Abstract
The focusing theorem fails for evaporating black holes because the null energy condition is violated by quantum effects. The quantum focusing conjecture is proposed so that it is satisfied even if the null energy condition is violated. The conjecture states that the derivative of the sum of the area of a cross-section of the null geodesic congruence and the entanglement entropy of matter outside it is non-increasing. Naively, it is expected that the quantum focusing conjecture is violated after the Page time as both the area of the horizon and the entanglement entropy of the Hawking radiation are decreasing. We calculate the entanglement entropy after the Page time by using the island rule, and find the following results: (i) the page time is given by an approximately null surface, (ii) the entanglement entropy is increasing along the outgoing null geodesic even after the Page time, and (iii) the quantum focusing conjecture is not violated.
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References
J.D. Bekenstein, Black holes and the second law, Lett. Nuovo Cim. 4 (1972) 737 [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
J.D. Bekenstein, Generalized second law of thermodynamics in black hole physics, Phys. Rev. D 9 (1974) 3292 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
S.W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].
A.C. Wall, The Generalized Second Law implies a Quantum Singularity Theorem, Class. Quant. Grav. 30 (2013) 165003 [Erratum ibid. 30 (2013) 199501] [arXiv:1010.5513] [INSPIRE].
R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [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].
D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].
D.N. Page, Time Dependence of Hawking Radiation Entropy, JCAP 09 (2013) 028 [arXiv:1301.4995] [INSPIRE].
G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [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].
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, R. Mahajan and J. Maldacena, Islands outside the horizon, arXiv:1910.11077 [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].
A. Almheiri et al., Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
A. Almheiri et al., The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002 [arXiv:2006.06872] [INSPIRE].
C. Akers, N. Engelhardt and D. Harlow, Simple holographic models of black hole evaporation, JHEP 08 (2020) 032 [arXiv:1910.00972] [INSPIRE].
H.Z. Chen et al., Information Flow in Black Hole Evaporation, JHEP 03 (2020) 152 [arXiv:1911.03402] [INSPIRE].
A. Almheiri, R. Mahajan and J.E. Santos, Entanglement islands in higher dimensions, SciPost Phys. 9 (2020) 001 [arXiv:1911.09666] [INSPIRE].
Y. Chen, Pulling Out the Island with Modular Flow, JHEP 03 (2020) 033 [arXiv:1912.02210] [INSPIRE].
C. Akers, N. Engelhardt, G. Penington and M. Usatyuk, Quantum Maximin Surfaces, JHEP 08 (2020) 140 [arXiv:1912.02799] [INSPIRE].
H. Liu and S. Vardhan, A dynamical mechanism for the Page curve from quantum chaos, JHEP 03 (2021) 088 [arXiv:2002.05734] [INSPIRE].
D. Marolf and H. Maxfield, Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information, JHEP 08 (2020) 044 [arXiv:2002.08950] [INSPIRE].
V. Balasubramanian et al., Geometric secret sharing in a model of Hawking radiation, JHEP 01 (2021) 177 [arXiv:2003.05448] [INSPIRE].
A. Bhattacharya, Multipartite purification, multiboundary wormholes, and islands in AdS3/CFT2, Phys. Rev. D 102 (2020) 046013 [arXiv:2003.11870] [INSPIRE].
H. Verlinde, ER = EPR revisited: On the Entropy of an Einstein-Rosen Bridge, arXiv:2003.13117 [INSPIRE].
Y. Chen, X.-L. Qi and P. Zhang, Replica wormhole and information retrieval in the SYK model coupled to Majorana chains, JHEP 06 (2020) 121 [arXiv:2003.13147] [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. Anegawa and N. Iizuka, Notes on islands in asymptotically flat 2d dilaton black holes, JHEP 07 (2020) 036 [arXiv:2004.01601] [INSPIRE].
K. Hashimoto, N. Iizuka and Y. Matsuo, Islands in Schwarzschild black holes, JHEP 06 (2020) 085 [arXiv:2004.05863] [INSPIRE].
J. Sully, M. Van Raamsdonk and D. Wakeham, BCFT entanglement entropy at large central charge and the black hole interior, JHEP 03 (2021) 167 [arXiv:2004.13088] [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].
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].
T. Banks, Microscopic Models of Linear Dilaton Gravity and Their Semi-classical Approximations, arXiv:2005.09479 [INSPIRE].
