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Singularity and differentiability at the origin of static and spherically symmetric black holes

Tommaso Antonelli* and Marco Sebastianutti†

  • *Contact author: t.antonelli@sussex.ac.uk
  • †Contact author: m.sebastianutti@sussex.ac.uk
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Phys. Rev. D 113, 064007 – Published 2 March, 2026

DOI: https://doi.org/10.1103/hf4r-19xh

Abstract

The divergence of curvature invariants at a given point signals the impossibility of extending the spacetime to that point, with the derivative order of these diverging invariants determining the differentiability class of the considered spacetime. We hereby focus on a general static and spherically symmetric geometry and determine, in the full nonlinear regime and in a model-independent way, the conditions that the metric functions must satisfy in order to achieve finiteness of all curvature invariants at the origin. Our findings have direct implications regarding the extendibility of such spacetimes, which we illustrate by making explicit examples of various black hole geometries. This work is structured around a central theorem, which relates the finiteness of curvature invariants at the origin to the leading order behavior and parity properties of the metric functions. The detailed proof of this theorem constitutes the main result of the paper.

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References (84)

  1. Roger Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14, 57 (1965).
  2. Stephen Hawking, The occurrence of singularities in cosmology, Proc. R. Soc. A 294, 511 (1966).
  3. S. W. Hawking and R. Penrose, The singularities of gravitational collapse and cosmology, Proc. R. Soc. A 314, 529 (1970).
  4. Stephen W. Hawking and George F. R. Ellis, The Large Scale Structure of Space-Time: 50th Anniversary Edition, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2023), 10.1017/9781009253161.
  5. José M. M. Senovilla, A critical appraisal of the singularity theorems, Phil. Trans. R. Soc. A 380, 20210174 (2022).
  6. W. Rindler, Hyperbolic motion in curved space time, Phys. Rev. 119, 2082 (1960).
  7. W. Rindler, Kruskal space and the uniformly accelerated frame, Am. J. Phys. 34, 1174 (1966).
  8. Robert M. Wald, General Relativity, (University of Chicago Press, Chicago, 1984), 10.7208/chicago/9780226870373.001.0001.
  9. G. F. R. Ellis and B. G. Schmidt, Singular space-times, Gen. Relativ. Gravit. 8, 915 (1977).
  10. Piotr T. Chruściel, Geometry of Black Holes, International Series of Monographs on Physics (Oxford University Press, New York, 2020), 10.1093/oso/9780198855415.001.0001.
  11. C. J. S. Clarke, Local extensions in singular space-times, Commun. Math. Phys. 32, 205 (1973).
  12. C. J. S. Clarke, Local extensions in singular space-times II, Commun. Math. Phys. 84, 329 (1982).
  13. C. J. S. Clarke, Space-times of low differentiability and singularities, J. Math. Anal. Appl. 88, 270 (1982).
  14. Erik Curiel, The analysis of singular spacetimes, Philos. Sci. 66, S119 (1999).
  15. James M. Bardeen, Non-singular general relativistic gravitational collapse, in Proceedings of the 5th International Conference on Gravitation and the Theory of Relativity (Tbilisi University Press, 1968), p. 87, https://ui.adsabs.harvard.edu/abs/1968qtr..conf...87B.
  16. Eric Poisson and W. Israel, Structure of the black hole nucleus, Classical Quantum Gravity 5, L201 (1988).
  17. Valeri P. Frolov, M. A. Markov, and Viatcheslav F. Mukhanov, Through a black hole into a new universe?, Phys. Lett. B 216, 272 (1989).
  18. I. Dymnikova, Vacuum nonsingular black hole, Gen. Relativ. Gravit. 24, 235 (1992).
  19. Marc Mars, M. Mercè Martín-Prats, and M. M. José Senovilla, Models of regular Schwarzschild black holes satisfying weak energy conditions, Classical Quantum Gravity 13, L51 (1996).
  20. Eloy Ayon-Beato and Alberto Garcia, Regular black hole in general relativity coupled to nonlinear electrodynamics, Phys. Rev. Lett. 80, 5056 (1998).
  21. Eloy Ayon-Beato and Alberto Garcia, The Bardeen model as a nonlinear magnetic monopole, Phys. Lett. B 493, 149 (2000).
  22. Alfio Bonanno and Martin Reuter, Renormalization group improved black hole space-times, Phys. Rev. D 62, 043008 (2000).
  23. Kirill A. Bronnikov, Regular magnetic black holes and monopoles from nonlinear electrodynamics, Phys. Rev. D 63, 044005 (2001).
