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Stability Properties of Regular Black Holes

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Regular Black Holes

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Abstract

Black holes encountered in general relativity are characterized by spacetime singularities hidden within an event horizon. These singularities provide a key motivation to go beyond general relativity and look for regular black holes where the spacetime curvature remains bounded everywhere. A prominent mechanism achieving this replaces the singularity by a regular patch of de Sitter space. The resulting regular geometries exhibit two horizons: the outer event horizon is supplemented by an inner Cauchy horizon. The latter could render the geometry unstable against perturbations through the so-called mass-inflation effect, i.e., an exponential growth of the mass function. This chapter reviews the mass-inflation effect for spherically symmetric black hole spacetimes contrasting the dynamics of the mass function for Reissner–Nordstöm and regular black holes. We also cover recent developments related to the late-time attractors induced by Hawking radiation which exorcise the exponential growth of the spacetime curvature encountered in the standard mass-inflation scenario. In order to make the exposition self-contained, we also briefly discuss basic properties of regular black holes including their thermodynamics.

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Notes

  1. 1.

    For \(a_3 < 0\) one encounters a regular anti-de Sitter core. Regular geometries of this type have recently been discussed in the context of Gauss black holes [37]. The loop black hole suggested in [38] also falls into this classification having \(a_3 = 0\).

  2. 2.

    For more details on the infinitely thin-shell formalism, see [57].

  3. 3.

    This makes the crucial assumption that the Cauchy horizon actually exists. While there is good evidence for forming such a horizon for the black holes encountered in general relativity, the question whether such a horizon actually forms in the collapse to a regular black hole is a largely open question, see [63, 64] for some recent work in this direction. For a first self-consistent dynamical calculation of mass-inflation without assuming the existence of a Cauchy horizon we refer to [65].

  4. 4.

    According to Tipler, a null singularity is called “strong” if there exists at least one component of the Riemann tensor (in a parallelly propagated frame) which does not converge when integrated with respect to the affine parameter \(\tau \) twice. The physical meaning of this requirement is that the tidal distortion is not finite as an observer crosses the singularity.

  5. 5.

    This conclusion has been challenged in [69, 70] (also see [60] for an earlier analysis based on the DTR-relations), claiming that the onset of the attractor behavior is preceded by a “fatal” phase of exponential growth. This conclusion is flawed for three reasons though. Firstly, it builds on an analysis of the system at “early times” outside the region of validity of the underlying assumptions: Price’s law holds at asymptotically late times only where the optical geometric limit is valid and one is allowed to neglect the “finite size” effects of the inner potential barrier. In a realistic collapse the early-time dynamics is significantly more complicated because of the presence of the flux from the collapsing star. Secondly, closing eyes to this difficulty and extrapolating the model to early times, Fig. 10.7 establishes that the analysis of [69, 70] is incomplete. There are initial conditions which actually reach the salient late-time attractor also from early times. Determining the precise initial data requires the analysis of the full dynamical process and is beyond the scope of a simplified model whose fundamental limitation is the assumption that the Cauchy Horizon always exist at \(v=\infty \). Thirdly, a divergence in the asymptotic mass \(m_+\) may just not be fatal for the geometry as the Misner-Sharp mass controlling the curvature of spacetime may remain finite. This is clear from Eq. (10.65) which shows that the limit \(m\rightarrow \infty \) has different physical implications for the Reissner–Nordström and Hayward geometry. Most remarkably, very similar conclusions have already been reached when analyzing the mass-inflation effect for loop black holes [59]. For a recent discussion of this viewpoint by the authors of [69, 70] also see [71].

References

  1. R. Penrose, Phys. Rev. Lett. 14, 57 (1965). https://doi.org/10.1103/PhysRevLett.14.57

    Article  ADS  MathSciNet  Google Scholar 

  2. R. Penrose, Riv. Nuovo Cim. 1, 252 (1969). https://doi.org/10.1023/A:1016578408204

    Article  ADS  Google Scholar 

  3. N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space. Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, UK, 1984). https://doi.org/10.1017/CBO9780511622632

  4. E. Poisson, W. Israel, Class. Quant. Grav. 5, L201 (1988). https://doi.org/10.1088/0264-9381/5/12/002

    Article  ADS  Google Scholar 

  5. J.M. Bardeen, in Proceedings of the International Conference GR5 Tbilisi, vol. 174 (1968)

    Google Scholar 

  6. M.A. Markov, ZhETF Pisma Redaktsiiu 36, 214 (1982)

    ADS  Google Scholar 

  7. I. Dymnikova, Gen. Rel. Grav. 24, 235 (1992). https://doi.org/10.1007/BF00760226

    Article  ADS  Google Scholar 

  8. V.P. Frolov, M. Markov, V.F. Mukhanov, Phys. Rev. D 41, 383 (1990). https://doi.org/10.1103/PhysRevD.41.383

    Article  ADS  MathSciNet  Google Scholar 

  9. S. Ansoldi, Spherical black holes with regular center: a Review of existing models including a recent realization with Gaussian sources

