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Quantum fate of the Choptuik naked singularity

Chih-Hung Wu*

  • *Contact author: chwu29@uw.edu
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Phys. Rev. D 114, 044092 – Published 28 August, 2026

DOI: https://doi.org/10.1103/75b7-wjrq

Abstract

Classical critical collapse provides a dynamical route from smooth initial data to a naked singularity, representing a sharper violation of predictability than ordinary black hole singularities. We argue that this distinction is erased by quantum backreaction. Building on the semiclassical interior analysis, where quantum self-energy of the collapsing matter generates a universal growing mode and a finite mass gap, we study the exterior naked singularity region that determines global visibility in the Einstein-scalar system. We analyze controlled exterior models in both 2+1 and 3+1 dimensions. In the former, smooth matching and physical boundary conditions analytically select a vacuum polarization state, whose backreaction cloaks the classically naked region by a quantum trapped branch. In the latter, numerical horizon tracing shows that near a quantum-shifted threshold the exterior develops finite-mass marginally trapped surfaces rather than a zero-mass naked endpoint. These results suggest a global quantum picture in which the Choptuik naked singularity shares the fate of an ordinary black hole singularity: quantum effects push the putative Cauchy horizon behind a quantum-generated horizon, thereby reducing the loss of predictability to the standard black hole evaporation problem.

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References (99)

