Journal of High Energy Physics

, 2018:119 | Cite as

Skyrmions around Kerr black holes and spinning BHs with Skyrme hair

  • C. Herdeiro
  • I. Perapechka
  • E. RaduEmail author
  • Ya. Shnir
Open Access
Regular Article - Theoretical Physics


We study solutions of the Einstein-Skyrme model. Firstly we consider test field Skyrmions on the Kerr background. These configurations — hereafter dubbed Skerrmions — can be in equilibrium with a Kerr black hole (BH) by virtue of a synchronisation condition. We consider two sectors for Skerrmions. In the sector with non-zero baryon charge, Skerrmions are akin to the known Skyrme solutions on the Schwarzschild background. These “topological” configurations reduce to flat spacetime Skyrmions in a vanishing BH mass limit; moreoever, they never become “small” perturbations on the Kerr background: the non-linearities of the Skyrme model are crucial for all such Skerrmions. In the non-topological sector, on the other hand, Skerrmions have no analogue on the Schwarzschild background. Non-topological Skerrmions carry not baryon charge and bifurcate from a subset of Kerr solutions defining an existence line. Therein the appropriate truncation of the Skyrme model yield a linear scalar field theory containing a complex plus a real field, both massive and decoupled, and the Skerrmions reduce to the known stationary scalar clouds around Kerr BHs. Moreover, non-topological Skerrmions trivialise in the vanishing BH mass limit. We then discuss the backreaction of these Skerrmions, that yield rotating BHs with synchronised Skyrme hair, which continously connect to the Kerr solution (self-gravitating Skyrmions) in the non-topological (topological) sector. In particular, the non-topological hairy BHs provide a non-linear realisation, within the Skyrme model, of the synchronous stationary scalar clouds around Kerr.


