Skip to main content
SpringerLink
Log in

Generalized hidden symmetries and the Kerr-Sen black hole

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We elaborate on basic properties of generalized Killing-Yano tensors which naturally extend Killing-Yano symmetry in the presence of skew-symmetric torsion. In particular, we discuss their relationship to Killing tensors and the separability of various field equations. We further demonstrate that the Kerr-Sen black hole spacetime of heterotic string theory, as well as its generalization to all dimensions, possesses a generalized closed conformal Killing-Yano 2-form with respect to a torsion identified with the 3-form occuring naturally in the theory. Such a 2-form is responsible for complete integrability of geodesic motion as well as for separability of the scalar and Dirac equations in these spacetimes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. K. Yano, Some remarks on tensor fields and curvature, Annals Math. 55 (1952) 328.

    Article  Google Scholar 

  2. R.C. Myers and M.J. Perry, Black holes in higher dimensional space-times, Annals Phys. 172 (1986) 304 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. G.W. Gibbons, H. Lü, D.N. Page and C.N. Pope, Rotating black holes in higher dimensions with a cosmological constant, Phys. Rev. Lett. 93 (2004) 171102 [hep-th/0409155] [SPIRES].

    Article  ADS  Google Scholar 

  4. W. Chen, H. Lü and C.N. Pope, General Kerr-NUT-AdS metrics in all dimensions, Class. Quant. Grav. 23 (2006) 5323 [hep-th/0604125] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  5. D. Kubizňák and V.P. Frolov, Hidden symmetry of higher dimensional Kerr-NUT-AdS spacetimes, Class. Quant. Grav. 24 (2007) F1 [gr-qc/0610144] [SPIRES].

    Article  ADS  Google Scholar 

  6. D.N. Page, D. Kubizňák, M. Vasudevan and P. Krtouš, Complete integrability of geodesic motion in general Kerr-NUT-AdS spacetimes, Phys. Rev. Lett. 98 (2007) 061102 [hep-th/0611083] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  7. P. Krtouš, D. Kubizňák, D.N. Page and V.P. Frolov, Killing-Yano tensors, rank-2 Killing tensors and conserved quantities in higher dimensions, JHEP 02 (2007) 004 [hep-th/0612029] [SPIRES].

    Article  ADS  Google Scholar 

  8. T. Houri, T. Oota and Y. Yasui, Closed conformal Killing-Yano tensor and geodesic integrability, J. Phys. A 41 (2008) 025204 [arXiv:0707.4039] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  9. V.P. Frolov, P. Krtouš and D. Kubizňák, Separability of Hamilton-Jacobi and Klein-Gordon equations in general Kerr-NUT-AdS spacetimes, JHEP 02 (2007) 005 [hep-th/0611245] [SPIRES].

    Article  ADS  Google Scholar 

  10. A. Sergyeyev and P. Krtouš, Complete set of commuting symmetry operators for the Klein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetimes, Phys. Rev. D 77 (2008) 044033 [arXiv:0711.4623] [SPIRES].

    ADS  Google Scholar 

  11. T. Oota and Y. Yasui, Separability of Dirac equation in higher dimensional Kerr-NUT-de Sitter spacetime, Phys. Lett. B 659 (2008) 688 [arXiv:0711.0078] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  12. S.-Q. Wu, Symmetry operators and separability of the massive Dirac’s equation in the general 5-dimensional Kerr-(anti-)de Sitter black hole background, Class. Quant. Grav. 26 (2009) 055001 [Erratum ibid. 26 (2009) 189801] [arXiv:0808.3435] [SPIRES].

