- 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

W˜1+ subclass: Extending the topological classification of black hole thermodynamics

Wangyu Ai (艾望余) and Di Wu (吴迪)*

  • *Contact author: wdcwnu@163.com
PDF

Phys. Rev. D 112, 124024 – Published 4 December, 2025

DOI: https://doi.org/10.1103/dy7y-j24r

Abstract

In this paper, we identify a novel topological subclass, dubbed W˜1+, in the thermodynamics of higher odd-dimensional, multiply rotating Kerr anti–de Sitter (AdS) black holes. This discovery extends the established topological classification beyond the five classes and two subclasses previously known. The W˜1+ subclass exhibits a unique and previously unreported stability profile: it admits a thermodynamically stable small black hole state in the low-temperature limit, while in the high-temperature limit, the phase space simultaneously contains one stable large black hole, one stable small black hole, and one unstable small black hole state. Our analysis, which treats black hole solutions as topological defects, reveals a richer landscape of black hole thermodynamics than previously understood and necessitates an expansion of the topological classification scheme to accommodate this new phenomenology.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (69)

  1. J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973).
  2. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975); 46, 206(E) (1976).
  3. J. D. E. Creighton and R. B. Mann, Quasilocal thermodynamics of dilaton gravity coupled to gauge fields, Phys. Rev. D 52, 4569 (1995).
  4. M. M. Caldarelli, G. Cognola, and D. Klemm, Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories, Classical Quantum Gravity 17, 399 (2000).
  5. S. Wang, S.-Q. Wu, F. Xie, and L. Dan, The first laws of thermodynamics of the (2+1)-dimensional BTZ black holes and Kerr-de Sitter spacetimes, Chin. Phys. Lett. 23, 1096 (2006).
  6. D. Kastor, S. Ray, and J. Traschen, Enthalpy and the mechanics of AdS black holes, Classical Quantum Gravity 26, 195011 (2009).
  7. M. Cvetič, G. W. Gibbons, D. Kubizňák, and C. N. Pope, Black hole enthalpy and an entropy inequality for the thermodynamic volume, Phys. Rev. D 84, 024037 (2011).
  8. D. Kubizňák, R. B. Mann, and M. Teo, Black hole chemistry: Thermodynamics with Lambda, Classical Quantum Gravity 34, 063001 (2017).
  9. Y. Xiao, Y. Tian, and Y.-X. Liu, Extended black hole thermodynamics from extended Iyer-Wald formalism, Phys. Rev. Lett. 132, 021401 (2024).
  10. K. Hajian and B. Tekin, Coupling constants as conserved charges in black hole thermodynamics, Phys. Rev. Lett. 132, 191401 (2024).
  11. W. Cong, D. Kubizňák, and R. B. Mann, Thermodynamics of AdS black holes: Critical behavior of the central charge, Phys. Rev. Lett. 127, 091301 (2021).
  12. W. Cong, D. Kubizňák, R. B. Mann, and M. R. Visser, Holographic CFT phase transitions and criticality for charged AdS black holes, J. High Energy Phys. 08 (2022) 174.
  13. Z. Gao and L. Zhao, Restricted phase space thermodynamics for AdS black holes via holography, Classical Quantum Gravity 39, 075019 (2022).
  14. M. B. Ahmed, W. Cong, D. Kubizňák, R. B. Mann, and M. R. Visser, Holographic dual of extended black hole thermodynamics, Phys. Rev. Lett. 130, 181401 (2023).
  15. T.-F. Gong, J. Jiang, and M. Zhang, Holographic thermodynamics of rotating black holes, J. High Energy Phys. 06 (2023) 105.
  16. M. B. Ahmed, W. Cong, D. Kubizňák, R. B. Mann, and M. R. Visser, Holographic CFT phase transitions and criticality for rotating AdS black holes, J. High Energy Phys. 08 (2023) 142.
  17. M. Zhang, W.-D. Tan, M. Lu, D. Bhattacharya, J. Yang, and R. B. Mann, Finite-cutoff holographic thermodynamics, arXiv:2507.01010.
  18. C. V. Johnson, Holographic heat engines, Classical Quantum Gravity 31, 205002 (2014).
  19. A. A. Balushi, R. A. Hennigar, H. K. Kunduri, and R. B. Mann, Holographic complexity and thermodynamic volume, Phys. Rev. Lett. 126, 101601 (2021).
  20. A. A. Balushi, R. A. Hennigar, H. K. Kunduri, and R. B. Mann, Holographic complexity of rotating black holes, J. High Energy Phys. 05 (2021) 226.
  21. Å. Folkestad, Penrose inequality as a constraint on the low energy limit of quantum gravity, Phys. Rev. Lett. 130, 121501 (2023).
  22. M. Amo, A. M. Frassino, and R. A. Hennigar, Entropy bounds for rotating AdS black holes, Phys. Rev. Lett. 131, 241401 (2023).
  23. A. M. Frassino, R. A. Hennigar, J. F. Pedraza, and A. Svesko, Quantum inequalities for quantum black holes, Phys. Rev. Lett. 133, 181501 (2024).
  24. S. W. Hawking and D. N. Page, Thermodynamics of black holes in anti-de Sitter space, Commun. Math. Phys. 87, 577 (1983).
  25. D. Kubizňák and R. B. Mann, P-V criticality of charged AdS black holes, J. High Energy Phys. 07 (2012) 033.
  26. N. Altamirano, D. Kubizňák, and R. B. Mann, Reentrant phase transitions in rotating anti-de Sitter black holes, Phys. Rev. D 88, 101502 (2013).
  27. R. A. Hennigar, R. B. Mann, and E. Tjoa, Superfluid black holes, Phys. Rev. Lett. 118, 021301 (2017).
  28. R. Li, K. Zhang, J. Yang, R. B. Mann, and J. Wang, Critical slowing down of black hole phase transition and kinetic crossover in supercritical regime, Phys. Rev. D 112, 064004 (2025).
  29. S. Wang, X. Li, Y. Jin, and L. Li, Analogous supercritical crossovers in black holes and water, arXiv:2506.10808.
  30. S.-W. Wei, S.-P. Wu, Y.-P. Zhang, and Y.-X. Liu, Universal exponents of black hole phase transition at zero-temperature limit, arXiv:2507.10028.
  31. S.-W. Wei, Y.-X. Liu, and R. B. Mann, Black hole solutions as topological thermodynamic defects, Phys. Rev. Lett. 129, 191101 (2022).
  32. S.-W. Wei, Y.-X. Liu, and R. B. Mann, Universal topological classifications of black hole thermodynamics, Phys. Rev. D 110, L081501 (2024).
  33. D. Wu, W. Liu, S.-Q. Wu, and R. B. Mann, Novel topological classes in black hole thermodynamics, Phys. Rev. D 111, L061501 (2025).
  34. S.-P. Wu, S.-J. Yang, and S.-W. Wei, Extended thermodynamical topology of black hole, arXiv:2508.01614.
  35. D. Wu, Topological classes of rotating black holes, Phys. Rev. D 107, 024024 (2023).
  36. C. H. Liu and J. Wang, The topological natures of the Gauss-Bonnet black hole in AdS space, Phys. Rev. D 107, 064023 (2023).
  37. D. Wu and S.-Q. Wu, Topological classes of thermodynamics of rotating AdS black holes, Phys. Rev. D 107, 084002 (2023).
  38. D. Wu, Classifying topology of consistent thermodynamics of the four-dimensional neutral Lorentzian NUT-charged spacetimes, Eur. Phys. J. C 83, 365 (2023).
  39. D. Wu, Consistent thermodynamics and topological classes for the four-dimensional Lorentzian charged Taub-NUT spacetimes, Eur. Phys. J. C 83, 589 (2023).
  40. D. Wu, Topological classes of thermodynamics of the four-dimensional static accelerating black holes, Phys. Rev. D 108, 084041 (2023).
  41. D. Wu, S.-Y. Gu, X.-D. Zhu, Q.-Q. Jiang, and S.-Z. Yang, Topological classes of thermodynamics of the static multi-charge AdS black holes in gauged supergravities: Novel temperature-dependent thermodynamic topological phase transition, J. High Energy Phys. 06 (2024) 213.
  42. X.-D. Zhu, D. Wu, and D. Wen, Topological classes of thermodynamics of the rotating charged AdS black holes in gauged supergravities, Phys. Lett. B 856, 138919 (2024).
  43. H. Chen, D. Wu, M.-Y. Zhang, H. Hassanabadi, and Z.-W. Long, Thermodynamic topology of phantom AdS black holes in massive gravity, Phys. Dark Universe 46, 101617 (2024).
  44. Z.-Q. Chen and S.-W. Wei, Thermodynamical topology with multiple defect curves for dyonic AdS black holes, Eur. Phys. J. C 84, 1294 (2024).
  45. W. Liu, L. Zhang, D. Wu, and J. Wang, Thermodynamic topological classes of the rotating, accelerating black holes, Classical Quantum Gravity 42, 125007 (2025).
  46. X.-D. Zhu, W. Liu, and D. Wu, Universal thermodynamic topological classes of rotating black holes, Phys. Lett. B 860, 139163 (2025).
  47. Y. Chen, X.-D. Zhu, and D. Wu, Universal thermodynamic topological classes of three-dimensional BTZ black holes, Phys. Lett. B 865, 139482 (2025).
  48. H. Chen, D. Wu, M.-Y. Zhang, S. Zare, H. Hassanabadi, B. C. Lütfüoğlu, and Z.-W. Long, Universal thermodynamic topological classes of static black holes in Conformal Killing Gravity, Eur. Phys. J. C 85, 828 (2025).
  49. D. Wu and S.-Q. Wu, Thermodynamics and topological classifications of static non-extremal four-charge AdS black hole in the five-dimensional N=2, STU−W2U gauged supergravity, arXiv:2510.20164.
  50. P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Light ring stability in ultra-compact objects, Phys. Rev. Lett. 119, 251102 (2017).
  51. P. V. P. Cunha and C. A. R. Herdeiro, Stationary black holes and light rings, Phys. Rev. Lett. 124, 181101 (2020).
  52. S.-W. Wei, Topological charge and black hole photon spheres, Phys. Rev. D 102, 0604039 (2020).
  53. M. Guo and S. Gao, Universal properties of light rings for stationary axisymmetric spacetimes, Phys. Rev. D 103, 104031 (2021).
  54. W. Liu, D. Wu, and J. Wang, Light rings and shadows of static black holes in effective quantum gravity, Phys. Lett. B 858, 139052 (2024).
  55. W. Liu, D. Wu, and J. Wang, Light rings and shadows of static black holes in effective quantum gravity II: A new solution without Cauchy horizons, Phys. Lett. B 868, 139742 (2025).
  56. S.-W. Wei and Y.-X. Liu, Topology of equatorial timelike circular orbits around stationary black holes, Phys. Rev. D 107, 064006 (2023).
  57. X. Ye and S.-W. Wei, Topological study of equatorial timelike circular orbit for spherically symmetric (hairy) black holes, J. Cosmol. Astropart. Phys. 07 (2023) 049.
  58. J. Yin, J. Jiang, and M. Zhang, Kinematic topologies of black holes, Phys. Rev. D 108, 044077 (2023).
  59. X. Ye and S.-W. Wei, Novel topological phenomena of timelike circular orbits for charged test particles, Classical Quantum Gravity 42, 025020 (2025).
  60. G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15, 2752 (1977).
  61. J. W. York, Black-hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D 33, 2092 (1986).
  62. S.-J. Yang, R. Zhou, S. W. Wei, and Y.-X. Liu, Dynamics and kinetics of phase transition for Kerr AdS black hole on free energy landscape, Phys. Rev. D 105, 084030 (2022).
  63. R. Li and J. Wang, Generalized free energy landscape of a black hole phase transition, Phys. Rev. D 106, 106015 (2022).
  64. Y.-S. Duan and M.-L. Ge, SU (2) gauge theory and electrodynamics of N moving magnetic monopoles, Sci. Sin. 9, 1072 (1979).
  65. Y.-S. Duan, S. Li, and G.-H. Yang, The bifurcation theory of the Gauss-Bonnet-Chern topological current and Morse function, Nucl. Phys. B514, 705 (1998).
  66. L.-B. Fu, Y.-S. Duan, and H. Zhang, Evolution of the Chern-Simons vortices, Phys. Rev. D 61, 045004 (2000).
  67. G. W. Gibbons, H. Lü, D. N. Page, and C. N. Pope, Rotating black holes in higher dimensions with a cosmological constant, Phys. Rev. Lett. 93, 171102 (2004).
  68. R. A. Hennigar, D. Kubizňák, R. B. Mann, and N. Musoke, Ultraspinning limits and super-entropic black holes, J. High Energy Phys. 06 (2015) 096.
  69. G. W. Gibbons, M. J. Perry, and C. N. Pope, The first law of thermodynamics for Kerr-anti-de Sitter black holes, Classical Quantum Gravity 22, 1503 (2005).

Outline

Information

Phys. Rev. D 112, 124024– Published 4 December, 2025
  • Received 3 September 2025
  • Accepted 17 November 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