# Weak cosmic censorship conjecture in Kerr-(anti-)de Sitter black hole with scalar field

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## Abstract

We investigate the weak cosmic censorship conjecture in Kerr-(anti-)de Sitter black holes under the scattering of a scalar field. We test the conjecture in terms of whether the black hole can exceed the extremal condition with respect to its change caused by the energy and angular momentum fluxes of the scalar field. Without imposing the laws of thermodynamics, we prove that the conjecture is valid in all the initial states of the black hole (non-extremal, near-extremal, and extremal black holes). The validity in the case of the near-extremal black hole is different from the results of similar tests conducted by adding a particle because the fluxes represent the energy and angular momentum transferred to the black hole during the time interval not included in the tests involving the particle. Using the time interval, we show that the angular velocity of the black hole with the scalar field of a constant state takes a long time for saturation to the frequency of the scalar field.

## Keywords

Black Holes Classical Theories of Gravity Spacetime Singularities## Notes

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## References

- [1]S.W. Hawking,
*Particle Creation by Black Holes*,*Commun. Math. Phys.***43**(1975) 199 [*Erratum ibid.***46**(1976) 206] [INSPIRE]. - [2]S.W. Hawking,
*Black Holes and Thermodynamics*,*Phys. Rev.***D 13**(1976) 191 [INSPIRE].ADSCrossRefGoogle Scholar - [3]D. Christodoulou,
*Reversible and irreversible transforations in black hole physics*,*Phys. Rev. Lett.***25**(1970) 1596 [INSPIRE].ADSCrossRefGoogle Scholar - [4]
- [5]D. Christodoulou and R. Ruffini,
*Reversible transformations of a charged black hole*,*Phys. Rev.***D 4**(1971) 3552 [INSPIRE].ADSCrossRefGoogle Scholar - [6]L. Smarr,
*Mass formula for Kerr black holes*,*Phys. Rev. Lett.***30**(1973) 71 [*Erratum ibid.***30**(1973) 521] [INSPIRE].ADSCrossRefGoogle Scholar - [7]J.D. Bekenstein,
*Black holes and entropy*,*Phys. Rev.***D 7**(1973) 2333 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [8]J.D. Bekenstein,
*Generalized second law of thermodynamics in black hole physics*,*Phys. Rev.***D 9**(1974) 3292 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [9]R. Penrose,
*Gravitational collapse and space-time singularities*,*Phys. Rev. Lett.***14**(1965) 57 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [10]R. Penrose,
*Gravitational collapse: The role of general relativity*,*Riv. Nuovo Cim.***1**(1969) 252 [INSPIRE].ADSGoogle Scholar - [11]R. Wald,
*Gedanken experiments to destroy a black hole*,*Annals Phys.***82**(1974) 548.ADSCrossRefGoogle Scholar - [12]T. Jacobson and T.P. Sotiriou,
*Over-spinning a black hole with a test body*,*Phys. Rev. Lett.***103**(2009) 141101 [*Erratum ibid.***103**(2009) 209903] [arXiv:0907.4146] [INSPIRE]. - [13]E. Barausse, V. Cardoso and G. Khanna,
*Test bodies and naked singularities: Is the self-force the cosmic censor?*,*Phys. Rev. Lett.***105**(2010) 261102 [arXiv:1008.5159] [INSPIRE].ADSCrossRefGoogle Scholar - [14]E. Barausse, V. Cardoso and G. Khanna,
*Testing the Cosmic Censorship Conjecture with point particles: the effect of radiation reaction and the self-force*,*Phys. Rev.***D 84**(2011) 104006 [arXiv:1106.1692] [INSPIRE]. - [15]M. Colleoni, L. Barack, A.G. Shah and M. van de Meent,
*Self-force as a cosmic censor in the Kerr overspinning problem*,*Phys. Rev.***D 92**(2015) 084044 [arXiv:1508.04031] [INSPIRE]. - [16]M. Colleoni and L. Barack,
*Overspinning a Kerr black hole: the effect of self-force*,*Phys. Rev.***D 91**(2015) 104024 [arXiv:1501.07330] [INSPIRE]. - [17]J. Sorce and R.M. Wald,