    Article  ADS  Google Scholar 

  13. N. Hamamoto, T. Houri, T. Oota and Y. Yasui, Kerr-NUT-de Sitter curvature in all dimensions, J. Phys. A 40 (2007) F177 [hep-th/0611285] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  14. L. Mason and A. Taghavi-Chabert, Killing-Yano tensors and multi-Hermitian structures, J. Geom. Phys. 60 (2010) 907 [arXiv:0805.3756] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  15. H.K. Kunduri, J. Lucietti and H.S. Reall, Gravitational perturbations of higher dimensional rotating black holes: tensor perturbations, Phys. Rev. D 74 (2006) 084021 [hep-th/0606076] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  16. K. Murata and J. Soda, A note on separability of field equations in Myers-Perry spacetimes, Class. Quant. Grav. 25 (2008) 035006 [arXiv:0710.0221] [SPIRES].

    Article  MathSciNet  Google Scholar 

  17. T. Oota and Y. Yasui, Separability of gravitational perturbation in generalized Kerr-NUT-de Sitter spacetime, Int. J. Mod. Phys. A 25 (2010) 3055 [arXiv:0812.1623] [SPIRES].

    ADS  Google Scholar 

  18. T. Houri, T. Oota and Y. Yasui, Closed conformal Killing-Yano tensor and Kerr-NUT-de Sitter spacetime uniqueness, Phys. Lett. B 656 (2007) 214 [arXiv:0708.1368] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  19. P. Krtouš, V.P. Frolov and D. Kubizňák, Hidden symmetries of higher dimensional black holes and uniqueness of the Kerr-NUT-(A)dS spacetime, Phys. Rev. D 78 (2008) 064022 [arXiv:0804.4705] [SPIRES].

    ADS  Google Scholar 

  20. T. Houri, T. Oota and Y. Yasui, Closed conformal Killing-Yano tensor and uniqueness of generalized Kerr-NUT-de Sitter spacetime, Class. Quant. Grav. 26 (2009) 045015 [arXiv:0805.3877] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  21. K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies 32, Princeton University Press, Princeton U.S.A. (1953).

    MATH  Google Scholar 

  22. S.-Q. Wu, Separability of a modified Dirac equation in a five-dimensional rotating, charged black hole in string theory, Phys. Rev. D 80 (2009) 044037 [Erratum ibid. D 80 (2009) 069902] [arXiv:0902.2823] [SPIRES].

    ADS  Google Scholar 

  23. D. Kubiznak, H.K. Kunduri and Y. Yasui, Generalized Killing-Yano equations in D = 5 gauged supergravity, Phys. Lett. B 678 (2009) 240 [arXiv:0905.0722] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  24. S.-Q. Wu, Separability of massive field equations for spin-0 and spin-1/2 charged particles in the general non-extremal rotating charged black holes in minimal five-dimensional gauged supergravity, Phys. Rev. D 80 (2009) 084009 [arXiv:0906.2049] [SPIRES].

    ADS  Google Scholar 

  25. Z.W. Chong, M. Cvetič, H. Lü and C.N. Pope, General non-extremal rotating black holes in minimal five-dimensional gauged supergravity, Phys. Rev. Lett. 95 (2005) 161301 [hep-th/0506029] [SPIRES].

    Article  ADS  Google Scholar 

  26. D. Kubiznak, Black hole spacetimes with Killing-Yano symmetries, arXiv:0909.1589 [SPIRES].

  27. P. Davis, H.K. Kunduri and J. Lucietti, Special symmetries of the charged Kerr-AdS black hole of D = 5 minimal gauged supergravity, Phys. Lett. B 628 (2005) 275 [hep-th/0508169] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  28. H. Ahmedov and A.N. Aliev, Uniqueness of rotating charged black holes in five-dimensional minimal gauged supergravity, Phys. Lett. B 679 (2009) 396 [arXiv:0907.1804] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  29. T. Houri, D. Kubiznak, C. Warnick and Y. Yasui, Symmetries of the Dirac operator with skew-symmetric torsion, arXiv:1002.3616 [SPIRES].

  30. A. Strominger, Superstrings with torsion, Nucl. Phys. B 274 (1986) 253 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  31. I. Agricola, The Srni lectures on non-integrable geometries with torsion, math.DG/0606705 [SPIRES].