H. Geng and A. Karch, Massive islands, JHEP 09 (2020) 121 [arXiv:2006.02438] [INSPIRE].
H.Z. Chen et al., Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane, JHEP 10 (2020) 166 [arXiv:2006.04851] [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].
T. Li, J. Chu and Y. Zhou, Reflected Entropy for an Evaporating Black Hole, JHEP 11 (2020) 155 [arXiv:2006.10846] [INSPIRE].
D. Bak, C. Kim, S.-H. Yi and J. Yoon, Unitarity of entanglement and islands in two-sided Janus black holes, JHEP 01 (2021) 155 [arXiv:2006.11717] [INSPIRE].
R. Bousso and E. Wildenhain, Gravity/ensemble duality, Phys. Rev. D 102 (2020) 066005 [arXiv:2006.16289] [INSPIRE].
T. Anous, J. Kruthoff and R. Mahajan, Density matrices in quantum gravity, SciPost Phys. 9 (2020) 045 [arXiv:2006.17000] [INSPIRE].
X. Dong, X.-L. Qi, Z. Shangnan and Z. Yang, Effective entropy of quantum fields coupled with gravity, JHEP 10 (2020) 052 [arXiv:2007.02987] [INSPIRE].
T.J. Hollowood, S. Prem Kumar and A. Legramandi, Hawking radiation correlations of evaporating black holes in JT gravity, J. Phys. A 53 (2020) 475401 [arXiv:2007.04877] [INSPIRE].
C. Krishnan, Critical Islands, JHEP 01 (2021) 179 [arXiv:2007.06551] [INSPIRE].
N. Engelhardt, S. Fischetti and A. Maloney, Free energy from replica wormholes, Phys. Rev. D 103 (2021) 046021 [arXiv:2007.07444] [INSPIRE].
A. Karlsson, Replica wormhole and island incompatibility with monogamy of entanglement, arXiv:2007.10523 [INSPIRE].
H.Z. Chen et al., Evaporating Black Holes Coupled to a Thermal Bath, JHEP 01 (2021) 065 [arXiv:2007.11658] [INSPIRE].
Y. Chen, V. Gorbenko and J. Maldacena, Bra-ket wormholes in gravitationally prepared states, JHEP 02 (2021) 009 [arXiv:2007.16091] [INSPIRE].
T. Hartman, Y. Jiang and E. Shaghoulian, Islands in cosmology, JHEP 11 (2020) 111 [arXiv:2008.01022] [INSPIRE].
H. Liu and S. Vardhan, Entanglement Entropies of Equilibrated Pure States in Quantum Many-Body Systems and Gravity, PRX Quantum 2 (2021) 010344 [arXiv:2008.01089] [INSPIRE].
C. Murdia, Y. Nomura and P. Rath, Coarse-Graining Holographic States: A Semiclassical Flow in General Spacetimes, Phys. Rev. D 102 (2020) 086001 [arXiv:2008.01755] [INSPIRE].
C. Akers and G. Penington, Leading order corrections to the quantum extremal surface prescription, JHEP 04 (2021) 062 [arXiv:2008.03319] [INSPIRE].
V. Balasubramanian, A. Kar and T. Ugajin, Islands in de Sitter space, JHEP 02 (2021) 072 [arXiv:2008.05275] [INSPIRE].
V. Balasubramanian, A. Kar and T. Ugajin, Entanglement between two disjoint universes, JHEP 02 (2021) 136 [arXiv:2008.05274] [INSPIRE].
W. Sybesma, Pure de Sitter space and the island moving back in time, Class. Quant. Grav. 38 (2021) 145012 [arXiv:2008.07994] [INSPIRE].
D. Stanford, More quantum noise from wormholes, arXiv:2008.08570 [INSPIRE].
H.Z. Chen et al., Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane, JHEP 12 (2020) 025 [arXiv:2010.00018] [INSPIRE].
Y. Ling, Y. Liu and Z.-Y. Xian, Island in Charged Black Holes, JHEP 03 (2021) 251 [arXiv:2010.00037] [INSPIRE].
D. Marolf and H. Maxfield, Observations of Hawking radiation: the Page curve and baby universes, JHEP 04 (2021) 272 [arXiv:2010.06602] [INSPIRE].
D. Harlow and E. Shaghoulian, Global symmetry, Euclidean gravity, and the black hole information problem, JHEP 04 (2021) 175 [arXiv:2010.10539] [INSPIRE].
