Regularity of a General Class of “Quantum Deformed” Black Holes
Abstract
:1. Introduction
- (Schwarzschild): Not regular;
- : Metric–regular;
- : Christoffel–symbol–regular;
- : Curvature–regular.
2. Geometric Analysis
2.1. Metric Components
2.2. Event Horizons
2.3. Christoffel Symbols of the Second Kind
2.4. Orthonormal Components
2.5. Riemann Tensor
2.6. Ricci Tensor
2.7. Ricci Scalar
2.8. Einstein Tensor
2.9. Weyl Tensor
2.10. Weyl Scalar
2.11. Kretschmann Scalar
3. Surface Gravity and Hawking Temperature
4. Stress-Energy Tensor
5. Energy Conditions
5.1. Null Energy Condition
5.2. Weak Energy Condition
5.3. Strong Energy Condition
5.4. Dominant Energy Condition
6. ISCO and Photon Sphere Analysis
6.1. Photon Orbits
6.2. ISCOs
6.3. Summary
- ;
- ;
- .
7. Regge–Wheeler Analysis
8. Discussion and Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Kazakov, D.; Solodukhin, S. On quantum deformation of the Schwarzschild solution. Nucl. Phys. B 1994, 429, 153–176. [Google Scholar] [CrossRef] [Green Version]
- Solodukhin, S.N. “Nongeometric” contribution to the entropy of a black hole due to quantum corrections. Phys. Rev. D 1995, 51, 618–621. [Google Scholar] [CrossRef] [Green Version]
- Solodukhin, S.N. Two-dimensional quantum-corrected eternal black hole. Phys. Rev. D 1996, 53, 824–835. [Google Scholar] [CrossRef] [Green Version]
- Ashtekar, A.; Olmedo, J.; Singh, P. Quantum extension of the Kruskal spacetime. Phys. Rev. D 2018, 98, 126003. [Google Scholar] [CrossRef] [Green Version]
- Nojiri, S.; Odintsov, S.D. Can quantum-corrected btz black hole anti-evaporate? Mod. Phys. Lett. A 1998, 13, 2695–2704. [Google Scholar] [CrossRef] [Green Version]
- Maluf, R.V.; Neves, J. Bardeen regular black hole as a quantum-corrected Schwarzschild black hole. Int. J. Mod. Phys. D 2019, 28, 1950048. [Google Scholar] [CrossRef] [Green Version]
- Zaslavskii, O.B. Near-extremal and extremal quantum-corrected two-dimensional charged black holes. Class. Quantum Gravity 2004, 21, 2687–2701. [Google Scholar] [CrossRef] [Green Version]
- Ali, A.F.; Khalil, M.M. Black hole with quantum potential. Nucl. Phys. B 2016, 909, 173–185. [Google Scholar] [CrossRef] [Green Version]
- Calmet, X.; El-Menoufi, B.K. Quantum corrections to Schwarzschild black hole. Eur. Phys. J. C 2017, 77, 243. [Google Scholar] [CrossRef]
- Shahjalal, M. Shahjalal Phase transition of quantum-corrected Schwarzschild black hole in rainbow gravity. Phys. Lett. B 2018, 784, 6–11. [Google Scholar] [CrossRef]
- Qi, D.-J.; Liu, L.; Liu, S.-X. Quantum tunneling and remnant from a quantum-modified Schwarzschild space–time close to Planck scale. Can. J. Phys. 2019, 97, 1012–1018. [Google Scholar] [CrossRef]
- Shahjalal, M. Thermodynamics of quantum-corrected Schwarzschild black hole surrounded by quintessence. Nucl. Phys. B 2019, 940, 63–77. [Google Scholar] [CrossRef]
- Eslamzadeh, S.; Nozari, K. Tunneling of massless and massive particles from a quantum deformed Schwarzschild black hole surrounded by quintessence. Nucl. Phys. B 2020, 959, 115136. [Google Scholar] [CrossRef]
- Good, M.R.; Linder, E.V. Schwarzschild Metric with Planck Length. arXiv 2020, arXiv:2003.01333. [Google Scholar]
- Nozari, K.; Hajebrahimi, M. Geodesic Structure of the Quantum-Corrected Schwarzschild Black Hole Surrounded by Quintessence. arXiv 2020, arXiv:2004.14775. [Google Scholar]
- Nozari, K.; Hajebrahimi, M.; Saghafi, S. Quantum corrections to the accretion onto a Schwarzschild black hole in the background of quintessence. Eur. Phys. J. C 2020, 80, 1–13. [Google Scholar] [CrossRef]
- Burger, D.J.; Moynihan, N.; Das, S.; Haque, S.S.; Underwood, B. Towards the Raychaudhuri equation beyond general relativity. Phys. Rev. D 2018, 98, 024006. [Google Scholar] [CrossRef] [Green Version]