    Google Scholar 

  10. S.A. Hayward, Phys. Rev. Lett. 96, 031103 (2006). https://doi.org/10.1103/PhysRevLett.96.031103

    Article  ADS  Google Scholar 

  11. A. Bonanno, M. Reuter, Phys. Rev. D 62, 043008 (2000). https://doi.org/10.1103/PhysRevD.62.043008

    Article  ADS  Google Scholar 

  12. A. Bonanno, M. Reuter, Phys. Rev. D 73, 083005 (2006). https://doi.org/10.1103/PhysRevD.73.083005

    Article  ADS  MathSciNet  Google Scholar 

  13. Y.F. Cai, D.A. Easson, JCAP 09, 002 (2010). https://doi.org/10.1088/1475-7516/2010/09/002

    Article  ADS  Google Scholar 

  14. M. Reuter, E. Tuiran, Phys. Rev. D 83, 044041 (2011). https://doi.org/10.1103/PhysRevD.83.044041

    Article  ADS  Google Scholar 

  15. B. Koch, F. Saueressig, Class. Quant. Grav. 31, 015006 (2014). https://doi.org/10.1088/0264-9381/31/1/015006

    Article  ADS  Google Scholar 

  16. B. Koch, F. Saueressig, Int. J. Mod. Phys. A 29(8), 1430011 (2014). https://doi.org/10.1142/S0217751X14300117

    Article  ADS  Google Scholar 

  17. R. Torres, Phys. Lett. B 733, 21 (2014). https://doi.org/10.1016/j.physletb.2014.04.010

    Article  ADS  Google Scholar 

  18. A. Bonanno, B. Koch, A. Platania, Found. Phys. 48(10), 1393 (2018). https://doi.org/10.1007/s10701-018-0195-7

    Article  ADS  MathSciNet  Google Scholar 

  19. J.M. Pawlowski, D. Stock, Phys. Rev. D 98(10), 106008 (2018). https://doi.org/10.1103/PhysRevD.98.106008

    Article  ADS  MathSciNet  Google Scholar 

  20. A. Adeifeoba, A. Eichhorn, A. Platania, Class. Quant. Grav. 35(22), 225007 (2018). https://doi.org/10.1088/1361-6382/aae6ef

    Article  ADS  Google Scholar 

  21. A. Platania, Eur. Phys. J. C 79(6), 470 (2019). https://doi.org/10.1140/epjc/s10052-019-6990-2

    Article  ADS  Google Scholar 

  22. C. Rovelli, F. Vidotto, Int. J. Mod. Phys. D 23(12), 1442026 (2014). https://doi.org/10.1142/S0218271814420267

    Article  ADS  Google Scholar 

  23. A. Barrau, C. Rovelli, Phys. Lett. B 739, 405 (2014). https://doi.org/10.1016/j.physletb.2014.11.020

    Article  ADS  Google Scholar 

  24. T. De Lorenzo, C. Pacilio, C. Rovelli, S. Speziale, Gen. Rel. Grav. 47(4), 41 (2015). https://doi.org/10.1007/s10714-015-1882-8

    Article  ADS  Google Scholar 

  25. M. Christodoulou, C. Rovelli, S. Speziale, I. Vilensky, Phys. Rev. D 94(8), 084035 (2016). https://doi.org/10.1103/PhysRevD.94.084035

    Article  ADS  MathSciNet  Google Scholar 

  26. A. Perez, Rept. Prog. Phys. 80(12), 126901 (2017). https://doi.org/10.1088/1361-6633/aa7e14

    Article  ADS  Google Scholar 

  27. R. Penrose, in Battelle Rencontres - 1967 Lectures in Mathematics and Physics, Seattle, WA, USA, 16–31 July 1967 (1968), pp. 121–235