  32. A. Sen, Rotating charged black hole solution in heterotic string theory, Phys. Rev. Lett. 69 (1992) 1006 [hep-th/9204046] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  33. M. Cvetič and D. Youm, Near-BPS-saturated rotating electrically charged black holes as string states, Nucl. Phys. B 477 (1996) 449 [hep-th/9605051] [SPIRES].

    Article  ADS  Google Scholar 

  34. D.D.K. Chow, Symmetries of supergravity black holes, arXiv:0811.1264 [SPIRES].

  35. I.M. Benn and P. Charlton, Dirac symmetry operators from conformal Killing-Yano tensors, Class. Quant. Grav. 14 (1997) 1037 [gr-qc/9612011] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  36. M. Walker and R. Penrose, On quadratic first integrals of the geodesic equations for type {22} spacetimes, Commun. Math. Phys. 18 (1970) 265 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. P. Stackel, Sur l’integration de l’équation differentielle de Hamilton, Comptes Rendus Acad. Sci. Paris Ser. IV 121 (1895) 489.

    Google Scholar 

  38. C.D. Collinson, On the relationship between Killing tensors and Killing-Yano tensors, Int. J. Theor. Phys. 15 (1976) 311.

    Article  MATH  MathSciNet  Google Scholar 

  39. J.J. Ferrando and J.A. Saez, A type-Rainich approach to the Killing-Yano tensors, gr-qc/0212085 [SPIRES].

  40. B. Carter, Killing tensor quantum numbers and conserved currents in curved space, Phys. Rev. D 16 (1977) 3395 [SPIRES].

    ADS  Google Scholar 

  41. B. Carter and R.G. Mclenaghan, Generalized total angular momentum operator for the Dirac equation in curved space-time, Phys. Rev. D 19 (1979) 1093 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  42. G.W. Gibbons, R.H. Rietdijk and J.W. van Holten, SUSY in the sky, Nucl. Phys. B 404 (1993) 42 [hep-th/9303112] [SPIRES].

    Article  ADS  Google Scholar 

  43. M. Cariglia, New quantum numbers for the Dirac equation in curved spacetime, Class. Quant. Grav. 21 (2004) 1051 [hep-th/0305153] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  44. M. Tanimoto, The role of Killing-Yano tensors in supersymmetric mechanics on a curved manifold, Nucl. Phys. B 442 (1995) 549 [gr-qc/9501006] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  45. R.H. Rietdijk and J.W. van Holten, Killing tensors and a new geometric duality, Nucl. Phys. B 472 (1996) 427 [hep-th/9511166] [SPIRES].

    Article  ADS  Google Scholar 

  46. F. De Jonghe, K. Peeters and K. Sfetsos, Killing-Yano supersymmetry in string theory, Class. Quant. Grav. 14 (1997) 35 [hep-th/9607203] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  47. I. Benmachiche, J. Louis and D. Martínez-Pedrera, The effective action of the heterotic string compactified on manifolds with SU(3) structure, Class. Quant. Grav. 25 (2008) 135006 [arXiv:0802.0410] [SPIRES].

    Article  ADS  Google Scholar 

  48. T. Houri, T. Oota and Y. Yasui, Generalized Kerr-NUT-de Sitter metrics in all dimensions, Phys. Lett. B 666 (2008) 391 [arXiv:0805.0838] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  49. V.P. Frolov and D. Kubizňák, Higher-dimensional black holes: hidden symmetries and separation of variables, Class. Quant. Grav. 25 (2008) 154005 [arXiv:0802.0322] [SPIRES].

