- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Effect of a nonextensive parameter on the Page curve

Ankit Anand1,*, Dinesh Kumar2,†, and Aditya Singh3,‡

  • *Contact author: ankitanadp94@gmail.com
  • †Contact author: kumar.dinesh@galgotiasuniversity.edu.in
  • ‡Contact author: 24pr0148@iitism.ac.in
PDF

Phys. Rev. D 112, 026014 – Published 8 July, 2025

DOI: https://doi.org/10.1103/njpw-wlf3

Abstract

This work employs the quantum extremal surface framework to compute the Page curve for black holes corrected by nonextensive entropy. The entropy of Hawking radiation increases linearly with time, leading to the persistence of the information paradox for nonextensive entropy-corrected black holes. At late time, we extremize the generalized entropy functional; incorporating contributions from both matter and the quantum extremal island, we establish that the entanglement entropy of Hawking radiation saturates to the nonextensive extension of the Bekenstein-Hawking entropy. Finally, we study the dependence of nonextensive parameters on the Page time.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (117)

  1. C. Tsallis and L. J. L. Cirto, Black hole thermodynamical entropy, Eur. Phys. J. C 73, 2487 (2013).
  2. A. Rényi, On measures of information and entropy, Proceedings IV Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, 1961), pp. 547–561.
  3. C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479 (1988).
  4. J. D. Barrow, The area of a rough black hole, Phys. Lett. B 808, 135643 (2020).
  5. A. Sayahian Jahromi, S. A. Moosavi, H. Moradpour, J. P. Morais Graça, I. P. Lobo, I. G. Salako, and A. Jawad, Generalized entropy formalism and a new holographic dark energy model, Phys. Lett. B 780, 21 (2018).
  6. G. Kaniadakis, Statistical mechanics in the context of special relativity. II., Phys. Rev. E 72, 036108 (2005).
  7. N. Drepanou, A. Lymperis, E. N. Saridakis, and K. Yesmakhanova, Kaniadakis holographic dark energy and cosmology, Eur. Phys. J. C 82, 449 (2022).
  8. S. Nojiri, S. D. Odintsov, and V. Faraoni, Area-law versus Rényi and Tsallis black hole entropies, Phys. Rev. D 104, 084030 (2021).
  9. J. F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50, 3874 (1994).
  10. X. Calmet and F. Kuipers, Quantum gravitational corrections to the entropy of a Schwarzschild black hole, Phys. Rev. D 104, 066012 (2021).
  11. R. C. Delgado, Quantum gravitational corrections to the entropy of a Reissner–Nordström black hole, Eur. Phys. J. C 82, 272 (2022); 83, 468(E) (2023).
  12. R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48, R3427 (1993).
  13. P. A. Cano, S. Chimento, R. Linares, T. Ortín, and P. F. Ramírez, α′ corrections of Reissner-Nordström black holes, J. High Energy Phys. 02 (2020) 031.
  14. M. Yoon, J. Ha, and W. Kim, Entropy of Reissner-Nordstrom black holes with minimal length revisited, Phys. Rev. D 76, 047501 (2007).
  15. A. Singh, P. Mukherjee, and C. Bhamidipati, Thermodynamic curvature of charged black holes with AdS2 horizons, Phys. Rev. D 108, 106011 (2023).
  16. M. M. Akbar and S. Das, Entropy corrections for Schwarzschild and Reissner-Nordström black holes, Classical Quantum Gravity 21, 1383 (2004).
  17. J. Sadeghi, B. Pourhassan, and F. Rahimi, Logarithmic corrections of charged hairy black holes in (2+1) dimensions, Can. J. Phys. 92, 1638 (2014).
  18. L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50, 2700 (1994).
  19. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975); 46, 206(E) (1976).
  20. S. W. Hawking, Black hole explosions, Nature (London) 248, 30 (1974).
  21. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975); 46, 206(E) (1976).
  22. S. W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14, 2460 (1976).
  23. J. M. Maldacena, A. Strominger, and E. Witten, Black hole entropy in M theory, J. High Energy Phys. 12 (1997) 002.
  24. S. N. Solodukhin, Entanglement entropy of black holes, Living Rev. Relativity 14, 8 (2011).
  25. S. N. Solodukhin, The conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51, 609 (1995).
