Islands in non-minimal dilaton gravity: exploring effective theories for black hole evaporation. (English) Zbl 07774699
Summary: We start from \((3 + 1)\)-dimensional Einstein gravity with minimally coupled massless scalar matter, through spherical dimensional reduction, the matter theory is non-minimally coupled with the dilaton in \((1 + 1)\)-dimensions. Despite its simplicity, constructing a self-consistent one-loop effective theory for this model remains a challenge, partially due to a Weyl-invariant ambiguity in the effective action. With a universal splitting property for the one-loop action, the ambiguity can be identified with the state-dependent part of the covariant quantum stress tensor. By introducing on-shell equivalent auxiliary fields to construct minimal candidates of Weyl-invariant terms, we derive a one-parameter family of one-loop actions with unique, regular, and physical stress tensors corresponding to the Boulware, Hartle-Hawking and Unruh states. We further study the back-reacted geometry and the corresponding quantum extremal islands that were inaccessible without a consistent one-loop theory. Along the way, we elaborate on the implications of our construction for the non-minimal dilaton gravity model.
MSC:
| 81-XX | Quantum theory |
References:
| [1] | Hawking, S., Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D., 14, 2460 (1976) · doi:10.1103/PhysRevD.14.2460 |
| [2] | Penington, G., Entanglement Wedge Reconstruction and the Information Paradox, JHEP, 09, 002 (2020) · Zbl 1454.81039 · doi:10.1007/JHEP09(2020)002 |
| [3] | Almheiri, A.; Engelhardt, N.; Marolf, D.; Maxfield, H., The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP, 12, 063 (2019) · Zbl 1431.83123 · doi:10.1007/JHEP12(2019)063 |
| [4] | Almheiri, A.; Mahajan, R.; Maldacena, J.; Zhao, Y., The Page curve of Hawking radiation from semiclassical geometry, JHEP, 03, 149 (2020) · Zbl 1435.83110 · doi:10.1007/JHEP03(2020)149 |
| [5] | Penington, G.; Shenker, SH; Stanford, D.; Yang, Z., Replica wormholes and the black hole interior, JHEP, 03, 205 (2022) · Zbl 1522.83227 · doi:10.1007/JHEP03(2022)205 |
| [6] | Almheiri, A., Replica Wormholes and the Entropy of Hawking Radiation, JHEP, 05, 013 (2020) · Zbl 1437.83084 · doi:10.1007/JHEP05(2020)013 |
| [7] | Page, DN, Information in black hole radiation, Phys. Rev. Lett., 71, 3743 (1993) · Zbl 0972.83567 · doi:10.1103/PhysRevLett.71.3743 |
| [8] | Page, DN, Time Dependence of Hawking Radiation Entropy, JCAP, 09, 028 (2013) · doi:10.1088/1475-7516/2013/09/028 |
| [9] | Almheiri, A.; Mahajan, R.; Santos, JE, Entanglement islands in higher dimensions, SciPost Phys., 9, 001 (2020) · doi:10.21468/SciPostPhys.9.1.001 |
| [10] | Gautason, FF; Schneiderbauer, L.; Sybesma, W.; Thorlacius, L., Page Curve for an Evaporating Black Hole, JHEP, 05, 091 (2020) · Zbl 1437.83080 · doi:10.1007/JHEP05(2020)091 |
| [11] | Anegawa, T.; Iizuka, N., Notes on islands in asymptotically flat 2d dilaton black holes, JHEP, 07, 036 (2020) · Zbl 1451.83056 · doi:10.1007/JHEP07(2020)036 |
| [12] | Hartman, T.; Shaghoulian, E.; Strominger, A., Islands in Asymptotically Flat 2D Gravity, JHEP, 07, 022 (2020) · Zbl 1455.83017 · doi:10.1007/JHEP07(2020)022 |
| [13] | Hashimoto, K.; Iizuka, N.; Matsuo, Y., Islands in Schwarzschild black holes, JHEP, 06, 085 (2020) · Zbl 1437.83060 · doi:10.1007/JHEP06(2020)085 |
