Abstract
In this work, we consider a very simple gravitational theory that contains a scalar field with its kinetic and potential terms minimally coupled to gravity, while the scalar field is assumed to have a coulombic form. In the context of this theory, we study an analytic, asymptotically flat, and regular (ultra-compact) black-hole solutions with non-trivial scalar hair of secondary type. At first, we examine the properties of the static and spherically symmetric black-hole solution — firstly appeared in [109] — and we find that in the causal region of the spacetime the stress-energy tensor, needed to support our solution, satisfies the strong energy conditions. Then, by using the slow-rotating approximation, we generalize the static solution into a slowly rotating one, and we determine explicitly its angular velocity ω(r). We also find that the angular velocity of our ultra-compact solution is always larger compared to the angular velocity of the corresponding equally massive slow-rotating Schwarzschild black hole. In addition, we investigate the axial perturbations of the derived solutions by determining the Schrödinger-like equation and the effective potential. We show that there is a region in the parameter space of the free parameters of our theory, which allows for the existence of stable ultra-compact black hole solutions. Specifically, we calculate that the most compact and stable black hole solution is 0.551 times smaller than the Schwarzschild one, while it rotates 2.491 times faster compared to the slow-rotating Schwarzschild black hole. Finally, we present without going into details the generalization of the derived asymptotically flat solutions to asymptotically (A)dS solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
LIGO Scientific and Virgo collaborations, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
Event Horizon Telescope collaboration, First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, Astrophys. J. Lett. 875 (2019) L1 [arXiv:1906.11238] [INSPIRE].
ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE].
CMS collaboration, Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
J. D. Bekenstein, Transcendence of the law of baryon-number conservation in black hole physics, Phys. Rev. Lett. 28 (1972) 452 [INSPIRE].
C. Teitelboim, Nonmeasurability of the lepton number of a black hole, Lett. Nuovo Cim. 3S2 (1972) 397 [INSPIRE].
M. S. Volkov and D. V. Galtsov, NonAbelian Einstein Yang-Mills black holes, JETP Lett. 50 (1989) 346 [INSPIRE].
P. Bizon, Colored black holes, Phys. Rev. Lett. 64 (1990) 2844 [INSPIRE].
B. R. Greene, S. D. Mathur and C. M. O’Neill, Eluding the no hair conjecture: Black holes in spontaneously broken gauge theories, Phys. Rev. D 47 (1993) 2242 [hep-th/9211007] [INSPIRE].
K.-I. Maeda, T. Tachizawa, T. Torii and T. Maki, Stability of nonAbelian black holes and catastrophe theory, Phys. Rev. Lett. 72 (1994) 450 [gr-qc/9310015] [INSPIRE].
H. Lückock and I. Moss, Black holes have Skyrmion hair, Phys. Lett. B 176 (1986) 341 [INSPIRE].
S. Droz, M. Heusler and N. Straumann, New black hole solutions with hair, Phys. Lett. B 268 (1991) 371 [INSPIRE].
J. D. Bekenstein, Exact solutions of Einstein conformal scalar equations, Annals Phys. 82 (1974) 535 [INSPIRE].
J. D. Bekenstein, Black Holes with Scalar Charge, Annals Phys. 91 (1975) 75 [INSPIRE].
J. D. Bekenstein, Novel “no-scalar-hair” theorem for black holes, Phys. Rev. D 51 (1995) R6608 [INSPIRE].
P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis and E. Winstanley, Dilatonic black holes in higher curvature string gravity, Phys. Rev. D 54 (1996) 5049 [hep-th/9511071] [INSPIRE].
G. W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10 (1974) 363 [INSPIRE].
A. Nicolis, R. Rattazzi and E. Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].
T. Torii and K.-i. Maeda, Black holes with nonAbelian hair and their thermodynamical properties, Phys. Rev. D 48 (1993) 1643 [INSPIRE].
O. Bechmann and O. Lechtenfeld, Exact black hole solution with selfinteracting scalar field, Class. Quant. Grav. 12 (1995) 1473 [gr-qc/9502011] [INSPIRE].
