Abstract
Recent studies have made key progress on the black hole/solitonic solutions of the Einstein-Proca system. Firstly, fully non-linear dynamical evolutions of the Kerr black hole superradiant instability, triggered by a Proca field, have shown the formation of a new equilibrium state, a spinning black hole with synchronised Proca hair. Secondly, non-linear evolutions of spinning Proca stars have established that they are dynamically stable, unlike their scalar cousins. Thirdly, separability of the Proca equation on the Kerr background has been achieved. Motivated by these results, in this paper we reconsider Kerr black holes with synchronised Proca hair. The separability of the Proca equation on the Kerr background allows us to examine the stationary Proca clouds in greater detail, in particular their dependence on the different quantum numbers. These stationary clouds occur at a set of existence lines in the Kerr parameter space, from which the black holes with synchronised Proca hair bifurcate. We construct the domain of existence of these black holes, comparing the fundamental states missed in the original study with the first excited states and with the cousin scalar model, giving illustrative examples of Kerr-like and non- Kerr-like BHs. In the vanishing event horizon limit, these hairy black holes connect to the fundamental states of spinning Proca stars, which include the dynamically stable solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Planck collaboration, Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 [INSPIRE].
A. Suárez, V.H. Robles and T. Matos, A Review on the Scalar Field/Bose-Einstein Condensate Dark Matter Model, Astrophys. Space Sci. Proc. 38 (2014) 107 [arXiv:1302.0903] [INSPIRE].
L. Hui, J.P. Ostriker, S. Tremaine and E. Witten, Ultralight scalars as cosmological dark matter, Phys. Rev. D 95 (2017) 043541 [arXiv:1610.08297] [INSPIRE].
R. Brito, V. Cardoso and P. Pani, Superradiance : Energy Extraction, Black-Hole Bombs and Implications for Astrophysics and Particle Physics, vol. 906, Springer (2015), [https://doi.org/10.1007/978-3-319-19000-6] [arXiv:1501.06570] [INSPIRE].
W.H. Press and S.A. Teukolsky, Floating Orbits, Superradiant Scattering and the Black-hole Bomb, Nature 238 (1972) 211 [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].
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 and E. Radu, Kerr black holes with scalar hair, Phys. Rev. Lett. 112 (2014) 221101 [arXiv:1403.2757] [INSPIRE].
C. Herdeiro, E. Radu and H. Rúnarsson, Kerr black holes with Proca hair, Class. Quant. Grav. 33 (2016) 154001 [arXiv:1603.02687] [INSPIRE].
R. Ruffini and J.A. Wheeler, Introducing the black hole, Phys. Today 24 (1971) 30 [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].
V. Cardoso and L. Gualtieri, Testing the black hole ‘no-hair’ hypothesis, Class. Quant. Grav. 33 (2016) 174001 [arXiv:1607.03133] [INSPIRE].
F.E. Schunck and E.W. Mielke, Rotating boson star as an effective mass torus in general relativity, Phys. Lett. A 249 (1998) 389 [INSPIRE].
S. Yoshida and Y. Eriguchi, Rotating boson stars in general relativity, Phys. Rev. D 56 (1997) 762 [INSPIRE].
R. Brito, V. Cardoso, C.A.R. Herdeiro and E. Radu, Proca stars: Gravitating Bose-Einstein condensates of massive spin 1 particles, Phys. Lett. B 752 (2016) 291 [arXiv:1508.05395] [INSPIRE].
N. Sanchis-Gual et al., Nonlinear Dynamics of Spinning Bosonic Stars: Formation and Stability, Phys. Rev. Lett. 123 (2019) 221101 [arXiv:1907.12565] [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, Stationary resonances of rapidly-rotating Kerr black holes, Eur. Phys. J. C 73 (2013) 2378 [arXiv:1311.5298] [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, Quasi-Bound States of Massive Scalar Fields in the Kerr Black-Hole Spacetime: Beyond the Hydrogenic Approximation, Phys. Lett. B 749 (2015) 167 [arXiv:1510.05649] [INSPIRE].
H.M. Siahaan, Instability of charged massive scalar fields in bound states around Kerr-Sen black holes, Int. J. Mod. Phys. D 24 (2015) 1550102 [arXiv:1506.03957] [INSPIRE].
