Abstract
We systematically calculate the quantum extremal surface (QES) associated with Hawking radiation for general D-dimensional (D ≥ 2) asymptotically flat (or AdS) eternal black holes using the island formula. We collect the Hawking radiation particles by a non-gravitational bath and find that a QES exists in the near-horizon region outside the black hole when c · G(D) is smaller enough where c is the central charge of the conformal matter and G(D) the D-dimensional Newton constant. The locations of the QES in these backgrounds are obtained and the late-time radiation entropy saturates the two times of black hole entropy. Finally, we numerically check that the no island configuration exists once c · G(D) exceeds a certain upper bound in two-dimensional generalized dilaton theories (GDT). When c · G(D) close to the upper bound, the backreaction of the matter field on the background can not be neglected. We also consider the conditions of existence of the island configuration with the backreaction and prove that the upper bound also exist for the Witten black hole and Weyl-related Witten black hole.
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S. W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].
S. D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
D. N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
D. N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].
S. Raju, Failure of the split property in gravity and the information paradox, Class. Quant. Grav. 39 (2022) 064002 [arXiv:2110.05470] [INSPIRE].
C. Chowdhury, V. Godet, O. Papadoulaki and S. Raju, Holography from the Wheeler-DeWitt equation, JHEP 03 (2022) 019 [arXiv:2107.14802] [INSPIRE].
H. Geng et al., Inconsistency of islands in theories with long-range gravity, JHEP 01 (2022) 182 [arXiv:2107.03390] [INSPIRE].
S. Raju, Lessons from the information paradox, Phys. Rept. 943 (2022) 1 [arXiv:2012.05770] [INSPIRE].
A. Laddha, S. G. Prabhu, S. Raju and P. Shrivastava, The holographic nature of null infinity, SciPost Phys. 10 (2021) 041 [arXiv:2002.02448] [INSPIRE].
A. Karch and L. Randall, Locally localized gravity, JHEP 05 (2001) 008 [hep-th/0011156] [INSPIRE].
M. Porrati, Mass and gauge invariance 4. Holography for the Karch-Randall model, Phys. Rev. D 65 (2002) 044015 [hep-th/0109017] [INSPIRE].
M. Porrati, Higgs phenomenon for 4D gravity in anti-de Sitter space, JHEP 04 (2002) 058 [hep-th/0112166] [INSPIRE].
H. Geng and A. Karch, Massive islands, JHEP 09 (2020) 121 [arXiv:2006.02438] [INSPIRE].
A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica wormholes and the entropy of Hawking radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
G. Penington, Entanglement wedge reconstruction and the information paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [INSPIRE].
A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149 [arXiv:1908.10996] [INSPIRE].
A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002 [arXiv:2006.06872] [INSPIRE].
A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063 [arXiv:1905.08762] [INSPIRE].
T. Barrella, X. Dong, S. A. Hartnoll and V. L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
N. Engelhardt and A. C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V. E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
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].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
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].
G. Penington, S. H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205 [arXiv:1911.11977] [INSPIRE].
C. Krishnan, Critical islands, JHEP 01 (2021) 179 [arXiv:2007.06551] [INSPIRE].
E. Caceres, A. Kundu, A. K. Patra and S. Shashi, Warped information and entanglement islands in AdS/WCFT, JHEP 07 (2021) 004 [arXiv:2012.05425] [INSPIRE].
H. Geng, Y. Nomura and H.-Y. Sun, Information paradox and its resolution in de Sitter holography, Phys. Rev. D 103 (2021) 126004 [arXiv:2103.07477] [INSPIRE].
H. Geng, S. Lüst, R. K. Mishra and D. Wakeham, Holographic BCFTs and communicating black holes, jhep 08 (2021) 003 [arXiv:2104.07039] [INSPIRE].
F. F. Gautason, L. Schneiderbauer, W. Sybesma and L. Thorlacius, Page curve for an evaporating black hole, JHEP 05 (2020) 091 [arXiv:2004.00598] [INSPIRE].
T. Hartman, E. Shaghoulian and A. Strominger, Islands in asymptotically flat 2D gravity, JHEP 07 (2020) 022 [arXiv:2004.13857] [INSPIRE].
T. J. Hollowood and S. P. Kumar, Islands and Page curves for evaporating black holes in JT gravity, JHEP 08 (2020) 094 [arXiv:2004.14944] [INSPIRE].
