Journal of High Energy Physics

, 2014:68 | Cite as

Geroch group description of black holes

Open Access
Regular Article - Theoretical Physics


On one hand the Geroch group allows one to associate spacetime independent matrices with gravitational configurations that effectively only depend on two coordinates. This class includes stationary axisymmetric four- and five-dimensional black holes. On the other hand, a recently developed inverse scattering method allows one to factorize these matrices to explicitly construct the corresponding spacetime configurations. In this work we demonstrate the construction as well as the factorization of Geroch group matrices for a wide class of black hole examples. In particular, we obtain the Geroch group SL(3, ℝ) matrices for the five-dimensional Myers-Perry and Kaluza-Klein black holes and the Geroch group SU(2, 1) matrix for the four-dimensional Kerr-Newman black hole. We also present certain non-trivial relations between the Geroch group matrices and charge matrices for these black holes.


Black Holes in String Theory String Duality 


Open Access

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  1. [1]
    B. Julia, Application of supergravity to graviational theories, in Unified field theories of more than four dimensions, V. De Sabbata and E. Schmutze eds., World Scientific, Singapore (1983).Google Scholar
  2. [2]
    P. Breitenlohner, D. Maison and G.W. Gibbons, Four-dimensional black holes from Kaluza-Klein theories, Commun. Math. Phys. 120 (1988) 295 [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    D. Youm, Black holes and solitons in string theory, Phys. Rept. 316 (1999) 1 [hep-th/9710046] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  4. [4]
    D.D.K. Chow and G. Compère, Seed for general rotating non-extremal black holes of \( \mathcal{N} \) =8 supergravity, Class. Quant. Grav. 31 (2014) 022001 [arXiv:1310.1925] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    D.D.K. Chow and G. Compère, Black holes in N =8 supergravity from SO(4, 4) hidden symmetries, Phys. Rev. D 90 (2014) 025029 [arXiv:1404.2602] [INSPIRE].ADSGoogle Scholar
  6. [6]
    S. Tomizawa, Y. Yasui and A. Ishibashi, Uniqueness theorem for charged rotating black holes in five-dimensional minimal supergravity, Phys. Rev. D 79 (2009) 124023 [arXiv:0901.4724] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    S. Hollands, Black hole uniqueness theorems and new thermodynamic identities in eleven dimensional supergravity, Class. Quant. Grav. 29 (2012) 205009 [arXiv:1204.3421] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    G. Bossard, H. Nicolai and K.S. Stelle, Universal BPS structure of stationary supergravity solutions, JHEP 07 (2009) 003 [arXiv:0902.4438] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    G. Bossard, Y. Michel and B. Pioline, Extremal black holes, nilpotent orbits and the true fake superpotential, JHEP 01 (2010) 038 [arXiv:0908.1742] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    G. Bossard and C. Ruef, Interacting non-BPS black holes, Gen. Rel. Grav. 44 (2012) 21 [arXiv:1106.5806] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  11. [11]
    G. Bossard, Octonionic black holes, JHEP 05 (2012) 113 [arXiv:1203.0530] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    G. Bossard and S. Katmadas, A bubbling bolt, JHEP 07 (2014) 118 [arXiv:1405.4325] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    R.P. Geroch, A Method for generating solutions of Einstein’s equations, J. Math. Phys. 12 (1971) 918 [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    R.P. Geroch, A method for generating new solutions of Einstein’s equation. 2, J. Math. Phys. 13 (1972) 394 [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    B. Julia, Infinite Lie algebras in physics, in the proceedings of Unified Field Theories and Beyond, Baltimore, U.S.A. (1981).Google Scholar
  16. [16]
    B. Julia, Group disintegrations, Conf. Proc. C 8006162 (1980) 331.Google Scholar
  17. [17]
    P. Breitenlohner and D. Maison, On the Geroch group, Annales Poincaré Phys. Theor. 46 (1987) 215 [INSPIRE].MATHMathSciNetGoogle Scholar
  18. [18]
    P. Breitenlohner and D. Maison, Solitons in Kaluza-Klein theories, unpublished notes (June 1986).Google Scholar
  19. [19]
    H. Nicolai, Two-dimensional gravities and supergravities as integrable system, in the proceeding of Recent aspects of quantum fields, Schladming, Austria (1991), DESY-91-038 (1991).Google Scholar
  20. [20]
    I. Bakas, O(2, 2) transformations and the string Geroch group, Nucl. Phys. B 428 (1994) 374 [hep-th/9402016] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  21. [21]
