Journal of High Energy Physics

, 2019:250 | Cite as

Phases of holographic Hawking radiation on spatially compact spacetimes

  • Donald MarolfEmail author
  • Jorge E. Santos
Open Access
Regular Article - Theoretical Physics


We study phases of equilibrium Hawking radiation in d-dimensional holo- graphic CFTs on spatially compact spacetimes with two black holes. In the particular phases chosen the dual (d + 1)-dimensional bulk solutions describe a variety of black fun- nels and droplets. In the former the CFT readily conducts heat between the two black holes, but it in the latter such conduction is highly suppressed. While the generic case can be understood in certain extreme limits of parameters on general grounds, we focus on CFTs on specific geometries conformally equivalent to a pair of d ≥ 4 AdSd-Schwarzschild black holes of radius R. Such cases allow perturbative analyses of non-uniform funnels associated with Gregory-Laflamme zero-modes. For d = 4 we construct a phase diagram for pure funnels and droplets by constructing the desired bulk solutions numerically. The fat non-uniform funnel is a particular interesting phase that dominates at small R (due to having lowest free energy) despite being sub-dominant in the perturbative regime. The uniform funnel dominates at large R, and droplets and thin funnels dominate at certain intermediate values. The thin funnel phase provides a mystery as it dominates over our other phases all that way to a critical Rturn beyond which it fails to exist. The free energy of the system thus appears to be discontinuous at Rturn, but such discontinuities are for- bidden by the 2nd law. A new more-dominant phase is thus required near Rturn but the nature of this phase remains unclear.