- Russo, J.G.; Tseytlin, A. Scalar-tensor quantum gravity in two dimensions. Nucl. Phys. B 1992, 382, 259–275. [Google Scholar] [CrossRef] [Green Version]
- Jacobson, T. When is gttgrr = −1? Class. Quantum Gravity 2007, 24, 5717–5719. [Google Scholar] [CrossRef]
- Kiselev, V.V. Quintessence and black holes. Class. Quantum Gravity 2003, 20, 1187–1197. [Google Scholar] [CrossRef]
- Visser, M. The Kiselev black hole is neither perfect fluid, nor is it quintessence. Class. Quantum Gravity 2019, 37, 045001. [Google Scholar] [CrossRef] [Green Version]
- Boonserm, P.; Ngampitipan, T.; Simpson, A.; Visser, M. Decomposition of the total stress energy for the generalized Kiselev black hole. Phys. Rev. D 2020, 101, 024022. [Google Scholar] [CrossRef] [Green Version]
- Bardeen, J.M. Non-singular general-relativistic gravitational collapse. In Proceedings of the GR5 Conference, Tbilisi, Georgia, 9–13 September 1968; p. 174. [Google Scholar]
- Roman, T.A.; Bergmann, P.G. Stellar collapse without singularities? Phys. Rev. D 1983, 28, 1265–1277. [Google Scholar] [CrossRef]
- Borde, A. Regular black holes and topology change. Phys. Rev. D 1997, 55, 7615–7617. [Google Scholar] [CrossRef] [Green Version]
- Bronnikov, K.A. Regular magnetic black holes and monopoles from nonlinear electrodynamics. Phys. Rev. D 2001, 63, 044005. [Google Scholar] [CrossRef] [Green Version]
- Moreno, C.; Sarbach, O. Stability properties of black holes in self-gravitating nonlinear electrodynamics. Phys. Rev. D 2003, 67, 024028. [Google Scholar] [CrossRef] [Green Version]
- Ayon-Beato, E.; Garcia, A. Four parameter regular black hole solution. Gen. Rel. Grav. 2005, 37, 635. [Google Scholar] [CrossRef]
- Hayward, S.A. Formation and Evaporation of Nonsingular Black Holes. Phys. Rev. Lett. 2006, 96, 031103. [Google Scholar] [CrossRef] [Green Version]
- Bronnikov, K.A.; Fabris, J.C. Regular Phantom Black Holes. Phys. Rev. Lett. 2006, 96, 251101. [Google Scholar] [CrossRef] [Green Version]
- Bronnikov, K.A.; Dehnen, H.; Melnikov, V.N. Regular black holes and black universes. Gen. Relativ. Gravit. 2007, 39, 973–987. [Google Scholar] [CrossRef] [Green Version]
- Lemos, J.P.S.; Zaslavskii, O.B. Quasi-black holes: Definition and general properties. Phys. Rev. D 2007, 76, 084030. [Google Scholar] [CrossRef] [Green Version]
- Ansoldi, S. Spherical black holes with regular center: A Review of existing models including a recent realization with Gaussian sources. arXiv 2008, arXiv:0802.0330. [Google Scholar]
- Lemos, J.P.S.; Zanchin, V.T. Regular black holes: Electrically charged solutions, Reissner-Nordström outside a de Sitter core. Phys. Rev. D 2011, 83, 124005. [Google Scholar] [CrossRef] [Green Version]
- Bronnikov, K.A.; Konoplya, R.A.; Zhidenko, A. Instabilities of wormholes and regular black holes supported by a phantom scalar field. Phys. Rev. D 2012, 86, 024028. [Google Scholar] [CrossRef] [Green Version]
- Bambi, C.; Modesto, L. Rotating regular black holes. Phys. Lett. B 2013, 721, 329–334. [Google Scholar] [CrossRef] [Green Version]
- Bardeen, J.M. Black hole evaporation without an event horizon. arXiv 2014, arXiv:1406.4098. [Google Scholar]
- Frolov, V.P. Information loss problem and a ‘black hole’ model with a closed apparent horizon. JHEP 2014, 5, 49. [Google Scholar] [CrossRef] [Green Version]
- Frolov, V.P. Do Black Holes Exist? arXiv 2014, arXiv:1411.6981. [Google Scholar]
- Balart, L.; Vagenas, E.C. Regular black holes with a nonlinear electrodynamics source. Phys. Rev. D 2014, 90, 124045. [Google Scholar] [CrossRef] [Green Version]