*Gedanken experiments to destroy a black hole. II. Kerr-Newman black holes cannot be overcharged or overspun*,*Phys. Rev.***D 96**(2017) 104014 [arXiv:1707.05862] [INSPIRE]. - [18]V.E. Hubeny,
*Overcharging a black hole and cosmic censorship*,*Phys. Rev.***D 59**(1999) 064013 [gr-qc/9808043] [INSPIRE]. - [19]S. Isoyama, N. Sago and T. Tanaka,
*Cosmic censorship in overcharging a Reissner-Nordström black hole via charged particle absorption*,*Phys. Rev.***D 84**(2011) 124024 [arXiv:1108.6207] [INSPIRE]. - [20]M. Bouhmadi-Lopez, V. Cardoso, A. Nerozzi and J.V. Rocha,
*Black holes die hard: can one spin-up a black hole past extremality?*,*Phys. Rev.***D 81**(2010) 084051 [arXiv:1003.4295] [INSPIRE]. - [21]B. Gwak and B.-H. Lee,
*Rotating Black Hole Thermodynamics with a Particle Probe*,*Phys. Rev.***D 84**(2011) 084049 [arXiv:1106.1483] [INSPIRE]. - [22]J.V. Rocha and V. Cardoso,
*Gravitational perturbation of the BTZ black hole induced by test particles and weak cosmic censorship in AdS spacetime*,*Phys. Rev.***D 83**(2011) 104037 [arXiv:1102.4352] [INSPIRE]. - [23]J. Crisostomo and R. Olea,
*Hamiltonian treatment of the gravitational collapse of thin shells*,*Phys. Rev.***D 69**(2004) 104023 [hep-th/0311054] [INSPIRE]. - [24]S. Gao and Y. Zhang,
*Destroying extremal Kerr-Newman black holes with test particles*,*Phys. Rev.***D 87**(2013) 044028 [arXiv:1211.2631] [INSPIRE]. - [25]S. Hod,
*Cosmic Censorship: Formation of a Shielding Horizon Around a Fragile Horizon*,*Phys. Rev.***D 87**(2013) 024037 [arXiv:1302.6658] [INSPIRE]. - [26]Y. Zhang and S. Gao,
*Testing cosmic censorship conjecture near extremal black holes with cosmological constants*,*Int. J. Mod. Phys.***D 23**(2014) 1450044 [arXiv:1309.2027] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [27]J.V. Rocha and R. Santarelli,
*Flowing along the edge: spinning up black holes in AdS spacetimes with test particles*,*Phys. Rev.***D 89**(2014) 064065 [arXiv:1402.4840] [INSPIRE]. - [28]B. McInnes and Y.C. Ong,
*A Note on Physical Mass and the Thermodynamics of AdS-Kerr Black Holes*,*JCAP***11**(2015) 004 [arXiv:1506.01248] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [29]V. Cardoso and L. Queimada,
*Cosmic Censorship and parametrized spinning black-hole geometries*,*Gen. Rel. Grav.***47**(2015) 150 [arXiv:1511.00690] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [30]H.M. Siahaan,
*Destroying Kerr-Sen black holes*,*Phys. Rev.***D 93**(2016) 064028 [arXiv:1512.01654] [INSPIRE]. - [31]J. Natario, L. Queimada and R. Vicente,
*Test fields cannot destroy extremal black holes*,*Class. Quant. Grav.***33**(2016) 175002 [arXiv:1601.06809] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [32]G.T. Horowitz, J.E. Santos and B. Way,
*Evidence for an Electrifying Violation of Cosmic Censorship*,*Class. Quant. Grav.***33**(2016) 195007 [arXiv:1604.06465] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [33]K. Düztas,
*Overspinning BTZ black holes with test particles and fields*,*Phys. Rev.***D 94**(2016) 124031 [arXiv:1701.07241] [INSPIRE]. - [34]Y. Song, R.-H. Yue, M. Zhang, D.-C. Zou and C.-Y. Sun,
*Destroying a Near-Extremal Kerr-Newman-AdS Black Hole with Test Particles*,*Commun. Theor. Phys.***69**(2018) 694 [arXiv:1705.01676] [INSPIRE].ADSCrossRefGoogle Scholar - [35]K.S. Revelar and I. Vega,
*Overcharging higher-dimensional black holes with point particles*,*Phys. Rev.***D 96**(2017) 064010 [arXiv:1706.07190] [INSPIRE]. - [36]K. Düztas,
*Can test fields destroy the event horizon in the Kerr-Taub-NUT spacetime?*,*Class. Quant. Grav.***35**(2018) 045008 [arXiv:1710.06610] [INSPIRE].ADSCrossRefGoogle Scholar - [37]T.-Y. Yu and W.-Y. Wen,
*Cosmic censorship and Weak Gravity Conjecture in the Einstein-Maxwell-dilaton theory*,*Phys. Lett.***B 781**(2018) 713 [arXiv:1803.07916] [INSPIRE].ADSCrossRefGoogle Scholar - [38]B. Liang, S.-W. Wei and Y.-X. Liu,