S. Hod, Spinning Kerr black holes with stationary massive scalar clouds: The large-coupling regime, JHEP 01 (2017) 030 [arXiv:1612.00014] [INSPIRE].
S. Hod, The large-mass limit of cloudy black holes, Class. Quant. Grav. 32 (2015) 134002 [arXiv:1607.00003] [INSPIRE].
Y. Huang and D.-J. Liu, Scalar clouds and the superradiant instability regime of Kerr-Newman black hole, Phys. Rev. D 94 (2016) 064030 [arXiv:1606.08913] [INSPIRE].
C. Bernard, Stationary charged scalar clouds around black holes in string theory, Phys. Rev. D 94 (2016) 085007 [arXiv:1608.05974] [INSPIRE].
I. Sakalli and G. Tokgoz, Stationary Scalar Clouds Around Maximally Rotating Linear Dilaton Black Holes, Class. Quant. Grav. 34 (2017) 125007 [arXiv:1610.09329] [INSPIRE].
H.R.C. Ferreira and C.A.R. Herdeiro, Stationary scalar clouds around a BTZ black hole, Phys. Lett. B 773 (2017) 129 [arXiv:1707.08133] [INSPIRE].
M. Richartz, C.A.R. Herdeiro and E. Berti, Synchronous frequencies of extremal Kerr black holes: resonances, scattering and stability, Phys. Rev. D 96 (2017) 044034 [arXiv:1706.01112] [INSPIRE].
Y. Huang, D.-J. Liu, X.-H. Zhai and X.-Z. Li, Scalar clouds around Kerr-Sen black holes, Class. Quant. Grav. 34 (2017) 155002 [arXiv:1706.04441] [INSPIRE].
Y. Huang, D.-J. Liu, X.-h. Zhai and X.-z. Li, Instability for massive scalar fields in Kerr-Newman spacetime, Phys. Rev. D 98 (2018) 025021 [arXiv:1807.06263] [INSPIRE].
G. García and M. Salgado, Obstructions towards a generalization of no-hair theorems: Scalar clouds around Kerr black holes, Phys. Rev. D 99 (2019) 044036 [arXiv:1812.05809] [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].
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].
G. García and M. Salgado, Existence or absence of superregular boson clouds around extremal Kerr black holes and its connection with number theory, Phys. Rev. D 101 (2020) 044040 [arXiv:1909.12987] [INSPIRE].
D. Baumann, H.S. Chia, J. Stout and L. ter Haar, The Spectra of Gravitational Atoms, JCAP 12 (2019) 006 [arXiv:1908.10370] [INSPIRE].
A. Arvanitaki and S. Dubovsky, Exploring the String Axiverse with Precision Black Hole Physics, Phys. Rev. D 83 (2011) 044026 [arXiv:1004.3558] [INSPIRE].
C.L. Benone, L.C.B. Crispino, C. Herdeiro and E. Radu, Acoustic clouds: standing sound waves around a black hole analogue, Phys. Rev. D 91 (2015) 104038 [arXiv:1412.7278] [INSPIRE].
C.L. Benone, C.B. Crispino, Luís, C.A.R. Herdeiro and M. Richartz, Synchronized stationary clouds in a static fluid, Phys. Lett. B 786 (2018) 442 [arXiv:1809.03952] [INSPIRE].
W.E. East, Superradiant instability of massive vector fields around spinning black holes in the relativistic regime, Phys. Rev. D 96 (2017) 024004 [arXiv:1705.01544] [INSPIRE].
V.P. Frolov, P. Krtouš, D. Kubizňák and J.E. Santos, Massive Vector Fields in Rotating Black-Hole Spacetimes: Separability and Quasinormal Modes, Phys. Rev. Lett. 120 (2018) 231103 [arXiv:1804.00030] [INSPIRE].
S.R. Dolan, Instability of the Proca field on Kerr spacetime, Phys. Rev. D 98 (2018) 104006 [arXiv:1806.01604] [INSPIRE].
R. Cayuso et al., Massive vector fields in Kerr-Newman and Kerr-Sen black hole spacetimes, JHEP 04 (2020) 159 [arXiv:1912.08224] [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].