- De Lorenzo, T.; Pacilio, C.; Rovelli, C.; Speziale, S. On the effective metric of a Planck star. Gen. Relativ. Gravit. 2015, 47, 41. [Google Scholar] [CrossRef] [Green Version]
- Frolov, V.P. Notes on nonsingular models of black holes. Phys. Rev. D 2016, 94, 104056. [Google Scholar] [CrossRef] [Green Version]
- Fan, Z.-Y.; Wang, X. Construction of regular black holes in general relativity. Phys. Rev. D 2016, 94, 124027. [Google Scholar] [CrossRef] [Green Version]
- Frolov, V.P.; Zelnikov, A. Quantum radiation from an evaporating nonsingular black hole. Phys. Rev. D 2017, 95, 124028. [Google Scholar] [CrossRef] [Green Version]
- Frolov, V.P. Remarks on non-singular black holes. EPJ Web Conf. 2018, 168, 01001. [Google Scholar] [CrossRef] [Green Version]
- Cano, P.A.; Chimento, S.; Ortín, T.; Ruipérez, A. Regular stringy black holes? Phys. Rev. D 2019, 99. [Google Scholar] [CrossRef] [Green Version]
- Bardeen, J.M. Models for the nonsingular transition of an evaporating black hole into a white hole. arXiv 2018, arXiv:1811.06683. [Google Scholar]
- Carballo-Rubio, R.; Di Filippo, F.; Liberati, S.; Pacilio, C.; Visser, M. On the viability of regular black holes. J. High Energy Phys. 2018, 2018, 23. [Google Scholar] [CrossRef] [Green Version]
- Carballo-Rubio, R.; Di Filippo, F.; Liberati, S.; Visser, M. Phenomenological aspects of black holes beyond general relativity. Phys. Rev. D 2018, 98, 124009. [Google Scholar] [CrossRef] [Green Version]
- Carballo-Rubio, R.; Di Filippo, F.; Liberati, S.; Visser, M. Opening the Pandora’s box at the core of black holes. Class. Quantum Gravity 2020, 37, 145005. [Google Scholar] [CrossRef] [Green Version]
- Carballo-Rubio, R.; Di Filippo, F.; Liberati, S.; Visser, M. Geodesically complete black holes. Phys. Rev. D 2020, 101, 084047. [Google Scholar] [CrossRef]
- Carballo-Rubio, R.; Di Filippo, F.; Liberati, S.; Pacilio, C.; Visser, M. Inner horizon instability and the unstable cores of regular black holes. J. High Energy Phys. 2021, 2021, 1–16. [Google Scholar] [CrossRef]
- Cardoso, V.; Hopper, S.; Macedo, C.F.B.; Palenzuela, C.; Pani, P. Gravitational-wave signatures of exotic compact objects and of quantum corrections at the horizon scale. Phys. Rev. D 2016, 94, 084031. [Google Scholar] [CrossRef] [Green Version]
- Visser, M.; Barceló, C.; Liberati, S.; Sonego, S. Small, dark, and heavy: But is it a black hole? PoS 2009, 75, 10. [Google Scholar] [CrossRef] [Green Version]
- Visser, M. Black holes in general relativity. PoS 2009, 75. [Google Scholar] [CrossRef] [Green Version]
- Visser, M.; Wiltshire, D.L. Stable gravastars—An alternative to black holes? Class. Quantum Gravity 2004, 21, 1135–1151. [Google Scholar] [CrossRef]
- Barceló, C.; Liberati, S.; Sonego, S.; Visser, M. Black Stars, Not Holes. Sci. Am. 2009, 301, 38–45. [Google Scholar] [CrossRef] [PubMed]
- Simpson, A.; Visser, M. Black-bounce to traversable wormhole. J. Cosmol. Astropart. Phys. 2019, 2019, 042. [Google Scholar] [CrossRef] [Green Version]
- Simpson, A.; Martín-Moruno, P.; Visser, M. Vaidya spacetimes, black-bounces, and traversable wormholes. Class. Quantum Gravity 2019, 36, 145007. [Google Scholar] [CrossRef] [Green Version]
- Lobo, F.S.N.; Simpson, A.; Visser, M. Dynamic thin-shell black-bounce traversable wormholes. Phys. Rev. D 2020, 101, 124035. [Google Scholar] [CrossRef]
- Simpson, A.; Visser, M. Regular Black Holes with Asymptotically Minkowski Cores. Universe 2019, 6, 8. [Google Scholar] [CrossRef] [Green Version]
- Berry, T.; Lobo, F.S.N.; Simpson, A.; Visser, M. Thin-shell traversable wormhole crafted from a regular black hole with asymptotically Minkowski core. Phys. Rev. D 2020, 102, 064054. [Google Scholar] [CrossRef]