AdS-CFT Correspondence Black Holes 


Open Access

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  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].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    D. Astefanesei and R.C. Myers, Boundary black holes and ads/cft correspondence, talk presented by R.C. Myers at Black Holes IV: Theory and Mathematical Aspects, Honey Harbor, Ontario, 25–28 May 2003.Google Scholar
  3. [3]
    T. Wiseman, Relativistic stars in Randall-Sundrum gravity, Phys. Rev. D 65 (2002) 124007 [hep-th/0111057] [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    T. Wiseman, Static axisymmetric vacuum solutions and nonuniform black strings, Class. Quant. Grav. 20 (2003) 1137 [hep-th/0209051] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  5. [5]
    R. Casadio and L. Mazzacurati, Bulk shape of brane world black holes, Mod. Phys. Lett. A 18 (2003) 651 [gr-qc/0205129] [INSPIRE].
  6. [6]
    D. Karasik, C. Sahabandu, P. Suranyi and L.C.R. Wijewardhana, Small black holes in Randall-Sundrum I scenario, Phys. Rev. D 69 (2004) 064022 [gr-qc/0309076] [INSPIRE].
  7. [7]
    H. Kudoh, T. Tanaka and T. Nakamura, Small localized black holes in brane world: Formulation and numerical method, Phys. Rev. D 68 (2003) 024035 [gr-qc/0301089] [INSPIRE].
  8. [8]
    H. Kudoh, Thermodynamical properties of small localized black hole, Prog. Theor. Phys. 110 (2004) 1059 [hep-th/0306067] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  9. [9]
    H. Kudoh, Six-dimensional localized black holes: Numerical solutions, Phys. Rev. D 69 (2004) 104019 [Erratum ibid. D 70 (2004) 029901] [hep-th/0401229] [INSPIRE].
  10. [10]
    D. Karasik, C. Sahabandu, P. Suranyi and L.C.R. Wijewardhana, Small black holes on branes: Is the horizon regular or singular?, Phys. Rev. D 70 (2004) 064007 [gr-qc/0404015] [INSPIRE].
  11. [11]
    H. Yoshino, On the existence of a static black hole on a brane, JHEP 01 (2009) 068 [arXiv:0812.0465] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    A.L. Fitzpatrick, L. Randall and T. Wiseman, On the existence and dynamics of braneworld black holes, JHEP 11 (2006) 033 [hep-th/0608208] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    V.E. Hubeny, D. Marolf and M. Rangamani, Hawking radiation in large N strongly-coupled field theories, Class. Quant. Grav. 27 (2010) 095015 [arXiv:0908.2270] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    D. Marolf, M. Rangamani and T. Wiseman, Holographic thermal field theory on curved spacetimes, Class. Quant. Grav. 31 (2014) 063001 [arXiv:1312.0612] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    K.S. Thorne, R. Price and D. Macdonald, Black Holes: The Membrane Paradigm, Yale University Press, New Haven (1986).zbMATHGoogle Scholar
  16. [16]
    S. Fischetti and D. Marolf, Flowing Funnels: Heat sources for field theories and the AdS3 dual of C F T2 Hawking radiation, Class. Quant. Grav. 29 (2012) 105004 [arXiv:1202.5069] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  17. [17]
    P. Figueras and T. Wiseman, Stationary holographic plasma quenches and numerical methods for non-Killing horizons, Phys. Rev. Lett. 110 (2013) 171602 [arXiv:1212.4498] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    S. Fischetti, D. Marolf and J.E. Santos, AdS flowing black funnels: Stationary AdS black holes with non-Killing horizons and heat transport in the dual CFT, Class. Quant. Grav. 30 (2013) 075001 [arXiv:1212.4820] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    F.M. Haehl, The Schwarzschild-Black String AdS Soliton: Instability and Holographic Heat Transport, Class. Quant. Grav. 30 (2013) 055002 [arXiv:1210.5763] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    R. Emparan and M. Martinez, Black String Flow, JHEP 09 (2013) 068 [arXiv:1307.2276] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    S. Fischetti and J.E. Santos, Rotating Black Droplet, JHEP 07 (2013) 156 [arXiv:1304.1156] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].ADSGoogle Scholar
  23. [23]
    D. Marolf, M. Rangamani and M. Van Raamsdonk, Holographic models of de Sitter QFTs, Class. Quant. Grav. 28 (2011) 105015 [arXiv:1007.3996] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    L.-Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic Calculations of Renyi Entropy, JHEP 12 (2011) 047 [arXiv:1110.1084] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  25. [25]
    V.E. Hubeny, D. Marolf and M. Rangamani, Hawking radiation from AdS black holes, Class. Quant. Grav. 27 (2010) 095018 [arXiv:0911.4144] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    R. Gregory, S.F. Ross and R. Zegers, Classical and quantum gravity of brane black holes, JHEP 09 (2008) 029 [arXiv:0802.2037] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    V.E. Hubeny, D. Marolf and M. Rangamani, Black funnels and droplets from the AdS C-metrics, Class. Quant. Grav. 27 (2010) 025001 [arXiv:0909.0005] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    M.M. Caldarelli, O.J.C. Dias, R. Monteiro and J.E. Santos, Black funnels and droplets in thermal equilibrium, JHEP 05 (2011) 116 [arXiv:1102.4337] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  29. [29]
    P. Figueras, J. Lucietti and T. Wiseman, Ricci solitons, Ricci flow and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua, Class. Quant. Grav. 28 (2011) 215018 [arXiv:1104.4489] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    P. Figueras and T. Wiseman, Gravity and large black holes in Randall-Sundrum II braneworlds, Phys. Rev. Lett. 107 (2011) 081101 [arXiv:1105.2558] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    J.E. Santos and B. Way, Black Funnels, JHEP 12 (2012) 060 [arXiv:1208.6291] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    P. Figueras and S. Tunyasuvunakool, CFTs in rotating black hole backgrounds, Class. Quant. Grav. 30 (2013) 125015 [arXiv:1304.1162] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    J.E. Santos and B. Way, Black Droplets, JHEP 08 (2014) 072 [arXiv:1405.2078] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    R. Emparan, T. Shiromizu, R. Suzuki, K. Tanabe and T. Tanaka, Effective theory of Black Holes in the 1/D expansion, JHEP 06 (2015) 159 [arXiv:1504.06489] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    E. Mefford, Entanglement Entropy in Jammed CFTs, JHEP 09 (2017) 006 [arXiv:1605.09369] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  36. [36]
