Journal of High Energy Physics

, 2019:181 | Cite as

Holographic Bjorken flow at large-D

  • Jorge Casalderrey-Solana
  • Christopher P. Herzog
  • Ben MeiringEmail author
Open Access
Regular Article - Theoretical Physics


We use gauge/gravity duality to study the dynamics of strongly coupled gauge theories undergoing boost invariant expansion in an arbitrary number of space-time dimensions (D). By keeping the scale of the late-time energy density fixed, we explore the infinite-D limit and study the first few corrections to this expansion. In agreement with other studies, we find that the large-D dynamics are controlled by hydrodynamics and we use our computation to constrain the leading large-D dependence of a certain combination of transport coefficients up to 6th order in gradients. Going beyond late time physics, we discuss how non-hydrodynamic modes appear in the large-D expansion in the form of a trans-series in D, identical to the non-perturbative contributions to the gradient expansion. We discuss the consequence of this trans-series in the non-convergence of the large-D expansion.


Holography and quark-gluon plasmas Classical Theories of Gravity 


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. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U.A. Wiedemann, Gauge/String Duality, Hot QCD and Heavy Ion Collisions, arXiv:1101.0618 [INSPIRE].
  2. [2]
    S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, arXiv:1612.07324 [INSPIRE].
  3. [3]
    P.M. Chesler and L.G. Yaffe, Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes, JHEP 07 (2014) 086 [arXiv:1309.1439] [INSPIRE].CrossRefGoogle Scholar
  4. [4]
    R. Emparan, R. Suzuki and K. Tanabe, The large D limit of General Relativity, JHEP 06 (2013) 009 [arXiv:1302.6382] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    R. Emparan, D. Grumiller and K. Tanabe, Large-D gravity and low-D strings, Phys. Rev. Lett. 110 (2013) 251102 [arXiv:1303.1995] [INSPIRE].CrossRefGoogle Scholar
  6. [6]
    R. Emparan and K. Tanabe, Holographic superconductivity in the large D expansion, JHEP 01 (2014) 145 [arXiv:1312.1108] [INSPIRE].CrossRefGoogle Scholar
  7. [7]
    A.M. García-García and A. Romero-Bermúdez, Conductivity and entanglement entropy of high dimensional holographic superconductors, JHEP 09 (2015) 033 [arXiv:1502.03616] [INSPIRE].
  8. [8]
    R. Emparan, R. Suzuki and K. Tanabe, Quasinormal modes of (Anti-)de Sitter black holes in the 1/D expansion, JHEP 04 (2015) 085 [arXiv:1502.02820] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    T. Andrade, S.A. Gentle and B. Withers, Drude in D major, JHEP 06 (2016) 134 [arXiv:1512.06263] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    C.P. Herzog, M. Spillane and A. Yarom, The holographic dual of a Riemann problem in a large number of dimensions, JHEP 08 (2016) 120 [arXiv:1605.01404] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M. Rozali, E. Sabag and A. Yarom, Holographic Turbulence in a Large Number of Dimensions, JHEP 04 (2018) 065 [arXiv:1707.08973] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    R. Emparan, K. Izumi, R. Luna, R. Suzuki and K. Tanabe, Hydro-elastic Complementarity in Black Branes at large D, JHEP 06 (2016) 117 [arXiv:1602.05752] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  13. [13]
    N. Iizuka, A. Ishibashi and K. Maeda, Cosmic Censorship at Large D: Stability analysis in polarized AdS black branes (holes), JHEP 03 (2018) 177 [arXiv:1801.07268] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    K. Tanabe, Black rings at large D, JHEP 02 (2016) 151 [arXiv:1510.02200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S. Bhattacharyya, M. Mandlik, S. Minwalla and S. Thakur, A Charged Membrane Paradigm at Large D, JHEP 04 (2016) 128 [arXiv:1511.03432] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  16. [16]
    K. Tanabe, Charged rotating black holes at large D, arXiv:1605.08854 [INSPIRE].
  17. [17]
    M. Rozali and A. Vincart-Emard, On Brane Instabilities in the Large D Limit, JHEP 08 (2016) 166 [arXiv:1607.01747] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Y. Dandekar, A. De, S. Mazumdar, S. Minwalla and A. Saha, The large D black hole Membrane Paradigm at first subleading order, JHEP 12 (2016) 113 [arXiv:1607.06475] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Y. Dandekar, S. Mazumdar, S. Minwalla and A. Saha, Unstable ‘black branes’ from scaled membranes at large D, JHEP 12 (2016) 140 [arXiv:1609.02912] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    S. Bhattacharyya et al., Currents and Radiation from the large D Black Hole Membrane, JHEP 05 (2017) 098 [arXiv:1611.09310] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  21. [21]
