Journal of High Energy Physics

, 2013:220 | Cite as

The entropy of a hole in spacetime

  • Vijay Balasubramanian
  • Borun D. Chowdhury
  • Bartlomiej Czech
  • Jan de Boer
Open Access


We compute the gravitational entropy of “spherical Rindler space”, a timedependent, spherically symmetric generalization of ordinary Rindler space, defined with reference to a family of observers traveling along non-parallel, accelerated trajectories. All these observers are causally disconnected from a spherical region H (a “hole”) located at the origin of Minkowski space. The entropy evaluates to S = \( \mathcal{A} \) /4G, where \( \mathcal{A} \) is the area of the spherical acceleration horizon, which coincides with the boundary of H. We propose that S is the entropy of entanglement between quantum gravitational degrees of freedom supporting the interior and the exterior of the sphere H.


AdS-CFT Correspondence Models of Quantum Gravity Space-Time Symmetries 


  1. [1]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
  4. [4]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  5. [5]
    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Rindler Quantum Gravity, Class. Quant. Grav. 29 (2012) 235025 [arXiv:1206.1323] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    S.L. Braunstein, S. Pirandola and K. Zyczkowski, Entangled black holes as ciphers of hidden information, Phys. Rev. Lett. 110 (2013) 101301 [arXiv:0907.1190] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    V. Balasubramanian, M.B. McDermott and M. Van Raamsdonk, Momentum-space entanglement and renormalization in quantum field theory, Phys. Rev. D 86 (2012) 045014 [arXiv:1108.3568] [INSPIRE].ADSGoogle Scholar
  9. [9]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    R. Laflamme, Entropy of a Rindler wedge, Phys. Lett. B 196 (1987) 449.MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].MathSciNetADSGoogle Scholar
  12. [12]
    J.M. Bardeen, B. Carter and S. Hawking, The Four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  13. [13]
    G. Gibbons and S. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].MathSciNetADSGoogle Scholar
  14. [14]
    S.W. Hawking, The path-integral approach to quantum gravity, Chapter 15 in General Relativity: An Einstein Centenary Survey, eds. S.W. Hawking and W. Israel, Cambridge U.K. (1979).Google Scholar
  15. [15]
    R. Kallosh, T. Ortín and A.W. Peet, Entropy and action of dilaton black holes, Phys. Rev. D 47 (1993) 5400 [hep-th/9211015] [INSPIRE].ADSGoogle Scholar
  16. [16]
    L. Susskind, Some speculations about black hole entropy in string theory, hep-th/9309145 [INSPIRE].
  17. [17]
    V. Iyer 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-qc/9503052] [INSPIRE].MathSciNetADSGoogle Scholar
  18. [18]
    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].MathSciNetADSGoogle Scholar
  19. [19]
    K. Parattu, B.R. Majhi and T. Padmanabhan, The Structure of the Gravitational Action and its relation with Horizon Thermodynamics and Emergent Gravity Paradigm, Phys. Rev. D 87 (2013) 124011 [arXiv:1303.1535] [INSPIRE].ADSGoogle Scholar
  20. [20]
    S.F. Ross, Black hole thermodynamics, hep-th/0502195 [INSPIRE].
  21. [21]
    D.V. Fursaev and S.N. Solodukhin, On the description of the Riemannian geometry in the presence of conical defects, Phys. Rev. D 52 (1995) 2133 [hep-th/9501127] [INSPIRE].MathSciNetADSGoogle Scholar
  22. [22]
    M. Bañados, C. Teitelboim and J. Zanelli, Black hole entropy and the dimensional continuation of the Gauss-Bonnet theorem, Phys. Rev. Lett. 72 (1994) 957 [gr-qc/9309026] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  23. [23]
    D.V. Fursaev, A. Patrushev and S.N. Solodukhin, Distributional Geometry of Squashed Cones, arXiv:1306.4000 [INSPIRE].
  24. [24]
    A.J. Bray and M.A. Moore, Replica-Symmetry Breaking in Spin-Glass Theories, Phys. Rev. Lett. 41 (1978) 1068.ADSCrossRefGoogle Scholar
  25. [25]
    T. Castellani and A. Cavagna, Spin-glass theory for pedestrians, J. Stat. Mech. (2005) P05012 [cond-mat/0505032] [INSPIRE].
  26. [26]
    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  27. [27]
    B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
  28. [28]
    E. Bianchi and R.C. Myers, On the Architecture of Spacetime Geometry, arXiv:1212.5183 [INSPIRE].
  29. [29]
    R.C. Myers, R. Pourhasan and M. Smolkin, On Spacetime Entanglement, JHEP 06 (2013) 013 [arXiv:1304.2030] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    D.V. Fursaev, Entanglement entropy in quantum gravity and the Plateau groblem, Phys. Rev. D 77 (2008) 124002 [arXiv:0711.1221] [INSPIRE].MathSciNetADSGoogle Scholar
  31. [31]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].ADSGoogle Scholar
  34. [34]
    D. Fursaev, Entanglement Renyi Entropies in Conformal Field Theories and Holography, JHEP 05 (2012) 080 [arXiv:1201.1702] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA 2013

Authors and Affiliations

  • Vijay Balasubramanian
    • 1
    • 2
  • Borun D. Chowdhury
    • 3
  • Bartlomiej Czech
    • 3
  • Jan de Boer
    • 3
  1. 1.David Rittenhouse LaboratoriesUniversity of PennsylvaniaPhiladelphiaU.S.A.
  2. 2.Laboratoire de Physics Théorique, École Normale SupérieureParisFrance
  3. 3.Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands

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