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. Balakrishna, E. Seidel and W.-M. Suen, Dynamical evolution of boson stars. 2. Excited states and selfinteracting fields, Phys. Rev. D 58 (1998) 104004 [gr-qc/9712064] [INSPIRE].
C. Herdeiro, I. Perapechka, E. Radu and Y. Shnir, Asymptotically flat spinning scalar, Dirac and Proca stars, Phys. Lett. B 797 (2019) 134845 [arXiv:1906.05386] [INSPIRE].
D. Griffiths, Introduction of Quantum Mechanics, Prentice Hall, Inc., (1995).
K.S. Thorne, Multipole Expansions of Gravitational Radiation, Rev. Mod. Phys. 52 (1980) 299 [INSPIRE].
M. Maggiore, Gravitational Waves. Volume 1: Theory and Experiments, Oxford University Press, Oxford, U.K. (2007).
A. Proca, Th́eorie non relativiste des particules à spin entier, J. Phys. Radium 9 (1938) 61.
B. Carter, Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations, Commun. Math. Phys. 10 (1968) 280 [INSPIRE].
B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. 174 (1968) 1559 [INSPIRE].
P. Krtouš, V.P. Frolov and D. Kubizňák, Separation of Maxwell equations in Kerr-NUT–(A)dS spacetimes, Nucl. Phys. B 934 (2018) 7 [arXiv:1803.02485] [INSPIRE].
O. Lunin, Maxwell’s equations in the Myers-Perry geometry, JHEP 12 (2017) 138 [arXiv:1708.06766] [INSPIRE].
V. Frolov, P. Krtouš and D. Kubizňák, Black holes, hidden symmetries and complete integrability, Living Rev. Rel. 20 (2017) 6 [arXiv:1705.05482] [INSPIRE].
P. Amaro-Seoane, J. Barranco, A. Bernal and L. Rezzolla, Constraining scalar fields with stellar kinematics and collisional dark matter, JCAP 11 (2010) 002 [arXiv:1009.0019] [INSPIRE].
C. Herdeiro and E. Radu, Ergosurfaces for Kerr black holes with scalar hair, Phys. Rev. D 89 (2014) 124018 [arXiv:1406.1225] [INSPIRE].
C. Herdeiro, I. Perapechka, E. Radu and Y. Shnir, Gravitating solitons and black holes with synchronised hair in the four dimensional O(3) σ-model, JHEP 02 (2019) 111 [arXiv:1811.11799] [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].
C.A.R. Herdeiro and E. Radu, How fast can a black hole rotate?, Int. J. Mod. Phys. D 24 (2015) 1544022 [arXiv:1505.04189] [INSPIRE].
N. Siemonsen and W.E. East, Gravitational wave signatures of ultralight vector bosons from black hole superradiance, Phys. Rev. D 101 (2020) 024019 [arXiv:1910.09476] [INSPIRE].
P.V.P. Cunha, C.A.R. Herdeiro, E. Radu and H.F. Rúnarsson, Shadows of Kerr black holes with scalar hair, Phys. Rev. Lett. 115 (2015) 211102 [arXiv:1509.00021] [INSPIRE].
P.V.P. Cunha, C.A.R. Herdeiro and E. Radu, EHT constraint on the ultralight scalar hair of the M87 supermassive black hole, Universe 5 (2019) 220 [arXiv:1909.08039] [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].
M. Zhou, C. Bambi, C.A.R. Herdeiro and E. Radu, Iron Kα line of Kerr black holes with Proca hair, Phys. Rev. D 95 (2017) 104035 [arXiv:1703.06836] [INSPIRE].
N. Franchini et al., Constraining black holes with light boson hair and boson stars using epicyclic frequencies and quasiperiodic oscillations, Phys. Rev. D 95 (2017) 124025 [arXiv:1612.00038] [INSPIRE].
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
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: 2004.09536
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
Santos, N.M., Benone, C.L., Crispino, L.C. et al. Black holes with synchronised Proca hair: linear clouds and fundamental non-linear solutions. J. High Energ. Phys. 2020, 10 (2020). https://doi.org/10.1007/JHEP07(2020)010
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2020)010