    B. Chen, P.-C. Li and Z.-z. Wang, Charged Black Rings at large D, JHEP 04 (2017) 167 [arXiv:1702.00886] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  22. [22]
    B. Chen and P.-C. Li, Static Gauss-Bonnet Black Holes at Large D, JHEP 05 (2017) 025 [arXiv:1703.06381] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  23. [23]
    S. Bhattacharyya, P. Biswas, B. Chakrabarty, Y. Dandekar and A. Dinda, The large D black hole dynamics in AdS/dS backgrounds, JHEP 10 (2018) 033 [arXiv:1704.06076] [INSPIRE].CrossRefzbMATHGoogle Scholar
  24. [24]
    U. Miyamoto, Non-linear perturbation of black branes at large D, JHEP 06 (2017) 033 [arXiv:1705.00486] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    B. Chen, P.-C. Li and C.-Y. Zhang, Einstein-Gauss-Bonnet Black Strings at Large D, JHEP 10 (2017) 123 [arXiv:1707.09766] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    C.P. Herzog and Y. Kim, The Large Dimension Limit of a Small Black Hole Instability in Anti-de Sitter Space, JHEP 02 (2018) 167 [arXiv:1711.04865] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Y. Dandekar, S. Kundu, S. Mazumdar, S. Minwalla, A. Mishra and A. Saha, An Action for and Hydrodynamics from the improved Large D membrane, JHEP 09 (2018) 137 [arXiv:1712.09400] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    R. Emparan, R. Luna, M. Martínez, R. Suzuki and K. Tanabe, Phases and Stability of Non-Uniform Black Strings, JHEP 05 (2018) 104 [arXiv:1802.08191] [INSPIRE].
  29. [29]
    T. Andrade, C. Pantelidou and B. Withers, Large D holography with metric deformations, JHEP 09 (2018) 138 [arXiv:1806.00306] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    T. Andrade, R. Emparan and D. Licht, Rotating black holes and black bars at large D, JHEP 09 (2018) 107 [arXiv:1807.01131] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    J.D. Bjorken, Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region, Phys. Rev. D 27 (1983) 140 [INSPIRE].Google Scholar
  32. [32]
    R.A. Janik and R.B. Peschanski, Asymptotic perfect fluid dynamics as a consequence of AdS/CFT, Phys. Rev. D 73 (2006) 045013 [hep-th/0512162] [INSPIRE].MathSciNetGoogle Scholar
  33. [33]
    M.P. Heller, R.A. Janik and P. Witaszczyk, Hydrodynamic Gradient Expansion in Gauge Theory Plasmas, Phys. Rev. Lett. 110 (2013) 211602 [arXiv:1302.0697] [INSPIRE].CrossRefGoogle Scholar
  34. [34]
    M.P. Heller and M. Spaliński, Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation, Phys. Rev. Lett. 115 (2015) 072501 [arXiv:1503.07514] [INSPIRE].CrossRefGoogle Scholar
  35. [35]
    W. Florkowski, M.P. Heller and M. Spaliński, New theories of relativistic hydrodynamics in the LHC era, Rept. Prog. Phys. 81 (2018) 046001 [arXiv:1707.02282] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  36. [36]
    P. Romatschke, Relativistic Fluid Dynamics Far From Local Equilibrium, Phys. Rev. Lett. 120 (2018) 012301 [arXiv:1704.08699] [INSPIRE].CrossRefGoogle Scholar
  37. [37]
    G.S. Denicol and J. Noronha, Analytical attractor and the divergence of the slow-roll expansion in relativistic hydrodynamics, Phys. Rev. D 97 (2018) 056021 [arXiv:1711.01657] [INSPIRE].MathSciNetGoogle Scholar
  38. [38]
    M. Strickland, J. Noronha and G. Denicol, Anisotropic nonequilibrium hydrodynamic attractor, Phys. Rev. D 97 (2018) 036020 [arXiv:1709.06644] [INSPIRE].Google Scholar
  39. [39]
    M. Spaliński, On the hydrodynamic attractor of Yang-Mills plasma, Phys. Lett. B 776 (2018) 468 [arXiv:1708.01921] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  40. [40]
    P. Romatschke, Relativistic Hydrodynamic Attractors with Broken Symmetries: Non-Conformal and Non-Homogeneous, JHEP 12 (2017) 079 [arXiv:1710.03234] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    M.P. Heller, A. Kurkela, M. Spaliński and V. Svensson, Hydrodynamization in kinetic theory: Transient modes and the gradient expansion, Phys. Rev. D 97 (2018) 091503 [arXiv:1609.04803] [INSPIRE].Google Scholar
  42. [42]
    J. Casalderrey-Solana, N.I. Gushterov and B. Meiring, Resurgence and Hydrodynamic Attractors in Gauss-Bonnet Holography, JHEP 04 (2018) 042 [arXiv:1712.02772] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    A. Behtash, S. Kamata, M. Martinez and C.N. Cruz-Camacho, Non-perturbative rheological behavior of a far-from-equilibrium expanding plasma, arXiv:1805.07881 [INSPIRE].