H. Dennhardt and O. Lechtenfeld, Scalar deformations of Schwarzschild holes and their stability, Int. J. Mod. Phys. A 13 (1998) 741 [gr-qc/9612062] [INSPIRE].
U. Nucamendi and M. Salgado, Scalar hairy black holes and solitons in asymptotically flat space-times, Phys. Rev. D 68 (2003) 044026 [gr-qc/0301062] [INSPIRE].
S. S. Gubser, Phase transitions near black hole horizons, Class. Quant. Grav. 22 (2005) 5121 [hep-th/0505189] [INSPIRE].
K. A. Bronnikov and J. C. Fabris, Regular phantom black holes, Phys. Rev. Lett. 96 (2006) 251101 [gr-qc/0511109] [INSPIRE].
V. V. Nikonov, J. V. Tchemarina and A. N. Tsirulev, A two-parameter family of exact asymptotically flat solutions to the Einstein-scalar field equations, Class. Quant. Grav. 25 (2008) 138001 [INSPIRE].
A. Anabalon and J. Oliva, Exact Hairy Black Holes and their Modification to the Universal Law of Gravitation, Phys. Rev. D 86 (2012) 107501 [arXiv:1205.6012] [INSPIRE].
A. Anabalon, D. Astefanesei and R. Mann, Exact asymptotically flat charged hairy black holes with a dilaton potential, JHEP 10 (2013) 184 [arXiv:1308.1693] [INSPIRE].
B. Kleihaus, J. Kunz, E. Radu and B. Subagyo, Axially symmetric static scalar solitons and black holes with scalar hair, Phys. Lett. B 725 (2013) 489 [arXiv:1306.4616] [INSPIRE].
E. Babichev and C. Charmousis, Dressing a black hole with a time-dependent Galileon, JHEP 08 (2014) 106 [arXiv:1312.3204] [INSPIRE].
T. P. Sotiriou and S.-Y. Zhou, Black hole hair in generalized scalar-tensor gravity: An explicit example, Phys. Rev. D 90 (2014) 124063 [arXiv:1408.1698] [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. Charmousis, T. Kolyvaris, E. Papantonopoulos and M. Tsoukalas, Black Holes in Bi-scalar Extensions of Horndeski Theories, JHEP 07 (2014) 085 [arXiv:1404.1024] [INSPIRE].
M. Astorino, Stationary axisymmetric spacetimes with a conformally coupled scalar field, Phys. Rev. D 91 (2015) 064066 [arXiv:1412.3539] [INSPIRE].
M. Cadoni and E. Franzin, Asymptotically flat black holes sourced by a massless scalar field, Phys. Rev. D 91 (2015) 104011 [arXiv:1503.04734] [INSPIRE].
C. Herdeiro and E. Radu, Construction and physical properties of Kerr black holes with scalar hair, Class. Quant. Grav. 32 (2015) 144001 [arXiv:1501.04319] [INSPIRE].
B. Kleihaus, J. Kunz and S. Yazadjiev, Scalarized Hairy Black Holes, Phys. Lett. B 744 (2015) 406 [arXiv:1503.01672] [INSPIRE].
T. Tahamtan and O. Svitek, Robinson-Trautman solution with scalar hair, Phys. Rev. D 91 (2015) 104032 [arXiv:1503.09080] [INSPIRE].
A. J. Tolley, D.-J. Wu and S.-Y. Zhou, Hairy black holes in scalar extended massive gravity, Phys. Rev. D 92 (2015) 124063 [arXiv:1510.05208] [INSPIRE].
S. Hod, Extremal Kerr-Newman black holes with extremely short charged scalar hair, Phys. Lett. B 751 (2015) 177 [arXiv:1707.06246] [INSPIRE].
C. Herdeiro, E. Radu and H. Rúnarsson, Kerr black holes with Proca hair, Class. Quant. Grav. 33 (2016) 154001 [arXiv:1603.02687] [INSPIRE].