  26. D. V. Fursaev, Temperature and entropy of a quantum black hole and conformal anomaly, Phys. Rev. D 51, 5352 (1995).
  27. A. Sen, Logarithmic corrections to Schwarzschild and other non-extremal black hole entropy in different dimensions, J. High Energy Phys. 04 (2013) 156.
  28. B. K. El-Menoufi, Quantum gravity of Kerr-Schild spacetimes and the logarithmic correction to Schwarzschild black hole entropy, J. High Energy Phys. 05 (2016) 035.
  29. B. K. El-Menoufi, Quantum gravity effects on the thermodynamic stability of 4D Schwarzschild black hole, J. High Energy Phys. 08 (2017) 068.
  30. R. G. Cai, L. M. Cao, and N. Ohta, Black holes in gravity with conformal anomaly and logarithmic term in black hole entropy, J. High Energy Phys. 04 (2010) 082.
  31. S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Classical Quantum Gravity 17, 4175 (2000).
  32. R. Banerjee and B. R. Majhi, Quantum tunneling beyond semiclassical approximation, J. High Energy Phys. 06 (2008) 095.
  33. R. Banerjee and B. R. Majhi, Quantum tunneling, trace anomaly and effective metric, Phys. Lett. B 674, 218 (2009).
  34. R. K. Kaul and P. Majumdar, Logarithmic correction to the Bekenstein-Hawking entropy, Phys. Rev. Lett. 84, 5255 (2000).
  35. D. N. Page, Information in black hole radiation, Phys. Rev. Lett. 71, 3743 (1993).
  36. D. N. Page, Time dependence of Hawking radiation entropy, J. Cosmol. Astropart. Phys. 09 (2013) 028.
  37. G. Penington, Entanglement wedge reconstruction and the information paradox, J. High Energy Phys. 09 (2020) 002.
  38. A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, J. High Energy Phys. 12 (2019) 063.
  39. A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, J. High Energy Phys. 03 (2020) 149.
  40. A. Almheiri, R. Mahajan, and J. Maldacena, Islands outside the horizon, arXiv:1910.11077.
  41. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006).
  42. V. E. Hubeny, M. Rangamani, and T. Takayanagi, A covariant holographic entanglement entropy proposal, J. High Energy Phys. 07 (2007) 062.
  43. N. Engelhardt and A. C. Wall, Quantum extremal surfaces: Holographic entanglement entropy beyond the classical regime, J. High Energy Phys. 01 (2015) 073.
  44. C. G. Callan, Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333, 55 (1994).
  45. C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B424, 443 (1994).
  46. P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42, 504005 (2009).
  47. H. Geng, Replica wormholes and entanglement islands in the Karch-Randall braneworld, J. High Energy Phys. 01 (2025) 063.
  48. G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, Replica wormholes and the black hole interior, J. High Energy Phys. 03 (2022) 205.
  49. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Replica wormholes and the entropy of Hawking radiation, J. High Energy Phys. 05 (2020) 013.
  50. C. Tsallis and L. J. L. Cirto, Black hole thermodynamical entropy, Eur. Phys. J. C 73, 2487 (2013).
  51. A. Bialas and W. Czyz, Rényi entropies of a black hole from Hawking radiation, Europhys. Lett. 83, 60009 (2008).
  52. E. M. C. Abreu and J. Ananias Neto, Black holes thermodynamics from a dual Kaniadakis entropy, Europhys. Lett. 133, 49001 (2021).
  53. R. Nakarachinda, E. Hirunsirisawat, L. Tannukij, and P. Wongjun, Effective thermodynamical system of Schwarzschild–de Sitter black holes from Rényi statistics, Phys. Rev. D 104, 064003 (2021).
  54. K. Mejrhit and R. Hajji, A semi-classical estimate for the q-parameter and decay time with Tsallis entropy of black holes in quantum geometry, Eur. Phys. J. C 80, 1060 (2020).
  55. D. Samart and P. Channuie, AdS to dS phase transition mediated by thermalon in Einstein-Gauss-Bonnet gravity from Rényi statistics, Nucl. Phys. B989, 116140 (2023).
  56. C. V. Johnson, Physical generalizations of the Rényi entropy, Int. J. Mod. Phys. D 28, 1950091 (2019).
  57. V. G. Czinner and H. Iguchi, Thermodynamics, stability and Hawking–Page transition of Kerr black holes from Rényi statistics, Eur. Phys. J. C 77, 892 (2017).
