacobi and Schrödinger separable solutions of Einsteins equations, Commun. Math. Phys. 10 (1968) 280 [INSPIRE].MATHGoogle Scholar
  21. [21]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic superconductor/insulator transition at zero temperature, JHEP 03 (2010) 131 [arXiv:0911.0962] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    G.T. Horowitz and B. Way, Complete phase diagrams for a holographic superconductor/insulator system, JHEP 11 (2010) 011 [arXiv:1007.3714] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    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].MathSciNetMATHGoogle Scholar
  24. [24]
    G.T. Horowitz and R.C. Myers, The AdS/CFT correspondence and a new positive energy conjecture for general relativity, Phys. Rev. D 59 (1998) 026005 [hep-th/9808079] [INSPIRE].MathSciNetADSGoogle Scholar
  25. [25]
    S. Hawking and D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    S. Surya, K. Schleich and D.M. Witt, Phase transitions for flat AdS black holes, Phys. Rev. Lett. 86 (2001) 5231 [hep-th/0101134] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    I. Robinson, A solution of the Maxwell-Einstein equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 7 (1959) 351.MATHGoogle Scholar
  28. [28]
    B. Bertotti, Uniform electromagnetic field in the theory of general relativity, Phys. Rev. 116 (1959) 1331 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  29. [29]
    J.M. Bardeen and G.T. Horowitz, The extreme Kerr throat geometry: a vacuum analog of AdS 2 × S 2, Phys. Rev. D 60 (1999) 104030 [hep-th/9905099] [INSPIRE].MathSciNetADSGoogle Scholar
  30. [30]
    A. Sen, Black hole entropy function and the attractor mechanism in higher derivative gravity, JHEP 09 (2005) 038 [hep-th/0506177] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    A. Sen, Entropy function and AdS 2 /CFT 1 correspondence, JHEP 11 (2008) 075 [arXiv:0805.0095] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    O.J. Dias and P.J. Silva, Euclidean analysis of the entropy functional formalism, Phys. Rev. D 77 (2008) 084011 [arXiv:0704.1405] [INSPIRE].ADSGoogle Scholar
  33. [33]
    O.J. Dias, R. Monteiro, H.S. Reall and J.E. Santos, A scalar field condensation instability of rotating Anti-de Sitter black holes, JHEP 11 (2010) 036 [arXiv:1007.3745] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    Y. Brihaye and B. Hartmann, Stability of Gauss-Bonnet black holes in Anti-de-Sitter space-time against scalar field condensation, Phys. Rev. D 84 (2011) 084008 [arXiv:1107.3384] [INSPIRE].ADSGoogle Scholar
  35. [35]
    O.J. Dias, P. Figueras, S. Minwalla, P. Mitra, R. Monteiro, et al., Hairy black holes and solitons in global AdS 5, arXiv:1112.4447 [INSPIRE].
  36. [36]
    P. Basu, J. Bhattacharya, S. Bhattacharyya, R. Loganayagam, S. Minwalla, et al., Small hairy black holes in global AdS spacetime, JHEP 10 (2010) 045 [arXiv:1003.3232] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    J. Fernandez-Gracia and B. Fiol, A no-hair theorem for extremal black branes, JHEP 11 (2009) 054 [arXiv:0906.2353] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    S. Hawking, C. Hunter and M. Taylor, Rotation and the AdS/CFT correspondence, Phys. Rev. D 59 (1999) 064005 [hep-th/9811056] [INSPIRE].MathSciNetADSGoogle Scholar
  39. [39]
    G. Gibbons, H. Lü, D.N. Page and C. Pope, Rotating black holes in higher dimensions with a cosmological constant, Phys. Rev. Lett. 93 (2004) 171102 [hep-th/0409155] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    R.C. Myers and M. Perry, Black holes in higher dimensional space-times, Annals Phys. 172 (1986) 304 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  41. [41]
    Z. Chong, M. Cvetič, H. Lü and C. Pope, Non-extremal rotating black holes in five-dimensional gauged supergravity, Phys. Lett. B 644 (2007) 192 [hep-th/0606213] [INSPIRE].ADSGoogle Scholar
  42. [42]