Y. Ni, M. Zhou, A. Cardenas-Avendano, C. Bambi, C. A. R. Herdeiro and E. Radu, Iron Kα line of Kerr black holes with scalar hair, JCAP 07 (2016) 049 [arXiv:1606.04654] [INSPIRE].
R. Benkel, T. P. Sotiriou and H. Witek, Black hole hair formation in shift-symmetric generalised scalar-tensor gravity, Class. Quant. Grav. 34 (2017) 064001 [arXiv:1610.09168] [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].
L. Heisenberg, R. Kase, M. Minamitsuji and S. Tsujikawa, Hairy black-hole solutions in generalized Proca theories, Phys. Rev. D 96 (2017) 084049 [arXiv:1705.09662] [INSPIRE].
G. Antoniou, A. Bakopoulos and P. Kanti, Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories, Phys. Rev. Lett. 120 (2018) 131102 [arXiv:1711.03390] [INSPIRE].
G. Antoniou, A. Bakopoulos and P. Kanti, Black-Hole Solutions with Scalar Hair in Einstein-Scalar-Gauss-Bonnet Theories, Phys. Rev. D 97 (2018) 084037 [arXiv:1711.07431] [INSPIRE].
C. A. R. Herdeiro and E. Radu, Spinning boson stars and hairy black holes with nonminimal coupling, Int. J. Mod. Phys. D 27 (2018) 1843009 [arXiv:1803.08149] [INSPIRE].
C. Pacilio, Scalar charge of black holes in Einstein-Maxwell-dilaton theory, Phys. Rev. D 98 (2018) 064055 [arXiv:1806.10238] [INSPIRE].
Y. Brihaye, T. Delplace, C. Herdeiro and E. Radu, An analytic effective model for hairy black holes, Phys. Lett. B 782 (2018) 124 [arXiv:1803.09089] [INSPIRE].
D. Astefanesei, D. Choque, F. Gómez and R. Rojas, Thermodynamically stable asymptotically flat hairy black holes with a dilaton potential, JHEP 03 (2019) 205 [arXiv:1901.01269] [INSPIRE].
Y. Brihaye and L. Ducobu, Hairy black holes, boson stars and non-minimal coupling to curvature invariants, Phys. Lett. B 795 (2019) 135 [arXiv:1812.07438] [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].
C. Herdeiro, I. Perapechka, E. Radu and Y. Shnir, Skyrmions around Kerr black holes and spinning BHs with Skyrme hair, JHEP 10 (2018) 119 [arXiv:1808.05388] [INSPIRE].
K. Van Aelst, E. Gourgoulhon, P. Grandclément and C. Charmousis, Hairy rotating black holes in cubic Galileon theory, Class. Quant. Grav. 37 (2020) 035007 [arXiv:1910.08451] [INSPIRE].
S. Hod, Spontaneous scalarization of Gauss-Bonnet black holes: Analytic treatment in the linearized regime, Phys. Rev. D 100 (2019) 064039 [arXiv:1912.07630] [INSPIRE].
J. Kunz, I. Perapechka and Y. Shnir, Kerr black holes with synchronised scalar hair and boson stars in the Einstein-Friedberg-Lee-Sirlin model, JHEP 07 (2019) 109 [arXiv:1904.13379] [INSPIRE].
P. V. P. Cunha, C. A. R. Herdeiro and E. Radu, Spontaneously Scalarized Kerr Black Holes in Extended Scalar-Tensor-Gauss-Bonnet Gravity, Phys. Rev. Lett. 123 (2019) 011101 [arXiv:1904.09997] [INSPIRE].
F. Filippini and G. Tasinato, On long range axion hairs for black holes, Class. Quant. Grav. 36 (2019) 215015 [arXiv:1903.02950] [INSPIRE].
D.-C. Zou and Y. S. Myung, Scalar hairy black holes in Einstein-Maxwell-conformally coupled scalar theory, Phys. Lett. B 803 (2020) 135332 [arXiv:1911.08062] [INSPIRE].
P. G. S. Fernandes, Einstein-Maxwell-scalar black holes with massive and self-interacting scalar hair, Phys. Dark Univ. 30 (2020) 100716 [arXiv:2003.01045] [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].
