Journal of Holography Applications in Physics
The BTZ Black Hole, Thermofield Double State, and SPT Phases: A Duality
Document Type : Regular article
Author
Centro de Ciências Exatas, Naturais e Tecnológicas, UEMASUL, 65901-480, Imperatriz, MA, Brazil
Abstract
This letter present an investigation of the relationship between the interior and exterior solutions of the BTZ black hole, emphasizing the effects of interchanging spatial and temporal roles. By deriving the interior BTZ metric and its associated thermofield double state, we uncover a duality that complements the exterior solution, providing a comprehensive perspective on the full BTZ black hole geometry. The bulk partition function is shown to correspond to a non-orientable spacetime, specifically a Klein bottle, which establishes links to symmetry-protected topological (SPT) phases characterized by orientation-reversing symmetries. These results align with recent developments in understanding entanglement and topological phases in non-orientable geometries, as well as the role of thermofield double states in the AdS/CFT framework. This work bridges black hole physics, quantum entanglement, and topological invariants, offering fresh insights into the geometric and physical properties of non-orientable spacetimes.
Keywords
Main Subjects
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[32] J. Erdmenger, M. Flory, M. Gerbershagen, M. P. Heller and A. L. Weigel, “Exact Gravity Duals for Simple Quantum Circuits”, SciPost Phys. 13(3), 061 (2022). DOI:10.21468/SciPostPhys.13.3.061 [arXiv:2112.12158 [hep-th]].
[33] D. A. Roberts and D. Stanford, “Two-dimensional conformal field theory and the butterfly effect”, Phys. Rev. Lett. 115(13), 131603 (2015). DOI:10.1103/PhysRevLett.115.131603 [arXiv:1412.5123 [hep-th]].
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[35] N. Pinto-Neto, F. T. Falciano, R. Pereira and E. S. Santini, “The Wheeler-DeWitt Quantization Can Solve the Singularity Problem”, Phys. Rev. D 86, 063504 (2012). DOI:10.1103/PhysRevD.86.063504 [arXiv:1206.4021 [gr-qc]].
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[2] T. Takayanagi, “Holographic Dual of BCFT”, Phys. Rev. Lett. 107, 101602 (2011). [arXiv:1105.5165 [hep-th]]. DOI:10.1103/PhysRevLett.107.101602
[3] F. F. Santos, B. Pourhassan, E. N. Saridakis, O. Sokoliuk, A. Baransky and E. O. Kahya, “Holographic boundary conformal field theory within Horndeski gravity”, JHEP 12, 217 (2025). DOI:10.1007/JHEP12(2024)217 [arXiv:2410.18781 [hep-th]].
[4] S. A. Hartnoll, “Lectures on holographic methods for condensed matter physics”, Class. Quant. Grav. 26, 224002 (2009). DOI:10.1088/0264-9381/26/22/224002 [arXiv:0903.3246 [hep-th]].
[5] M. Fujita, M. Kaminski and A. Karch, “SL(2,Z) Action on AdS/BCFT and Hall Conductivities”, JHEP 07, 150 (2012). DOI:10.1007/JHEP07(2012)150 [arXiv:1204.0012 [hep-th]].
[6] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, “Building a Holographic Superconductor”, Phys. Rev. Lett. 101, 031601 (2008). DOI:10.1103/PhysRevLett.101.031601 [arXiv:0803.3295 [hep-th]].
[7] F. F. Santos and H. Boschi-Filho, “Holographic complexity and residual entropy of a rotating BTZ black hole within Horndeski gravity”, DOI:10.22128/jhap.2025.991.1114 [arXiv:2407.10004 [hep-th]].
[8] J. M. Maldacena, “Eternal black holes in anti-de Sitter”, JHEP 04, 021 (2003). DOI:10.1088/1126-6708/2003/04/021 [arXiv:hep-th/0106112 [hep-th]].
[9] J. Maldacena and L. Susskind, “Cool horizons for entangled black holes”, Fortsch. Phys. 61, 781 (2013). DOI:10.1002/prop.201300020 [arXiv:1306.0533 [hep-th]].
[10] K. Shiozaki, H. Shapourian and S. Ryu, “Many-body topological invariants in fermionic symmetry-protected topological phases: Cases of point group symmetries”, Phys. Rev. B 95(20), 205139 (2017). DOI:10.1103/PhysRevB.95.205139 [arXiv:1609.05970 [condmat.str-el]].
[11] O. Racorean, “The non-orientable spacetime of the eternal black hole”, Phys. Lett. B 868, 139767 (2025). DOI:10.1016/j.physletb.2025.139767
[12] D. Harlow, “Jerusalem Lectures on Black Holes and Quantum Information”, Rev. Mod. Phys. 88, 015002 (2016). DOI:10.1103/RevModPhys.88.015002 [arXiv:1409.1231 [hepth]].
[13] M. Guica and S. F. Ross, “Behind the geon horizon”, Class. Quant. Grav. 32, no.5, 055014 (2015). DOI:10.1088/0264-9381/32/5/055014 [arXiv:1412.1084 [hep-th]].
[14] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, “Holographic Complexity Equals Bulk Action?”, Phys. Rev. Lett. 116(19), 191301 (2016). DOI:10.1103/PhysRevLett.116.191301 [arXiv:1509.07876 [hep-th]].
[15] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, “Complexity, action, and black holes”, Phys. Rev. D 93(8), 086006 (2016). DOI:10.1103/PhysRevD.93.086006 [arXiv:1512.04993 [hep-th]].
[16] D. Stanford and L. Susskind, “Complexity and Shock Wave Geometries”, Phys. Rev. D 90(12), 126007 (2014). DOI:10.1103/PhysRevD.90.126007 [arXiv:1406.2678 [hep-th]].