  1. R. Penrose, Gravitational collapse: The role of general relativity, Riv. Nuovo Cimento 1, 252 (1969).
  2. S. W. Hawking, Nature of space and time, arXiv:hep-th/9409195.
  3. M. W. Choptuik, Universality and scaling in gravitational collapse of a massless scalar field, Phys. Rev. Lett. 70, 9 (1993).
  4. D. Christodoulou, Examples of naked singularity formation in the gravitational collapse of a scalar field, Ann. Math. 140, 607 (1994).
  5. C. Gundlach, D. Hilditch, and J. M. Martín-García, Critical phenomena in gravitational collapse, arXiv:2507.07636.
  6. A. Strominger and L. Thorlacius, Universality and scaling at the onset of quantum black hole formation, Phys. Rev. Lett. 72, 1584 (1994).
  7. J. G. Zhou, H. J. W. Mueller-Kirsten, and M.-Z. Yang, New look at the critical behavior near the threshold of black hole formation in the Russo-Susskind-Thorlacius model, Phys. Rev. D 51, R314 (1995).
  8. S. Bose, L. Parker, and Y. Peleg, Predictability and semiclassical approximation at the onset of black hole formation, Phys. Rev. D 54, 7490 (1996).
  9. Y. Peleg, S. Bose, and L. Parker, Choptuik scaling and quantum effects in 2-d dilaton gravity, Phys. Rev. D 55, R4525 (1997).
  10. S. Ayal and T. Piran, Spherical collapse of a massless scalar field with semiclassical corrections, Phys. Rev. D 56, 4768 (1997).
  11. P. R. Brady and A. C. Ottewill, Quantum corrections to critical phenomena in gravitational collapse, Phys. Rev. D 58, 024006 (1998).
  12. T. Chiba and M. Siino, Disappearance of black hole criticality in semiclassical general relativity, Mod. Phys. Lett. A 12, 709 (1997).
  13. V. Husain, Critical behaviour in quantum gravitational collapse, Adv. Sci. Lett. 2, 214 (2009).
  14. J. Ziprick and G. Kunstatter, Dynamical singularity resolution in spherically symmetric black hole formation, Phys. Rev. D 80, 024032 (2009).
  15. F. Benitez, R. Gambini, L. Lehner, S. Liebling, and J. Pullin, Critical collapse of a scalar field in semiclassical loop quantum gravity, Phys. Rev. Lett. 124, 071301 (2020).
  16. F. Benítez, R. Gambini, S. L. Liebling, and J. Pullin, Criticality in the collapse of spherically symmetric massless scalar fields in semiclassical loop quantum gravity, Phys. Rev. D 104, 024008 (2021).
  17. B. Berczi, P. M. Saffin, and S.-Y. Zhou, Gravitational collapse with quantum fields, Phys. Rev. D 104, L041703 (2021).
  18. J. N. Guenther, C. Hoelbling, and L. Varnhorst, Semiclassical gravitational collapse of a radially symmetric massless scalar quantum field, Phys. Rev. D 105, 105010 (2022).
  19. B. Berczi, P. M. Saffin, and S.-Y. Zhou, Gravitational collapse of quantum fields and Choptuik scaling, J. High Energy Phys. 02 (2022) 183.
  20. C. Hoelbling, J. N. Guenther, and L. Varnhorst, Real time dynamics of a semiclassical gravitational collapse of a scalar quantum field, Proc. Sci. LATTICE2021 (2022) 156 [arXiv:2111.15562].
  21. L. Varnhorst, C. Hoelbling, and J. N. Guenther, Real time evolution of scalar fields in semiclassical gravity, Proc. Sci. LATTICE2022 (2023) 391.
  22. U. Moitra, Self-similar gravitational dynamics, singularities and criticality in 2D, J. High Energy Phys. 06 (2023) 194.
  23. J. Zahn, Quantum fields on self-similar spacetimes, Classical Quantum Gravity 43, 025011 (2026).
  24. M. Tomašević and C.-H. Wu, Unveiling horizons in quantum critical collapse, J. High Energy Phys. 04 (2026) 151.
  25. M. Tomašević and C.-H. Wu, Quantum critical collapse abhors a naked singularity, Phys. Rev. D 113, 104029 (2026).
  26. I. Rodnianski and Y. Shlapentokh-Rothman, Naked singularities for the Einstein vacuum equations: The exterior solution, Ann. Math. 198, 231 (2023).
  27. Y. Shlapentokh-Rothman, Naked singularities for the Einstein vacuum equations: The interior solution, arXiv:2204.09891.
  28. S. Cicortas and C. Kehle, Discretely self-similar exterior-naked singularities for the Einstein-scalar field system, Arch. Ration. Mech. Anal. 250, 37 (2026).
  29. S. Cicortas, Weak cosmic censorship for the circularly symmetric Einstein-scalar field system in 2+1 dimensions, arXiv:2605.19143.
  30. K. Jay, K. N. Solanki, and P. S. Joshi, Obstructions to global visibility of singularities in asymptotically flat spacetimes, arXiv:2601.04152.
  31. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975).
  32. S. W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14, 2460 (1976).