    Google Scholar 

  28. W.A. Hiscock, Physics Letters A 83(3), 110 (1981). https://doi.org/10.1016/0375-9601(81)90508-9

    Article  ADS  MathSciNet  Google Scholar 

  29. E. Poisson, W. Israel, Phys. Rev. Lett. 63, 1663 (1989). https://doi.org/10.1103/PhysRevLett.63.1663

    Article  ADS  MathSciNet  Google Scholar 

  30. A. Ori, Phys. Rev. Lett. 67, 789 (1991). https://doi.org/10.1103/PhysRevLett.67.789

    Article  ADS  MathSciNet  Google Scholar 

  31. L.M. Burko, A. Ori, Phys. Rev. Lett. 74, 1064 (1995). https://doi.org/10.1103/PhysRevLett.74.1064

    Article  ADS  Google Scholar 

  32. R.H. Price, Phys. Rev. D 5, 2419 (1972). https://doi.org/10.1103/PhysRevD.5.2419

    Article  ADS  MathSciNet  Google Scholar 

  33. R. Balbinot, P.R. Brady, W. Israel, E. Poisson, Phys. Lett. A 161(3), 223 (1991). https://doi.org/10.1016/0375-9601(91)90007-U

    Article  ADS  MathSciNet  Google Scholar 

  34. A. Bonanno, A.P. Khosravi, F. Saueressig, Phys. Rev. D 103(12), 124027 (2021). https://doi.org/10.1103/PhysRevD.103.124027

    Article  ADS  Google Scholar 

  35. A. Bonanno, A.P. Khosravi, F. Saueressig, Phys. Rev. D 107(2), 024005 (2023). https://doi.org/10.1103/PhysRevD.107.024005

    Article  ADS  Google Scholar 

  36. V.P. Frolov, Phys. Rev. D 94(10), 104056 (2016). https://doi.org/10.1103/PhysRevD.94.104056

    Article  ADS  MathSciNet  Google Scholar 

  37. J. Boos, (2021)

    Google Scholar 

  38. L. Modesto, Int. J. Theor. Phys. 49, 1649 (2010). https://doi.org/10.1007/s10773-010-0346-x

    Article  Google Scholar 

  39. E. Ayon-Beato, A. Garcia, Phys. Lett. B 493, 149 (2000). https://doi.org/10.1016/S0370-2693(00)01125-4

    Article  ADS  MathSciNet  Google Scholar 

  40. I. Dymnikova, Class. Quant. Grav. 19, 725 (2002). https://doi.org/10.1088/0264-9381/19/4/306

    Article  ADS  Google Scholar 

  41. F. Saueressig, N. Alkofer, G. D’Odorico, F. Vidotto, PoS FFP14, 174 (2016). https://doi.org/10.22323/1.224.0174

  42. M. Markov, Pis’ma v ZHETF 36, 214 (1982)

    Google Scholar 

  43. V.P. Frolov, M. Markov, V.F. Mukhanov, Phys. Lett. B 216(3–4), 272 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  44. V.P. Frolov, M. Markov, V.F. Mukhanov, Phys. Rev. D 41(2), 383 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  45. R. Percacci, An Introduction to Covariant Quantum Gravity and Asymptotic Safety, 100 Years of General Relativity, vol. 3 (World Scientific, 2017). https://doi.org/10.1142/10369

  46. M. Reuter, F. Saueressig, Quantum Gravity and the Functional Renormalization Group: The Road Towards Asymptotic Safety (Cambridge University Press, 2019)

    Google Scholar 

  47. M. Reuter, Phys. Rev. D 57, 971 (1998). https://doi.org/10.1103/PhysRevD.57.971

    Article  ADS  MathSciNet  Google Scholar 

  48. O. Lauscher, M. Reuter, Phys. Rev. D 65, 025013 (2002). https://doi.org/10.1103/PhysRevD.65.025013

    Article  ADS  MathSciNet  Google Scholar 

  49. M. Reuter, F. Saueressig, Phys. Rev. D 65, 065016 (2002). https://doi.org/10.1103/PhysRevD.65.065016

    Article  ADS  MathSciNet  Google Scholar 

  50. S.W. Hawking, Commun. Math. Phys. 43, 199 (1975). https://doi.org/10.1007/BF02345020. [Erratum: Commun. Math. Phys. 46, 206 (1976)]

  51. B.L. Hu, T.A. Jacobson, M.P. Ryan, C.V. Vishveshwara (eds.). Directions in general relativity. Proceedings, 1993 International Symposium, College Park, USA, May 27–29, 1993. Vol. 1: Papers in honor of Charles Misner. Vol. 2: Papers in honor of Dieter Brill (1995)