    Article  ADS  Google Scholar 

  50. S.F. Hassan and A. Sen, Twisting classical solutions in heterotic string theory, Nucl. Phys. B 375 (1992) 103 [hep-th/9109038] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  51. R.P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11 (1963) 237 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  52. A. Burinskii, Some properties of the Kerr solution to low-energy string theory, Phys. Rev. D 52 (1995) 5826 [hep-th/9504139] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  53. T. Okai, Global structure and thermodynamic property of the four-dimensional twisted Kerr solution, Prog. Theor. Phys. 92 (1994) 47 [hep-th/9402149] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  54. P.A. Blaga and C. Blaga, Bounded radial geodesics around a Kerr-Sen black hole, Class. Quant. Grav. 18 (2001) 3893 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  55. S.Q. Wu and X. Cai, Massive complex scalar field in the Kerr-Sen geometry: exact solution of wave equation and Hawking radiation, J. Math. Phys. 44 (2003) 1084 [gr-qc/0303075] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  56. K. Hioki and U. Miyamoto, Hidden symmetries, null geodesics and photon capture in the Sen black hole, Phys. Rev. D 78 (2008) 044007 [arXiv:0805.3146] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  57. R. Floyd, The dynamics of Kerr fields, Ph.D. thesis, London University, London U.K. (1973).

  58. R. Penrose, Naked singularities, Annals N.Y. Acad. Sci. 224 (1973) 125 [SPIRES].

    Article  ADS  Google Scholar 

  59. B. Carter, A new family of Einstein spaces, Phys. Lett. A 26 (1968) 399.

    ADS  Google Scholar 

  60. J.F. Plebański, A class of solutions of Einstein-Maxwell equations, Annals Phys. 90 (1975) 196.

    Article  MATH  ADS  Google Scholar 

  61. V.P. Frolov, V. Skarzhinsky, A. Zelnikov and O. Heinrich, Equilibrium configurations of a cosmic string near a rotating black hole, Phys. Lett. B 224 (1989) 255 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  62. Z.W. Chong, M. Cvetic, H. Lu and C.N. Pope, Charged rotating black holes in four-dimensional gauged and ungauged supergravities, Nucl. Phys. B 717 (2005) 246 [hep-th/0411045] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  63. Z.W. Chong, M. Cvetic, H. Lu and C.N. Pope, Five-dimensional gauged supergravity black holes with independent rotation parameters, Phys. Rev. D 72 (2005) 041901 [hep-th/0505112] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  64. D.D.K. Chow, Charged rotating black holes in six-dimensional gauged supergravity, Class. Quant. Grav. 27 (2010) 065004 [arXiv:0808.2728] [SPIRES].

    Article  ADS  Google Scholar 

  65. D.D.K. Chow, Equal charge black holes and seven dimensional gauged supergravity, Class. Quant. Grav. 25 (2008) 175010 [arXiv:0711.1975] [SPIRES].

    Article  ADS  Google Scholar 

  66. E.G. Kalnins and W. Miller Jr., Killing-Yano tensors and variable separation in Kerr geometry, J. Math. Phys. 30 (1989) 2630.

    Google Scholar 

  67. E.G. Kalnins, G.C. Williams and W. Miller, Intrinsic characterization of the separation constant for spin one and gravitational perturbations in Kerr geometry, Proc. Roy. Soc. Lond. A 452 (1996) 997.

    MathSciNet  ADS  Google Scholar 

  68. I.M. Benn, P. Charlton and J.M. Kress, Debye potentials for Maxwell and Dirac fields from a generalisation of the Killing-Yano equation, J. Math. Phys. 38 (1997) 4504 [gr-qc/9610037] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi, Osaka, 558-8585, Japan

    Tsuyoshi Houri

  2. DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, U.K.

    David Kubizňák & Claude M. Warnick

  3. Queens’ College, Cambridge, CB3 9ET, U.K.

    Claude M. Warnick

  4. Department of Mathematics and Physics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi, Osaka, 558-8585, Japan

    Yukinori Yasui

Authors
  1. Tsuyoshi Houri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Kubizňák
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Claude M. Warnick
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yukinori Yasui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Kubizňák.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Houri, T., Kubizňák, D., Warnick, C.M. et al. Generalized hidden symmetries and the Kerr-Sen black hole. J. High Energ. Phys. 2010, 55 (2010). https://doi.org/10.1007/JHEP07(2010)055

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP07(2010)055

Keywords

Search

Navigation