J. Sultana, Hairy black holes in Einstein-Weyl gravity, Phys. Rev. D 101 (2020) 084027 [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. A. R. Herdeiro, E. Radu, H. O. Silva, T. P. Sotiriou and N. Yunes, Spin-induced scalarized black holes, Phys. Rev. Lett. 126 (2021) 011103 [arXiv:2009.03904] [INSPIRE].
D. Astefanesei, J. Luis Blázquez-Salcedo, F. Gómez and R. Rojas, Thermodynamically stable asymptotically flat hairy black holes with a dilaton potential: the general case, JHEP 02 (2021) 233 [arXiv:2009.01854] [INSPIRE].
Y. Shnir, Black holes with Skyrmion-anti-Skyrmion hairs, Phys. Lett. B 810 (2020) 135847 [arXiv:2008.09452] [INSPIRE].
S. Hod, Onset of spontaneous scalarization in spinning Gauss-Bonnet black holes, Phys. Rev. D 102 (2020) 084060 [arXiv:2006.09399] [INSPIRE].
J. Ovalle, R. Casadio, E. Contreras and A. Sotomayor, Hairy black holes by gravitational decoupling, Phys. Dark Univ. 31 (2021) 100744 [arXiv:2006.06735] [INSPIRE].
Y. Brihaye and Y. Verbin, Scalarized dyonic black holes in vector-tensor Horndeski gravity, Phys. Rev. D 104 (2021) 024047 [arXiv:2105.11402] [INSPIRE].
J. F. M. Delgado, C. A. R. Herdeiro and E. Radu, Kerr black holes with synchronized axionic hair, Phys. Rev. D 103 (2021) 104029 [arXiv:2012.03952] [INSPIRE].
Y. S. Myung and D.-C. Zou, Scalarized black holes in the Einstein-Maxwell-scalar theory with a quasitopological term, Phys. Rev. D 103 (2021) 024010 [arXiv:2011.09665] [INSPIRE].
D. D. Doneva and S. S. Yazadjiev, New Gauss-Bonnet Black Holes with Curvature-Induced Scalarization in Extended Scalar-Tensor Theories, Phys. Rev. Lett. 120 (2018) 131103 [arXiv:1711.01187] [INSPIRE].
H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou and E. Berti, Spontaneous scalarization of black holes and compact stars from a Gauss-Bonnet coupling, Phys. Rev. Lett. 120 (2018) 131104 [arXiv:1711.02080] [INSPIRE].
C. Martinez, R. Troncoso and J. Zanelli, Exact black hole solution with a minimally coupled scalar field, Phys. Rev. D 70 (2004) 084035 [hep-th/0406111] [INSPIRE].
C. Martinez and R. Troncoso, Electrically charged black hole with scalar hair, Phys. Rev. D 74 (2006) 064007 [hep-th/0606130] [INSPIRE].
A. Anabalon, Exact Black Holes and Universality in the Backreaction of non-linear Sigma Models with a potential in (A)dS4, JHEP 06 (2012) 127 [arXiv:1204.2720] [INSPIRE].
C. Charmousis and D. Iosifidis, Self tuning scalar tensor black holes, J. Phys. Conf. Ser. 600 (2015) 012003 [arXiv:1501.05167] [INSPIRE].
E. Babichev, C. Charmousis and M. Hassaine, Charged Galileon black holes, JCAP 05 (2015) 031 [arXiv:1503.02545] [INSPIRE].
Z.-Y. Fan and H. Lü, Charged Black Holes with Scalar Hair, JHEP 09 (2015) 060 [arXiv:1507.04369] [INSPIRE].
I. Perapechka and Y. Shnir, Generalized Skyrmions and hairy black holes in asymptotically AdS4 spacetime, Phys. Rev. D 95 (2017) 025024 [arXiv:1612.01914] [INSPIRE].
A. Bakopoulos, G. Antoniou and P. Kanti, Novel Black-Hole Solutions in Einstein-Scalar-Gauss-Bonnet Theories with a Cosmological Constant, Phys. Rev. D 99 (2019) 064003 [arXiv:1812.06941] [INSPIRE].
