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Journal of High Energy Physics

, 2019:90 | Cite as

Extensions of the generalized hedgehog ansatz for the Einstein-nonlinear σ-model system: black holes with NUT, black strings and time-dependent solutions

  • Alex Giacomini
  • Marcello OrtaggioEmail author
Open Access
Regular Article - Theoretical Physics
  • 12 Downloads

Abstract

We consider a class of ansätze for the construction of exact solutions of the Einstein-nonlinear σ-model system with an arbitrary cosmological constant in (3+1) dimensions. Exploiting a geometric interplay between the SU(2) field and Killing vectors of the spacetime reduces the matter field equations to a single scalar equation (identically satisfied in some cases) and simultaneously simplifies Einstein’s equations. This is then exemplified over various classes of spacetimes, which allows us to construct stationary black holes with a NUT parameter and uniform black strings, as well as time-dependent solutions such as Robinson-Trautman and Kundt spacetimes, Vaidya-type radiating black holes and certain Bianchi IX cosmologies. In addition to new solutions, some previously known ones are rederived in a more systematic way.

Keywords

Black Holes Sigma Models 

Notes

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.

References

  1. [1]
    N.S. Manton and P. Sutcliffe, Topological solitons, Cambridge University Press, Cambridge U.K. (2004).Google Scholar
  2. [2]
    V.P. Nair, Quantum field theory: a modern perspective, Springer, Germany (2005).Google Scholar
  3. [3]
    H. Lückock and I. Moss, Black holes have Skyrmion hair, Phys. Lett. B 176 (1986) 341 [INSPIRE].
  4. [4]
    S. Droz, M. Heusler and N. Straumann, New black hole solutions with hair, Phys. Lett. B 268 (1991) 371 [INSPIRE].
  5. [5]
    M. Heusler, S. Droz and N. Straumann, Stability analysis of selfgravitating skyrmions, Phys. Lett. B 271 (1991) 61 [INSPIRE].
  6. [6]
    P. Bizon, Gravitating solitons and hairy black holes, Acta Phys. Polon. B 25 (1994) 877 [gr-qc/9402016] [INSPIRE].
  7. [7]
    T. Ioannidou, B. Kleihaus and J. Kunz, Spinning gravitating skyrmions, Phys. Lett. B 643 (2006) 213 [gr-qc/0608110] [INSPIRE].
  8. [8]
    T. Ioannidou, B. Kleihaus and J. Kunz, Platonic gravitating skyrmions, Phys. Lett. B 635 (2006) 161 [gr-qc/0601103] [INSPIRE].
  9. [9]
    B. Kleihaus, J. Kunz and A. Sood, SU(3) Einstein-Skyrme solitons and black holes, Phys. Lett. B 352 (1995) 247 [hep-th/9503087] [INSPIRE].
  10. [10]
    S.B. Gudnason, M. Nitta and S. Sasaki, A supersymmetric Skyrme model, JHEP 02 (2016) 074 [arXiv:1512.07557] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M. Wachla, Gravitating gauged BPS baby Skyrmions, Phys. Rev. D 99 (2019) 065006 [arXiv:1803.10690] [INSPIRE].
  12. [12]
    P. Bizon and T. Chmaj, Gravitating Skyrmions, Phys. Lett. B 297 (1992) 55 [INSPIRE].
  13. [13]
    M. Heusler, S. Droz and N. Straumann, Linear stability of Einstein Skyrme black holes, Phys. Lett. B 285 (1992) 21 [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]
    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].
  16. [16]
    N.S. Manton and P.J. Ruback, Skyrmions in flat space and curved space, Phys. Lett. B 181 (1986) 137 [INSPIRE].
  17. [17]
    M.S. Volkov and D.V. Gal’tsov, Gravitating non-Abelian solitons and black holes with Yang-Mills fields, Phys. Rept. 319 (1999) 1 [hep-th/9810070] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    N. Shiiki and N. Sawado, Black holes with skyrme hair, gr-qc/0501025 [INSPIRE].
  19. [19]
    A. Gußmann, Aspects of Skyrmion black hole hair, PoS(CORFU2016)089.
  20. [20]
    F. Canfora and H. Maeda, Hedgehog ansatz and its generalization for self-gravitating Skyrmions, Phys. Rev. D 87 (2013) 084049 [arXiv:1302.3232] [INSPIRE].
  21. [21]
    F. Canfora and P. Salgado-Rebolledo, Generalized hedgehog ansatz and Gribov copies in regions with nontrivial topologies, Phys. Rev. D 87 (2013) 045023 [arXiv:1302.1264] [INSPIRE].
  22. [22]
    E. Ayon-Beato, F. Canfora and J. Zanelli, Analytic self-gravitating Skyrmions, cosmological bounces and AdS wormholes, Phys. Lett. B 752 (2016) 201 [arXiv:1509.02659] [INSPIRE].
  23. [23]
    H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt., Exact solutions of Einsteins field equations, 2nd edition, Cambridge University Press, Cambridge U.K. (2003).Google Scholar