*Weak cosmic censorship conjecture in Kerr black holes of modified gravity*, arXiv:1804.06966 [INSPIRE]. - [39]B. Gwak and B.-H. Lee,
*Cosmic Censorship of Rotating Anti-de Sitter Black Hole*,*JCAP***02**(2016) 015 [arXiv:1509.06691] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [40]B. Gwak,
*Thermodynamics with Pressure and Volume under Charged Particle Absorption*,*JHEP***11**(2017) 129 [arXiv:1709.08665] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [41]S. Hod,
*Weak Cosmic Censorship: As Strong as Ever*,*Phys. Rev. Lett.***100**(2008) 121101 [arXiv:0805.3873] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [42]İ. Semiz,
*Dyonic Kerr-Newman black holes, complex scalar field and cosmic censorship*,*Gen. Rel. Grav.***43**(2011) 833 [gr-qc/0508011] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [43]G.Z. Toth,
*Test of the weak cosmic censorship conjecture with a charged scalar field and dyonic Kerr-Newman black holes*,*Gen. Rel. Grav.***44**(2012) 2019 [arXiv:1112.2382] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [44]K. Düztas and İ. Semiz,
*Cosmic Censorship, Black Holes and Integer-spin Test Fields*,*Phys. Rev.***D 88**(2013) 064043 [arXiv:1307.1481] [INSPIRE]. - [45]İ. Semiz and K. Düztas,
*Weak Cosmic Censorship, Superradiance and Quantum Particle Creation*,*Phys. Rev.***D 92**(2015) 104021 [arXiv:1507.03744] [INSPIRE]. - [46]
- [47]Y.B. Zel’dovich,
*Amplification of cylindrical electromagnetic waves reflected from a rotating body*,*JETP***35**(1972) 1085.ADSGoogle Scholar - [48]R. Brito, V. Cardoso and P. Pani,
*Superradiance: Energy Extraction, Black-Hole Bombs and Implications for Astrophysics and Particle Physics*,*Lect. Notes Phys.***906**(2015) 1 [arXiv:1501.06570] [INSPIRE].CrossRefGoogle Scholar - [49]S.W. Hawking and H.S. Reall,
*Charged and rotating AdS black holes and their CFT duals*,*Phys. Rev.***D 61**(2000) 024014 [hep-th/9908109] [INSPIRE]. - [50]V. Cardoso and O.J.C. Dias,
*Small Kerr-anti-de Sitter black holes are unstable*,*Phys. Rev.***D 70**(2004) 084011 [hep-th/0405006] [INSPIRE]. - [51]V. Cardoso, O.J.C. Dias and S. Yoshida,
*Classical instability of Kerr-AdS black holes and the issue of final state*,*Phys. Rev.***D 74**(2006) 044008 [hep-th/0607162] [INSPIRE]. - [52]N. Uchikata, S. Yoshida and T. Futamase,
*Scalar perturbations of Kerr-AdS black holes*,*Phys. Rev.***D 80**(2009) 084020 [INSPIRE]. - [53]L.C.B. Crispino, A. Higuchi, E.S. Oliveira and J.V. Rocha,
*Greybody factors for nonminimally coupled scalar fields in Schwarzschild-de Sitter spacetime*,*Phys. Rev.***D 87**(2013) 104034 [arXiv:1304.0467] [INSPIRE]. - [54]P. Kanti, T. Pappas and N. Pappas,
*Greybody factors for scalar fields emitted by a higher-dimensional Schwarzschild-de Sitter black hole*,*Phys. Rev.***D 90**(2014) 124077 [arXiv:1409.8664] [INSPIRE]. - [55]T. Pappas, P. Kanti and N. Pappas,
*Hawking radiation spectra for scalar fields by a higher-dimensional Schwarzschild-de Sitter black hole*,*Phys. Rev.***D 94**(2016) 024035 [arXiv:1604.08617] [INSPIRE]. - [56]T. Pappas and P. Kanti,
*Schwarzschild-de Sitter spacetime: The role of temperature in the emission of Hawking radiation*,*Phys. Lett.***B 775**(2017) 140 [arXiv:1707.04900] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [57]S.A. Teukolsky,
*Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations*,*Astrophys. J.***185**(1973) 635 [INSPIRE].ADSCrossRefGoogle Scholar - [58]H. Suzuki, E. Takasugi and H. Umetsu,
*Perturbations of Kerr-de Sitter black hole and Heun’s equations*,*Prog. Theor. Phys.***100**(1998) 491 [gr-qc/9805064] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [59]H. Suzuki, E. Takasugi and H. Umetsu,
*Analytic solutions of Teukolsky equation in Kerr-de Sitter and Kerr-Newman-de Sitter geometries*,*Prog. Theor. Phys.***102**(1999) 253 [gr-qc/9905040] [INSPIRE]. - [60]H.T. Cho, A.S. Cornell, J. Doukas and W. Naylor,
