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On spacetime entanglement

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Abstract

We examine the idea that in quantum gravity, the entanglement entropy of a general region should be finite and the leading contribution is given by the Bekenstein-Hawking area law. Using holographic entanglement entropy calculations, we show that this idea is realized in the Randall-Sundrum II braneworld for sufficiently large regions in smoothly curved backgrounds. Extending the induced gravity action on the brane to include the curvature-squared interactions, we show that the Wald entropy closely matches the expression describing the entanglement entropy. The difference is that for a general region, the latter includes terms involving the extrinsic curvature of the entangling surface, which do not appear in the Wald entropy. We also consider various limitations on the validity of these results.

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References

  1. J. Bekenstein, Black holes and the second law, Lett. Nuovo Cim. 4 (1972) 737 [INSPIRE].

    Article  ADS  Google Scholar 

  2. J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  3. J.D. Bekenstein, Generalized second law of thermodynamics in black hole physics, Phys. Rev. D 9 (1974) 3292 [INSPIRE].

    ADS  Google Scholar 

  4. S. Hawking, Black holes in general relativity, Commun. Math. Phys. 25 (1972) 152 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge U.K. (1973).

    Book  MATH  Google Scholar 

  6. S. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].

    Article  ADS  Google Scholar 

  7. S. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206-206] [INSPIRE].

  8. J.M. Bardeen, B. Carter and S. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. G. Gibbons and S. Hawking, Cosmological Event Horizons, Thermodynamics and Particle Creation, Phys. Rev. D 15 (1977) 2738 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  10. R. Laflamme, Entropy of a rindler wedge, Phys. Lett. B 196 (1987) 449 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  11. R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  12. T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  13. 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].

    MathSciNet  ADS  Google Scholar 

  14. E. Bianchi and R.C. Myers, On the Architecture of Spacetime Geometry, arXiv:1212.5183 [INSPIRE].

  15. J.H. Cooperman and M.A. Luty, Renormalization of Entanglement Entropy and the Gravitational Effective Action, arXiv:1302.1878 [INSPIRE].

  16. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  17. S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].

    MathSciNet  Google Scholar 

  19. L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  20. M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. T. Jacobson, Thermodynamics of space-time: the Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. T. Jacobson, Gravitation and vacuum entanglement entropy, Int. J. Mod. Phys. D 21 (2012) 1242006 [arXiv:1204.6349] [INSPIRE].

    ADS  Google Scholar 

  23. A. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Sov. Phys. Dokl. 12 (1968) 1040 [Dokl. Akad. Nauk Ser. Fiz. 177 (1967) 70] [Sov. Phys. Usp. 34 (1991) 394] [Gen. Rel. Grav. 32 (2000) 365] [INSPIRE].

  24. R. Emparan, Black hole entropy as entanglement entropy: a holographic derivation, JHEP 06 (2006) 012 [hep-th/0603081] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  25. D.V. Fursaev, Entanglement entropy in critical phenomena and analogue models of quantum gravity, Phys. Rev. D 73 (2006) 124025 [hep-th/0602134] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  26. D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690 [hep-th/9906064] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. S. Hawking, J.M. Maldacena and A. Strominger, de Sitter entropy, quantum entanglement and AdS/CFT, JHEP 05 (2001) 001 [hep-th/0002145] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. Y. Iwashita, T. Kobayashi, T. Shiromizu and H. Yoshino, Holographic entanglement entropy of de Sitter braneworld, Phys. Rev. D 74 (2006) 064027 [hep-th/0606027] [INSPIRE].

    ADS  Google Scholar 

  30. S.N. Solodukhin, Entanglement entropy of black holes and AdS/CFT correspondence, Phys. Rev. Lett. 97 (2006) 201601 [hep-th/0606205] [INSPIRE].

    Article  ADS  Google Scholar 

  31. G. Dvali and S.N. Solodukhin, Black Hole Entropy and Gravity Cutoff, arXiv:0806.3976 [INSPIRE].

  32. R. Emparan, G.T. Horowitz and R.C. Myers, Exact description of black holes on branes, JHEP 01 (2000) 007 [hep-th/9911043] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. H.L. Verlinde, Holography and compactification, Nucl. Phys. B 580 (2000) 264 [hep-th/9906182] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  34. S.S. Gubser, AdS/CFT and gravity, Phys. Rev. D 63 (2001) 084017 [hep-th/9912001] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  35. D. Fursaev, Black hole thermodynamics, induced gravity and gravity in brane worlds, appeared in proceedings of International Conference on Quantization, Gauge Theory, and Strings: Conference Dedicated to the Memory of Professor Efim Fradkin, A. Semikhatov, M. Vasiliev and V. Zaikin eds., Moscow Russia (2000), Scientific World, Singapore (2001), pg. 1180. [hep-th/0009164] [INSPIRE].

