Abstract
We study the Page curve of an evaporating black hole using a toy model given by Jackiw-Teitelboim gravity with Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) antibranes. We treat the anti-FZZT branes as dynamical objects, taking their back-reaction into account. We construct the entanglement entropy from the dual matrix model and study its behavior as a function of the ’t Hooft coupling t proportional to the number of branes, which plays the role of time. By numerical computation we observe that the entropy first increases and then decreases as t grows, reproducing the well-known behavior of the Page curve of an evaporating black hole. The system finally exhibits a phase transition, which may be viewed as the end of the evaporation. We study the critical behavior of the entropy near the phase transition. We also make a conjecture about the late-time monotonically decreasing behavior of the entropy. We prove it in a certain limit as well as give an intuitive explanation by means of the dual matrix model.
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].
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.M. Maldacena and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149 [arXiv:1908.10996] [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.M. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [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.M. 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].
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.M. Maldacena, E. Shaghoulian and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
A. Almheiri, T. Hartman, J.M. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002 [arXiv:2006.06872] [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].
A. Goel, L.V. Iliesiu, J. Kruthoff and Z. Yang, Classifying boundary conditions in JT gravity: from energy-branes to α-branes, JHEP 04 (2021) 069 [arXiv:2010.12592] [INSPIRE].
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
K. Okuyama, Capacity of entanglement in random pure state, Phys. Lett. B 820 (2021) 136600 [arXiv:2103.08909] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
V. Fateev, A.B. Zamolodchikov and A.B. Zamolodchikov, Boundary Liouville field theory. 1. Boundary state and boundary two point function, hep-th/0001012 [INSPIRE].
J. Teschner, Remarks on Liouville theory with boundary, PoS tmr2000 (2000) 041 [hep-th/0009138] [INSPIRE].
K. Okuyama and K. Sakai, FZZT branes in JT gravity and topological gravity, JHEP 09 (2021) 191 [arXiv:2108.03876] [INSPIRE].
P. Gao, D.L. Jafferis and D.K. Kolchmeyer, An effective matrix model for dynamical end of the world branes in Jackiw-Teitelboim gravity, JHEP 01 (2022) 038 [arXiv:2104.01184] [INSPIRE].
K. Goto, T. Hartman and A. Tajdini, Replica wormholes for an evaporating 2D black hole, JHEP 04 (2021) 289 [arXiv:2011.09043] [INSPIRE].
M. Cadoni and A.P. Sanna, Unitarity and Page curve for evaporation of 2D AdS black holes, arXiv:2106.14738 [INSPIRE].
H. Maxfield and G.J. Turiaci, The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral, JHEP 01 (2021) 118 [arXiv:2006.11317] [INSPIRE].
E. Witten, Matrix Models and Deformations of JT Gravity, Proc. Roy. Soc. Lond. A 476 (2020) 20200582 [arXiv:2006.13414] [INSPIRE].
E. Witten, Deformations of JT Gravity and Phase Transitions, arXiv:2006.03494 [INSPIRE].
S. Sugiura and A. Shimizu, Canonical Thermal Pure Quantum State, Phys. Rev. Lett. 111 (2013) 010401 [arXiv:1302.3138] [INSPIRE].
K. Goto, Y. Kusuki, K. Tamaoka and T. Ugajin, Product of random states and spatial (half-)wormholes, JHEP 10 (2021) 205 [arXiv:2108.08308] [INSPIRE].
M. Mulase and B. Safnuk, Mirzakhani’s recursion relations, Virasoro constraints and the KdV hierarchy, math/0601194 [INSPIRE].
R. Dijkgraaf and E. Witten, Developments in Topological Gravity, Int. J. Mod. Phys. A 33 (2018) 1830029 [arXiv:1804.03275] [INSPIRE].
K. Okuyama and K. Sakai, JT gravity, KdV equations and macroscopic loop operators, JHEP 01 (2020) 156 [arXiv:1911.01659] [INSPIRE].
N. Seiberg and D. Starnford, unpublished.
S. Hirano and T. Kuroki, Replica wormholes from Liouville theory, JHEP 01 (2022) 094 [arXiv:2109.12539] [INSPIRE].
E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Transformation groups for soliton equations — Euclidean Lie algebras and reduction of the KP hierarchy, Publ. Res. Inst. Math. Sci. Kyoto 18 (1982) 1077.
C.V. Johnson and F. Rosso, Solving Puzzles in Deformed JT Gravity: Phase Transitions and Non-Perturbative Effects, JHEP 04 (2021) 030 [arXiv:2011.06026] [INSPIRE].
C. Itzykson and J.B. Zuber, Combinatorics of the modular group. 2. The Kontsevich integrals, Int. J. Mod. Phys. A 7 (1992) 5661 [hep-th/9201001] [INSPIRE].
Y. Nakaguchi and T. Nishioka, A holographic proof of Rényi entropic inequalities, JHEP 12 (2016) 129 [arXiv:1606.08443] [INSPIRE].
X. Dong, The Gravity Dual of Renyi Entropy, Nature Commun. 7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].
P. Gregori and R. Schiappa, From Minimal Strings towards Jackiw-Teitelboim Gravity: On their Resurgence, Resonance, and Black Holes, arXiv:2108.11409 [INSPIRE].
G.T. Horowitz and J.M. Maldacena, The Black hole final state, JHEP 02 (2004) 008 [hep-th/0310281] [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].
J. McNamara and C. Vafa, Baby Universes, Holography, and the Swampland, arXiv:2004.06738 [INSPIRE].
P. Saad, S.H. Shenker, D. Stanford and S. Yao, Wormholes without averaging, arXiv:2103.16754 [INSPIRE].
P. Saad, S. Shenker and S. Yao, Comments on wormholes and factorization, arXiv:2107.13130 [INSPIRE].
A. Blommaert, L.V. Iliesiu and J. Kruthoff, Gravity factorized, arXiv:2111.07863 [INSPIRE].
J.J. Heckman, A.P. Turner and X. Yu, Disorder Averaging and its UV (Dis)Contents, arXiv:2111.06404 [INSPIRE].
A. Blommaert and M. Usatyuk, Microstructure in matrix elements, arXiv:2108.02210 [INSPIRE].
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: 2111.09551
Rights and permissions
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.
About this article
Cite this article
Okuyama, K., Sakai, K. Page curve from dynamical branes in JT gravity. J. High Energ. Phys. 2022, 87 (2022). https://doi.org/10.1007/JHEP02(2022)087
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2022)087