  33. D. Garfinkle, An exact solution for 2+1 dimensional critical collapse, Phys. Rev. D 63, 044007 (2001).
  34. M. D. Roberts, Scalar field counterexamples to the cosmic censorship hypothesis, Gen. Relativ. Gravit. 21, 907 (1989).
  35. P. R. Brady, Analytic example of critical behaviour in scalar field collapse, Classical Quantum Gravity 11, 1255 (1994).
  36. Y. Oshiro, K. Nakamura, and A. Tomimatsu, Critical behavior of black hole formation in a scalar wave collapse, Prog. Theor. Phys. 91, 1265 (1994).
  37. P. R. Brady, Self-similar scalar field collapse: Naked singularities and critical behavior, Phys. Rev. D 51, 4168 (1995).
  38. F. Pretorius and M. W. Choptuik, Gravitational collapse in (2+1)-dimensional AdS space-time, Phys. Rev. D 62, 124012 (2000).
  39. V. Husain and M. Olivier, Scalar field collapse in three-dimensional AdS space-time, Classical Quantum Gravity 18, L1 (2001).
  40. D. Garfinkle and C. Gundlach, Perturbations of an exact solution for (2+1)-dimensional critical collapse, Phys. Rev. D 66, 044015 (2002).
  41. M. Cavaglia, G. Clement, and A. Fabbri, Approximately self-similar critical collapse in (2+1)-dimensions, Phys. Rev. D 70, 044010 (2004).
  42. J. Jałmużna, C. Gundlach, and T. Chmaj, Scalar field critical collapse in 2+1 dimensions, Phys. Rev. D 92, 124044 (2015).
  43. A. V. Frolov, Perturbations and critical behavior in the self-similar gravitational collapse of a massless scalar field, Phys. Rev. D 56 (1997) 6433.
  44. A. V. Frolov, Critical collapse beyond spherical symmetry: General perturbations of the Roberts solution, Phys. Rev. D 59, 104011 (1999).
  45. A. V. Frolov, Continuous self-similarity breaking in critical collapse, Phys. Rev. D 61, 084006 (2000).
  46. C. Gundlach, The Choptuik space-time as an eigenvalue problem, Phys. Rev. Lett. 75, 3214 (1995).
  47. C. Gundlach, Understanding critical collapse of a scalar field, Phys. Rev. D 55, 695 (1997).
  48. J. M. Martin-Garcia and C. Gundlach, All nonspherical perturbations of the Choptuik space-time decay, Phys. Rev. D 59, 064031 (1999).
  49. X. Shi, G. J. Turiaci, and C.-H. Wu, The fate of nucleated black holes in de Sitter quantum gravity, arXiv:2605.03015.
  50. V. F. Mukhanov, A. Wipf, and A. Zelnikov, On 4-D Hawking radiation from effective action, Phys. Lett. B 332, 283 (1994).
  51. R. Bousso and S. W. Hawking, Trace anomaly of dilaton coupled scalars in two-dimensions, Phys. Rev. D 56, 7788 (1997).
  52. W. Kummer and D. V. Vassilevich, Effective action and Hawking radiation for dilaton coupled scalars in two-dimensions, Phys. Rev. D 60, 084021 (1999).
  53. W. Kummer and D. V. Vassilevich, Hawking radiation from dilaton gravity in (1+1)-dimensions: A pedagogical review, Ann. Phys. (Berlin) 511, 801 (1999).
  54. R. Balbinot, A. Fabbri, V. P. Frolov, P. Nicolini, P. Sutton, and A. Zelnikov, Vacuum polarization in the Schwarzschild space-time and dimensional reduction, Phys. Rev. D 63, 084029 (2001).
  55. A. Fabbri, S. Farese, and J. Navarro-Salas, Generalized Virasoro anomaly and stress tensor for dilaton coupled theories, Phys. Lett. B 574, 309 (2003).
  56. D. Hofmann and W. Kummer, Effective action and Hawking flux from covariant perturbation theory, Eur. Phys. J. C 40, 275 (2005).
  57. C.-H. Wu and J. Xu, Islands in non-minimal dilaton gravity: Exploring effective theories for black hole evaporation, J. High Energy Phys. 10 (2023) 094.
  58. S. M. Christensen and S. A. Fulling, Trace anomalies and the Hawking effect, Phys. Rev. D 15, 2088 (1977).
  59. A. M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. 103B, 207 (1981).
  60. R. M. Wald, The back reaction effect in particle creation in curved space-time, Commun. Math. Phys. 54, 1 (1977).
  61. R. M. Wald, Trace anomaly of a conformally invariant quantum field in curved spacetime, Phys. Rev. D 17, 1477 (1978).
  62. R. M. Wald, Axiomatic renormalization of the stress tensor of a conformally invariant field in conformally flat space-times, Ann. Phys. (N.Y.) 110, 472 (1978).
  63. D. G. Boulware, Quantum field theory in Schwarzschild and Rindler spaces, Phys. Rev. D 11, 1404 (1975).
  64. W. Israel, Singular hypersurfaces and thin shells in general relativity, Nuovo Cimento B 44, 1 (1966).
  65. C. Barrabes and W. Israel, Thin shells in general relativity and cosmology: The lightlike limit, Phys. Rev. D 43, 1129 (1991).