| [14] | Matsuo, Y., Islands and stretched horizon, JHEP, 07, 051 (2021) · Zbl 1468.83027 · doi:10.1007/JHEP07(2021)051 |
| [15] | X. Wang, R. Li and J. Wang, Page curves for a family of exactly solvable evaporating black holes, Phys. Rev. D103 (2021) 126026 [arXiv:2104.00224] [INSPIRE]. |
| [16] | Tian, J., Islands in Generalized Dilaton Theories, Symmetry, 15, 1402 (2023) · doi:10.3390/sym15071402 |
| [17] | Gan, W-C; Du, D-H; Shu, F-W, Island and Page curve for one-sided asymptotically flat black hole, JHEP, 07, 020 (2022) · Zbl 1522.83170 · doi:10.1007/JHEP07(2022)020 |
| [18] | S. Djordjević, A. Gočanin, D. Gočanin and V. Radovanović, Page curve for an eternal Schwarzschild black hole in a dimensionally reduced model of dilaton gravity, Phys. Rev. D106 (2022) 105015 [arXiv:2207.07409] [INSPIRE]. |
| [19] | M.-H. Yu and X.-H. Ge, Entanglement islands in generalized two-dimensional dilaton black holes, Phys. Rev. D107 (2023) 066020 [arXiv:2208.01943] [INSPIRE]. |
| [20] | Guo, C-Z; Gan, W-C; Shu, F-W, Page curves and entanglement islands for the step-function Vaidya model of evaporating black holes, JHEP, 05, 042 (2023) · Zbl 07701857 · doi:10.1007/JHEP05(2023)042 |
| [21] | Krishnan, C., Critical Islands, JHEP, 01, 179 (2021) · Zbl 1459.83048 · doi:10.1007/JHEP01(2021)179 |
| [22] | Hartman, T.; Jiang, Y.; Shaghoulian, E., Islands in cosmology, JHEP, 11, 111 (2020) · doi:10.1007/JHEP11(2020)111 |
| [23] | S.E. Aguilar-Gutierrez et al., Islands in Multiverse Models, JHEP11 (2021) 212 [Addendum ibid.05 (2022) 137] [arXiv:2108.01278] [INSPIRE]. · Zbl 1521.83037 |
| [24] | Maldacena, JM, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys., 2, 231 (1998) · Zbl 0914.53047 · doi:10.4310/ATMP.1998.v2.n2.a1 |
| [25] | Gubser, SS; Klebanov, IR; Polyakov, AM, Gauge theory correlators from noncritical string theory, Phys. Lett. B, 428, 105 (1998) · Zbl 1355.81126 · doi:10.1016/S0370-2693(98)00377-3 |
| [26] | Witten, E., Anti-de Sitter space and holography, Adv. Theor. Math. Phys., 2, 253 (1998) · Zbl 0914.53048 · doi:10.4310/ATMP.1998.v2.n2.a2 |
| [27] | Goto, K.; Hartman, T.; Tajdini, A., Replica wormholes for an evaporating 2D black hole, JHEP, 04, 289 (2021) · Zbl 1462.83015 · doi:10.1007/JHEP04(2021)289 |
| [28] | Jackiw, R., Lower Dimensional Gravity, Nucl. Phys. B, 252, 343 (1985) · doi:10.1016/0550-3213(85)90448-1 |
| [29] | C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B126 (1983) 41 [INSPIRE]. |
| [30] | Callan, CG Jr; Giddings, SB; Harvey, JA; Strominger, A., Evanescent black holes, Phys. Rev. D, 45, R1005 (1992) · doi:10.1103/PhysRevD.45.R1005 |
| [31] | Russo, JG; Susskind, L.; Thorlacius, L., The Endpoint of Hawking radiation, Phys. Rev. D, 46, 3444 (1992) · doi:10.1103/PhysRevD.46.3444 |
| [32] | S.M. Christensen and S.A. Fulling, Trace Anomalies and the Hawking Effect, Phys. Rev. D15 (1977) 2088 [INSPIRE]. |
| [33] | A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett. B103 (1981) 207 [INSPIRE]. |
| [34] | W. Kummer and D.V. Vassilevich, Effective action and Hawking radiation for dilaton coupled scalars in two-dimensions, Phys. Rev. D60 (1999) 084021 [hep-th/9811092] [INSPIRE]. |
| [35] | W. Kummer and D.V. Vassilevich, Hawking radiation from dilaton gravity in (1+1)-dimensions: A Pedagogical review, Annalen Phys.8 (1999) 801 [gr-qc/9907041] [INSPIRE]. · Zbl 0948.83045 |