K. Goto, T. Hartman and A. Tajdini, Replica wormholes for an evaporating 2D black hole, JHEP 04 (2021) 289 [arXiv:2011.09043] [INSPIRE].
H. Z. Chen, Z. Fisher, J. Hernandez, R. C. Myers and S.-M. Ruan, Evaporating black holes coupled to a thermal bath, JHEP 01 (2021) 065 [arXiv:2007.11658] [INSPIRE].
X. Wang, R. Li and J. Wang, Page curves for a family of exactly solvable evaporating black holes, Phys. Rev. D 103 (2021) 126026 [arXiv:2104.00224] [INSPIRE].
A. Almheiri, R. Mahajan and J. E. Santos, Entanglement islands in higher dimensions, SciPost Phys. 9 (2020) 001 [arXiv:1911.09666] [INSPIRE].
K. Hashimoto, N. Iizuka and Y. Matsuo, Islands in Schwarzschild black holes, JHEP 06 (2020) 085 [arXiv:2004.05863] [INSPIRE].
X. Wang, R. Li and J. Wang, Islands and Page curves of Reissner-Nordström black holes, JHEP 04 (2021) 103 [arXiv:2101.06867] [INSPIRE].
M.-H. Yu and X.-H. Ge, Islands and Page curves in charged dilaton black holes, Eur. Phys. J. C 82 (2022) 14 [arXiv:2107.03031] [INSPIRE].
B. Ahn, S.-E. Bak, H.-S. Jeong, K.-Y. Kim and Y.-W. Sun, Islands in charged linear dilaton black holes, Phys. Rev. D 105 (2022) 046012 [arXiv:2107.07444] [INSPIRE].
G. K. Karananas, A. Kehagias and J. Taskas, Islands in linear dilaton black holes, JHEP 03 (2021) 253 [arXiv:2101.00024] [INSPIRE].
Y. Lu and J. Lin, Islands in Kaluza-Klein black holes, Eur. Phys. J. C 82 (2022) 132 [arXiv:2106.07845] [INSPIRE].
C. Krishnan, V. Patil and J. Pereira, Page curve and the information paradox in flat space, arXiv:2005.02993 [INSPIRE].
M. Alishahiha, A. Faraji Astaneh and A. Naseh, Island in the presence of higher derivative terms, JHEP 02 (2021) 035 [arXiv:2005.08715] [INSPIRE].
J. Hernandez, R. C. Myers and S.-M. Ruan, Quantum extremal islands made easy. Part III. Complexity on the brane, JHEP 02 (2021) 173 [arXiv:2010.16398] [INSPIRE].
I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi and Z. Wei, Entanglement entropy in a holographic moving mirror and the Page curve, Phys. Rev. Lett. 126 (2021) 061604 [arXiv:2011.12005] [INSPIRE].
I. A. Reyes, Moving mirrors, Page curves, and bulk entropies in AdS2, Phys. Rev. Lett. 127 (2021) 051602 [arXiv:2103.01230] [INSPIRE].
I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi and Z. Wei, Holographic moving mirrors, Class. Quant. Grav. 38 (2021) 224001 [arXiv:2106.11179] [INSPIRE].
K. Kawabata, T. Nishioka, Y. Okuyama and K. Watanabe, Probing Hawking radiation through capacity of entanglement, JHEP 05 (2021) 062 [arXiv:2102.02425] [INSPIRE].
J. Kumar Basak, D. Basu, V. Malvimat, H. Parihar and G. Sengupta, Islands for entanglement negativity, SciPost Phys. 12 (2022) 003 [arXiv:2012.03983] [INSPIRE].
S. Choudhury et al., Circuit complexity from cosmological islands, Symmetry 13 (2021) 1301 [arXiv:2012.10234] [INSPIRE].
Y. Ling, P. Liu, Y. Liu, C. Niu, Z.-Y. Xian and C.-Y. Zhang, Reflected entropy in double holography, JHEP 02 (2022) 037 [arXiv:2109.09243] [INSPIRE].
T. Li, M.-K. Yuan and Y. Zhou, Defect extremal surface for reflected entropy, JHEP 01 (2022) 018 [arXiv:2108.08544] [INSPIRE].
V. Chandrasekaran, M. Miyaji and P. Rath, Including contributions from entanglement islands to the reflected entropy, Phys. Rev. D 102 (2020) 086009 [arXiv:2006.10754] [INSPIRE].
K. Kawabata, T. Nishioka, Y. Okuyama and K. Watanabe, Replica wormholes and capacity of entanglement, JHEP 10 (2021) 227 [arXiv:2105.08396] [INSPIRE].
