Lumpy AdS5× S5 black holes and black belts

  • Óscar J. C. Dias
  • Jorge E. Santos
  • Benson Way
Open Access
Regular Article - Theoretical Physics


Sufficiently small Schwarzschild black holes in global AdS5×S5 are Gregory-Laflamme unstable. We construct new families of black hole solutions that bifurcate from the onset of this instability and break the full SO(6) symmetry group of the S5 down to SO(5). These new “lumpy” solutions are labelled by the harmonics . We find evidence that the = 1 branch never dominates the microcanonical/canonical ensembles and connects through a topology-changing merger to a localised black hole solution with S8 topology. We argue that these S8 black holes should become the dominant phase in the microcanonical ensemble for small enough energies, and that the transition to Schwarzschild black holes is first order. Furthermore, we find two branches of solutions with = 2. We expect one of these branches to connect to a solution containing two localised black holes, while the other branch connects to a black hole solution with horizon topology S4 × S4 which we call a “black belt”.


Gauge-gravity correspondence Black Holes in String Theory AdS-CFT Correspondence Black Holes 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    D. Marolf, M. Rangamani and T. Wiseman, Holographic thermal field theory on curved spacetimes, Class. Quant. Grav. 31 (2014) 063001 [arXiv:1312.0612] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    S. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-De Sitter Space, Commun.Math.Phys. 87 (1983) 577.ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
  6. [6]
    A.W. Peet and S.F. Ross, Microcanonical phases of string theory on AdS m × S n, JHEP 12 (1998) 020 [hep-th/9810200] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    V.E. Hubeny and M. Rangamani, Unstable horizons, JHEP 05 (2002) 027 [hep-th/0202189] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    R. Gregory and R. Laflamme, Black strings and p-branes are unstable, Phys. Rev. Lett. 70 (1993) 2837 [hep-th/9301052] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    R. Gregory, Black string instabilities in Anti-de Sitter space, Class. Quant. Grav. 17 (2000) L125 [hep-th/0004101] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    G.T. Horowitz and K. Maeda, Fate of the black string instability, Phys. Rev. Lett. 87 (2001) 131301 [hep-th/0105111] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    S.S. Gubser, On nonuniform black branes, Class. Quant. Grav. 19 (2002) 4825 [hep-th/0110193] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  12. [12]
    B. Kol, Topology change in general relativity and the black hole black string transition, JHEP 10 (2005) 049 [hep-th/0206220] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    T. Wiseman, Static axisymmetric vacuum solutions and nonuniform black strings, Class. Quant. Grav. 20 (2003) 1137 [hep-th/0209051] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    B. Kol and T. Wiseman, Evidence that highly nonuniform black strings have a conical waist, Class. Quant. Grav. 20 (2003) 3493 [hep-th/0304070] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  15. [15]
    T. Harmark and N.A. Obers, Black holes on cylinders, JHEP 05 (2002) 032 [hep-th/0204047] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    T. Harmark, Small black holes on cylinders, Phys. Rev. D 69 (2004) 104015 [hep-th/0310259] [INSPIRE].ADSMathSciNetGoogle Scholar
  17. [17]
    D. Gorbonos and B. Kol, A Dialogue of multipoles: Matched asymptotic expansion for caged black holes, JHEP 06 (2004) 053 [hep-th/0406002] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    H. Kudoh and T. Wiseman, Connecting black holes and black strings, Phys. Rev. Lett. 94 (2005) 161102 [hep-th/0409111] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    E. Sorkin, A Critical dimension in the black string phase transition, Phys. Rev. Lett. 93 (2004) 031601 [hep-th/0402216] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    V. Asnin, B. Kol and M. Smolkin, Analytic evidence for continuous self similarity of the critical merger solution, Class. Quant. Grav. 23 (2006) 6805 [hep-th/0607129] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    T. Harmark, V. Niarchos and N.A. Obers, Instabilities of black strings and branes, Class. Quant. Grav. 24 (2007) R1 [hep-th/0701022] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    G.T. Horowitz and T. Wiseman, General black holes in Kaluza-Klein theory, arXiv:1107.5563 [INSPIRE].
  23. [23]