| [36] | Mukhanov, VF; Wipf, A.; Zelnikov, A., On 4-D Hawking radiation from effective action, Phys. Lett. B, 332, 283 (1994) · doi:10.1016/0370-2693(94)91255-6 |
| [37] | M. Burić, V. Radovanović and A.R. Mikovic, One loop correction for Schwarzschild black hole via 2-D dilaton gravity, Phys. Rev. D59 (1999) 084002 [gr-qc/9804083] [INSPIRE]. |
| [38] | R. Balbinot and A. Fabbri, Hawking radiation by effective two-dimensional theories, Phys. Rev. D59 (1999) 044031 [hep-th/9807123] [INSPIRE]. · Zbl 1170.83402 |
| [39] | R. Balbinot and A. Fabbri, 4-D quantum black hole physics from 2-D models?, Phys. Lett. B459 (1999) 112 [gr-qc/9904034] [INSPIRE]. · Zbl 0992.83027 |
| [40] | M. Burić and V. Radovanović, Quantum corrections for anti-evaporating black hole, Phys. Rev. D63 (2001) 044020 [hep-th/0007172] [INSPIRE]. · Zbl 0939.83030 |
| [41] | F.C. Lombardo, F.D. Mazzitelli and J.G. Russo, Energy momentum tensor for scalar fields coupled to the dilaton in two-dimensions, Phys. Rev. D59 (1999) 064007 [gr-qc/9808048] [INSPIRE]. |
| [42] | Y.V. Gusev and A.I. Zelnikov, Two-dimensional effective action for matter fields coupled to the dilaton, Phys. Rev. D61 (2000) 084010 [hep-th/9910198] [INSPIRE]. |
| [43] | D. Hofmann and W. Kummer, Effective action and Hawking flux from covariant perturbation theory, Eur. Phys. J. C40 (2005) 275 [gr-qc/0408088] [INSPIRE]. |
| [44] | D. Hofmann and W. Kummer, IR renormalisation of general effective actions and Hawking flux in 2-D gravity theories, Eur. Phys. J. C48 (2006) 291 [gr-qc/0512163] [INSPIRE]. |
| [45] | R. Balbinot et al., Vacuum polarization in the Schwarzschild space-time and dimensional reduction, Phys. Rev. D63 (2001) 084029 [hep-th/0012048] [INSPIRE]. |
| [46] | R. Balbinot, A. Fabbri, P. Nicolini and P.J. Sutton, Vacuum polarization in two-dimensional static space-times and dimensional reduction, Phys. Rev. D66 (2002) 024014 [hep-th/0202036] [INSPIRE]. |
| [47] | Fabbri, A.; Farese, S.; Navarro-Salas, J., Generalized Virasoro anomaly and stress tensor for dilaton coupled theories, Phys. Lett. B, 574, 309 (2003) · Zbl 1058.83525 · doi:10.1016/j.physletb.2003.09.012 |
| [48] | D.G. Boulware, Quantum Field Theory in Schwarzschild and Rindler Spaces, Phys. Rev. D11 (1975) 1404 [INSPIRE]. |
| [49] | J.B. Hartle and S.W. Hawking, Path Integral Derivation of Black Hole Radiance, Phys. Rev. D13 (1976) 2188 [INSPIRE]. |
| [50] | Israel, W., Thermo field dynamics of black holes, Phys. Lett. A, 57, 107 (1976) · doi:10.1016/0375-9601(76)90178-X |
| [51] | Davies, PCW; Fulling, SA; Unruh, WG, Energy Momentum Tensor Near an Evaporating Black Hole, Phys. Rev. D, 13, 2720 (1976) · doi:10.1103/PhysRevD.13.2720 |
| [52] | W.A. Hiscock, Models of Evaporating Black Holes, Phys. Rev. D23 (1981) 2813 [INSPIRE]. |
| [53] | W.A. Hiscock, Models of Evaporating Black Holes. II. Effects of the Outgoing Created Radiation, Phys. Rev. D23 (1981) 2823 [INSPIRE]. |
| [54] | A. Fabbri and A. J. Navarro-Salas, Modeling Black Hole Evaporation, Imperial College Press and World Scientific (2005) [doi:10.1142/p378]. |
| [55] | W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D14 (1976) 870 [INSPIRE]. |
| [56] | Grumiller, D.; Kummer, W.; Vassilevich, DV, Dilaton gravity in two-dimensions, Phys. Rept., 369, 327 (2002) · Zbl 0998.83038 · doi:10.1016/S0370-1573(02)00267-3 |