  24. [24]
    R.M. Wald, General relativity, University of Chicago Press, Chicago U.S.A. (1984).Google Scholar
  25. [25]
    E. Schrödinger, Expanding universes, Cambridge University Press, Cambridge U.K. (1956).Google Scholar
  26. [26]
    M. Astorino, F. Canfora, A. Giacomini and M. Ortaggio, Hairy AdS black holes with a toroidal horizon in 4D Einstein-nonlinear σ-model system, Phys. Lett. B 776 (2018) 236 [arXiv:1711.08100] [INSPIRE].
  27. [27]
    M.M. Caldarelli, A. Christodoulou, I. Papadimitriou and K. Skenderis, Phases of planar AdS black holes with axionic charge, JHEP 04 (2017) 001 [arXiv:1612.07214] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    F. Canfora, N. Dimakis, A. Giacomini and A. Paliathanasis, Bianchi IX cosmologies in the Einstein-Skyrme system in a sector with nontrivial topological charge, Phys. Rev. D 99 (2019) 044035 [arXiv:1902.00400] [INSPIRE].
  29. [29]
    J.B. Griffiths and J. Podolský, Exact space-times in Einsteins general relativity, Cambridge University Press, Cambridge U.K. (2009).Google Scholar
  30. [30]
    J. Plebanski and A. Krasinski, An introduction to general relativity and cosmology, Cambridge University Press, Cambridge U.K. (2006).Google Scholar
  31. [31]
    Y. Bardoux, M.M. Caldarelli and C. Charmousis, Shaping black holes with free fields, JHEP 05 (2012) 054 [arXiv:1202.4458] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  32. [32]
    H. Tan, J. Yang, J. Zhang and T. He, The global monopole spacetime and its topological charge, Chin. Phys. B 27 (2018) 030401.Google Scholar
  33. [33]
    S. Chen, L. Wang, C. Ding and J. Jing, Holographic superconductors in the AdS black hole spacetime with a global monopole, Nucl. Phys. B 836 (2010) 222 [arXiv:0912.2397] [INSPIRE].
  34. [34]
    J.F. Plebanski and S. Hacyan, Some exceptional electrovac type D metrics with cosmological constant, J. Math. Phys. 20 (1979) 1004.Google Scholar
  35. [35]
    H.K. Kunduri and J. Lucietti, Classification of near-horizon geometries of extremal black holes, Living Rev. Rel. 16 (2013) 8 [arXiv:1306.2517].
  36. [36]
    E. Newman, L. Tamburino and T. Unti, Empty space generalization of the Schwarzschild metric, J. Math. Phys. 4 (1963) 915 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    B. Carter, Hamilton-Jacobi and Schrödinger separable solutions of Einsteins equations, Commun. Math. Phys. 10 (1968) 280 [INSPIRE].
  38. [38]
    M. Cahen and L. Defrise, Lorentzian 4 dimensional manifolds withlocal isotropy”, Commun. Math. Phys. 11 (1968) 5.Google Scholar
  39. [39]
    A.D. García and M.C. Alvarez, Shear-free special electrovac type-II solutions with cosmological constant, Nuovo Cim. B 79 (1984) 266.Google Scholar
  40. [40]
    V.I. Khlebnikov, Gravitational radiation in electromagnetic universes, Class. Quant. Grav. 3 (1986) 169 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    M. Ortaggio and J. Podolský, Impulsive waves in electrovac direct product space-times with Lambda, Class. Quant. Grav. 19 (2002) 5221 [gr-qc/0209068] [INSPIRE].
  42. [42]
    J. Podolský and M. Ortaggio, Explicit Kundt type-II and N solutions as gravitational waves in various type D and O universes, Class. Quant. Grav. 20 (2003) 1685 [gr-qc/0212073] [INSPIRE].
  43. [43]
    H. Kadlecová, A. Zelnikov, P. Krtouš and J. Podolský, Gyratons on direct-product spacetimes, Phys. Rev. D 80 (2009) 024004 [arXiv:0905.2476] [INSPIRE].
  44. [44]
    M. Barriola and A. Vilenkin, Gravitational field of a global monopole, Phys. Rev. Lett. 63 (1989) 341 [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    G.W. Gibbons, Selfgravitating magnetic monopoles, global monopoles and black holes, Lect. Notes Phys. 383 (1991) 110 [arXiv:1109.3538] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    E.I. Guendelman and A. Rabinowitz, The gravitational field of a hedgehog and the evolution of vacuum bubbles, Phys. Rev. D 44 (1991) 3152 [INSPIRE].
  47. [47]
    T. Levi-Civita, Realtà fisica di alcuni spazi normali del Bianchi, Rend. Acc. Lincei 26 (1917) 519.zbMATHGoogle Scholar
  48. [48]
    H. Nariai, On a new cosmological solution of Einsteins field equations of gravitation, Sci. Rep. Tôhoku Univ. 35 (1951) 62.Google Scholar
  49. [49]