  58. W. Y. Wen, Thermodynamic metric of deformed Schwarzschild black holes, Int. J. Mod. Phys. D 26, 1750106 (2017).
  59. X. Dong, The gravity dual of Rényi entropy, Nat. Commun. 7, 12472 (2016).
  60. V. G. Czinner and H. Iguchi, Rényi entropy and the thermodynamic stability of black holes, Phys. Lett. B 752, 306 (2016).
  61. T. Nishioka, The gravity dual of supersymmetric Rényi entropy, J. High Energy Phys. 07 (2014) 061.
  62. X. Huang and Y. Zhou, N=4 Super-Yang-Mills on conic space as hologram of STU topological black hole, J. High Energy Phys. 02 (2015) 068.
  63. J. Ren, Analytic critical points of charged Rényi entropies from hyperbolic black holes, J. High Energy Phys. 05 (2021) 080.
  64. D. Amati, M. Ciafaloni, and G. Veneziano, Can space-time be probed below the string size?, Phys. Lett. B 216, 41 (1989).
  65. S. Hossenfelder, Minimal length scale scenarios for quantum gravity, Living Rev. Relativity 16, 2 (2013).
  66. K. Konishi, G. Paffuti, and P. Provero, Minimum physical length and the generalized uncertainty principle in string theory, Phys. Lett. B 234, 276 (1990).
  67. A. Kempf, G. Mangano, and R. B. Mann, Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D 52, 1108 (1995).
  68. C. Rovelli, Black hole entropy from loop quantum gravity, Phys. Rev. Lett. 77, 3288 (1996).
  69. R. J. Adler and D. I. Santiago, On gravity and the uncertainty principle, Mod. Phys. Lett. A 14, 1371 (1999).
  70. Z. W. Feng, X. Zhou, S. Q. Zhou, and D. D. Feng, Rainbow gravity corrections to the information flux of a black hole and the sparsity of Hawking radiation, Ann. Phys. (Amsterdam) 416, 168144 (2020).
  71. S. Schuster, Black hole evaporation: Sparsity in analogue and general relativistic space-times, arXiv:1901.05648.
  72. S. Schuster, Sparsity of Hawking radiation in D+1 space-time dimensions for massless and massive particles, Classical Quantum Gravity 38, 4 (2021).
  73. A. Paul and B. R. Majhi, Hawking evaporation cascade in the presence of backreaction effect, Int. J. Mod. Phys. A 32, 1750088 (2017).
  74. A. Chowdhury and N. Banerjee, Greybody factor and sparsity of Hawking radiation from a charged spherical black hole with scalar hair, Phys. Lett. B 805, 135417 (2020).
  75. A. Alonso-Serrano, M. P. Dkabrowski, and H. Gohar, Generalized uncertainty principle impact onto the black holes information flux and the sparsity of Hawking radiation, Phys. Rev. D 97, 044029 (2018).
  76. J. T. Firouzjaee and G. F. R. Ellis, Particle creation rate for dynamical black holes, Eur. Phys. J. C 76, 620 (2016).
  77. D. N. Page, Particle emission rates from a black hole. 3. Charged leptons from a nonrotating hole, Phys. Rev. D 16, 2402 (1977).
  78. D. N. Page, Particle emission rates from a black hole. 2. Massless particles from a rotating hole, Phys. Rev. D 14, 3260 (1976).
  79. D. N. Page, Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole, Phys. Rev. D 13, 198 (1976).
  80. F. Gray, S. Schuster, A. Van-Brunt, and M. Visser, The Hawking cascade from a black hole is extremely sparse, Classical Quantum Gravity 33, 115003 (2016).
  81. L. H. Wang and M. S. Ma, Barrow black holes and the minimal length, Phys. Lett. B 831, 137181 (2022).
  82. G. E. Volovik, Tsallis-Cirto entropy of black hole and black hole atom, JETP Lett. 121, 243 (2025).
  83. G. G. Luciano, Tsallis statistics and generalized uncertainty principle, Eur. Phys. J. C 81, 672 (2021).
  84. R. Nakarachinda, C. Promsiri, L. Tannukij, and P. Wongjun, Thermodynamics of black holes with Rényi entropy from classical gravity, Nucl. Phys. B1011, 116796 (2025).
  85. T. S. Biró and V. G. Czinner, A q-parameter bound for particle spectra based on black hole thermodynamics with Rényi entropy, Phys. Lett. B 726, 861 (2013).
  86. A. Alonso-Serrano, M. P. Dabrowski, and H. Gohar, Nonextensive black hole entropy and quantum gravity effects at the last stages of evaporation, Phys. Rev. D 103, 026021 (2021).
