Abstract
A mechanism for circumventing the Mayo-Bekenstein no-hair theorem allows endowing four dimensional (D = 4) asymptotically flat, spherical, electro-vacuum black holes with a minimally coupled U (1)-gauged scalar field profile: Q-hair. The scalar field must be massive, self-interacting and obey a resonance condition at the threshold of (charged) superradiance. We establish generality for this mechanism by endowing three different types of static black objects with scalar hair, within a D = 5 Einstein-Maxwell-gauged scalar field model: asymptotically flat black holes and black rings; and black strings which asymptote to a Kaluza-Klein vacuum. These D = 5 Q-hairy black objects share many of the features of their D = 4 counterparts. In particular, the scalar field is subject to a resonance condition and possesses a Q-ball type potential. For the static black ring, the charged scalar hair can balance it, yielding solutions that are singularity free on and outside the horizon.
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J.D. Bekenstein, Black hole hair: 25 years after, in proceedings of the 2nd International Sakharov Conference on Physics, Moscow, Russian Federation, 20–23 May 1996, pp. 216–219 [gr-qc/9605059] [INSPIRE].
C.A.R. Herdeiro and E. Radu, Asymptotically flat black holes with scalar hair: a review, Int. J. Mod. Phys. D 24 (2015) 1542014 [arXiv:1504.08209] [INSPIRE].
T.P. Sotiriou, Black Holes and Scalar Fields, Class. Quant. Grav. 32 (2015) 214002 [arXiv:1505.00248] [INSPIRE].
J.D. Bekenstein, Transcendence of the law of baryon-number conservation in black hole physics, Phys. Rev. Lett. 28 (1972) 452 [INSPIRE].
A.E. Mayo and J.D. Bekenstein, No hair for spherical black holes: Charged and nonminimally coupled scalar field with selfinteraction, Phys. Rev. D 54 (1996) 5059 [gr-qc/9602057] [INSPIRE].
C.A.R. Herdeiro and E. Radu, Kerr black holes with scalar hair, Phys. Rev. Lett. 112 (2014) 221101 [arXiv:1403.2757] [INSPIRE].
C.A.R. Herdeiro and E. Radu, A new spin on black hole hair, Int. J. Mod. Phys. D 23 (2014) 1442014 [arXiv:1405.3696] [INSPIRE].
I. Smolić, Symmetry inheritance of scalar fields, Class. Quant. Grav. 32 (2015) 145010 [arXiv:1501.04967] [INSPIRE].
O.J.C. Dias, G.T. Horowitz and J.E. Santos, Black holes with only one Killing field, JHEP 07 (2011) 115 [arXiv:1105.4167] [INSPIRE].
O.J.C. Dias, J.E. Santos and B. Way, Black holes with a single Killing vector field: black resonators, JHEP 12 (2015) 171 [arXiv:1505.04793] [INSPIRE].
T. Ishii and K. Murata, Black resonators and geons in AdS5, Class. Quant. Grav. 36 (2019) 125011 [arXiv:1810.11089] [INSPIRE].
Y. Brihaye, C. Herdeiro and E. Radu, Myers-Perry black holes with scalar hair and a mass gap, Phys. Lett. B 739 (2014) 1 [arXiv:1408.5581] [INSPIRE].
C. Herdeiro, J. Kunz, E. Radu and B. Subagyo, Myers-Perry black holes with scalar hair and a mass gap: Unequal spins, Phys. Lett. B 748 (2015) 30 [arXiv:1505.02407] [INSPIRE].
C.A.R. Herdeiro, E. Radu and H. Rúnarsson, Kerr black holes with self-interacting scalar hair: hairier but not heavier, Phys. Rev. D 92 (2015) 084059 [arXiv:1509.02923] [INSPIRE].
J.F.M. Delgado, C.A.R. Herdeiro, E. Radu and H. Runarsson, Kerr-Newman black holes with scalar hair, Phys. Lett. B 761 (2016) 234 [arXiv:1608.00631] [INSPIRE].
C. Herdeiro, J. Kunz, E. Radu and B. Subagyo, Probing the universality of synchronised hair around rotating black holes with Q-clouds, Phys. Lett. B 779 (2018) 151 [arXiv:1712.04286] [INSPIRE].
Y.-Q. Wang, Y.-X. Liu and S.-W. Wei, Excited Kerr black holes with scalar hair, Phys. Rev. D 99 (2019) 064036 [arXiv:1811.08795] [INSPIRE].
J. Kunz, I. Perapechka and Y. Shnir, Kerr black holes with parity-odd scalar hair, Phys. Rev. D 100 (2019) 064032 [arXiv:1904.07630] [INSPIRE].