J. Ben Achour and H. Liu, Hairy Schwarzschild-(A)dS black hole solutions in degenerate higher order scalar-tensor theories beyond shift symmetry, Phys. Rev. D 99 (2019) 064042 [arXiv:1811.05369] [INSPIRE].
Y. Brihaye, C. Herdeiro and E. Radu, Black Hole Spontaneous Scalarisation with a Positive Cosmological Constant, Phys. Lett. B 802 (2020) 135269 [arXiv:1910.05286] [INSPIRE].
G. Guo, P. Wang, H. Wu and H. Yang, Scalarized Einstein-Maxwell-scalar black holes in anti-de Sitter spacetime, Eur. Phys. J. C 81 (2021) 864 [arXiv:2102.04015] [INSPIRE].
V. Faraoni, A. Giusti and B. H. Fahim, Spherical inhomogeneous solutions of Einstein and scalar-tensor gravity: A map of the land, Phys. Rept. 925 (2021) 1 [arXiv:2101.00266] [INSPIRE].
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 (1988) 395 [INSPIRE].
J. A. Wheeler, Geons, Phys. Rev. 97 (1955) 511 [INSPIRE].
C. W. Misner and J. A. Wheeler, Classical physics as geometry: Gravitation, electromagnetism, unquantized charge, and mass as properties of curved empty space, Annals Phys. 2 (1957) 525 [INSPIRE].
P. Kanti, B. Kleihaus and J. Kunz, Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory, Phys. Rev. Lett. 107 (2011) 271101 [arXiv:1108.3003] [INSPIRE].
P. Kanti, B. Kleihaus and J. Kunz, Stable Lorentzian Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory, Phys. Rev. D 85 (2012) 044007 [arXiv:1111.4049] [INSPIRE].
G. Antoniou, A. Bakopoulos, P. Kanti, B. Kleihaus and J. Kunz, Novel Einstein-scalar-Gauss-Bonnet wormholes without exotic matter, Phys. Rev. D 101 (2020) 024033 [arXiv:1904.13091] [INSPIRE].
A. I. Janis, E. T. Newman and J. Winicour, Reality of the Schwarzschild Singularity, Phys. Rev. Lett. 20 (1968) 878 [INSPIRE].
I. Z. Fisher, Scalar mesostatic field with regard for gravitational effects, Zh. Eksp. Teor. Fiz. 18 (1948) 636 [gr-qc/9911008] [INSPIRE].
M. Wyman, Static Spherically Symmetric Scalar Fields in General Relativity, Phys. Rev. D 24 (1981) 839 [INSPIRE].
A. G. Agnese and M. La Camera, Gravitation without black holes, Phys. Rev. D 31 (1985) 1280 [INSPIRE].
M. D. Roberts, Scalar Field Counterexamples to the Cosmic Censorship Hypothesis, Gen. Rel. Grav. 21 (1989) 907 [INSPIRE].
B. Kleihaus, J. Kunz and P. Kanti, Particle-like ultracompact objects in Einstein-scalar-Gauss-Bonnet theories, Phys. Lett. B 804 (2020) 135401 [arXiv:1910.02121] [INSPIRE].
B. Kleihaus, J. Kunz and P. Kanti, Properties of ultracompact particlelike solutions in Einstein-scalar-Gauss-Bonnet theories, Phys. Rev. D 102 (2020) 024070 [arXiv:2005.07650] [INSPIRE].
O. Baake, C. Charmousis, M. Hassaine and M. San Juan, Regular black holes and gravitational particle-like solutions in generic DHOST theories, JCAP 06 (2021) 021 [arXiv:2104.08221] [INSPIRE].
C. A. R. Herdeiro, J. M. S. Oliveira and E. Radu, A class of solitons in Maxwell-scalar and Einstein-Maxwell-scalar models, Eur. Phys. J. C 80 (2020) 23 [arXiv:1910.11021] [INSPIRE].