- Berry, T.; Simpson, A.; Visser, M. Photon Spheres, ISCOs, and OSCOs: Astrophysical Observables for Regular Black Holes with Asymptotically Minkowski Cores. Universe 2020, 7, 2. [Google Scholar] [CrossRef]
- Boonserm, P.; Ngampitipan, T.; Simpson, A.; Visser, M. Exponential metric represents a traversable wormhole. Phys. Rev. D 2018, 98, 084048. [Google Scholar] [CrossRef] [Green Version]
- Barausse, E.; Berti, E.; Hertog, T.; Hughes, S.A.; Jetzer, P.; Pani, P.; Sotiriou, T.P.; Tamanini, N.; Witek, H.; Yagi, K.; et al. Prospects for fundamental physics with LISA. Gen. Relativ. Gravit. 2020, 52, 1–33. [Google Scholar] [CrossRef]
- Visser, M. Physical observability of horizons. Phys. Rev. D 2014, 90, 127502. [Google Scholar] [CrossRef] [Green Version]
- Hawking, S.W. Information Preservation and Weather Forecasting for Black Holes. arxiv 2014, arXiv:1401.5761. [Google Scholar]
- Lobo, F.S.N.; Rodrigues, M.E.; Silva, M.V.D.S.; Simpson, A.; Visser, M. Novel black-bounce geometries. arXiv 2021, arXiv:2009.12057. [Google Scholar]
- Visser, M. Dirty black holes: Thermodynamics and horizon structure. Phys. Rev. D 1992, 46, 2445–2451. [Google Scholar] [CrossRef] [Green Version]
- Available online: https://en.wikipedia.org/wiki/Energy_condition (accessed on 26 May 2021).
- Tipler, F.J. Energy conditions and spacetime singularities. Phys. Rev. D 1978, 17, 2521–2528. [Google Scholar] [CrossRef]
- Borde, A. Geodesic focusing, energy conditions and singularities. Class. Quantum Gravity 1987, 4, 343–356. [Google Scholar] [CrossRef]
- Klinkhammer, G. Averaged energy conditions for free scalar fields in flat spacetime. Phys. Rev. D 1991, 43, 2542–2548. [Google Scholar] [CrossRef]
- Ford, L.H.; Roman, T.A. Averaged energy conditions and quantum inequalities. Phys. Rev. D 1995, 51, 4277–4286. [Google Scholar] [CrossRef] [Green Version]
- Visser, M. Lorentzian Wormholes: From Einstein to Hawking; Springer: New York, NY, USA, 1995. [Google Scholar]
- Fewster, C.J.; Roman, T.A. Null energy conditions in quantum field theory. Phys. Rev. D 2003, 67, 044003. [Google Scholar] [CrossRef] [Green Version]
- Barceló, C.; Visser, M. Twilight for the energy conditions? Int. J. Mod. Phys. D 2002, 11, 1553–1560. [Google Scholar] [CrossRef] [Green Version]
- Visser, M. Energy Conditions in the Epoch of Galaxy Formation. Science 1997, 276, 88–90. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Visser, M.; Barceló, C. Energy conditions and their cosmological implications. arXiv 1999, arXiv:gr-qc/0001099. [Google Scholar]
- Visser, M. Gravitational vacuum polarization. II. Energy conditions in the Boulware vacuum. Phys. Rev. D 1996, 54, 5116–5122. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Roman, T.A. Some thoughts on energy conditions and wormholes. arXiv 2004, arXiv:gr-qc/0409090. [Google Scholar]
- Cattoen, C.; Visser, M. Cosmological milestones and energy conditions. J. Phys. Conf. Ser. 2007, 68, 012011. [Google Scholar] [CrossRef] [Green Version]
- Visser, M. Energy conditions and galaxy formation. arXiv 1997, arXiv:gr-qc/9710010. [Google Scholar] [CrossRef] [Green Version]
- Fewster, C.J.; Galloway, G.J. Singularity theorems from weakened energy conditions. Class. Quantum Gravity 2011, 28, 125009. [Google Scholar] [CrossRef] [Green Version]
- Zaslavskii, O. Regular black holes and energy conditions. Phys. Lett. B 2010, 688, 278–280. [Google Scholar] [CrossRef] [Green Version]
- Martín-Moruno, P.; Visser, M. Classical and Semi-classical Energy Conditions. Black Hole Phys. 2017, 189, 193–213. [Google Scholar] [CrossRef] [Green Version]