| [57] | Bousso, R.; Hawking, SW, Trace anomaly of dilaton coupled scalars in two-dimensions, Phys. Rev. D, 56, 7788 (1997) · doi:10.1103/PhysRevD.56.7788 |
| [58] | Mikovic, AR; Radovanović, V., One loop effective action for spherical scalar field collapse, Class. Quant. Grav., 15, 827 (1998) · Zbl 0941.83016 · doi:10.1088/0264-9381/15/4/010 |
| [59] | Elizalde, E.; Naftulin, S.; Odintsov, SD, Covariant effective action and one loop renormalization of 2-D dilaton gravity with fermionic matter, Phys. Rev. D, 49, 2852 (1994) · doi:10.1103/PhysRevD.49.2852 |
| [60] | Ichinose, S., Weyl anomaly of 2-D dilaton-scalar gravity and hermiticity of system operator, Phys. Rev. D, 57, 6224 (1998) · doi:10.1103/PhysRevD.57.6224 |
| [61] | Dowker, JS, Conformal anomaly in 2-d dilaton scalar theory, Class. Quant. Grav., 15, 1881 (1998) · Zbl 1063.81645 · doi:10.1088/0264-9381/15/7/006 |
| [62] | M.O. Katanaev, W. Kummer, H. Liebl and D.V. Vassilevich, Generalized 2d-dilaton models, the true black hole and quantum integrability, gr-qc/9709010. |
| [63] | S. Nojiri, O. Obregon and S.D. Odintsov, Unified approach to study quantum properties of primordial black holes, wormholes and of quantum cosmology, Mod. Phys. Lett. A14 (1999) 1309 [gr-qc/9907008] [INSPIRE]. |
| [64] | R.M. Wald, Trace Anomaly of a Conformally Invariant Quantum Field in Curved Space-Time, Phys. Rev. D17 (1978) 1477 [INSPIRE]. |
| [65] | B.S. DeWitt, Dynamical theory of groups and fields, Conf. Proc. C630701 (1964) 585 [INSPIRE]. · Zbl 0148.46102 |
| [66] | I.L. Shapiro, Effective Action of Vacuum: Semiclassical Approach, Class. Quant. Grav.25 (2008) 103001 [arXiv:0801.0216] [INSPIRE]. · Zbl 1140.83301 |
| [67] | L.E. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Field and Gravity, Cambridge University Press (2009) [doi:10.1017/CBO9780511813924] [INSPIRE]. · Zbl 1180.81001 |
| [68] | Vassilevich, DV, Heat kernel expansion: User’s manual, Phys. Rept., 388, 279 (2003) · Zbl 1042.81093 · doi:10.1016/j.physrep.2003.09.002 |
| [69] | Barvinsky, AO; Vilkovisky, GA, Beyond the Schwinger-Dewitt Technique: Converting Loops Into Trees and In-In Currents, Nucl. Phys. B, 282, 163 (1987) · doi:10.1016/0550-3213(87)90681-X |
| [70] | Barvinsky, AO; Vilkovisky, GA, Covariant perturbation theory. II. Second order in the curvature. General algorithms, Nucl. Phys. B, 333, 471 (1990) · doi:10.1016/0550-3213(90)90047-H |
| [71] | Barvinsky, AO; Vilkovisky, GA, Covariant perturbation theory. III. Spectral representations of the third order form-factors, Nucl. Phys. B, 333, 512 (1990) · doi:10.1016/0550-3213(90)90048-I |
| [72] | A.O. Barvinsky, Y.V. Gusev, V.V. Zhytnikov and G.A. Vilkovisky, Covariant perturbation theory. IV. Third order in the curvature, arXiv:0911.1168 [INSPIRE]. · Zbl 0809.53075 |
| [73] | A.O. Barvinsky, Y.V. Gusev, G.A. Vilkovisky and V.V. Zhytnikov, Asymptotic behaviors of the heat kernel in covariant perturbation theory, J. Math. Phys.35 (1994) 3543 [gr-qc/9404063] [INSPIRE]. · Zbl 0809.53075 |
| [74] | A.O. Barvinsky and V.F. Mukhanov, New nonlocal effective action, Phys. Rev. D66 (2002) 065007 [hep-th/0203132] [INSPIRE]. |
| [75] | A.O. Barvinsky, Y.V. Gusev, V.F. Mukhanov and D.V. Nesterov, Nonperturbative late time asymptotics for heat kernel in gravity theory, Phys. Rev. D68 (2003) 105003 [hep-th/0306052] [INSPIRE]. |