    S. Fischetti, J.E. Santos and B. Way, Dissonant Black Droplets and Black Funnels, Class. Quant. Grav. 34 (2017) 155001 [arXiv:1611.09363] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    W. Bunting, Z. Fu and D. Marolf, A coarse-grained generalized second law for holographic conformal field theories, Class. Quant. Grav. 33 (2016) 055008 [arXiv:1509.00074] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    B. Kol, Topology change in general relativity and the black hole black string transition, JHEP 10 (2005) 049 [hep-th/0206220] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    E. Sorkin, B. Kol and T. Piran, Caged black holes: Black holes in compactified space-times. 2. 5-D numerical implementation, Phys. Rev. D 69 (2004) 064032 [hep-th/0310096] [INSPIRE].
  40. [40]
    H. Kudoh and T. Wiseman, Properties of Kaluza-Klein black holes, Prog. Theor. Phys. 111 (2004) 475 [hep-th/0310104] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    H. Kudoh and T. Wiseman, Connecting black holes and black strings, Phys. Rev. Lett. 94 (2005) 161102 [hep-th/0409111] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    B. Kol, Choptuik scaling and the merger transition, JHEP 10 (2006) 017 [hep-th/0502033] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    M. Kalisch and M. Ansorg, Highly Deformed Non-uniform Black Strings in Six Dimensions, in Proceedings, 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories (MG14) (in 4 Volumes), Rome, Italy, 12–18 July 2015, vol. 2, pp. 1799–1804 (2017) [DOI:10.1142/9789813226609 0185] [arXiv:1509.03083] [INSPIRE].
  45. [45]
    M. Kalisch and M. Ansorg, Pseudo-spectral construction of non-uniform black string solutions in five and six spacetime dimensions, Class. Quant. Grav. 33 (2016) 215005 [arXiv:1607.03099] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    M. Kalisch, S. Möckel and M. Ammon, Critical behavior of the black hole/black string transition, JHEP 08 (2017) 049 [arXiv:1706.02323] [INSPIRE].
  47. [47]
    S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    S.S. Gubser, On nonuniform black branes, Class. Quant. Grav. 19 (2002) 4825 [hep-th/0110193] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  49. [49]
    E. Sorkin, A Critical dimension in the black string phase transition, Phys. Rev. Lett. 93 (2004) 031601 [hep-th/0402216] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  51. [51]
    M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [Erratum ibid. D 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].
  52. [52]
    R. Emparan, G.T. Horowitz and R.C. Myers, Exact description of black holes on branes. 2. Comparison with BTZ black holes and black strings, JHEP 01 (2000) 021 [hep-th/9912135] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    T. Hirayama and G. Kang, Stable black strings in anti-de Sitter space, Phys. Rev. D 64 (2001) 064010 [hep-th/0104213] [INSPIRE].ADSMathSciNetGoogle Scholar
  54. [54]
    A. Chamblin and A. Karch, Hawking and Page on the brane, Phys. Rev. D 72 (2005) 066011 [hep-th/0412017] [INSPIRE].ADSMathSciNetGoogle Scholar
  55. [55]
    R. Gregory and R. Laflamme, Black strings and p-branes are unstable, Phys. Rev. Lett. 70 (1993) 2837 [hep-th/9301052] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    T. Prestidge, Dynamic and thermodynamic stability and negative modes in Schwarzschild-anti-de Sitter, Phys. Rev. D 61 (2000) 084002 [hep-th/9907163] [INSPIRE].ADSMathSciNetGoogle Scholar
  57. [57]
    S.S. Gubser and I. Mitra, Instability of charged black holes in Anti-de Sitter space, Clay Math. Proc. 1 (2002) 221 [hep-th/0009126] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  58. [58]
    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].ADSMathSciNetCrossRefGoogle Scholar
  59. [59]
    Ó .J.C. Dias, J.E. Santos and B. Way, Numerical Methods for Finding Stationary Gravitational Solutions, Class. Quant. Grav. 33 (2016) 133001 [arXiv:1510.02804] [INSPIRE].
  60. [60]
    V. Asnin, D. Gorbonos, S. Hadar, B. Kol, M. Levi and U. Miyamoto, High and Low Dimensions in The Black Hole Negative Mode, Class. Quant. Grav. 24 (2007) 5527 [arXiv:0706.1555] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  61. [61]
    Ó .J.C. Dias, G.T. Horowitz and J.E. Santos, Gravitational Turbulent Instability of Anti-de Sitter Space, Class. Quant. Grav. 29 (2012) 194002 [arXiv:1109.1825] [INSPIRE].
  62. [62]
    H. Kodama and A. Ishibashi, A Master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions, Prog. Theor. Phys. 110 (2003) 701 [hep-th/0305147] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  63. [63]
    A. Adam, S. Kitchen and T. Wiseman, A numerical approach to finding general stationary vacuum black holes, Class. Quant. Grav. 29 (2012) 165002 [arXiv:1105.6347] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  64. [64]
    T. Wiseman, Numerical construction of static and stationary black holes, in Black holes in higher dimensions, G.T. Horowitz ed., pp. 233–270 (2012), arXiv:1107.5513 [INSPIRE].
  65. [65]
    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].ADSzbMATHCrossRefGoogle Scholar
  66. [66]
    M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  67. [67]
    V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  68. [68]
    D. Marolf, W. Kelly and S. Fischetti, Conserved Charges in Asymptotically (Locally) AdS Spacetimes, in Springer Handbook of Spacetime, A. Ashtekar and V. Petkov eds., pp. 381–407 (2014) [DOI:10.1007/978-3-642-41992-8 19] [arXiv:1211.6347] [INSPIRE].
  69. [69]
    T. Wiseman, From black strings to black holes, Class. Quant. Grav. 20 (2003) 1177 [hep-th/0211028] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  70. [70]
    Ó .J.C. Dias, J.E. Santos and B. Way, Localised and nonuniform thermal states of super-Yang-Mills on a circle, JHEP 06 (2017) 029 [arXiv:1702.07718] [INSPIRE].
  71. [71]
    M. Kalisch, Numerical construction and critical behavior of Kaluza-Klein black holes, Ph.D. Thesis, Jena U. (2018) [arXiv:1802.06596] [INSPIRE].
  72. [72]
    B. Cardona and P. Figueras, Critical Kaluza-Klein black holes and black strings in D = 10, JHEP 11 (2018) 120 [arXiv:1806.11129] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  73. [73]
    M. Ammon, M. Kalisch and S. Moeckel, Notes on ten-dimensional localized black holes and deconfined states in two-dimensional SYM, JHEP 11 (2018) 090 [arXiv:1806.11174] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  74. [74]
    R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
  75. [75]
    V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
  76. [76]
    V. lyer and R.M. Wald, A Comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52 (1995) 4430 [gr-q c/9503052] [INSPIRE].
  77. [77]
    R.M. Wald and A. Zoupas, A General definition of 'conserved quantities' in general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc /9911095] [INSPIRE].

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of California at Santa BarbaraSanta BarbaraU.S.A.
  2. 2.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeU.K.

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