  87. I. Cimidiker, M. P. Daabrowski, and H. Gohar, Generalized uncertainty principle impact on nonextensive black hole thermodynamics, Classical Quantum Gravity 40, 145001 (2023).
  88. G. Kaniadakis, Statistical mechanics in the context of special relativity, Phys. Rev. E 66, 056125 (2002).
  89. G. G. Luciano, Gravity and cosmology in kaniadakis statistics: Current status and future challenges, Entropy 24, 1712 (2022).
  90. G. G. Luciano and E. Saridakis, P−v criticalities, phase transitions and geometrothermodynamics of charged AdS black holes from Kaniadakis statistics, J. High Energy Phys. 12 (2023) 114.
  91. Y. Ling, B. Hu, and X. Li, Modified dispersion relations and black hole physics, Phys. Rev. D 73, 087702 (2006).
  92. A. Sheykhi, Entropic corrections to Friedmann equations, Phys. Rev. D 81, 104011 (2010).
  93. Y. Liu, Non-extensive statistical mechanics and the thermodynamic stability of FRW Universe, Europhys. Lett. 138, 39001 (2022).
  94. E. P. Verlinde, Emergent gravity and the dark Universe, SciPost Phys. 2, 016 (2017).
  95. A. Majhi, Non-extensive statistical mechanics and black hole entropy from quantum geometry, Phys. Lett. B 775, 32 (2017).
  96. G. Landi, An introduction to noncommutative spaces and their geometry, Lect. Notes Phys., M: Monogr. 51, 1 (1997).
  97. J. Magueijo and L. Smolin, Gravity’s rainbow, Classical Quantum Gravity 21, 1725 (2004).
  98. L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34, 373 (1986).
  99. M. Srednicki, Entropy and area, Phys. Rev. Lett. 71, 666 (1993).
  100. T. Faulkner, A. Lewkowycz, and J. Maldacena, Quantum corrections to holographic entanglement entropy, J. High Energy Phys. 11 (2013) 074.
  101. X. Dong, A. Lewkowycz, and M. Rangamani, Deriving covariant holographic entanglement, J. High Energy Phys. 11 (2016) 028.
  102. T. Nutma, xTras: A field-theory inspired xact package for Mathematica, Comput. Phys. Commun. 185, 1719 (2014).
  103. K. Hashimoto, N. Iizuka, and Y. Matsuo, Islands in Schwarzschild black holes, J. High Energy Phys. 06 (2020) 085.
  104. S. Di Gennaro, H. Xu, and Y. C. Ong, How barrow entropy modifies gravity: With comments on Tsallis entropy, Eur. Phys. J. C 82, 1066 (2022).
  105. A. Anand and R. Campos Delgado, Modified gravity theories from the Barrow hypothesis, Europhys. Lett. 146, 69001 (2024).
  106. K. Jusufi, M. Azreg-Aïnou, M. Jamil, and E. N. Saridakis, Constraints on barrow entropy from M87 and S2 star observations, Universe 8, 102 (2022).
  107. E. M. C. Abreu, Surface gravity analysis in Gauss-Bonnet and Barrow black holes, Phys. Lett. B 861, 139236 (2025).
  108. S. Nojiri, S. D. Odintsov, and V. Faraoni, From nonextensive statistics and black hole entropy to the holographic dark Universe, Phys. Rev. D 105, 044042 (2022).
  109. M. Masi, A step beyond Tsallis and Rényi entropies, Phys. Lett. A 338, 217 (2005).
  110. E. Akturk, G. B. Bagci, and R. Sever, Is Sharma-Mittal entropy really a step beyond Tsallis and Rényi entropies?, arXiv:cond-mat/0703277.
  111. A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, J. High Energy Phys. 08 (2013) 090.
  112. S. Nojiri, S. D. Odintsov, and T. Paul, Early and late Universe holographic cosmology from a new generalized entropy, Phys. Lett. B 831, 137189 (2022).
  113. M. Alishahiha, A. Faraji Astaneh, and A. Naseh, Island in the presence of higher derivative terms, J. High Energy Phys. 02 (2021) 035.
  114. A. Anand, Page curve and island in EGB gravity, Nucl. Phys. B993, 116284 (2023).
  115. G. Yadav, Page curves of Reissner–Nordström black hole in HD gravity, Eur. Phys. J. C 82, 904 (2022).
  116. O. Trivedi, Landauer’s principle and information at the cosmological horizon, arXiv:2407.15231.
  117. S. D. Odintsov, T. Paul, and S. SenGupta, Natural validation of the second law of thermodynamics in cosmology, Phys. Rev. D 111, 043544 (2025).

Outline

Information

Phys. Rev. D 112, 026014– Published 8 July, 2025
  • Received 8 February 2025
  • Accepted 16 June 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation