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Frozen states of charged boson stars

Yves Brihaye1 and Betti Hartmann2

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Phys. Rev. D 113, 065023 – Published 25 March, 2026

DOI: https://doi.org/10.1103/djk9-pjtb

Abstract

In this paper, we study frozen states of charged boson stars. These solutions are globally regular and exist in a U(1) gauged scalar field model minimally coupled to gravity for suitable choices of the coupling constants. These configurations are field theoretical realizations of the Mazur-Mottola solution with a de Sitter interior, a black hole exterior and a thin shell that interpolates between the two and replaces the event horizon. We demonstrate that standard electrodynamics is sufficient to find these frozen states, but that the self-interaction of the scalar field is crucial. For a sixth-order self-interaction of the scalar field, solutions exist that possess k=2n, n∈N nodes in the scalar field. The interval in which these nodes exist corresponds to an additional intermediate shell that corresponds to vacuum. Adding Horndeski vector-tensor gravity to the model allows the frozen states to exist without self-interaction though. The frozen states possess one stable and one unstable lightring, the former inside the thin shell, the latter in the black hole exterior.

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References (34)

  1. R. Penrose, Gravitational collapse: The role of general relativity, Riv. Nuovo Cimento 1, 252 (1969).
  2. C. Bambi, R. Brustein, V. Cardoso, A. Chael, U. Danielsson, S. Giri, A. Gupta, P. Heidmann, L. Lehner, S. Liebling et al., Black hole mimickers: from theory to observation, arXiv:2505.09014.
  3. D. J. Kaup, Klein-Gordon Geon, Phys. Rev. 172, 1331 (1968).
  4. F. E. Schunck and E. W. Mielke, Boson stars: Rotation, formation, and evolution, Gen. Relativ. Gravit. 31, 787 (1999).
  5. R. Friedberg, T. D. Lee, and Y. Pang, Mini—soliton stars, Phys. Rev. D 35, 3640 (1987).
  6. P. Jetzer, Boson stars, Phys. Rep. 220, 163 (1992).
  7. F. E. Schunck and E. W. Mielke, General relativistic boson stars, Classical Quantum Gravity 20, R301 (2003).
  8. S. L. Liebling and C. Palenzuela, Dynamical boson stars, Living Rev. Relativity 26, 1 (2023).
  9. J. Bardeen, Nonsingular general relativistic gravitational collapse, in Proceedings of the 5th International Conference on Gravitation and the Theory of Relativity (Tbilissi University Press, Tbilisi, 1968).
  10. E. Ayon-Beato and A. Garcia, Regular black hole in general relativity coupled to nonlinear electrodynamics, Phys. Rev. Lett. 80, 5056 (1998).
  11. E. Ayon-Beato and A. Garcia, The Bardeen model as a nonlinear magnetic monopole, Phys. Lett. B 493, 149 (2000).
  12. P. O. Mazur and E. Mottola, Gravitational condensate stars: An alternative to black holes, Universe 9, 88 (2023).
  13. R. Brustein, A. J. M. Medved, and T. Simhon, Black holes as frozen stars, Phys. Rev. D 105, 024019 (2022).
  14. R. Brustein, A. J. M. Medved, T. Shindelman, and T. Simhon, Black holes as frozen stars: Regular interior geometry, Fortschr. Phys. 72, 2300188 (2024).
  15. Z. H. Zhao, Y. N. Gu, S. C. Liu, L. X. Huang, and Y. Q. Wang, Non-topological soliton Bardeen boson stars and their frozen states, Eur. Phys. J. C 85, 511 (2025).
  16. X. E. Wang, From Bardeen-boson stars to black holes without event horizon, arXiv:2305.19057.
  17. Y. Yue and Y. Q. Wang, Frozen Hayward-boson stars, J. Cosmol. Astropart. Phys. 05 (2025) 066.
  18. C. A. Herdeiro and E. Radu, Spherical electro-vacuum black holes with resonant, scalar Q-hair, Eur. Phys. J. C 80, 390 (2020).
  19. J. P. Hong, M. Suzuki, and M. Yamada, Spherically symmetric scalar hair for charged black holes, Phys. Rev. Lett. 125, 111104 (2020).
  20. Y. Brihaye and B. Hartmann, Strong gravity effects of charged Q-clouds and inflating black holes, Classical Quantum Gravity 38, 06LT01 (2021).
  21. Y. Brihaye and B. Hartmann, Boson stars and black holes with wavy scalar hair, Phys. Rev. D 105, 104063 (2022).
  22. G. W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10, 363 (1974).
  23. G. W. Horndeski, Conservation of charge and the Einstein-Maxwell field equations, J. Math. Phys. (N.Y.) 17, 1980 (1976).
  24. F. Mueller-Hoissen and R. Sippel, Spherically symmetric solutions of the nonminimally coupled Einstein-Maxwell equations, Classical Quantum Gravity 5, 1473 (1988).
  25. Y. Verbin, Magnetic and electric black holes in the vector-tensor Horndeski theory, Phys. Rev. D 106, 024057 (2022).
  26. E. J. Copeland and M. I. Tsumagari, Q-balls in flat potentials, Phys. Rev. D 80, 025016 (2009).
  27. D. Pugliese, H. Quevedo, J. A. Rueda H., and R. Ruffini, On charged boson stars, Phys. Rev. D 88, 024053 (2013).
  28. J. D. López and M. Alcubierre, Charged boson stars revisited, Gen. Relativ. Gravit. 55, 72 (2023).
  29. Y. Brihaye and B. Hartmann, Note on super-critical charged boson stars, Gen. Relativ. Gravit. 56, 6 (2024).
  30. P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Light-ring stability for ultracompact objects, Phys. Rev. Lett. 119, 251102 (2017).
  31. V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa, and P. Pani, Light rings as observational evidence for event horizons: Long-lived modes, ergoregions and nonlinear instabilities of ultracompact objects, Phys. Rev. D 90, 044069 (2014).
  32. P. 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).
  33. P. Jetzer, Stability of charged boson stars, Phys. Lett. B 231, 433 (1989).
  34. V. Jaramillo, D. Núñez, M. Ruiz, and M. Zilhão, Full 3D nonlinear dynamics of charged and magnetized boson stars, Phys. Rev. D 111, 024070 (2025).

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Phys. Rev. D 113, 065023– Published 25 March, 2026
  • Received 30 October 2025
  • Accepted 18 February 2026
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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.

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