    A. Sen, Duality symmetry group of two-dimensional heterotic string theory, Nucl. Phys. B 447 (1995) 62 [hep-th/9503057] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    I. Bakas, Solitons of axion-dilaton gravity, Phys. Rev. D 54 (1996) 6424 [hep-th/9605043] [INSPIRE].ADSMathSciNetGoogle Scholar
  23. [23]
    A.K. Das, J. Maharana and A. Melikyan, Duality, monodromy and integrability of two-dimensional string effective action, Phys. Rev. D 65 (2002) 126001 [hep-th/0203144] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    V.A. Belinsky and V.E. Zakharov, Integration of the Einstein equations by the inverse scattering problem technique and the calculation of the exact soliton solutions, Sov. Phys. JETP 48 (1978) 985 [Zh. Eksp. Teor. Fiz. 75 (1978) 1953] [INSPIRE].
  25. [25]
    V.A. Belinsky and V.E. Sakharov, Stationary gravitational solitons with axial symmetry, Sov. Phys. JETP 50 (1979) 1 [Zh. Eksp. Teor. Fiz. 77 (1979) 3] [INSPIRE].
  26. [26]
    V. Belinski and E. Verdaguer, Gravitational solitons, Cambridge University Press, Cambridge U.K. (2001).CrossRefMATHGoogle Scholar
  27. [27]
    D. Rasheed, The Rotating dyonic black holes of Kaluza-Klein theory, Nucl. Phys. B 454 (1995) 379 [hep-th/9505038] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  28. [28]
    F. Larsen, Rotating Kaluza-Klein black holes, Nucl. Phys. B 575 (2000) 211 [hep-th/9909102] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    R.C. Myers and M.J. Perry, Black holes in higher dimensional space-times, Annals Phys. 172 (1986) 304 [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  30. [30]
    D. Katsimpouri, A. Kleinschmidt and A. Virmani, Inverse scattering and the Geroch group, JHEP 02 (2013) 011 [arXiv:1211.3044] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  31. [31]
    D. Katsimpouri, A. Kleinschmidt and A. Virmani, An inverse scattering formalism for STU supergravity, JHEP 03 (2014) 101 [arXiv:1311.7018] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  32. [32]
    C.N. Pope, Lectures on Kaluza-Klein theory,
  33. [33]
    S. Hollands and S. Yazadjiev, Uniqueness theorem for 5-dimensional black holes with two axial Killing fields, Commun. Math. Phys. 283 (2008) 749 [arXiv:0707.2775] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  34. [34]
    S. Giusto and A. Saxena, Stationary axisymmetric solutions of five dimensional gravity, Class. Quant. Grav. 24 (2007) 4269 [arXiv:0705.4484] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  35. [35]
    R. Emparan and A. Maccarrone, Statistical description of rotating Kaluza-Klein black holes, Phys. Rev. D 75 (2007) 084006 [hep-th/0701150] [INSPIRE].ADSMathSciNetGoogle Scholar
  36. [36]
    T. Harmark, Stationary and axisymmetric solutions of higher-dimensional general relativity, Phys. Rev. D 70 (2004) 124002 [hep-th/0408141] [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    L. Andrianopoli, A. Gallerati and M. Trigiante, On extremal limits and duality orbits of stationary black holes, JHEP 01 (2014) 053 [arXiv:1310.7886] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    W. Kinnersley, Generation of stationary Einstein-Maxwell fields, J. Math. Phys. 14 (1973) 651.ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    J.L. Hornlund and A. Virmani, Extremal limits of the Cvetič-Youm black hole and nilpotent orbits of G 2(2), JHEP 11 (2010) 062 [Erratum ibid. 1205 (2012) 038] [arXiv:1008.3329] [INSPIRE].
  40. [40]
    M. Cvetič and D. Youm, General rotating five-dimensional black holes of toroidally compactified heterotic string, Nucl. Phys. B 476 (1996) 118 [hep-th/9603100] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    M. Cvetič and D. Youm, Entropy of nonextreme charged rotating black holes in string theory, Phys. Rev. D 54 (1996) 2612 [hep-th/9603147] [INSPIRE].ADSGoogle Scholar
  42. [42]
    P. Figueras, E. Jamsin, J.V. Rocha and A. Virmani, Integrability of five dimensional minimal supergravity and charged rotating black holes, Class. Quant. Grav. 27 (2010) 135011 [arXiv:0912.3199] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  43. [43]
    R.G. Leigh, A.C. Petkou, P.M. Petropoulos and P.K. Tripathy, The Geroch group in Einstein spaces, arXiv:1403.6511 [INSPIRE].
  44. [44]
    L. Houart, A. Kleinschmidt, J. Lindman Hornlund, D. Persson and N. Tabti, Finite and infinite-dimensional symmetries of pure N =2 supergravity in D =4, JHEP 08 (2009) 098 [arXiv:0905.4651] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Institute of Physics, Sachivalaya MargBhubaneswarIndia

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