  44. [44]
    G.S. Denicol and J. Noronha, Hydrodynamic attractor and the fate of perturbative expansions in Gubser flow, arXiv:1804.04771 [INSPIRE].
  45. [45]
    M.P. Heller and V. Svensson, How does relativistic kinetic theory remember about initial conditions?, Phys. Rev. D 98 (2018) 054016 [arXiv:1802.08225] [INSPIRE].Google Scholar
  46. [46]
    A. Behtash, C.N. Cruz-Camacho and M. Martinez, Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow, Phys. Rev. D 97 (2018) 044041 [arXiv:1711.01745] [INSPIRE].MathSciNetGoogle Scholar
  47. [47]
    R. Baier, P. Romatschke, D.T. Son, A.O. Starinets and M.A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance and holography, JHEP 04 (2008) 100 [arXiv:0712.2451] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    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].CrossRefGoogle Scholar
  49. [49]
    S. Bhattacharyya, R. Loganayagam, I. Mandal, S. Minwalla and A. Sharma, Conformal Nonlinear Fluid Dynamics from Gravity in Arbitrary Dimensions, JHEP 12 (2008) 116 [arXiv:0809.4272] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    M.P. Heller, R.A. Janik and P. Witaszczyk, The characteristics of thermalization of boost-invariant plasma from holography, Phys. Rev. Lett. 108 (2012) 201602 [arXiv:1103.3452] [INSPIRE].CrossRefGoogle Scholar
  51. [51]
    M.P. Heller, R.A. Janik and P. Witaszczyk, A numerical relativity approach to the initial value problem in asymptotically Anti-de Sitter spacetime for plasma thermalization - an ADM formulation, Phys. Rev. D 85 (2012) 126002 [arXiv:1203.0755] [INSPIRE].Google Scholar
  52. [52]
    P.M. Chesler and L.G. Yaffe, Boost invariant flow, black hole formation and far-from-equilibrium dynamics in N = 4 supersymmetric Yang-Mills theory, Phys. Rev. D 82 (2010) 026006 [arXiv:0906.4426] [INSPIRE].Google Scholar
  53. [53]
    S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].CrossRefGoogle Scholar
  54. [54]
    S. Kinoshita, S. Mukohyama, S. Nakamura and K.-y. Oda, A Holographic Dual of Bjorken Flow, Prog. Theor. Phys. 121 (2009) 121 [arXiv:0807.3797] [INSPIRE].CrossRefzbMATHGoogle Scholar
  55. [55]
    S. Bhattacharyya, P. Biswas and M. Patra, A leading-order comparison between fluid-gravity and membrane-gravity dualities, arXiv:1807.05058 [INSPIRE].
  56. [56]
    R. Emparan and K. Tanabe, Universal quasinormal modes of large D black holes, Phys. Rev. D 89 (2014) 064028 [arXiv:1401.1957] [INSPIRE].Google Scholar
  57. [57]
    R. Emparan, R. Suzuki and K. Tanabe, Decoupling and non-decoupling dynamics of large D black holes, JHEP 07 (2014) 113 [arXiv:1406.1258] [INSPIRE].CrossRefzbMATHGoogle Scholar
  58. [58]
    G. Basar and G.V. Dunne, Hydrodynamics, resurgence and transasymptotics, Phys. Rev. D 92 (2015) 125011 [arXiv:1509.05046] [INSPIRE].Google Scholar
  59. [59]
    M. Spaliński, Universal behaviour, transients and attractors in supersymmetric Yang-Mills plasma, Phys. Lett. B 784 (2018) 21 [arXiv:1805.11689] [INSPIRE].CrossRefGoogle Scholar
  60. [60]
    I. Aniceto, J. Jankowski, B. Meiring and M. Spaliński, The large proper-time expansion of Yang-Mills plasma as a resurgent transseries, arXiv:1810.07130 [INSPIRE].
  61. [61]
    S. Grozdanov and N. Kaplis, Constructing higher-order hydrodynamics: The third order, Phys. Rev. D 93 (2016) 066012 [arXiv:1507.02461] [INSPIRE].MathSciNetGoogle Scholar
  62. [62]
    P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72 (2005) 086009 [hep-th/0506184] [INSPIRE].Google Scholar
  63. [63]
    J. Natario and R. Schiappa, On the classification of asymptotic quasinormal frequencies for d-dimensional black holes and quantum gravity, Adv. Theor. Math. Phys. 8 (2004) 1001 [hep-th/0411267] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    P. Betzios, U. Gürsoy, M. Järvinen and G. Policastro, Fluctuations in non-conformal holographic plasma at criticality, arXiv:1807.01718 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Departament de Física Quántica i Astrofísica and Institut de Ciències del Cosmos (ICC)Universitat de BarcelonaBarcelonaSpain
  2. 2.Rudolf Peierls Centre for Theoretical PhysicsUniversity of OxfordOxfordUnited Kingdom
  3. 3.Mathematics DepartmentKing’s College LondonLondonU.K.
  4. 4.C.N. Yang Institute for Theoretical Physics, Department of Physics and AstronomyStony Brook UniversityStony BrookU.S.A.

Personalised recommendations