A. Bakopoulos, P. Kanti and N. Pappas, Large and ultracompact Gauss-Bonnet black holes with a self-interacting scalar field, Phys. Rev. D 101 (2020) 084059 [arXiv:2003.02473] [INSPIRE].
L. Kreidberg, C. D. Bailyn, W. M. Farr and V. Kalogera, Mass Measurements of Black Holes in X-Ray Transients: Is There a Mass Gap?, Astrophys. J. 757 (2012) 36 [arXiv:1205.1805] [INSPIRE].
LIGO Scientific and VIRGO collaborations, Predictions for the Rates of Compact Binary Coalescences Observable by Ground-based Gravitational-wave Detectors, Class. Quant. Grav. 27 (2010) 173001 [arXiv:1003.2480] [INSPIRE].
LIGO Scientific and Virgo collaborations, GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs, Phys. Rev. X 9 (2019) 031040 [arXiv:1811.12907] [INSPIRE].
LIGO Scientific and Virgo collaborations, GW190425: Observation of a Compact Binary Coalescence with Total Mass ∼ 3.4M⨀, Astrophys. J. Lett. 892 (2020) L3 [arXiv:2001.01761] [INSPIRE].
H. G. Ellis, Ether flow through a drainhole — a particle model in general relativity, J. Math. Phys. 14 (1973) 104 [INSPIRE].
H. G. Ellis, The evolving, flowless drain hole: a nongravitating particle model in general relativity theory, Gen. Rel. Grav. 10 (1979) 105 [INSPIRE].
K. A. Bronnikov, Scalar-tensor theory and scalar charge, Acta Phys. Polon. B 4 (1973) 251 [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].
R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
M. Visser, Lorentzian Wormholes: From Einstein to Hawking, Computational and Mathematical Physics, American Institute of Physics, Melville, NY, U.S.A. (1995).
P. Pani and V. Cardoso, Are black holes in alternative theories serious astrophysical candidates? The Case for Einstein-Dilaton-Gauss-Bonnet black holes, Phys. Rev. D 79 (2009) 084031 [arXiv:0902.1569] [INSPIRE].
J. B. Hartle, Slowly rotating relativistic stars. 1. Equations of structure, Astrophys. J. 150 (1967) 1005 [INSPIRE].
T. Regge and J. A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108 (1957) 1063 [INSPIRE].
F. J. Zerilli, Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett. 24 (1970) 737 [INSPIRE].
C. V. Vishveshwara, Stability of the Schwarzschild metric, Phys. Rev. D 1 (1970) 2870 [INSPIRE].
F. J. Zerilli, Gravitational field of a particle falling in a Schwarzschild geometry analyzed in tensor harmonics, Phys. Rev. D 2 (1970) 2141 [INSPIRE].
J. Mathews, Gravitational multipole radiation, J. Soc. Indust. Appl. Math. 10 (1962) 768.
F. J. Zerilli, Tensor harmonics in canonical form for gravitational radiation and other applications, J. Math. Phys. 11 (1970) 2203.
V. Moncrief, Gravitational perturbations of spherically symmetric systems. I. The exterior problem, Annals Phys. 88 (1974) 323 [INSPIRE].
J. M. Stewart and M. Walker, Perturbations of spacetimes in general relativity, Proc. Roy. Soc. Lond. A 341 (1974) 49 [INSPIRE].
K. S. Thorne, Multipole Expansions of Gravitational Radiation, Rev. Mod. Phys. 52 (1980) 299 [INSPIRE].
W. F. Buell and B. A. Shadwick, Potentials and bound states, Am. J. Phys. 63 (1995) 256.
E. Berti, K. Yagi, H. Yang and N. Yunes, Extreme Gravity Tests with Gravitational Waves from Compact Binary Coalescences: (II) Ringdown, Gen. Rel. Grav. 50 (2018) 49 [arXiv:1801.03587] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2107.05656
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Bakopoulos, A., Nakas, T. Analytic and asymptotically flat hairy (ultra-compact) black-hole solutions and their axial perturbations. J. High Energ. Phys. 2022, 96 (2022). https://doi.org/10.1007/JHEP04(2022)096
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2022)096