J.F.M. Delgado, C.A.R. Herdeiro and E. Radu, Kerr black holes with synchronised scalar hair and higher azimuthal harmonic index, Phys. Lett. B 792 (2019) 436 [arXiv:1903.01488] [INSPIRE].
C. Herdeiro, E. Radu and H. Rúnarsson, Kerr black holes with Proca hair, Class. Quant. Grav. 33 (2016) 154001 [arXiv:1603.02687] [INSPIRE].
N.M. Santos, C.L. Benone, L.C.B. Crispino, C.A.R. Herdeiro and E. Radu, Black holes with synchronised Proca hair: linear clouds and fundamental non-linear solutions, JHEP 07 (2020) 010 [arXiv:2004.09536] [INSPIRE].
S. Hod, Stationary Scalar Clouds Around Rotating Black Holes, Phys. Rev. D 86 (2012) 104026 [Erratum ibid. 86 (2012) 129902] [arXiv:1211.3202] [INSPIRE].
S. Hod, Kerr-Newman black holes with stationary charged scalar clouds, Phys. Rev. D 90 (2014) 024051 [arXiv:1406.1179] [INSPIRE].
C.L. Benone, L.C.B. Crispino, C. Herdeiro and E. Radu, Kerr-Newman scalar clouds, Phys. Rev. D 90 (2014) 104024 [arXiv:1409.1593] [INSPIRE].
S. Hod, Spinning Kerr black holes with stationary massive scalar clouds: The large-coupling regime, JHEP 01 (2017) 030 [arXiv:1612.00014] [INSPIRE].
R. Brito, V. Cardoso and P. Pani, Superradiance: New Frontiers in Black Hole Physics, in Lecture Notes in Physics 971, Springer (2015) [arXiv:1501.06570] [INSPIRE].
W.E. East and F. Pretorius, Superradiant Instability and Backreaction of Massive Vector Fields around Kerr Black Holes, Phys. Rev. Lett. 119 (2017) 041101 [arXiv:1704.04791] [INSPIRE].
C.A.R. Herdeiro and E. Radu, Dynamical Formation of Kerr Black Holes with Synchronized Hair: An Analytic Model, Phys. Rev. Lett. 119 (2017) 261101 [arXiv:1706.06597] [INSPIRE].
C.A.R. Herdeiro, Black holes: on the universality of the Kerr hypothesis, arXiv:2204.05640 [INSPIRE].
P. Basu, J. Bhattacharya, S. Bhattacharyya, R. Loganayagam, S. Minwalla and V. Umesh, Small Hairy Black Holes in Global AdS Spacetime, JHEP 10 (2010) 045 [arXiv:1003.3232] [INSPIRE].
O.J.C. Dias, P. Figueras, S. Minwalla, P. Mitra, R. Monteiro and J.E. Santos, Hairy black holes and solitons in global AdS5, JHEP 08 (2012) 117 [arXiv:1112.4447] [INSPIRE].
O.J.C. Dias and R. Masachs, Hairy black holes and the endpoint of AdS4 charged superradiance, JHEP 02 (2017) 128 [arXiv:1610.03496] [INSPIRE].
N. Sanchis-Gual, J.C. Degollado, P.J. Montero, J.A. Font and C. Herdeiro, Explosion and Final State of an Unstable Reissner-Nordström Black Hole, Phys. Rev. Lett. 116 (2016) 141101 [arXiv:1512.05358] [INSPIRE].
N. Sanchis-Gual, J.C. Degollado, C. Herdeiro, J.A. Font and P.J. Montero, Dynamical formation of a Reissner-Nordström black hole with scalar hair in a cavity, Phys. Rev. D 94 (2016) 044061 [arXiv:1607.06304] [INSPIRE].
S. Bhattacharyya, S. Minwalla and K. Papadodimas, Small Hairy Black Holes in AdS5 × S5, JHEP 11 (2011) 035 [arXiv:1005.1287] [INSPIRE].
J. Markeviciute and J.E. Santos, Hairy black holes in AdS5 × S5, JHEP 06 (2016) 096 [arXiv:1602.03893] [INSPIRE].
O.J.C. Dias, P. Mitra and J.E. Santos, New phases of \( \mathcal{N} \) = 4 SYM at finite chemical potential, arXiv:2207.07134 [INSPIRE].
S. Hod, Stability of the extremal Reissner-Nordström black hole to charged scalar perturbations, Phys. Lett. B 713 (2012) 505 [arXiv:1304.6474] [INSPIRE].
S. Hod, No-bomb theorem for charged Reissner-Nordström black holes, Phys. Lett. B 718 (2013) 1489 [INSPIRE].
J.C. Degollado and C.A.R. Herdeiro, Stationary scalar configurations around extremal charged black holes, Gen. Rel. Grav. 45 (2013) 2483 [arXiv:1303.2392] [INSPIRE].