  36. R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  37. 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].

    Article  ADS  MATH  Google Scholar 

  38. K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  39. D.P. Jatkar and A. Sinha, New massive gravity and AdS 4 counterterms, Phys. Rev. Lett. 106 (2011) 171601 [arXiv:1101.4746] [INSPIRE].

    Article  ADS  Google Scholar 

  40. K. Sen, A. Sinha and N.V. Suryanarayana, Counterterms, critical gravity and holography, Phys. Rev. D 85 (2012) 124017 [arXiv:1201.1288] [INSPIRE].

    ADS  Google Scholar 

  41. E.P. Verlinde and H.L. Verlinde, RG flow, gravity and the cosmological constant, JHEP 05 (2000) 034 [hep-th/9912018] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  42. A. Kehagias, Exponential and power law hierarchies from supergravity, Phys. Lett. B 469 (1999) 123 [hep-th/9906204] [INSPIRE].

    ADS  Google Scholar 

  43. A. Karch and L. Randall, Localized gravity in string theory, Phys. Rev. Lett. 87 (2001) 061601 [hep-th/0105108] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  44. O. Aharony, O. DeWolfe, D. Z. Freedman and A. Karch, Defect conformal field theory and locally localized gravity, JHEP 07 (2003) 030 [hep-th/0303249] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  45. T. Jacobson, Black hole entropy and induced gravity, gr-qc/9404039 [INSPIRE].

  46. V.P. Frolov, D. Fursaev and A. Zelnikov, Statistical origin of black hole entropy in induced gravity, Nucl. Phys. B 486 (1997) 339 [hep-th/9607104] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. L.-Y. Hung, R.C. Myers and M. Smolkin, On holographic entanglement entropy and higher curvature gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [INSPIRE].

    Article  ADS  Google Scholar 

  48. H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  49. M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].

    ADS  Google Scholar 

  50. J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Entanglement Entropy in Lovelock Gravities, JHEP 07 (2011) 109 [arXiv:1101.5781] [INSPIRE].

    Article  ADS  Google Scholar 

  51. C. Fefferman and C. R. Graham, Conformal Invariants, in Elie Cartan et les Mathématiques d’aujourd hui, Astérisque, Société Mathématique de France, Paris France (1985), pg. 95.

  52. C. Fefferman and C.R. Graham, The ambient metric, arXiv:0710.0919 [INSPIRE].

  53. L.-Y. Hung, R.C. Myers and M. Smolkin, Some calculable contributions to holographic entanglement entropy, JHEP 08 (2011) 039 [arXiv:1105.6055] [INSPIRE].

    Article  ADS  Google Scholar 

  54. C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz, Diffeomorphisms and holographic anomalies, Class. Quant. Grav. 17 (2000) 1129 [hep-th/9910267] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. A. Schwimmer and S. Theisen, Entanglement Entropy, Trace Anomalies and Holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  56. T. Jacobson and R.C. Myers, Black hole entropy and higher curvature interactions, Phys. Rev. Lett. 70 (1993) 3684 [hep-th/9305016] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  57. R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].

    ADS  Google Scholar 

  58. R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  59. A. Buchel et al., Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [INSPIRE].

    Article  ADS  Google Scholar 

  60. S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  61. H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  62. H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys. A 40 (2007) 7031 [cond-mat/0610375] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  63. H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].

    ADS  Google Scholar 

  64. H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, arXiv:1202.2070 [INSPIRE].

  65. R.C. Myers and A. Singh, comments on holographic entanglement entropy and RG flows, JHEP 04 (2012) 122 [arXiv:1202.2068] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  66. A. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  67. E. Lieb and M. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14 (1973) 1938 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  68. D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-theorem: N = 2 field theories on the three-sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  69. I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  70. T. Hirata and T. Takayanagi, AdS/CFT and strong subadditivity of entanglement entropy, JHEP 02 (2007) 042 [hep-th/0608213] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  71. C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  72. L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700 [hep-th/9401070] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  73. J.D. Bekenstein, A Universal Upper Bound on the Entropy to Energy Ratio for Bounded Systems, Phys. Rev. D 23 (1981) 287 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  74. R. Bousso, A covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  75. R. Bousso, The holographic principle, Rev. Mod. Phys. 74 (2002) 825 [hep-th/0203101] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  76. E.E. Flanagan, D. Marolf and R.M. Wald, Proof of classical versions of the Bousso entropy bound and of the generalized second law, Phys. Rev. D 62 (2000) 084035 [hep-th/9908070] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  77. H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  78. D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, in preparation.

