Intermediate behavior of Kerr tails

  • Anıl ZenginoğluEmail author
  • Gaurav Khanna
  • Lior M. Burko
Research Article


The numerical investigation of wave propagation in the asymptotic domain of Kerr spacetime has only recently been possible thanks to the construction of suitable hyperboloidal coordinates. The asymptotics revealed an apparent puzzle in the decay rates of scalar fields: the late-time rates seemed to depend on whether finite distance observers are in the strong field domain or far away from the rotating black hole, an apparent phenomenon dubbed ‘splitting.’ We discuss far-field ‘splitting’ in the full field and near-horizon ‘splitting’ in certain projected modes using horizon-penetrating, hyperboloidal coordinates. For either case we propose an explanation to the cause of the ‘splitting’ behavior, and we determine uniquely decay rates that previous studies found to be ambiguous or immeasurable. The far-field ‘splitting’ is explained by competition between projected modes. The near-horizon ‘splitting’ is due to excitation of lower multipole modes that back excite the multipole mode for which ‘splitting’ is observed. In both cases ‘splitting’ is an intermediate effect, such that asymptotically in time strong field rates are valid at all finite distances. At any finite time, however, there are three domains with different decay rates whose boundaries move outwards during evolution. We then propose a formula for the decay rate of tails that takes into account the inter-mode excitation effect that we study.


Black hole perturbation theory Wave equations Tail decay Hyperboloidal compactification 



We thank Gábor Zs Tóth and István Rácz for discussions. AZ is supported by the NSF Grant PHY-1068881, and by a Sherman Fairchild Foundation grant to Caltech. GK acknowledges research support from NSF Grant Nos. PHY-1016906, PHY-113566 and PHY-1303724. LMB is supported by a NASA EPSCoR RID grant and by NSF Grants PHY-0757344, PHY-1249302, DUE-0941327 and DUE-1300717. Initial work on this project was done while LMB was at the University of Alabama in Huntsville. Most of the data presented in this work were generated on the Air Force Research Laboratory CONDOR supercomputer. GK also acknowledges support from AFRL under CRADA No. 10-RI-CRADA-09.