I. Akal, Universality, intertwiners and black hole information, arXiv:2010.12565 [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].
Y. Chen and H.W. Lin, Signatures of global symmetry violation in relative entropies and replica wormholes, JHEP 03 (2021) 040 [arXiv:2011.06005] [INSPIRE].
Y. Matsuo, Islands and stretched horizon, JHEP 07 (2021) 051 [arXiv:2011.08814] [INSPIRE].
K. Goto, T. Hartman and A. Tajdini, Replica wormholes for an evaporating 2D black hole, JHEP 04 (2021) 289 [arXiv:2011.09043] [INSPIRE].
P.-S. Hsin, L.V. Iliesiu and Z. Yang, A violation of global symmetries from replica wormholes and the fate of black hole remnants, Class. Quant. Grav. 38 (2021) 194004 [arXiv:2011.09444] [INSPIRE].
I. Akal et al., Entanglement Entropy in a Holographic Moving Mirror and the Page Curve, Phys. Rev. Lett. 126 (2021) 061604 [arXiv:2011.12005] [INSPIRE].
S. Colin-Ellerin et al., Real-time gravitational replicas: Formalism and a variational principle, JHEP 05 (2021) 117 [arXiv:2012.00828] [INSPIRE].
J. Kumar Basak et al., Islands for entanglement negativity, SciPost Phys. 12 (2022) 003 [arXiv:2012.03983] [INSPIRE].
H. Geng et al., Information Transfer with a Gravitating Bath, SciPost Phys. 10 (2021) 103 [arXiv:2012.04671] [INSPIRE].
G.K. Karananas, A. Kehagias and J. Taskas, Islands in linear dilaton black holes, JHEP 03 (2021) 253 [arXiv:2101.00024] [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].
D. Marolf and J.E. Santos, AdS Euclidean wormholes, Class. Quant. Grav. 38 (2021) 224002 [arXiv:2101.08875] [INSPIRE].
R. Bousso and A. Shahbazi-Moghaddam, Island Finder and Entropy Bound, Phys. Rev. D 103 (2021) 106005 [arXiv:2101.11648] [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].
K. Ghosh and C. Krishnan, Dirichlet baths and the not-so-fine-grained Page curve, JHEP 08 (2021) 119 [arXiv:2103.17253] [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].
C.F. Uhlemann, Islands and Page curves in 4d from Type IIB, JHEP 08 (2021) 104 [arXiv:2105.00008] [INSPIRE].
K. Kawabata, T. Nishioka, Y. Okuyama and K. Watanabe, Replica wormholes and capacity of entanglement, JHEP 10 (2021) 227 [arXiv:2105.08396] [INSPIRE].
I. Akal et al., Holographic moving mirrors, Class. Quant. Grav. 38 (2021) 224001 [arXiv:2106.11179] [INSPIRE].
H. Omiya and Z. Wei, Causal structures and nonlocality in double holography, JHEP 07 (2022) 128 [arXiv:2107.01219] [INSPIRE].
H. Geng et al., Inconsistency of islands in theories with long-range gravity, JHEP 01 (2022) 182 [arXiv:2107.03390] [INSPIRE].
D. Stanford, Z. Yang and S. Yao, Subleading Weingartens, JHEP 02 (2022) 200 [arXiv:2107.10252] [INSPIRE].
V. Balasubramanian, B. Craps, M. Khramtsov and E. Shaghoulian, Submerging islands through thermalization, JHEP 10 (2021) 048 [arXiv:2107.14746] [INSPIRE].
C. Akers, S. Hernández-Cuenca and P. Rath, Quantum Extremal Surfaces and the Holographic Entropy Cone, JHEP 11 (2021) 177 [arXiv:2108.07280] [INSPIRE].
M. Miyaji, Island for gravitationally prepared state and pseudo entanglement wedge, JHEP 12 (2021) 013 [arXiv:2109.03830] [INSPIRE].
S. Azarnia, R. Fareghbal, A. Naseh and H. Zolfi, Islands in flat-space cosmology, Phys. Rev. D 104 (2021) 126017 [arXiv:2109.04795] [INSPIRE].
S. He, Y. Sun, L. Zhao and Y.-X. Zhang, The universality of islands outside the horizon, JHEP 05 (2022) 047 [arXiv:2110.07598] [INSPIRE].