  66. E. Poisson, A reformulation of the Barrabes-Israel null shell formalism, arXiv:gr-qc/0207101.
  67. W. G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14, 870 (1976).
  68. P. Davies, S. Fulling, and W. Unruh, Energy momentum tensor near an evaporating black hole, Phys. Rev. D 13, 2720 (1976).
  69. W. A. Hiscock, Models of evaporating black holes, Phys. Rev. D 23, 2813 (1981).
  70. W. A. Hiscock, Models of evaporating black holes. II. Effects of the outgoing created radiation, Phys. Rev. D 23, 2823 (1981).
  71. S. Hawking, Gravitational radiation in an expanding universe, J. Math. Phys. (N.Y.) 9, 598 (1968).
  72. S. A. Hayward, Quasilocal gravitational energy, Phys. Rev. D 49, 831 (1994).
  73. C. W. Misner and D. H. Sharp, Relativistic equations for adiabatic, spherically symmetric gravitational collapse, Phys. Rev. 136, B571 (1964).
  74. W. C. Hernandez and C. W. Misner, Observer time as a coordinate in relativistic spherical hydrodynamics, Astrophys. J. 143, 452 (1966).
  75. M. Bañados, C. Teitelboim, and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69, 1849 (1992).
  76. H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. R. Soc. A 269, 21 (1962).
  77. R. K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. R. Soc. A 270, 103 (1962).
  78. P. C. Vaidya, The gravitational field of a radiating star, Proc. Indian Acad. Sci. A 33, 264 (1951).
  79. A. Wang and H. P. de Oliveira, Critical phenomena of collapsing massless scalar wave packets, Phys. Rev. D 56, 753 (1997).
  80. N. Engelhardt and I. Nagar, A quantum singularity theorem for the evaporating black hole, arXiv:2605.05326.
  81. E.-A. Kontou, The sufficiently trapped surface, arXiv:2605.14798.
  82. C. R. Clark and G. L. Pimentel, Black hole critical collapse in infinite dimensions: Continuous self-similar solutions, J. High Energy Phys. 11 (2025) 017.
  83. C. Ecker, F. Ecker, and D. Grumiller, Analytic discrete self-similar solutions of Einstein-Klein-Gordon at large D, Phys. Rev. Lett. 136, 191401 (2026).
  84. C. Ecker, F. Ecker, D. Grumiller, and T. Jechtl, Critical spacetime crystals in continuous dimensions, SciPost Phys. 20, 176 (2026).
  85. C. Barcelo, S. Liberati, S. Sonego, and M. Visser, Hawking-like radiation does not require a trapped region, Phys. Rev. Lett. 97, 171301 (2006).
  86. C. Barcelo, S. Liberati, S. Sonego, and M. Visser, Minimal conditions for the existence of a Hawking-like flux, Phys. Rev. D 83, 041501 (2011).
  87. P. Chen, W. G. Unruh, C.-H. Wu, and D.-H. Yeom, Pre-Hawking radiation cannot prevent the formation of apparent horizon, Phys. Rev. D 97, 064045 (2018).
  88. W. G. Unruh, Prehawking radiation, arXiv:1802.09107.
  89. R. Gregory and R. Laflamme, Black strings and p-branes are unstable, Phys. Rev. Lett. 70, 2837 (1993).
  90. G. T. Horowitz and K. Maeda, Fate of the black string instability, Phys. Rev. Lett. 87, 131301 (2001).
  91. P. Figueras, T. França, C. Gu, and T. Andrade, Endpoint of the Gregory-Laflamme instability of black strings revisited, Phys. Rev. D 107, 044028 (2023).
  92. S. Hod, Hawking radiation may violate the Penrose cosmic censorship conjecture, Int. J. Mod. Phys. D 28, 1944023 (2019).
  93. M. Casals, A. Fabbri, C. Martínez, and J. Zanelli, Quantum dress for a naked singularity, Phys. Lett. B 760, 244 (2016).
  94. A. M. Frassino, R. A. Hennigar, J. F. Pedraza, and A. Svesko, Quantum censors: Backreaction builds horizons, Int. J. Mod. Phys. D 34, 2544024 (2025).
  95. G. T. Horowitz and J. Polchinski, A correspondence principle for black holes and strings, Phys. Rev. D 55, 6189 (1997).
  96. G. T. Horowitz and J. Polchinski, Self-gravitating fundamental strings, Phys. Rev. D 57, 2557 (1998).
  97. N. Čeplak, R. Emparan, A. Puhm, and M. Tomašević, The correspondence between rotating black holes and fundamental strings, J. High Energy Phys. 11 (2023) 226.
  98. J. Seitz and E. Y. Urbach, A spin on Hagedorn temperatures and string stars, J. High Energy Phys. 06 (2026) 245.
  99. R. Emparan, M. Sanchez-Garitaonandia, and M. Tomašević, String theory in a pinch: Resolving the Gregory-Laflamme singularity, J. High Energy Phys. 02 (2025) 104.

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Phys. Rev. D 114, 044092– Published 28 August, 2026
  • Received 8 July 2026
  • Accepted 12 August 2026
Article part of CHORUS
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