  1. 1.
    Regge, T., Wheeler, J.A.: Phys. Rev. 108, 1063 (1957). doi: 10.1103/PhysRev.108.1063
  2. 2.
    Price, R.H.: Phys. Rev. D5, 2419 (1972). doi: 10.1103/PhysRevD.5.2419 ADSGoogle Scholar
  3. 3.
    Barack, L.: Phys. Rev. D61, 024026 (2000). doi: 10.1103/PhysRevD.61.024026 ADSMathSciNetGoogle Scholar
  4. 4.
    Barack, L., Ori, A.: Phys. Rev. Lett. 82, 4388 (1999). doi: 10.1103/PhysRevLett.82.4388 Google Scholar
  5. 5.
    Hod, S.: Phys. Rev. D61, 024033 (1999). doi: 10.1103/PhysRevD.61.024033 ADSMathSciNetGoogle Scholar
  6. 6.
    Hod, S.: Phys. Rev. D 61, 064018 (2000)ADSCrossRefMathSciNetGoogle Scholar
  7. 7.
    Poisson, E.: Phys. Rev. D66, 044008 (2002). doi: 10.1103/PhysRevD.66.044008 ADSMathSciNetGoogle Scholar
  8. 8.
    Krivan, W.: Phys. Rev. D60, 101501 (1999). doi: 10.1103/PhysRevD.60.101501 ADSMathSciNetGoogle Scholar
  9. 9.
    Burko, L.M., Khanna, G.: Phys. Rev. D67, 081502 (2003). doi: 10.1103/PhysRevD.67.081502 ADSMathSciNetGoogle Scholar
  10. 10.
    Tiglio, M., Kidder, L.E., Teukolsky, S.A.: Class. Quant. Gravit. 25, 105022 (2008). doi: 10.1088/0264-9381/25/10/105022 ADSCrossRefGoogle Scholar
  11. 11.
    Burko, L.M., Khanna, G.: Class. Quant. Gravit. 26, 015014 (2009). doi: 10.1088/0264-9381/26/1/015014 ADSCrossRefMathSciNetGoogle Scholar
  12. 12.
    Burko, L.M., Khanna, G.: Class. Quant. Gravit. 28, 025012 (2011). doi: 10.1088/0264-9381/28/2/025012 ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gundlach, C., Price, R.H., Pullin, J.: Phys. Rev. D49, 883 (1994). doi: 10.1103/PhysRevD.49.883 ADSGoogle Scholar
  14. 14.
    Zenginoglu, A., Tiglio, M.: Phys. Rev. D80, 024044 (2009). doi: 10.1103/PhysRevD.80.024044 ADSMathSciNetGoogle Scholar
  15. 15.
    Racz, I., Toth, G.Z.: Class. Quant. Gravit. 28, 195003 (2011). doi: 10.1088/0264-9381/28/19/195003 ADSCrossRefMathSciNetGoogle Scholar
  16. 16.
    Jasiulek, M.: Class. Quant. Gravit. 29, 015008 (2012). doi: 10.1088/0264-9381/29/1/015008 ADSCrossRefMathSciNetGoogle Scholar
  17. 17.
    Purrer, M., Husa, S., Aichelburg, P.C.: Phys. Rev. D71, 104005 (2005). doi: 10.1103/PhysRevD.71.104005 ADSGoogle Scholar
  18. 18.
    Zenginoglu, A.: Class. Quant. Gravit. 25, 175013 (2008). doi: 10.1088/0264-9381/25/17/175013 ADSCrossRefMathSciNetGoogle Scholar
  19. 19.
    Pretorius, F., Israel, W.: Class. Quant. Gravit. 15, 2289 (1998). doi: 10.1088/0264-9381/15/8/012 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Fletcher, S.J., Lun, A.W.C.: Class. Quant. Gravit. 20, 4153 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Venter, L.R., Bishop, N.T.: Phys. Rev. D73, 084023 (2006). doi: 10.1103/PhysRevD.73.084023 ADSMathSciNetGoogle Scholar
  22. 22.
    Zenginoglu, A.: Class. Quant. Gravit. 25, 145002 (2008). doi: 10.1088/0264-9381/25/14/145002 ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    Harms, E., Bernuzzi, S., Bruegmann, B.: Class. Quant. Gravit. 30, 115013 (2013). doi: 10.1088/0264-9381/30/11/115013 ADSCrossRefGoogle Scholar
  24. 24.
    Strauss, W., Tsutaya, K.: Discr. Cont. Dyn. Syst. 3, 175 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Szpak, N., Bizon, P., Chmaj, T., Rostworowski, A.: J. Hyperbol. Diff. Equat. 6, 107 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Harms, B.B.E., Bernuzzi, S.: Class. Quantum Gravit. 30, 115013 (2013)ADSCrossRefMathSciNetGoogle Scholar
  27. 27.
    Mino, Y., Sasaki, M., Tanaka, T.: Phys. Rev. D55, 3457 (1997). doi: 10.1103/PhysRevD.55.3457 ADSGoogle Scholar
  28. 28.
    Quinn, T.C., Wald, R.M.: Phys. Rev. D56, 3381 (1997). doi: 10.1103/PhysRevD.56.3381 ADSMathSciNetGoogle Scholar
  29. 29.
    Zenginoglu, A., Galley, C.R.: Phys. Rev. D86, 064030 (2012). doi: 10.1103/PhysRevD.86.064030 ADSGoogle Scholar
  30. 30.
    Casals, M., Dolan, S., Ottewill, A.C., Wardell, B.: Phys. Rev. D88, 044022 (2013). doi: 10.1103/PhysRevD.88.044022
  31. 31.
    Zenginoglu, A.: J. Comput. Phys. 230, 2286 (2011). doi: 10.1016/ ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Zenginoglu, A., Khanna, G.: Phys. Rev. X1, 021017 (2011). doi: 10.1103/PhysRevX.1.021017 Google Scholar
  33. 33.
    Bernuzzi, S., Nagar, A., Zenginoglu, A.: Phys. Rev. D86, 104038 (2012). doi: 10.1103/PhysRevD.86.104038 ADSGoogle Scholar
  34. 34.
    Burko, L.M., Ori, A.: Phys. Rev. D56, 7820 (1997). doi: 10.1103/PhysRevD.56.7820 ADSGoogle Scholar
  35. 35.
    Gleiser, R., Price, R., Pullin, J.: Class. Quant. Gravit. 25, 072001 (2008)ADSCrossRefMathSciNetGoogle Scholar
  36. 36.
    Burko, L.M., Khanna, G.: Phys. Rev. D. arXiv:1312.5247 (in press)
  37. 37.
    Csizmadia, P., Laszlo, A., Racz, I.: Class. Quant. Gravit. 30, 015010 (2013). doi: 10.1088/0264-9381/30/1/015010 ADSCrossRefMathSciNetGoogle Scholar
  38. 38.
    Aretakis, S: Horizon instability of extremal black holes. arXiv:1206.6598 [gr-qc] (2012)

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Anıl Zenginoğlu
    • 1
    Email author
  • Gaurav Khanna
    • 2
  • Lior M. Burko
    • 3
    • 4
    • 5
  1. 1.Theoretical AstrophysicsCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Department of PhysicsUniversity of MassachusettsDartmouthUSA
  3. 3.Observatoire des Sciences de l’Univers en Region CentreUniversité d’OrléansOrléansFrance
  4. 4.Department of Physics, Chemistry, and MathematicsAlabama A&M UniversityNormalUSA
  5. 5.Theiss ResearchLa JollaUSA

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