Black Holes Sigma Models Solitons Monopoles and Instantons 


Open Access

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  1. [1]
    R. Ruffini and J.A. Wheeler, Introducing the black hole, Phys. Today 24 (1971) 30.ADSCrossRefGoogle Scholar
  2. [2]
    J.D. Bekenstein, Black hole hair: 25 years after, in Physics. Proceedings, 2nd International A.D. Sakharov Conference, Moscow, Russia, 20-24 May 1996, pp. 216-219, gr-qc/9605059 [INSPIRE].
  3. [3]
    C. 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].
  4. [4]
    T.P. Sotiriou, Black Holes and Scalar Fields, Class. Quant. Grav. 32 (2015) 214002 [arXiv:1505.00248] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    M.S. Volkov, Hairy black holes in the XX-th and XXI-st centuries, in proceedings of 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories (MG14) (In 4 Volumes), Rome, Italy, 12-18 July 2015, volume 2, pp. 1779-1798, [] [arXiv:1601.08230] [INSPIRE].
  6. [6]
    T.H.R. Skyrme, A Nonlinear field theory, Proc. Roy. Soc. Lond. A 260 (1961) 127 [INSPIRE].
  7. [7]
    T.H.R. Skyrme, A unified field theory of mesons and baryons, Nucl. Phys. A 31 (1962) 556.Google Scholar
  8. [8]
    E. Witten, Global Aspects of Current Algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].
  9. [9]
    E. Witten, Current Algebra, Baryons and Quark Confinement, Nucl. Phys. B 223 (1983) 433 [INSPIRE].
  10. [10]
    C.G. Callan Jr. and E. Witten, Monopole Catalysis of Skyrmion Decay, Nucl. Phys. B 239 (1984) 161 [INSPIRE].
  11. [11]
    H. Lückock and I. Moss, Black holes have skyrmion hair, Phys. Lett. B 176 (1986) 341 [INSPIRE].
  12. [12]
    H. Luckock, Black hole skyrmions, in String Theory, Quantum Cosmology and Quantum Gravity, Integrable and Conformal Integrable Theories, H.J. De Vega and N. Sanches eds., World Scientific (1987), p. 455.Google Scholar
  13. [13]
    S. Droz, M. Heusler and N. Straumann, New black hole solutions with hair, Phys. Lett. B 268 (1991) 371 [INSPIRE].
  14. [14]
    C. Adam, O. Kichakova, Ya. Shnir and A. Wereszczynski, Hairy black holes in the general Skyrme model, Phys. Rev. D 94 (2016) 024060 [arXiv:1605.07625] [INSPIRE].
  15. [15]
    G. Dvali and A. Gußmann, Skyrmion Black Hole Hair: Conservation of Baryon Number by Black Holes and Observable Manifestations, Nucl. Phys. B 913 (2016) 1001 [arXiv:1605.00543] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    S.B. Gudnason, M. Nitta and N. Sawado, Black hole Skyrmion in a generalized Skyrme model, JHEP 09 (2016) 055 [arXiv:1605.07954] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    N.K. Glendenning, T. Kodama and F.R. Klinkhamer, Skyrme topological soliton coupled to gravity, Phys. Rev. D 38 (1988) 3226 [INSPIRE].
  18. [18]
    P. Bizon and T. Chmaj, Gravitating skyrmions, Phys. Lett. B 297 (1992) 55 [INSPIRE].
  19. [19]
    M. Heusler, S. Droz and N. Straumann, Stability analysis of selfgravitating skyrmions, Phys. Lett. B 271 (1991) 61 [INSPIRE].
  20. [20]
    M. Heusler, N. Straumann and Z.-h. Zhou, Selfgravitating solutions of the Skyrme model and their stability, Helv. Phys. Acta 66 (1993) 614 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  21. [21]
    R.A. Battye, S. Krusch and P.M. Sutcliffe, Spinning skyrmions and the skyrme parameters, Phys. Lett. B 626 (2005) 120 [hep-th/0507279] [INSPIRE].
  22. [22]
    T. Ioannidou, B. Kleihaus and J. Kunz, Spinning gravitating skyrmions, Phys. Lett. B 643 (2006) 213 [gr-qc/0608110] [INSPIRE].
  23. [23]
    I. Perapechka and Ya. Shnir, Spinning gravitating Skyrmions in a generalized Einstein-Skyrme model, Phys. Rev. D 96 (2017) 125006 [arXiv:1710.06334] [INSPIRE].
  24. [24]
    S. Hod, Stationary Scalar Clouds Around Rotating Black Holes, Phys. Rev. D 86 (2012) 104026 [Erratum ibid. D 86 (2012) 129902] [arXiv:1211.3202] [INSPIRE].
  25. [25]
    C. Herdeiro and E. Radu, Kerr black holes with scalar hair, Phys. Rev. Lett. 112 (2014) 221101 [arXiv:1403.2757] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    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].
  28. [28]
    B. Kleihaus, J. Kunz and S. Yazadjiev, Scalarized Hairy Black Holes, Phys. Lett. B 744 (2015) 406 [arXiv:1503.01672] [INSPIRE].
  29. [29]
    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].
  30. [30]
    C. 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].
  31. [31]
    C. Herdeiro, E. Radu and H. Runarsson, Kerr black holes with Proca hair, Class. Quant. Grav. 33 (2016) 154001 [arXiv:1603.02687] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    J.F.M. Delgado, C. Herdeiro, E. Radu and H. Runarsson, Kerr-Newman black holes with scalar hair, Phys. Lett. B 761 (2016) 234 [arXiv:1608.00631] [INSPIRE].
  33. [33]
    C. 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].
  34. [34]
    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].
  35. [35]
    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].
  36. [36]
    R. Emparan and H.S. Reall, A Rotating black ring solution in five-dimensions, Phys. Rev. Lett. 88 (2002) 101101 [hep-th/0110260] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    S.R. Coleman, Q Balls, Nucl. Phys. B 262 (1985) 263 [Erratum ibid. B 269 (1986) 744] [INSPIRE].
