Abstract
We study the Page curve for asymptotically flat eternal Schwarzschild black holes in four and higher spacetime dimensions. Before the Page time, the entanglement entropy grows linearly in time. After the Page time, the entanglement entropy of a given region outside the black hole is largely modified by the emergence of an island, which extends to the outer vicinity of the event horizon. As a result, it remains a constant value which reproduces the Bekenstein-Hawking entropy, consistent with the finiteness of the von Neumann entropy for an eternal black hole.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [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, 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].
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].
N. Engelhardt and A.C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, arXiv:1911.11977 [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].
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. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
H.Z. Chen, Z. Fisher, J. Hernandez, R.C. Myers and S.-M. Ruan, Information flow in black hole evaporation, JHEP 03 (2020) 152 [arXiv:1911.03402] [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, arXiv:1912.02799 [INSPIRE].
H. Liu and S. Vardhan, A dynamical mechanism for the Page curve from quantum chaos, 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, arXiv:2002.08950 [INSPIRE].
V. Balasubramanian, A. Kar, O. Parrikar, G. S´arosi and T. Ugajin, Geometric secret sharing in a model of Hawking radiation, arXiv:2003.05448 [INSPIRE].
A. Bhattacharya, Multipartite purification, multiboundary wormholes and islands in AdS3 /CFT2 , 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, 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, arXiv:2004.01601 [INSPIRE].
A. Almheiri, R. Mahajan and J.E. Santos, Entanglement islands in higher dimensions, arXiv:1911.09666 [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].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [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].
D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
L. Susskind and Y. Zhao, Switchbacks and the bridge to nowhere, arXiv:1408.2823 [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].
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].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
D.N. Page, Particle emission rates from a black hole: massless particles from an uncharged, nonrotating hole, Phys. Rev. D 13 (1976) 198 [INSPIRE].
Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
D. Carmi, S. Chapman, H. Marrochio, R.C. Myers and S. Sugishita, On the time dependence of holographic complexity, JHEP 11 (2017) 188 [arXiv:1709.10184] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2004.05863
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Hashimoto, K., Iizuka, N. & Matsuo, Y. Islands in Schwarzschild black holes. J. High Energ. Phys. 2020, 85 (2020). https://doi.org/10.1007/JHEP06(2020)085
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2020)085