| [76] | A.O. Barvinsky and D.V. Nesterov, Nonperturbative heat kernel and nonlocal effective action, hep-th/0402043 [INSPIRE]. · Zbl 1178.81195 |
| [77] | V.P. Frolov, P. Sutton and A. Zelnikov, The Dimensional reduction anomaly, Phys. Rev. D61 (2000) 024021 [hep-th/9909086] [INSPIRE]. · Zbl 1097.81672 |
| [78] | G. Cognola and S. Zerbini, On the dimensional reduced theories, in the proceedings of the Conference on Geometrical Aspects of Quantum Fields, Londrina Brazil, April 17-22 (2000), p. 64-72 [doi:10.1142/9789812810366_0006] [hep-th/0008137] [INSPIRE]. · Zbl 0981.81059 |
| [79] | Cognola, G.; Zerbini, S., On the dimensional reduction procedure, Nucl. Phys. B, 602, 383 (2001) · Zbl 0989.83041 · doi:10.1016/S0550-3213(01)00091-8 |
| [80] | Karakhanian, DR; Manvelyan, RP; Mkrtchian, RL, Area preserving structure of 2-d gravity, Phys. Lett. B, 329, 185 (1994) · doi:10.1016/0370-2693(94)90758-7 |
| [81] | R. Jackiw, Another view on massless matter-gravity fields in two-dimensions, hep-th/9501016 [INSPIRE]. |
| [82] | Navarro-Salas, J.; Navarro, M.; Talavera, CF, Weyl invariance and black hole evaporation, Phys. Lett. B, 356, 217 (1995) · doi:10.1016/0370-2693(95)00848-F |
| [83] | J.M. Bardeen, Trace anomaly effective actions — a critique, arXiv:1808.09629 [INSPIRE]. |
| [84] | Bose, S.; Parker, L.; Peleg, Y., Semiinfinite throat as the end state geometry of two-dimensional black hole evaporation, Phys. Rev. D, 52, 3512 (1995) · doi:10.1103/PhysRevD.52.3512 |
| [85] | S. Bose, L. Parker and Y. Peleg, Hawking radiation and unitary evolution, Phys. Rev. Lett.76 (1996) 861 [gr-qc/9508027] [INSPIRE]. · Zbl 0944.81524 |
| [86] | Fabbri, A.; Russo, JG, Soluble models in 2-d dilaton gravity, Phys. Rev. D, 53, 6995 (1996) · doi:10.1103/PhysRevD.53.6995 |
| [87] | Cruz, J.; Navarro-Salas, J., Solvable models for radiating black holes and area preserving diffeomorphisms, Phys. Lett. B, 375, 47 (1996) · Zbl 0997.81564 · doi:10.1016/0370-2693(96)00246-8 |
| [88] | O.B. Zaslavsky, Exactly solvable models of two-dimensional dilaton gravity and quantum eternal black holes, Phys. Rev. D59 (1999) 084013 [hep-th/9804089] [INSPIRE]. |
| [89] | Engelhardt, N.; Wall, AC, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP, 01, 073 (2015) · doi:10.1007/JHEP01(2015)073 |
| [90] | A. Almheiri et al., The entropy of Hawking radiation, Rev. Mod. Phys.93 (2021) 035002 [arXiv:2006.06872] [INSPIRE]. |
| [91] | Fiola, TM; Preskill, J.; Strominger, A.; Trivedi, SP, Black hole thermodynamics and information loss in two-dimensions, Phys. Rev. D, 50, 3987 (1994) · doi:10.1103/PhysRevD.50.3987 |
| [92] | S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys.43 (1975) 199 [Erratum ibid.46 (1976) 206] [INSPIRE]. · Zbl 1378.83040 |
| [93] | A. Almheiri, R. Mahajan and J. Maldacena, Islands outside the horizon, arXiv:1910.11077 [INSPIRE]. · Zbl 1435.83110 |
| [94] | P.-M. Ho and Y. Matsuo, Static Black Holes With Back Reaction From Vacuum Energy, Class. Quant. Grav.35 (2018) 065012 [arXiv:1703.08662] [INSPIRE]. · Zbl 1386.83083 |
| [95] | 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]. |
| [96] | H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A42 (2009) 504007 [arXiv:0905.2562] [INSPIRE]. · Zbl 1186.81017 |