    B. Bertotti, Uniform electromagnetic field in the theory of general relativity, Phys. Rev. 116 (1959) 1331 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    I. Robinson, A solution of the Maxwell-Einstein equations, Bull. Acad. Polon. 7 (1959) 351.MathSciNetzbMATHGoogle Scholar
  51. [51]
    A.H. Taub, Empty space-times admitting a three parameter group of motions, Ann. Math. 53 (1951) 472.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    M. Ortaggio, Impulsive waves in the Nariai universe, Phys. Rev. D 65 (2002) 084046 [gr-qc/0110126] [INSPIRE].
  53. [53]
    S. Deser and P.O. Mazur, Static solutions in D = 3 Einstein-Maxwell theory, Class. Quant. Grav. 2 (1985) L51 [INSPIRE].
  54. [54]
    M.A. Melvin, Exterior solutions for electric and magnetic stars in 2 + 1 dimensions, Class. Quant. Grav. 3 (1986) 117.Google Scholar
  55. [55]
    J.R. Gott, J.Z. Simon and M. Alpert, General relativity in a (2 + 1)-dimensional space-time: an electrically charged solution, Gen. Rel. Grav. 18 (1986) 1019 [INSPIRE].
  56. [56]
    A. Giacomini, M. Lagos, J. Oliva and A. Vera, Charged black strings and black branes in Lovelock theories, Phys. Rev. D 98 (2018) 044019 [arXiv:1804.03130].
  57. [57]
    M. Astorino, F. Canfora, M. Lagos and A. Vera, Black hole and BTZ black string in the Einstein-SU(2) Skyrme model, Phys. Rev. D 97 (2018) 124032 [arXiv:1805.12252].
  58. [58]
    M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    A. Cisterna and J. Oliva, Exact black strings and p-branes in general relativity, Class. Quant. Grav. 35 (2018) 035012 [arXiv:1708.02916] [INSPIRE].
  60. [60]
    A. Cisterna, C. Corral and S. del Pino, Static and rotating black strings in dynamical Chern-Simons modified gravity, Eur. Phys. J. C 79 (2019) 400 [arXiv:1809.02903] [INSPIRE].
  61. [61]
    R. Emparan, G.T. Horowitz and R.C. Myers, Exact description of black holes on branes. 2. Comparison with BTZ black holes and black strings, JHEP 01 (2000) 021 [hep-th/9912135] [INSPIRE].
  62. [62]
    M. Ortaggio, V. Pravda and A. Pravdová, On higher dimensional Einstein spacetimes with a warped extra dimension, Class. Quant. Grav. 28 (2011) 105006 [arXiv:1011.3153] [INSPIRE].
  63. [63]
    E.R. Harrison, Classification of uniform cosmological models, Mon. Not. Roy. Astron. Soc. 137 (1967) 69 [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    F. Canfora, A. Paliathanasis, T. Taves and J. Zanelli, Cosmological Einstein-Skyrme solutions with nonvanishing topological charge, Phys. Rev. D 95 (2017) 065032 [arXiv:1703.04860].
  65. [65]
    S.A. Pavluchenko, Dynamics of gravitating hadron matter in a Bianchi-IX cosmological model, Phys. Rev. D 94 (2016) 044046 [arXiv:1607.00536] [INSPIRE].
  66. [66]
    M.P. Ryan and L.C. Shepley, Homogeneous relativistic cosmologies, Princeton University Press, Princeton U.S.A. (1975).Google Scholar
  67. [67]
    G.F.R. Ellis and M.A.H. MacCallum, A class of homogeneous cosmological models, Commun. Math. Phys. 12 (1969) 108 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    M.A.H. MacCallum, J.M. Stewart and B.G. Schmidt, Anisotropic stresses in homogeneous cosmologies, Commun. Math. Phys. 17 (1970) 343 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  69. [69]
    M.A.H. MacCallum, Cosmological models from a geometric point of view, in Cargese Lectures in Physics. Volume 6, E. Schatzman ed., Gordon and Breach, New York U.S.A. (1973).Google Scholar
  70. [70]
    R.P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11 (1963) 237 [INSPIRE].
  71. [71]
    C. Herdeiro, I. Perapechka, E. Radu and Ya. Shnir, Skyrmions around Kerr black holes and spinning BHs with Skyrme hair, JHEP 10 (2018) 119 [arXiv:1808.05388] [INSPIRE].
  72. [72]
    E.T. Newman, R. Couch, K. Chinnapared, A. Exton, A. Prakash and R. Torrence, Metric of a rotating, charged mass, J. Math. Phys. 6 (1965) 918 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    F. Canfora, F. Correa, A. Giacomini and J. Oliva, Exact meron black holes in four dimensional SU(2) Einstein-Yang-Mills theory, Phys. Lett. B 722 (2013) 364 [arXiv:1208.6042] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Instituto de Ciencias Físicas y MatemáticasUniversidad Austral de ChileValdiviaChile
  2. 2.Institute of Mathematics of the Czech Academy of SciencesPrague 1Czech Republic

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