  24. Irina Dymnikova, Cosmological term as a source of mass, Classical Quantum Gravity 19, 725 (2002).
  25. Piero Nicolini, Anais Smailagic, and Euro Spallucci, Noncommutative geometry inspired Schwarzschild black hole, Phys. Lett. B 632, 547 (2006).
  26. Sean A. Hayward, Formation and evaporation of regular black holes, Phys. Rev. Lett. 96, 031103 (2006).
  27. K. A. Bronnikov and J. C. Fabris, Regular phantom black holes, Phys. Rev. Lett. 96, 251101 (2006).
  28. Jose P. S. Lemos and Vilson T. Zanchin, Regular black holes: Electrically charged solutions, Reissner-Nordström outside a de Sitter core, Phys. Rev. D 83, 124005 (2011).
  29. Cosimo Bambi and Leonardo Modesto, Rotating regular black holes, Phys. Lett. B 721, 329 (2013).
  30. Zhong-Ying Fan and Xiaobao Wang, Construction of regular black holes in general relativity, Phys. Rev. D 94, 124027 (2016).
  31. Valeri P. Frolov and Andrei Zelnikov, Quantum radiation from an evaporating nonsingular black hole, Phys. Rev. D 95, 124028 (2017).
  32. Alex Simpson and Matt Visser, Regular black holes with asymptotically Minkowski cores, Universe 6, 8 (2019).
  33. Hristu Culetu, On a regular modified Schwarzschild spacetime, arXiv:1305.5964.
  34. Hristu Culetu, On a regular charged black hole with a nonlinear electric source, Int. J. Theor. Phys. 54, 2855 (2015).
  35. Leonardo Balart and Elias C. Vagenas, Regular black holes with a nonlinear electrodynamics source, Phys. Rev. D 90, 124045 (2014).
  36. Antonio De Felice and Shinji Tsujikawa, Instability of nonsingular black holes in nonlinear electrodynamics, Phys. Rev. Lett. 134, 081401 (2025).
  37. Alex Simpson and Matt Visser, Black-bounce to traversable wormhole, J. Cosmol. Astropart. Phys. 02 (2019) 042.
  38. Francisco S. N. Lobo, Manuel E. Rodrigues, Marcos V. de Sousa Silva, Alex Simpson, and Matt Visser, Novel black-bounce spacetimes: Wormholes, regularity, energy conditions, and causal structure, Phys. Rev. D 103, 084052 (2021).
  39. Edgardo Franzin, Stefano Liberati, Jacopo Mazza, Alex Simpson, and Matt Visser, Charged black-bounce spacetimes, J. Cosmol. Astropart. Phys. 07 (2021) 036.
  40. Kirill A. Bronnikov and Rahul Kumar Walia, Field sources for Simpson-Visser spacetimes, Phys. Rev. D 105, 044039 (2022).
  41. Kirill A. Bronnikov, Regular black holes as an alternative to black bounce, Phys. Rev. D 110, 024021 (2024).
  42. Tommaso De Lorenzo, Costantino Pacilio, Carlo Rovelli, and Simone Speziale, On the effective metric of a Planck star, Gen. Relativ. Gravit. 47, 41 (2015).
  43. Piero Nicolini, Euro Spallucci, and Michael F. Wondrak, Quantum corrected black holes from string T-duality, Phys. Lett. B 797, 134888 (2019).
  44. Bob Holdom, On the fate of singularities and horizons in higher derivative gravity, Phys. Rev. D 66, 084010 (2002).
  45. Li Xiang, Yi Ling, and You Gen Shen, Singularities and the finale of black hole evaporation, Int. J. Mod. Phys. D 22, 1342016 (2013).
  46. Raúl Carballo-Rubio, Francesco Di Filippo, Stefano Liberati, Costantino Pacilio, and Matt Visser, Regular black holes without mass inflation instability, J. High Energy Phys. 09 (2022) 118.
  47. Xavier Calmet, Roberto Casadio, and Folkert Kuipers, Singularities in quantum corrected space-times, Phys. Lett. B 807, 135605 (2020).
  48. Jorge Ovalle, Roberto Casadio, and Andrea Giusti, Regular hairy black holes through Minkowski deformation, Phys. Lett. B 844, 138085 (2023).
  49. J. Ovalle, Schwarzschild black hole revisited: Before the complete collapse, Phys. Rev. D 109, 104032 (2024).
  50. Roberto Casadio, Alexander Kamenshchik, and Jorge Ovalle, Regular Schwarzschild black holes and cosmological models, Phys. Rev. D 111, 064036 (2025).
  51. Jens Boos, What happens to topological invariants and black holes in singularity-free theories?, Phys. Rev. D 111, 084063 (2025).
  52. Roberto Casadio, Andrea Giugno, and Andrea Giusti, Matter and gravitons in the gravitational collapse, Phys. Lett. B 763, 337 (2016).
  53. Roberto Casadio, Andrea Giugno, Andrea Giusti, and Michele Lenzi, Quantum corpuscular corrections to the Newtonian potential, Phys. Rev. D 96, 044010 (2017).
  54. Roberto Casadio, Geometry and thermodynamics of coherent quantum black holes, Int. J. Mod. Phys. D 31, 2250128 (2022).
  55. Roberto Casadio, Andrea Giusti, and Jorge Ovalle, Quantum Reissner-Nordström geometry: Singularity and Cauchy horizon, Phys. Rev. D 105, 124026 (2022).