J.-P. Hong, M. Suzuki and M. Yamada, Charged black holes in non-linear Q-clouds with O(3) symmetry, Phys. Lett. B 803 (2020) 135324 [arXiv:1907.04982] [INSPIRE].
C.A.R. Herdeiro and E. Radu, Spherical electro-vacuum black holes with resonant, scalar Q-hair, Eur. Phys. J. C 80 (2020) 390 [arXiv:2004.00336] [INSPIRE].
J.-P. Hong, M. Suzuki and M. Yamada, Spherically Symmetric Scalar Hair for Charged Black Holes, Phys. Rev. Lett. 125 (2020) 111104 [arXiv:2004.03148] [INSPIRE].
C. Herdeiro, E. Radu and H. Runarsson, Non-linear Q-clouds around Kerr black holes, Phys. Lett. B 739 (2014) 302 [arXiv:1409.2877] [INSPIRE].
R. Emparan and H.S. Reall, Black Holes in Higher Dimensions, Living Rev. Rel. 11 (2008) 6 [arXiv:0801.3471] [INSPIRE].
K.-i. Maeda, T. Shiromizu and T. Tanaka, Higher Dimensional Black Holes, in Progress in Theoretical Physics Supplement 189, Oxford University Press, Oxford, U.K. (2011).
G.T. Horowitz ed., Black Holes in Higher Dimensions, Cambridge University Press, Cambridge, U.K. (2012).
H.S. Reall, Higher dimensional black holes, Int. J. Mod. Phys. D 21 (2012) 1230001 [arXiv:1210.1402] [INSPIRE].
C. Herdeiro, B. Kleihaus, J. Kunz and E. Radu, On the Bekenstein-Hawking area law for black objects with conical singularities, Phys. Rev. D 81 (2010) 064013 [arXiv:0912.3386] [INSPIRE].
C. Herdeiro, E. Radu and C. Rebelo, Thermodynamical description of stationary, asymptotically flat solutions with conical singularities, Phys. Rev. D 81 (2010) 104031 [arXiv:1004.3959] [INSPIRE].
D. Astefanesei, M.J. Rodriguez and S. Theisen, Quasilocal equilibrium condition for black ring, JHEP 12 (2009) 040 [arXiv:0909.0008] [INSPIRE].
J.H. Traschen and D. Fox, Tension perturbations of black brane space-times, Class. Quant. Grav. 21 (2004) 289 [gr-qc/0103106] [INSPIRE].
J.H. Traschen, A Positivity theorem for gravitational tension in brane space-times, Class. Quant. Grav. 21 (2004) 1343 [hep-th/0308173] [INSPIRE].
T. Harmark and N.A. Obers, New phase diagram for black holes and strings on cylinders, Class. Quant. Grav. 21 (2004) 1709 [hep-th/0309116] [INSPIRE].
B. Kol, E. Sorkin and T. Piran, Caged black holes: Black holes in compactified space-times. Part 1. Theory, Phys. Rev. D 69 (2004) 064031 [hep-th/0309190] [INSPIRE].
U. Ascher, J. Christiansen and R.D. Russell, A Collocation Solver for Mixed Order Systems of Boundary Value Problems, Math. Comput. 33 (1979) 659 [INSPIRE].
U. Asher, J. Christiansen and R.D. Russel, Collocation Software for Boundary-Value ODEs, ACM Trans. Math. Software 7 (1981) 209.
Y. Brihaye and B. Hartmann, Spherically symmetric charged black holes with wavy scalar hair, Class. Quant. Grav. 39 (2022) 015010 [arXiv:2108.02248] [INSPIRE].
Y. Brihaye and B. Hartmann, Boson stars and black holes with wavy scalar hair, Phys. Rev. D 105 (2022) 104063 [arXiv:2112.12830] [INSPIRE].
P. Jetzer and J.J. van der Bij, Charged Boson Stars, Phys. Lett. B 227 (1989) 341 [INSPIRE].
P. Jetzer, P. Liljenberg and B.S. Skagerstam, Charged boson stars and vacuum instabilities, Astropart. Phys. 1 (1993) 429 [astro-ph/9305014] [INSPIRE].
D. Pugliese, H. Quevedo, J.A. Rueda Hernández and R. Ruffini, On charged boson stars, Phys. Rev. D 88 (2013) 024053 [arXiv:1305.4241] [INSPIRE].
A. Prikas, q stars and charged q stars, Phys. Rev. D 66 (2002) 025023 [hep-th/0205197] [INSPIRE].
Y. Brihaye, V. Diemer and B. Hartmann, Charged Q-balls and boson stars and dynamics of charged test particles, Phys. Rev. D 89 (2014) 084048 [arXiv:1402.1055] [INSPIRE].