[17] B. Swingle, “Entanglement Renormalization and Holography”, Phys. Rev. D 86, 065007 (2012). DOI:10.1103/PhysRevD.86.065007 [arXiv:0905.1317 [cond-mat.str-el]].
[18] G. Evenbly and G. Vidal, “Tensor Network States and Geometry”, J. Statist. Phys. 145(4), 891 (2011). DOI:10.1007/s10955-011-0237-4
[19] T. Hartman and J. Maldacena, “Time Evolution of Entanglement Entropy from Black Hole Interiors”, JHEP 05, 014 (2013). DOI:10.1007/JHEP05(2013)014 [arXiv:1303.1080 [hep-th]].
[20] L. Susskind, “Entanglement is not enough”, Fortsch. Phys. 64, 49 (2016). DOI:10.1002/prop.201500095 [arXiv:1411.0690 [hep-th]].
[21] Franzoni, G. “The Klein bottle in its classical shape: a further step towards a good parametrization”, (2009). [arXiv preprint arXiv:0909.5354].
[22] El Mir, C. “Bavard’s systolically extremal Klein bottles and three dimensional applications”, Differential Geometry and its Applications, 81, 101850 (2022).
[23] F. F. Santos, “Probing the Black Hole Interior with Holographic Entanglement Entropy and the Role of AdS/BCFT Correspondence”, DOI:10.1002/prop.70031 [arXiv:2508.21224 [hep-th]].
[24] F. F. Santos and H. Boschi-Filho, “Geometric Josephson junction”, JHEP 01, 135 (2025). DOI:10.1007/JHEP01(2025)135 [arXiv:2407.10008 [hep-th]].
[25] F. F. Santos, M. Bravo-Gaete, O. Sokoliuk and A. Baransky, “AdS/BCFT Correspondence and Horndeski Gravity in the Presence of Gauge Fields: Holographic Paramagnetism/Ferromagnetism Phase Transition”, Fortsch. Phys. 71(12), 2300008 (2023). DOI:10.1002/prop.202300008 [arXiv:2301.03121 [hep-th]].
[26] F. F. Santos, M. Bravo-Gaete, M. M. Ferreira and R. Casana, “Magnetized AdS/BCFT Correspondence in Horndeski Gravity”, Fortsch. Phys. 72(7-8), 2400088 (2024). DOI:10.1002/prop.202400088 [arXiv:2310.17092 [hep-th]].
[27] S. Carlip, “The (2+1)-Dimensional black hole”, Class. Quant. Grav. 12, 2853 (1995). DOI:10.1088/0264-9381/12/12/005 [arXiv:gr-qc/9506079 [gr-qc]].
[28] M. Banados, C. Teitelboim and J. Zanelli, “The Black hole in three-dimensional spacetime”, Phys. Rev. Lett. 69, 1849 (1992). DOI:10.1103/PhysRevLett.69.1849 [arXiv:hepth/9204099 [hep-th]].
[29] T. G. Mertens and G. J. Turiaci, “Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity”, Living Rev. Rel. 26(1), 4 (2023). DOI:10.1007/s41114- 023-00046-1 [arXiv:2210.10846 [hep-th]].
[30] J. Erdmenger, A. L. Weigel, M. Gerbershagen and M. P. Heller, “From complexity geometry to holographic spacetime”, Phys. Rev. D 108(10), 106020 (2023). DOI:10.1103/PhysRevD.108.106020 [arXiv:2212.00043 [hep-th]].
[31] E. Cáceres, R. Carrasco, V. Patil, J. F. Pedraza and A. Svesko, “The landscape of complexity measures in 2D gravity”, [arXiv:2503.20943 [hep-th]].
[32] J. Erdmenger, M. Flory, M. Gerbershagen, M. P. Heller and A. L. Weigel, “Exact Gravity Duals for Simple Quantum Circuits”, SciPost Phys. 13(3), 061 (2022). DOI:10.21468/SciPostPhys.13.3.061 [arXiv:2112.12158 [hep-th]].
[33] D. A. Roberts and D. Stanford, “Two-dimensional conformal field theory and the butterfly effect”, Phys. Rev. Lett. 115(13), 131603 (2015). DOI:10.1103/PhysRevLett.115.131603 [arXiv:1412.5123 [hep-th]].
[34] S. H. Shenker and D. Stanford, “Black holes and the butterfly effect”, JHEP 03, 067 (2014). DOI:10.1007/JHEP03(2014)067 [arXiv:1306.0622 [hep-th]].
[35] N. Pinto-Neto, F. T. Falciano, R. Pereira and E. S. Santini, “The Wheeler-DeWitt Quantization Can Solve the Singularity Problem”, Phys. Rev. D 86, 063504 (2012). DOI:10.1103/PhysRevD.86.063504 [arXiv:1206.4021 [gr-qc]].
[36] H. J. Matschull, “Solutions to the Wheeler-DeWitt constraint of canonical gravity coupled to scalar matter fields”, Class. Quant. Grav. 10, L149 (1993). DOI:10.1088/0264- 9381/10/9/007 [arXiv:gr-qc/9305025 [gr-qc]].
[37] C. H. Chien, W. Song, G. Tumurtushaa and D. h. Yeom, “Quantum Resolution of Mass Inflation in Reissner-Nordström Interiors via Wheeler-DeWitt Equation”, [arXiv:2509.13483 [gr-qc]].
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Volume 5, Issue 4
October 2025Pages 82-94
- Receive Date: 19 August 2025
- Revise Date: 27 September 2025
- Accept Date: 28 September 2025