    M. Headrick, S. Kitchen and T. Wiseman, A New approach to static numerical relativity and its application to Kaluza-Klein black holes, Class. Quant. Grav. 27 (2010) 035002 [arXiv:0905.1822] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    P. Figueras, K. Murata and H.S. Reall, Stable non-uniform black strings below the critical dimension, JHEP 11 (2012) 071 [arXiv:1209.1981] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    R. Emparan and R.C. Myers, Instability of ultra-spinning black holes, JHEP 09 (2003) 025 [hep-th/0308056] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    R. Emparan, T. Harmark, V. Niarchos, N.A. Obers and M.J. Rodriguez, The Phase Structure of Higher-Dimensional Black Rings and Black Holes, JHEP 10 (2007) 110 [arXiv:0708.2181] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    O.J.C. Dias, P. Figueras, R. Monteiro, J.E. Santos and R. Emparan, Instability and new phases of higher-dimensional rotating black holes, Phys. Rev. D 80 (2009) 111701 [arXiv:0907.2248] [INSPIRE].ADSGoogle Scholar
  28. [28]
    O.J.C. Dias, P. Figueras, R. Monteiro, H.S. Reall and J.E. Santos, An instability of higher-dimensional rotating black holes, JHEP 05 (2010) 076 [arXiv:1001.4527] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    O.J.C. Dias, P. Figueras, R. Monteiro and J.E. Santos, Ultraspinning instability of rotating black holes, Phys. Rev. D 82 (2010) 104025 [arXiv:1006.1904] [INSPIRE].ADSMATHGoogle Scholar
  30. [30]
    O.J.C. Dias, P. Figueras, R. Monteiro and J.E. Santos, Ultraspinning instability of anti-de Sitter black holes, JHEP 12 (2010) 067 [arXiv:1011.0996] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    O.J.C. Dias, R. Monteiro and J.E. Santos, Ultraspinning instability: the missing link, JHEP 08 (2011) 139 [arXiv:1106.4554] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  32. [32]
    R. Emparan and N. Haddad, Self-similar critical geometries at horizon intersections and mergers, JHEP 10 (2011) 064 [arXiv:1109.1983] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    O.J.C. Dias, J.E. Santos and B. Way, Rings, Ripples and Rotation: Connecting Black Holes to Black Rings, JHEP 07 (2014) 045 [arXiv:1402.6345] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    R. Emparan, P. Figueras and M. Martinez, Bumpy black holes, JHEP 12 (2014) 072 [arXiv:1410.4764] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    H.J. Kim, L.J. Romans and P. van Nieuwenhuizen, The Mass Spectrum of Chiral N = 2 D = 10 Supergravity on S 5, Phys. Rev. D 32 (1985) 389 [INSPIRE].ADSGoogle Scholar
  36. [36]
    M. Günaydin and N. Marcus, The Spectrum of the S 5 Compactification of the Chiral N = 2, D = 10 Supergravity and the Unitary Supermultiplets of U(2, 2/4),Class. Quant. Grav. 2 (1985) L11 [INSPIRE].CrossRefMATHGoogle Scholar
  37. [37]
    S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large-N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    S. Lee, AdS 5 /CFT 4 four point functions of chiral primary operators: Cubic vertices, Nucl. Phys. B 563 (1999) 349 [hep-th/9907108] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    G. Arutyunov and S. Frolov, Some cubic couplings in type IIB supergravity on AdS 5 × S 5 and three point functions in SYM(4) at large-N , Phys. Rev. D 61 (2000) 064009 [hep-th/9907085] [INSPIRE].ADSGoogle Scholar
  40. [40]
    K. Skenderis and M. Taylor, Kaluza-Klein holography, JHEP 05 (2006) 057 [hep-th/0603016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    K. Skenderis and M. Taylor, Holographic Coulomb branch vevs, JHEP 08 (2006) 001 [hep-th/0604169] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    K. Skenderis and M. Taylor, Anatomy of bubbling solutions, JHEP 09 (2007) 019 [arXiv:0706.0216] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    S. Bhattacharyya, S. Minwalla and K. Papadodimas, Small Hairy Black Holes in AdS 5 × S 5, JHEP 11 (2011) 035 [arXiv:1005.1287] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  44. [44]
    O.J.C. Dias et al., Hairy black holes and solitons in global AdS 5, JHEP 08 (2012) 117 [arXiv:1112.4447] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    S.A. Gentle, M. Rangamani and B. Withers, A Soliton Menagerie in AdS, JHEP 05 (2012) 106 [arXiv:1112.3979] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    O.J. Dias, J.E. Santos and B. Way, Localized AdS 5 × S 5 black holes, in preparation (2014).Google Scholar
  47. [47]
    O.J.C. Dias, T. Harmark, R.C. Myers and N.A. Obers, Multi-black hole configurations on the cylinder, Phys. Rev. D 76 (2007) 104025 [arXiv:0706.3645] [INSPIRE].ADSMathSciNetGoogle Scholar
  48. [48]
    R. Suzuki, S. Kinoshita and T. Shiromizu, Caged black hole with Maxwell charge, Phys. Rev. D 86 (2012) 044018 [arXiv:1205.4596] [INSPIRE].ADSGoogle Scholar
  49. [49]
    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].MathSciNetCrossRefMATHGoogle Scholar
  50. [50]
    S.S. Gubser and I.R. Klebanov, A Universal result on central charges in the presence of double trace deformations, Nucl. Phys. B 656 (2003) 23 [hep-th/0212138] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].ADSCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Óscar J. C. Dias
    • 1
  • Jorge E. Santos
    • 2
  • Benson Way
    • 2
  1. 1.STAG research centre and Mathematical SciencesUniversity of SouthamptonSouthamptonUnited Kingdom
  2. 2.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUnited Kingdom

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