F.R. Tangherlini, Schwarzschild field in n dimensions and the dimensionality of space problem, Nuovo Cim. 27 (1963) 636 [INSPIRE].
R.C. Myers and M.J. Perry, Black Holes in Higher Dimensional Space-Times, Annals Phys. 172 (1986) 304 [INSPIRE].
R. Emparan and H.S. Reall, A Rotating black ring solution in five-dimensions, Phys. Rev. Lett. 88 (2002) 101101 [hep-th/0110260] [INSPIRE].
R. Emparan and H.S. Reall, Generalized Weyl solutions, Phys. Rev. D 65 (2002) 084025 [hep-th/0110258] [INSPIRE].
S.S. Yazadjiev, Asymptotically and non-asymptotically flat static black rings in charged dilaton gravity, hep-th/0507097 [INSPIRE].
S.S. Yazadjiev, Magnetized black holes and black rings in the higher dimensional dilaton gravity, Phys. Rev. D 73 (2006) 064008 [gr-qc/0511114] [INSPIRE].
B. Kleihaus, J. Kunz and E. Radu, Generalized Weyl solutions in d = 5 Einstein-Gauss-Bonnet theory: The Static black ring, JHEP 02 (2010) 092 [arXiv:0912.1725] [INSPIRE].
H.K. Kunduri and J. Lucietti, Electrically charged dilatonic black rings, Phys. Lett. B 609 (2005) 143 [hep-th/0412153] [INSPIRE].
B. Kleihaus, J. Kunz and E. Radu, Balancing a static black ring with a phantom scalar field, Phys. Lett. B 797 (2019) 134892 [arXiv:1906.06372] [INSPIRE].
B. Kleihaus, J. Kunz, E. Radu and M.J. Rodriguez, New generalized nonspherical black hole solutions, JHEP 02 (2011) 058 [arXiv:1010.2898] [INSPIRE].
B. Kleihaus, J. Kunz and K. Schnulle, Charged Balanced Black Rings in Five Dimensions, Phys. Lett. B 699 (2011) 192 [arXiv:1012.5044] [INSPIRE].
B. Kleihaus, J. Kunz and E. Radu, Black rings in six dimensions, Phys. Lett. B 718 (2013) 1073 [arXiv:1205.5437] [INSPIRE].
B. Kleihaus, J. Kunz and E. Radu, Black ringoids: spinning balanced black objects in d 5 dimensions — the codimension-two case, JHEP 01 (2015) 117 [arXiv:1410.0581] [INSPIRE].
W. Schönauer and R. Weiß, Efficient vectorizable PDE solvers, J. Comput. Appl. Math. 27 (1989) 279.
M. Schauder, R. Weiß and W. Schönauer, The CADSOL Program Package, in Interner Bericht 46, Universität Karlsruhe (1992).
I. Salazar Landea and F. García, Charged Proca Stars, Phys. Rev. D 94 (2016) 104006 [arXiv:1608.00011] [INSPIRE].
S.S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].
H. Kudoh and T. Wiseman, Connecting black holes and black strings, Phys. Rev. Lett. 94 (2005) 161102 [hep-th/0409111] [INSPIRE].
M. Kalisch, Numerical construction and critical behavior of Kaluza-Klein black holes, Dissertation, Friedrich-Schiller-Universität Jena (2018) [https://doi.org/10.22032/dbt.34074] [arXiv:1802.06596] [INSPIRE].
H. Ishihara and K. Matsuno, Kaluza-Klein black holes with squashed horizons, Prog. Theor. Phys. 116 (2006) 417 [hep-th/0510094] [INSPIRE].
R. Gregory and R. Laflamme, Black strings and p-branes are unstable, Phys. Rev. Lett. 70 (1993) 2837 [hep-th/9301052] [INSPIRE].
T. Wiseman, Static axisymmetric vacuum solutions and nonuniform black strings, Class. Quant. Grav. 20 (2003) 1137 [hep-th/0209051] [INSPIRE].
S.S. Gubser, On nonuniform black branes, Class. Quant. Grav. 19 (2002) 4825 [hep-th/0110193] [INSPIRE].
B. Kleihaus, J. Kunz and E. Radu, New nonuniform black string solutions, JHEP 06 (2006) 016 [hep-th/0603119] [INSPIRE].
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Brihaye, Y., Herdeiro, C. & Radu, E. D = 5 static, charged black holes, strings and rings with resonant, scalar Q-hair. J. High Energ. Phys. 2022, 153 (2022). https://doi.org/10.1007/JHEP10(2022)153
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DOI: https://doi.org/10.1007/JHEP10(2022)153