  79. D.V. Fursaev, Entanglement entropy in quantum gravity and the Plateau groblem, Phys. Rev. D 77 (2008) 124002 [arXiv:0711.1221] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  80. D.V. Fursaev, ‘Thermodynamics’ of Minimal Surfaces and Entropic Origin of Gravity, Phys. Rev. D 82 (2010) 064013 [Erratum ibid. D 86 (2012) 049903] [arXiv:1006.2623] [INSPIRE].

  81. H. Casini, Geometric entropy, area and strong subadditivity, Class. Quant. Grav. 21 (2004) 2351 [hep-th/0312238] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  82. D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  83. D. Lovelock, Divergence-free tensorial concomitants, Aequationes Math. 4 (1970) 127.

    Article  MathSciNet  MATH  Google Scholar 

  84. S. Nojiri and S.D. Odintsov, On the conformal anomaly from higher derivative gravity in AdS/CFT correspondence, Int. J. Mod. Phys. A 15 (2000) 413 [hep-th/9903033] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  85. M. Blau, K. Narain and E. Gava, On subleading contributions to the AdS/CFT trace anomaly, JHEP 09 (1999) 018 [hep-th/9904179] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  86. M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, Viscosity bound violation in higher derivative gravity, Phys. Rev. D 77 (2008) 126006 [arXiv:0712.0805] [INSPIRE].

    ADS  Google Scholar 

  87. A. Buchel and R.C. Myers, Causality of holographic hydrodynamics, JHEP 08 (2009) 016 [arXiv:0906.2922] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  88. J. de Boer, M. Kulaxizi and A. Parnachev, AdS 7 /CF T 6 , Gauss-Bonnet gravity and viscosity bound, JHEP 03 (2010) 087 [arXiv:0910.5347] [INSPIRE].

    Article  Google Scholar 

  89. X.O. Camanho and J.D. Edelstein, Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity, JHEP 04 (2010) 007 [arXiv:0911.3160] [INSPIRE].

    Article  ADS  Google Scholar 

  90. J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Lovelock gravities and black holes, JHEP 06 (2010) 008 [arXiv:0912.1877] [INSPIRE].

    Article  Google Scholar 

  91. D.M. Hofman, Higher Derivative Gravity, Causality and Positivity of Energy in a UV complete QFT, Nucl. Phys. B 823 (2009) 174 [arXiv:0907.1625] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  92. X.O. Camanho and J.D. Edelstein, Causality in AdS/CFT and Lovelock theory, JHEP 06 (2010) 099 [arXiv:0912.1944] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  93. R.C. Myers, Higher derivative gravity, surface terms and string theory, Phys. Rev. D 36 (1987) 392 [INSPIRE].

    ADS  Google Scholar 

  94. R. Olea, Mass, angular momentum and thermodynamics in four-dimensional Kerr-AdS black holes, JHEP 06 (2005) 023 [hep-th/0504233] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  95. D.G. Boulware and S. Deser, String generated gravity models, Phys. Rev. Lett. 55 (1985) 2656 [INSPIRE].

    Article  ADS  Google Scholar 

  96. R.C. Myers and B. Robinson, Black holes in quasi-topological gravity, JHEP 08 (2010) 067 [arXiv:1003.5357] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  97. R.C. Myers, M.F. Paulos and A. Sinha, Holographic studies of quasi-topological gravity, JHEP 08 (2010) 035 [arXiv:1004.2055] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  98. A. Yale, Simple counterterms for asymptotically AdS spacetimes in Lovelock gravity, Phys. Rev. D 84 (2011) 104036 [arXiv:1107.1250] [INSPIRE].

    ADS  Google Scholar 

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  1. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada

    Robert C. Myers, Razieh Pourhasan & Michael Smolkin

  2. Department of Physics & Astronomy and Guelph-Waterloo Physics Institute, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Razieh Pourhasan

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  1. Robert C. Myers
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Myers, R.C., Pourhasan, R. & Smolkin, M. On spacetime entanglement. J. High Energ. Phys. 2013, 13 (2013). https://doi.org/10.1007/JHEP06(2013)013

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