- Martin-Moruno, P.; Visser, M. Semiclassical energy conditions for quantum vacuum states. J. High Energy Phys. 2013, 2013, 1–36. [Google Scholar] [CrossRef] [Green Version]
- Martín-Moruno, P.; Visser, M. Classical and quantum flux energy conditions for quantum vacuum states. Phys. Rev. D 2013, 88. [Google Scholar] [CrossRef] [Green Version]
- Curiel, E. A Primer on Energy Conditions. Einstein Stud. 2017, 13, 43–104. [Google Scholar] [CrossRef] [Green Version]
- Martín-Moruno, P.; Visser, M. Semi-classical and nonlinear energy conditions. In Proceedings of the 14th Marcel Grossmann Meeting, Rome, Italy, 12–18 July 2015. [Google Scholar] [CrossRef] [Green Version]
- Deng, X.-M. Geodesics and periodic orbits around quantum-corrected black holes. Phys. Dark Universe 2020, 30, 100629. [Google Scholar] [CrossRef]
- Peng, J.; Guo, M.; Feng, X.H. Influence of Quantum Correction on the Black Hole Shadows, Photon Rings and Lensing Rings. arXiv 2020, arXiv:2008.00657. [Google Scholar]
- Boonserm, P.; Ngampitipan, T.; Visser, M. Regge-Wheeler equation, linear stability, and greybody factors for dirty black holes. Phys. Rev. D 2013, 88. [Google Scholar] [CrossRef] [Green Version]
- Flachi, A.; Lemos, J.P.S. Quasinormal modes of regular black holes. Phys. Rev. D 2013, 87, 24034. [Google Scholar] [CrossRef] [Green Version]
- Fernando, S.; Correa, J. Quasinormal modes of the Bardeen black hole: Scalar perturbations. Phys. Rev. D 2012, 86. [Google Scholar] [CrossRef] [Green Version]
- Rincón, Á.; Panotopoulos, G. Quasinormal modes of an improved Schwarzschild black hole. Phys. Dark Universe 2020, 30, 100639. [Google Scholar] [CrossRef]
- Konoplya, R. Quantum corrected black holes: Quasinormal modes, scattering, shadows. Phys. Lett. B 2020, 804, 135363. [Google Scholar] [CrossRef]
- Saleh, M.; Bouetou, B.T.; Kofane, T.C. Quasinormal modes of a quantum-corrected Schwarzschild black hole: Gravitational and Dirac perturbations. Astrophys. Space Sci. 2016, 361, 1–8. [Google Scholar] [CrossRef] [Green Version]
- Zhang, Y.; Lu, X. Electromagnetic quasinormal mode of quantum corrected Schwarzschild black hole. J. Kunming Univ. Sci. Technol. Nat. Sci. Ed. 2016, 6, 139–144. [Google Scholar] [CrossRef]
- Saleh, M.; Thomas, B.B.; Kofané, T.C. Quasinormal modes of scalar perturbation around a quantum-corrected Schwarzschild black hole. Astrophys. Space Sci. 2014, 350, 721–726. [Google Scholar] [CrossRef]
n | ||||
---|---|---|---|---|
0 | ||||
1 | globally violated | globally violated | globally violated | globally violated |
3 | globally violated | |||
5 | ||||
7 | ||||
9 | ||||
11 | ||||
⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
n | NEC | WEC | SEC | DEC |
---|---|---|---|---|
0 | ||||
1 | globally violated | globally violated | globally violated | globally violated |
3 | same as NEC | globally violated | ||
5 | same as NEC | |||
7 | same as NEC | |||
9 | same as NEC | |||
11 | same as NEC | |||
⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
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Berry, T.; Simpson, A.; Visser, M. Regularity of a General Class of “Quantum Deformed” Black Holes. Universe 2021, 7, 165. https://doi.org/10.3390/universe7060165
Berry T, Simpson A, Visser M. Regularity of a General Class of “Quantum Deformed” Black Holes. Universe. 2021; 7(6):165. https://doi.org/10.3390/universe7060165
Chicago/Turabian StyleBerry, Thomas, Alex Simpson, and Matt Visser. 2021. "Regularity of a General Class of “Quantum Deformed” Black Holes" Universe 7, no. 6: 165. https://doi.org/10.3390/universe7060165
APA StyleBerry, T., Simpson, A., & Visser, M. (2021). Regularity of a General Class of “Quantum Deformed” Black Holes. Universe, 7(6), 165. https://doi.org/10.3390/universe7060165