| [97] | Holzhey, C.; Larsen, F.; Wilczek, F., Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B, 424, 443 (1994) · Zbl 0990.81564 · doi:10.1016/0550-3213(94)90402-2 |
| [98] | P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A42 (2009) 504005 [arXiv:0905.4013] [INSPIRE]. · Zbl 1179.81026 |
| [99] | Geng, H., Information Transfer with a Gravitating Bath, SciPost Phys., 10, 103 (2021) · doi:10.21468/SciPostPhys.10.5.103 |
| [100] | Geng, H., Inconsistency of islands in theories with long-range gravity, JHEP, 01, 182 (2022) · Zbl 1521.83109 · doi:10.1007/JHEP01(2022)182 |
| [101] | Karananas, GK; Kehagias, A.; Taskas, J., Islands in linear dilaton black holes, JHEP, 03, 253 (2021) · Zbl 1461.83032 · doi:10.1007/JHEP03(2021)253 |
| [102] | B. Ahn et al., Islands in charged linear dilaton black holes, Phys. Rev. D105 (2022) 046012 [arXiv:2107.07444] [INSPIRE]. |
| [103] | Almheiri, A.; Polchinski, J., Models of AdS_2backreaction and holography, JHEP, 11, 014 (2015) · Zbl 1388.83079 · doi:10.1007/JHEP11(2015)014 |
| [104] | J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE]. · Zbl 1361.81112 |
| [105] | Engelsöy, J.; Mertens, TG; Verlinde, H., An investigation of AdS_2backreaction and holography, JHEP, 07, 139 (2016) · Zbl 1390.83104 · doi:10.1007/JHEP07(2016)139 |
| [106] | Mertens, TG; Turiaci, GJ, Solvable models of quantum black holes: a review on Jackiw-Teitelboim gravity, Living Rev. Rel., 26, 4 (2023) · doi:10.1007/s41114-023-00046-1 |
| [107] | H. Geng et al., Jackiw-Teitelboim Gravity from the Karch-Randall Braneworld, Phys. Rev. Lett.129 (2022) 231601 [arXiv:2206.04695] [INSPIRE]. |
| [108] | Geng, H., Aspects of AdS_2quantum gravity and the Karch-Randall braneworld, JHEP, 09, 024 (2022) · Zbl 1531.83158 · doi:10.1007/JHEP09(2022)024 |
| [109] | P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE]. |
| [110] | D.L. Jafferis, D.K. Kolchmeyer, B. Mukhametzhanov and J. Sonner, Jackiw-Teitelboim gravity with matter, generalized eigenstate thermalization hypothesis, and random matrices, Phys. Rev. D108 (2023) 066015 [arXiv:2209.02131] [INSPIRE]. |
| [111] | D.L. Jafferis, D.K. Kolchmeyer, B. Mukhametzhanov and J. Sonner, Matrix Models for Eigenstate Thermalization, Phys. Rev. X13 (2023) 031033 [arXiv:2209.02130] [INSPIRE]. |
| [112] | Ho, P-M; Kawai, H.; Matsuo, Y.; Yokokura, Y., Back Reaction of 4D Conformal Fields on Static Geometry, JHEP, 11, 056 (2018) · Zbl 1404.83031 · doi:10.1007/JHEP11(2018)056 |
| [113] | P.-M. Ho, Y. Matsuo and Y. Yokokura, Analytic description of semiclassical black-hole geometry, Phys. Rev. D102 (2020) 024090 [arXiv:1912.12855] [INSPIRE]. |
| [114] | A. Fabbri et al., Semiclassical zero-temperature corrections to Schwarzschild spacetime and holography, Phys. Rev. D73 (2006) 104023 [hep-th/0512167] [INSPIRE]. |
| [115] | Fabbri, A., Static quantum corrections to the Schwarzschild spacetime, J. Phys. Conf. Ser., 33, 457 (2006) · doi:10.1088/1742-6596/33/1/059 |
| [116] | J. Arrechea, C. Barceló, R. Carballo-Rubio and L.J. Garay, Schwarzschild geometry counterpart in semiclassical gravity, Phys. Rev. D101 (2020) 064059 [arXiv:1911.03213] [INSPIRE]. · Zbl 1481.83056 |
| [117] | P. Beltrán-Palau, A. del Río and J. Navarro-Salas, Quantum corrections to the Schwarzschild metric from vacuum polarization, Phys. Rev. D107 (2023) 085023 [arXiv:2212.08089] [INSPIRE]. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