    Google Scholar 

  52. J.E. Chase, Nuovo Cimento Serie B 67(2), 136 (1970). https://doi.org/10.1007/BF02711502

    Article  ADS  Google Scholar 

  53. S.K. Blau, Phys. Rev. D 39, 2901 (1989). https://doi.org/10.1103/PhysRevD.39.2901

    Article  ADS  Google Scholar 

  54. T. Dray, G. ’t Hooft, Commun. Math. Phys. 99, 613 (1985). https://doi.org/10.1007/BF01215912

  55. I.H. Redmount, Progr. Theor. Phys. 73(6), 1401 (1985). https://doi.org/10.1143/PTP.73.1401

    Article  ADS  MathSciNet  Google Scholar 

  56. C. Barrabes, W. Israel, Phys. Rev. D 43, 1129 (1991). https://doi.org/10.1103/PhysRevD.43.1129

    Article  ADS  MathSciNet  Google Scholar 

  57. W. Israel, Nuovo Cim. B 44S10, 1 (1966). https://doi.org/10.1007/BF02710419. [Erratum: Nuovo Cim. B 48, 463 (1967)]

  58. A.J.S. Hamilton, P.P. Avelino, Phys. Rept. 495, 1 (2010). https://doi.org/10.1016/j.physrep.2010.06.002

    Article  ADS  Google Scholar 

  59. E.G. Brown, R.B. Mann, L. Modesto, Phys. Rev. D 84, 104041 (2011). https://doi.org/10.1103/PhysRevD.84.104041

    Article  ADS  Google Scholar 

  60. R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, M. Visser, JHEP 07, 023 (2018). https://doi.org/10.1007/JHEP07(2018)023

    Article  ADS  Google Scholar 

  61. M. Bertipagani, M. Rinaldi, L. Sebastiani, S. Zerbini, Phys. Dark Univ. 33, 100853 (2021). https://doi.org/10.1016/j.dark.2021.100853

    Article  Google Scholar 

  62. E. Poisson, W. Israel, Phys. Rev. D 41, 1796 (1990). https://doi.org/10.1103/PhysRevD.41.1796

    Article  ADS  MathSciNet  Google Scholar 

  63. J. Ziprick, G. Kunstatter, Phys. Rev. D 82, 044031 (2010). https://doi.org/10.1103/PhysRevD.82.044031

    Article  ADS  Google Scholar 

  64. A. Biasi, O. Evnin, S. Sypsas, (2022)

    Google Scholar 

  65. A. Bonanno, S. Droz, W. Israel, S.M. Morsink, Proc. Roy. Soc. Lond. A 450, 553 (1995). https://doi.org/10.1098/rspa.1995.0100

    Article  ADS  Google Scholar 

  66. L.M. Burko, Phys. Rev. D 60, 104033 (1999). https://doi.org/10.1103/PhysRevD.60.104033

    Article  ADS  MathSciNet  Google Scholar 

  67. F.J. Tipler, Phys. Lett. A 64, 8 (1977). https://doi.org/10.1016/0375-9601(77)90508-4

    Article  ADS  MathSciNet  Google Scholar 

  68. A. Krolak, J. Math. Phys. 28(11), 2685 (1987). https://doi.org/10.1063/1.527829

    Article  ADS  MathSciNet  Google Scholar 

  69. R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, M. Visser, JHEP 05, 132 (2021). https://doi.org/10.1007/JHEP05(2021)132

    Article  ADS  Google Scholar 

  70. F. Di Filippo, R. Carballo-Rubio, S. Liberati, C. Pacilio, M. Visser, Universe 8(4), 204 (2022). https://doi.org/10.3390/universe8040204

    Article  ADS  Google Scholar 

  71. R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, M. Visser, arXiv:2212.07458 [gr-qc]

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Acknowledgements

We thank our collaborators N. Alkhofer, J. Daas, M. Galis, G. d’Odorico, I. van der Pas, A. Platania, A. Khosravi, B. Koch,, S. Silveravalle, F. Vidotto, M. Wondrak, and, foremost, M. Reuter for many inspiring discussions developing our understanding of spacetime singularities, regular black holes, and the stability properties of Cauchy horizons. The work of F.S. is supported by the Dutch Black Hole Consortium.

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Authors and Affiliations

  1. INAF, Osservatorio Astrofisico di Catania, via S.Sofia 78, 95123, Catania, Italy

    Alfio Bonanno

  2. INFN, Sezione di Catania, via S.Sofia 64, 95123, Catania, Italy

    Alfio Bonanno

  3. Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands

    Frank Saueressig

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  1. Alfio Bonanno
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Correspondence to Alfio Bonanno or Frank Saueressig .

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  1. Department of Physics, Fudan University, Shanghai, China

    Cosimo Bambi

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Bonanno, A., Saueressig, F. (2023). Stability Properties of Regular Black Holes. In: Bambi, C. (eds) Regular Black Holes. Springer Series in Astrophysics and Cosmology. Springer, Singapore. https://doi.org/10.1007/978-981-99-1596-5_10

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