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A membrane paradigm at large D

  • Sayantani Bhattacharyya
  • Anandita De
  • Shiraz Minwalla
  • Ravi Mohan
  • Arunabha Saha
Open Access
Regular Article - Theoretical Physics
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Abstract

We study SO(d + 1) invariant solutions of the classical vacuum Einstein equations in p + d + 3 dimensions. In the limit d → ∞ with p held fixed we construct a class of solutions labelled by the shape of a membrane (the event horizon), together with a ‘velocity’ field that lives on this membrane. We demonstrate that our metrics can be corrected to nonsingular solutions at first sub-leading order in d if and only if the membrane shape and ‘velocity’ field obey equations of motion which we determine. These equations define a well posed initial value problem for the membrane shape and this ‘velocity’ and so completely determine the dynamics of the black hole. They may be viewed as governing the non-linear dynamics of the light quasi normal modes of Emparan, Suzuki and Tanabe.

Keywords

Black Holes Classical Theories of Gravity 
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References

  1. [1]
    G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
  2. [2]
    E. Witten, Quarks, atoms, and the 1/N expansion, Phys. Today 33 (1980) 38.CrossRefGoogle Scholar
  3. [3]
    R. Emparan, R. Suzuki and K. Tanabe, The large D limit of General Relativity, JHEP 06 (2013) 009 [arXiv:1302.6382] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    R. Emparan, D. Grumiller and K. Tanabe, Large-D gravity and low-D strings, Phys. Rev. Lett. 110 (2013) 251102 [arXiv:1303.1995] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    R. Emparan and K. Tanabe, Holographic superconductivity in the large D expansion, JHEP 01 (2014) 145 [arXiv:1312.1108] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    R. Emparan and K. Tanabe, Universal quasinormal modes of large D black holes, Phys. Rev. D 89 (2014) 064028 [arXiv:1401.1957] [INSPIRE].ADSGoogle Scholar
  7. [7]
    R. Emparan, R. Suzuki and K. Tanabe, Instability of rotating black holes: large D analysis, JHEP 06 (2014) 106 [arXiv:1402.6215] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    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].ADSCrossRefGoogle Scholar
  9. [9]
    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].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    R. Emparan, Black Holes in 1/D expansion, talk given at Eurostrings 2015, Cambridge, U.K., 23-27 March 2015.Google Scholar
  11. [11]
    S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    M. Rangamani, Gravity and Hydrodynamics: Lectures on the fluid-gravity correspondence, Class. Quant. Grav. 26 (2009) 224003 [arXiv:0905.4352] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    V.E. Hubeny, S. Minwalla and M. Rangamani, The fluid/gravity correspondence, arXiv:1107.5780 [INSPIRE].
  14. [14]
    J. Armas, (Non)-Dissipative Hydrodynamics on Embedded Surfaces, JHEP 09 (2014) 047 [arXiv:1312.0597] [INSPIRE].
  15. [15]
    R. Emparan, T. Harmark, V. Niarchos and N.A. Obers, Blackfolds in Supergravity and String Theory, JHEP 08 (2011) 154 [arXiv:1106.4428] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    J. Camps and R. Emparan, Derivation of the blackfold effective theory, JHEP 03 (2012) 038 [Erratum ibid. 1206 (2012) 155] [arXiv:1201.3506] [INSPIRE].
  17. [17]
    J. Armas, How Fluids Bend: the Elastic Expansion for Higher-Dimensional Black Holes, JHEP 09 (2013) 073 [arXiv:1304.7773] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    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].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    T. Damour, Black Hole Eddy Currents, Phys. Rev. D 18 (1978) 3598 [INSPIRE].ADSGoogle Scholar
  20. [20]
    R.H. Price and K.S. Thorne, The membrane paradigm for black holes, Sci. Am. 258 (1988) 69.ADSCrossRefGoogle Scholar
  21. [21]
    I. Bredberg, C. Keeler, V. Lysov and A. Strominger, Wilsonian Approach to Fluid/Gravity Duality, JHEP 03 (2011) 141 [arXiv:1006.1902] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    S. Bhattacharyya et al., Local Fluid Dynamical Entropy from Gravity, JHEP 06 (2008) 055 [arXiv:0803.2526] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    R.C. Myers and M.J. Perry, Black Holes in Higher Dimensional Space-Times, Annals Phys. 172 (1986) 304 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    R.C. Myers, Myers-Perry black holes, arXiv:1111.1903 [INSPIRE].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Sayantani Bhattacharyya
    • 1
  • Anandita De
    • 2
  • Shiraz Minwalla
    • 3
  • Ravi Mohan
    • 3
  • Arunabha Saha
    • 3
  1. 1.Department of PhysicsIndian Institute of Technology KanpurKanpurIndia
  2. 2.Department of PhysicsIndian Institute of Science Education and ResearchPuneIndia
  3. 3.Department of Theoretical PhysicsTata Institute of Fundamental ResearchMumbaiIndia

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