A. Saha, S. Gangopadhyay and J. P. Saha, Mutual information, islands in black holes and the Page curve, arXiv:2109.02996 [INSPIRE].
J. F. Pedraza, A. Svesko, W. Sybesma and M. R. Visser, Semi-classical thermodynamics of quantum extremal surfaces in Jackiw-Teitelboim gravity, JHEP 12 (2021) 134 [arXiv:2107.10358] [INSPIRE].
A. Almheiri, R. Mahajan and J. Maldacena, Islands outside the horizon, arXiv:1910.11077 [INSPIRE].
T. Anegawa and N. Iizuka, Notes on islands in asymptotically flat 2d dilaton black holes, JHEP 07 (2020) 036 [arXiv:2004.01601] [INSPIRE].
W. Kim and M. Nam, Entanglement entropy of asymptotically flat non-extremal and extremal black holes with an island, Eur. Phys. J. C 81 (2021) 869 [arXiv:2103.16163] [INSPIRE].
D. Grumiller, W. Kummer and D. V. Vassilevich, Dilaton gravity in two-dimensions, Phys. Rept. 369 (2002) 327 [hep-th/0204253] [INSPIRE].
N. H. Cao, Entanglement entropy and Page curve of black holes with island in massive gravity, Eur. Phys. J. C 82 (2022) 381 [arXiv:2108.10144] [INSPIRE].
S. Hemming and E. Keski-Vakkuri, Hawking radiation from AdS black holes, Phys. Rev. D 64 (2001) 044006 [gr-qc/0005115] [INSPIRE].
H. Casini, C. D. Fosco and M. Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. 0507 (2005) P07007 [cond-mat/0505563] [INSPIRE].
R.-G. Cai, S. He, S.-J. Wang and Y.-X. Zhang, Revisit on holographic complexity in two-dimensional gravity, JHEP 08 (2020) 102 [arXiv:2001.11626] [INSPIRE].
D. Grumiller and R. McNees, Thermodynamics of black holes in two (and higher) dimensions, JHEP 04 (2007) 074 [hep-th/0703230] [INSPIRE].
G. Mandal, A. M. Sengupta and S. R. Wadia, Classical solutions of two-dimensional string theory, Mod. Phys. Lett. A 6 (1991) 1685 [INSPIRE].
E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].
C. G. Callan, Jr., S. B. Giddings, J. A. Harvey and A. Strominger, Evanescent black holes, Phys. Rev. D 45 (1992) R1005 [hep-th/9111056] [INSPIRE].
K. Schwarzschild, On the gravitational field of a mass point according to Einstein’s theory, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916 (1916) 189 [physics/9905030] [INSPIRE].
D. Grumiller, Long time black hole evaporation with bounded Hawking flux, JCAP 05 (2004) 005 [gr-qc/0307005] [INSPIRE].
R. Jackiw, Lower dimensional gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian structure in two space-time dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
S. W. Hawking and D. N. Page, Thermodynamics of black holes in anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
M. Socolovsky, Schwarzschild black hole in anti-de Sitter space, Adv. Appl. Clifford Algebras 28 (2018) 18 [arXiv:1711.02744] [INSPIRE].
A. Fabbri and J. Navarro-Salas, Modeling black hole evaporation, World Scientific, Singapore (2005).
A. M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981) 207 [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
L. Anderson, O. Parrikar and R. M. Soni, Islands with gravitating baths: towards ER = EPR, JHEP 10 (2021) 226 [arXiv:2103.14746] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
T. Banks and M. O’Loughlin, Two-dimensional quantum gravity in Minkowski space, Nucl. Phys. B 362 (1991) 649 [INSPIRE].
D. Louis-Martinez, J. Gegenberg and G. Kunstatter, Exact Dirac quantization of all 2D dilaton gravity theories, Phys. Lett. B 321 (1994) 193 [gr-qc/9309018] [INSPIRE].
M. Cavaglià, A note on Weyl transformations in two-dimensional dilaton gravity, Mod. Phys. Lett. A 15 (2000) 2113 [hep-th/0011136] [INSPIRE].
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He, S., Sun, Y., Zhao, L. et al. The universality of islands outside the horizon. J. High Energ. Phys. 2022, 47 (2022). https://doi.org/10.1007/JHEP05(2022)047
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DOI: https://doi.org/10.1007/JHEP05(2022)047