*Asymptotic iteration method for spheroidal harmonics of higher-dimensional Kerr-(A)dS black holes*,*Phys. Rev.***D 80**(2009) 064022 [arXiv:0904.1867] [INSPIRE]. - [61]S. Yoshida, N. Uchikata and T. Futamase,
*Quasinormal modes of Kerr-de Sitter black holes*,*Phys. Rev.***D 81**(2010) 044005 [INSPIRE]. - [62]O.J.C. Dias and J.E. Santos,
*Boundary Conditions for Kerr-AdS Perturbations*,*JHEP***10**(2013) 156 [arXiv:1302.1580] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [63]V. Cardoso, O.J.C. Dias, G.S. Hartnett, L. Lehner and J.E. Santos,
*Holographic thermalization, quasinormal modes and superradiance in Kerr-AdS*,*JHEP***04**(2014) 183 [arXiv:1312.5323] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [64]C.-Y. Zhang, S.-J. Zhang and B. Wang,
*Superradiant instability of Kerr-de Sitter black holes in scalar-tensor theory*,*JHEP***08**(2014) 011 [arXiv:1405.3811] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [65]O. Delice and T. Durğut,
*Superradiance Instability of Small Rotating AdS Black Holes in Arbitrary Dimensions*,*Phys. Rev.***D 92**(2015) 024053 [arXiv:1503.05818] [INSPIRE]. - [66]
- [67]J. Ahmed and K. Saifullah,
*Greybody factor of a scalar field from Reissner-Nordström-de Sitter black hole*,*Eur. Phys. J.***C 78**(2018) 316 [arXiv:1610.06104] [INSPIRE]. - [68]S.W. Hawking, C.J. Hunter and M. Taylor,
*Rotation and the AdS/CFT correspondence*,*Phys. Rev.***D 59**(1999) 064005 [hep-th/9811056] [INSPIRE]. - [69]M.M. Caldarelli, G. Cognola and D. Klemm,
*Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories*,*Class. Quant. Grav.***17**(2000) 399 [hep-th/9908022] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [70]G.W. Gibbons, M.J. Perry and C.N. Pope,
*The First law of thermodynamics for Kerr-anti-de Sitter black holes*,*Class. Quant. Grav.***22**(2005) 1503 [hep-th/0408217] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [71]B.P. Dolan, D. Kastor, D. Kubiznak, R.B. Mann and J. Traschen,
*Thermodynamic Volumes and Isoperimetric Inequalities for de Sitter Black Holes*,*Phys. Rev.***D 87**(2013) 104017 [arXiv:1301.5926] [INSPIRE]. - [72]D. Kubiznak and F. Simovic,
*Thermodynamics of horizons: de Sitter black holes and reentrant phase transitions*,*Class. Quant. Grav.***33**(2016) 245001 [arXiv:1507.08630] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [73]D. Kubiznak, R.B. Mann and M. Teo,
*Black hole chemistry: thermodynamics with Lambda*,*Class. Quant. Grav.***34**(2017) 063001 [arXiv:1608.06147] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [74]M. Vasudevan, K.A. Stevens and D.N. Page,
*Separability of the Hamilton-Jacobi and Klein-Gordon equations in Kerr-de Sitter metrics*,*Class. Quant. Grav.***22**(2005) 339 [gr-qc/0405125] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [75]D.N. Page, D. Kubiznak, M. Vasudevan and P. Krtous,
*Complete integrability of geodesic motion in general Kerr-NUT-AdS spacetimes*,*Phys. Rev. Lett.***98**(2007) 061102 [hep-th/0611083] [INSPIRE]. - [76]R. Penrose and R.M. Floyd,
*Extraction of rotational energy from a black hole*,*Nature***229**(1971) 177 [INSPIRE].ADSCrossRefGoogle Scholar - [77]U. Khanal,
*Further investigations of the Kerr-de Sitter space*,*Phys. Rev.***D 32**(1985) 879 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [78]S. Gao and R.M. Wald,
*The ‘Physical process’ version of the first law and the generalized second law for charged and rotating black holes*,*Phys. Rev.***D 64**(2001) 084020 [gr-qc/0106071] [INSPIRE]. - [79]B. Gwak,
*Cosmic Censorship Conjecture in Kerr-Sen Black Hole*,*Phys. Rev.***D 95**(2017) 124050 [arXiv:1611.09640] [INSPIRE]. - [80]G. Chirco, S. Liberati and T.P. Sotiriou,
*Gedanken experiments on nearly extremal black holes and the Third Law*,*Phys. Rev.***D 82**(2010) 104015 [arXiv:1006.3655] [INSPIRE].