X. Dong, S. McBride and W.W. Weng, Replica wormholes and holographic entanglement negativity, JHEP 06 (2022) 094 [arXiv:2110.11947] [INSPIRE].
E. Shaghoulian, The central dogma and cosmological horizons, JHEP 01 (2022) 132 [arXiv:2110.13210] [INSPIRE].
Y. Matsuo, Entanglement entropy and vacuum states in Schwarzschild geometry, JHEP 06 (2022) 109 [arXiv:2110.13898] [INSPIRE].
C.F. Uhlemann, Information transfer with a twist, JHEP 01 (2022) 126 [arXiv:2111.11443] [INSPIRE].
F. Omidi, Entropy of Hawking radiation for two-sided hyperscaling violating black branes, JHEP 04 (2022) 022 [arXiv:2112.05890] [INSPIRE].
A. Bhattacharya, A. Bhattacharyya, P. Nandy and A.K. Patra, Bath deformations, islands, and holographic complexity, Phys. Rev. D 105 (2022) 066019 [arXiv:2112.06967] [INSPIRE].
M.-H. Yu, C.-Y. Lu, X.-H. Ge and S.-J. Sin, Island, Page curve, and superradiance of rotating BTZ black holes, Phys. Rev. D 105 (2022) 066009 [arXiv:2112.14361] [INSPIRE].
N. Engelhardt and Å. Folkestad, Canonical purification of evaporating black holes, Phys. Rev. D 105 (2022) 086010 [arXiv:2201.08395] [INSPIRE].
C. Akers, T. Faulkner, S. Lin and P. Rath, The Page curve for reflected entropy, JHEP 06 (2022) 089 [arXiv:2201.11730] [INSPIRE].
K. Suzuki and T. Takayanagi, BCFT and Islands in two dimensions, JHEP 06 (2022) 095 [arXiv:2202.08462] [INSPIRE].
W.-C. Gan, D.-H. Du and F.-W. Shu, Island and Page curve for one-sided asymptotically flat black hole, JHEP 07 (2022) 020 [arXiv:2203.06310] [INSPIRE].
A. Rolph, Island mirages, JHEP 08 (2022) 142 [arXiv:2206.06144] [INSPIRE].
R. Bousso and A. Shahbazi-Moghaddam, Quantum singularities, Phys. Rev. D 107 (2023) 066002 [arXiv:2206.07001] [INSPIRE].
C. Akers et al., The black hole interior from non-isometric codes and complexity, arXiv:2207.06536 [INSPIRE].
V. Balasubramanian, A. Lawrence, J.M. Magan and M. Sasieta, Microscopic origin of the entropy of black holes in general relativity, arXiv:2212.02447 [INSPIRE].
P.C.W. Davies, S.A. Fulling and W.G. Unruh, Energy Momentum Tensor Near an Evaporating Black Hole, Phys. Rev. D 13 (1976) 2720 [INSPIRE].
P.C.W. Davies and S.A. Fulling, Radiation from a moving mirror in two-dimensional space-time conformal anomaly, Proc. Roy. Soc. Lond. A 348 (1976) 393 [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].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [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].
X. Dong, A. Lewkowycz and M. Rangamani, Deriving covariant holographic entanglement, JHEP 11 (2016) 028 [arXiv:1607.07506] [INSPIRE].
X. Dong and A. Lewkowycz, Entropy, Extremality, Euclidean Variations, and the Equations of Motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [INSPIRE].
P.-M. Ho and Y. Matsuo, On the Near-Horizon Geometry of an Evaporating Black Hole, JHEP 07 (2018) 047 [arXiv:1804.04821] [INSPIRE].
P.-M. Ho, Y. Matsuo and Y. Yokokura, Analytic description of semiclassical black-hole geometry, Phys. Rev. D 102 (2020) 024090 [arXiv:1912.12855] [INSPIRE].
H. Casini and M. Huerta, Entanglement and alpha entropies for a massive scalar field in two dimensions, J. Stat. Mech. 0512 (2005) P12012 [cond-mat/0511014] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [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].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
Acknowledgments
The author would like to thank Akihiro Ishibashi for fruitful discussions. This work is supported in part by JSPS KAKENHI Grant No. JP20K03930 and JP21H05186.
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Matsuo, Y. Quantum focusing conjecture and the Page curve. J. High Energ. Phys. 2023, 50 (2023). https://doi.org/10.1007/JHEP12(2023)050
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DOI: https://doi.org/10.1007/JHEP12(2023)050