  56. Wenbin Feng, Andrea Giusti, and Roberto Casadio, Horizon quantum mechanics for coherent quantum black holes, Eur. Phys. J. Plus 140, 145 (2025).
  57. Tommaso Antonelli, Marco Sebastianutti, and Andrea Giusti, Coherent electrically-charged quantum black holes, Eur. Phys. J. C 85, 1219 (2025).
  58. Regular Black Holes. Towards a New Paradigm of Gravitational Collapse, edited by Cosimo Bambi, Springer Series in Astrophysics and Cosmology (Springer, New York, 2023), 10.1007/978-981-99-1596-5.
  59. Raúl Carballo-Rubio et al., Towards a non-singular paradigm of black hole physics, J. Cosmol. Astropart. Phys. 05 (2025) 003.
  60. Nicolò Burzillà, Breno L. Giacchini, Tibério de Paula Netto, and Leonardo Modesto, Higher-order regularity in local and nonlocal quantum gravity, Eur. Phys. J. C 81, 462 (2021).
  61. Breno L. Giacchini, Tibério de Paula Netto, and Leonardo Modesto, Action principle selection of regular black holes, Phys. Rev. D 104, 084072 (2021).
  62. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 1982), 10.1017/CBO9780511622632.
  63. M. Asorey, J. L. Lopez, and I. L. Shapiro, Some remarks on high derivative quantum gravity, Int. J. Mod. Phys. A 12, 5711 (1997).
  64. Leonardo Modesto, Super-renormalizable quantum gravity, Phys. Rev. D 86, 044005 (2012).
  65. John D. Barrow and Frank J. Tipler, Action principles in nature, Nature (London) 331, 31 (1988).
  66. John D. Barrow, Finite action principle revisited, Phys. Rev. D 101, 023527 (2020).
  67. Jean-Luc Lehners and K. S. Stelle, A safe beginning for the universe?, Phys. Rev. D 100, 083540 (2019).
  68. Caroline Jonas, Jean-Luc Lehners, and Jerome Quintin, Cosmological consequences of a principle of finite amplitudes, Phys. Rev. D 103, 103525 (2021).
  69. Johanna N. Borissova and Astrid Eichhorn, Towards black-hole singularity-resolution in the Lorentzian gravitational path integral, Universe 7, 48 (2021).
  70. Johanna N. Borissova, Suppression of spacetime singularities in quantum gravity, Classical Quantum Gravity 41, 127002 (2024).
  71. Johanna N. Borissova, Astrid Eichhorn, and Shouryya Ray, A non-local way around the no-global-symmetries conjecture in quantum gravity?, Classical Quantum Gravity 42, 037001 (2025).
  72. Bernard F. Schutz, A First Course in General Relativity, 3rd ed. (Cambridge University Press, Cambridge, England, 2022), 10.1017/9781108610865.
  73. M. S. Morris and K. S. Thorne, Wormholes in space-time and their use for interstellar travel: A tool for teaching general relativity, Am. J. Phys. 56, 395 (1988).
  74. Cosimo Bambi, Leonardo Modesto, and Leslaw Rachwał, Spacetime completeness of non-singular black holes in conformal gravity, J. Cosmol. Astropart. Phys. 05 (2017) 003.
  75. José M. Martín-García, xact: Efficient tensor computer algebra for the wolfram language, http://www.xact.es/ (2021).
  76. Terence Tao, Analysis II, 4th ed., Texts and Readings in Mathematics (Springer, Singapore, 2022), 10.1007/978-981-19-7284-3.
  77. L. Hörmander, The Analysis of Linear Partial Differential Operators I, 2nd ed., Classics in Mathematics (Springer, Berlin, Heidelberg, 2003), 10.1007/978-3-642-61497-2.
  78. Jan Sbierski, The C0-inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian geometry, J. Diff. Geom. 108, 319 (2018).
  79. Vladimir N. Lukash and Vladimir N. Strokov, Space-times with integrable singularity, Int. J. Mod. Phys. A 28, 1350007 (2013).
  80. Julio Arrechea, Stefano Liberati, Hooman Neshat, and Vania Vellucci, Physical and theoretical challenges to integrable singularities, Phys. Rev. D 112, 044024 (2025).
  81. Tian Zhou and Leonardo Modesto, Geodesic incompleteness of some popular regular black holes, Phys. Rev. D 107, 044016 (2023).
  82. Alexey S. Koshelev and Anna Tokareva, Nonperturbative quantum gravity denounces singular black holes, Phys. Rev. D 111, 086026 (2025).
  83. Warren P. Johnson, The curious history of Faà di Bruno’s formula, Am. Math. Mon. 109, 217 (2002).
  84. Richard P. Stanley, Enumerative Combinatorics, 2nd ed., Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, England, 2011), Vol. I, 10.1017/CBO9781139058520.

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Phys. Rev. D 113, 064007– Published 2 March, 2026
  • Received 8 October 2025
  • Revised 14 January 2026
  • Accepted 28 January 2026
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