  38. [38]
    M.S. Volkov and E. Wohnert, Spinning Q balls, Phys. Rev. D 66 (2002) 085003 [hep-th/0205157] [INSPIRE].
  39. [39]
    B. Kleihaus, J. Kunz and M. List, Rotating boson stars and Q-balls, Phys. Rev. D 72 (2005) 064002 [gr-qc/0505143] [INSPIRE].
  40. [40]
    S. Hod, Kerr-Newman black holes with stationary charged scalar clouds, Phys. Rev. D 90 (2014) 024051 [arXiv:1406.1179] [INSPIRE].
  41. [41]
    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].
  42. [42]
    J. Wilson-Gerow and A. Ritz, Black hole energy extraction via a stationary scalar analog of the Blandford-Znajek mechanism, Phys. Rev. D 93 (2016) 044043 [arXiv:1509.06681] [INSPIRE].
  43. [43]
    C. Bernard, Stationary charged scalar clouds around black holes in string theory, Phys. Rev. D 94 (2016) 085007 [arXiv:1608.05974] [INSPIRE].
  44. [44]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    S. Hod, Spinning Kerr black holes with stationary massive scalar clouds: The large-coupling regime, JHEP 01 (2017) 030 [arXiv:1612.00014] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    H.R.C. Ferreira and C. Herdeiro, Stationary scalar clouds around a BTZ black hole, Phys. Lett. B 773 (2017) 129 [arXiv:1707.08133] [INSPIRE].
  47. [47]
    G.H. Derrick, Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys. 5 (1964) 1252 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A Skyrme-type proposal for baryonic matter, Phys. Lett. B 691 (2010) 105 [arXiv:1001.4544] [INSPIRE].
  49. [49]
    C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A BPS Skyrme model and baryons at large N c, Phys. Rev. D 82 (2010) 085015 [arXiv:1007.1567] [INSPIRE].
  50. [50]
    I. Perapechka and Ya. Shnir, Crystal structures in generalized Skyrme model, Phys. Rev. D 96 (2017) 045013 [arXiv:1703.10673] [INSPIRE].
  51. [51]
    M.J. Esteban, A Direct Variational Approach to Skyrme’s Model for Meson Fields, Commun. Math. Phys. 105 (1986) 571 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    N.S. Manton and P. Sutcliffe, Topological solitons, Cambridge University Press (2004).Google Scholar
  53. [53]
    S. Krusch and P. Sutcliffe, Sphalerons in the Skyrme model, J. Phys. A 37 (2004) 9037 [hep-th/0407002] [INSPIRE].
  54. [54]
    Ya. Shnir and D.H. Tchrakian, Skyrmion-Anti-Skyrmion Chains, J. Phys. A 43 (2010) 025401 [arXiv:0906.5583] [INSPIRE].
  55. [55]
    I. Smolić, Symmetry inheritance of scalar fields, Class. Quant. Grav. 32 (2015) 145010 [arXiv:1501.04967] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    H. Boutaleb-Joutei, A. Chakrabarti and A. Comtet, Gauge Field Configurations in Curved Space-times, Phys. Rev. D 20 (1979) 1884 [INSPIRE].
  57. [57]
    S.B. Gudnason and M. Nitta, Higher-order Skyrme hair of black holes, JHEP 05 (2018) 071 [arXiv:1803.10786] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    O. Schenk and K. Gärtner Solving unsymmetric sparse systems of linear equations with PARDISO, Future Gener. Comp. Sy. 20(3) (2004) 475.Google Scholar
  59. [59]
    W. Schönauer and R. Weiß, Efficient vectorizable PDE solvers, J. Comput. Appl. Math. 27 (1989) 279.Google Scholar
  60. [60]
    M. Schauder, R. Weiß and W. Schönauer, The CADSOL Program Package, Universität Karlsruhe, Interner Bericht Nr. 46/92 (1992).Google Scholar
  61. [61]
    R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory, North-Holland Publishing Company (1982).Google Scholar
  62. [62]
    E. Radu and M.S. Volkov, Existence of stationary, non-radiating ring solitons in field theory: knots and vortons, Phys. Rept. 468 (2008) 101 [arXiv:0804.1357] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    J.F.M. Delgado, C. Herdeiro and E. Radu, Violations of the Kerr and Reissner-Nordström bounds: Horizon versus asymptotic quantities, Phys. Rev. D 94 (2016) 024006 [arXiv:1606.07900] [INSPIRE].
  64. [64]
    M.S. Volkov and D.V. Gal’tsov, Gravitating nonAbelian solitons and black holes with Yang-Mills fields, Phys. Rept. 319 (1999) 1 [hep-th/9810070] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    S.R. Brandt and E. Seidel, The Evolution of distorted rotating black holes. 3: Initial data, Phys. Rev. D 54 (1996) 1403 [gr-qc/9601010] [INSPIRE].
  66. [66]
    S. Yoshida and Y. Eriguchi, Rotating boson stars in general relativity, Phys. Rev. D 56 (1997) 762 [INSPIRE].
  67. [67]
    Y. Brihaye, C. Herdeiro, E. Radu and D.H. Tchrakian, Skyrmions, Skyrme stars and black holes with Skyrme hair in five spacetime dimension, JHEP 11 (2017) 037 [arXiv:1710.03833] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • C. Herdeiro
    • 1
    • 2
  • I. Perapechka
    • 3
  • E. Radu
    • 1
    Email author
  • Ya. Shnir
    • 4
  1. 1.Departamento de Física da Universidade de Aveiro and CIDMAAveiroPortugal
  2. 2.CENTRA, Departamento de Física, Instituto Superior Técnico — ISTUniversidade de Lisboa — ULLisboaPortugal
  3. 3.Department of Theoretical Physics and AstrophysicsBelarusian State UniversityMinskBelarus
  4. 4.DubnaRussia

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