    Z.-W. Chong, M. Cvetič, H. Lü and C. Pope, General non-extremal rotating black holes in minimal five-dimensional gauged supergravity, Phys. Rev. Lett. 95 (2005) 161301 [hep-th/0506029] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    Z. Chong, M. Cvetič, H. Lü and C. Pope, Five-dimensional gauged supergravity black holes with independent rotation parameters, Phys. Rev. D 72 (2005) 041901 [hep-th/0505112] [INSPIRE].ADSGoogle Scholar
  44. [44]
    Z.-W. Chong, M. Cvetič, H. Lü and C. Pope, Non-extremal charged rotating black holes in seven-dimensional gauged supergravity, Phys. Lett. B 626 (2005) 215 [hep-th/0412094] [INSPIRE].ADSGoogle Scholar
  45. [45]
    M. Cvetič, H. Lü and C. Pope, Charged rotating black holes in five dimensional U(1)3 gauged N =2 supergravity, Phys. Rev. D 70(2004) 081502 [hep-th/0407058][INSPIRE].ADSGoogle Scholar
  46. [46]
    M. Cvetič, H. Lü and C. Pope, Charged Kerr-de Sitter black holes in five dimensions, Phys. Lett. B 598 (2004) 273 [hep-th/0406196] [INSPIRE].ADSGoogle Scholar
  47. [47]
    J. Kunz, F. Navarro-Lerida and A.K. Petersen, Five-dimensional charged rotating black holes, Phys. Lett. B 614 (2005) 104 [gr-qc/0503010] [INSPIRE].MathSciNetADSGoogle Scholar
  48. [48]
    J. Kunz, F. Navarro-Lerida and E. Radu, Higher dimensional rotating black holes in Einstein-Maxwell theory with negative cosmological constant, Phys. Lett. B 649 (2007) 463 [gr-qc/0702086] [INSPIRE].MathSciNetADSGoogle Scholar
  49. [49]
    Y. Brihaye and E. Radu, Five-dimensional rotating black holes in Einstein-Gauss-Bonnet theory, Phys. Lett. B 661 (2008) 167 [arXiv:0801.1021] [INSPIRE].MathSciNetADSGoogle Scholar
  50. [50]
    Y. Brihaye, Charged, rotating black holes in Einstein-Gauss-Bonnet gravity, arXiv:1108.2779 [INSPIRE].
  51. [51]
    K. Behrndt, M. Cvetič and W. Sabra, Nonextreme black holes of five-dimensional N = 2 AdS supergravity, Nucl. Phys. B 553 (1999) 317 [hep-th/9810227] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    M. Cvetič and S.S. Gubser, Phases of R charged black holes, spinning branes and strongly coupled gauge theories, JHEP 04 (1999) 024 [hep-th/9902195] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    S.S. Gubser and I. Mitra, The evolution of unstable black holes in Anti-de Sitter space, JHEP 08 (2001) 018 [hep-th/0011127] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    S.S. Gubser and I. Mitra, Instability of charged black holes in Anti-de Sitter space, hep-th/0009126 [INSPIRE].
  55. [55]
    U. Ascher, J. Christiansen and R.D. Russell, A collocation solver for mixed order systems of boundary value problems, Math. Comput. 33 (1979) 659.MathSciNetMATHCrossRefGoogle Scholar
  56. [56]
    U. Ascher, J. Christiansen and R.D. Russell, Collocation Software for Boundary-Value ODEs, ACM Trans. Math. Softw. 7 (1981) 209.MATHCrossRefGoogle Scholar
  57. [57]
    R. Monteiro and J.E. Santos, Negative modes and the thermodynamics of Reissner-Nordstrom black holes, Phys. Rev. D 79 (2009) 064006 [arXiv:0812.1767] [INSPIRE].MathSciNetADSGoogle Scholar
  58. [58]
    G.T. Horowitz and M.M. Roberts, Zero temperature limit of holographic superconductors, JHEP 11 (2009) 015 [arXiv:0908.3677] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    R. Gregory, S. Kanno and J. Soda, Holographic superconductors with higher curvature corrections, JHEP 10 (2009) 010 [arXiv:0907.3203] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  60. [60]
    Y. Brihaye and B. Hartmann, Holographic superconductors in 3 + 1 dimensions away from the probe limit, Phys. Rev. D 81 (2010) 126008 [arXiv:1003.5130] [INSPIRE].ADSGoogle Scholar
  61. [61]
    L. Barclay, R. Gregory, S. Kanno and P. Sutcliffe, Gauss-Bonnet holographic superconductors, JHEP 12 (2010) 029 [arXiv:1009.1991] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Physique-MathématiqueUniversite de Mons-HainautMonsBelgium
  2. 2.School of Engineering and ScienceJacobs University BremenBremenGermany

Personalised recommendations