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Chinese Physics C> 2018, Vol. 42> Issue(10) : 105102 DOI: 10.1088/1674-1137/42/10/105102

Quasinormal modes of a Schwarzschild black hole immersed in an electromagnetic universe

  • 1.

     Instituto de Fí

  • 2.

     Physics Department, Arts and Sciences Faculty, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey

  • 3.

     School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

  • 4.

      Instituto de Fí

  • We study the quasinormal modes (QNMs) of a Schwarzschild black hole immersed in an electromagnetic (EM) universe. The immersed Schwarzschild black hole (ISBH) originates from the metric of colliding EM waves with double polarization[Class. Quantum Grav. 12, 3013 (1995)]. The perturbation equations of the scalar fields for the ISBH geometry are written in the form of separable equations. We show that these equations can be transformed to the confluent Heun's equations, for which we are able to use known techniques to perform analytical quasinormal (QNM) analysis of the solutions. Furthermore, we employ numerical methods (Mashhoon and 6th-order Wentzel-Kramers-Brillouin (WKB)) to derive the QNMs. The results obtained are discussed and depicted with the appropriate plots.
      PCAS:

  • References

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    [45] X. Rao, B. Wang, and G. Yang, Phys. Lett. B, 649:472 (2007)
    [46] S. Chen, B. Wang and R. Su, Class. Quantum Grav., 23:7581 (2006)
    [47] X. He, Songbai-Chen, B. Wang, R. Cai, and C. Lin, Phys. Lett. B, 665:392 (2008)
    [48] X. He, B. Wang, and S. Chen, Phys. Rev. D, 79:084005 (2009)
    [49] B. Wang, C. Lin and E. Abdalla, Phys. Lett. B, 481:79 (2000)
    [50] B. Wang, C. Lin, and C. Molina, Phys. Rev. D, 70:064025 (2004)
    [51] A. Jansen, Eur. Phys. J. Plus, 132(12):546 (2017)
    [52] M. Halilsoy and A. Al-Badawi, IL Nuovo Cimento B, 113:761 (1998)
    [53] M. Halilsoy, Gen. Relativ. Gravit., 25(3):275 (1992)
    [54] M. Halilsoy and A. Al-Badawi, Class. Quantum Grav., 12:3013 (1995)
    [55] A. Ovgun, Int. J. Theor. Phys., 55(6):2919 (2016)
    [56] H. Reissner, Annalen der Physik (in German), 50:106 (1916)
    [57] G. Nordstrm, Verhandl. Koninkl. Ned. Akad. Wetenschap., Afdel. Natuurk., Amsterdam., 26:1201 (1918)
    [58] S. Chandrasekhar, The Mathematical Theory of Black Holes (New York:Oxford University Press, 1983)
    [59] R. M. Wald, General Relativity (The University of Chicago Press, Chicago and London, 1984)
    [60] S. Q. Wu and X. Cai, J. Math. Phys. (N.Y.), 44:1084 (2003)
    [61] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965)
    [62] H. S. Vieira, V. B. Bezerra, and G. V. Silva, Ann. Phys. (Amsterdam), 362:576 (2015)
    [63] K. Heun, Math. Ann., 33:161 (1888)
    [64] A. Ronveaux, Heun's differential equations:(New York:Oxford University Press, 1995)
    [65] R. S. Maier, The 192 Solutions of Heun Equation:(Preprint math CA/0408317, 2004)
    [66] P. P. Fiziev, J. Phys. A:Math. Theor., 43:035203 (2010)
    [67] I. Sakalli and M. Halilsoy, Phys. Rev. D, 69:124012 (2004)
    [68] A. Al-Badawi and I. Sakalli, J. Math. Phys., 49:052501 (2008)
    [69] T. Birkandan and M. Hortacsu, EPL, 119(2):20002 (2017)
    [70] I. Sakalli, Phys. Rev. D, 94:084040 (2016)
    [71] G. V. Kraniotis, 2016 Class. Quantum Grav., 33:225011 (2016)
    [72] Here, the computer package, Maple 2017 is used for solving the confluent Heun differential equation
    [73] H. S. Vieira and V. B. Bezerra, Ann. Phys. (NY), 373:28 (2016)
    [74] H. S. Vieira, J. P. Morais Graca, and V. B. Bezerra, Chin. Phys. C, 41:095102 (2017)
    [75] V. Frolov and I. Novikov, Black Hole Physics:Basic Concepts and New Developments. Fundamental Theories of Physics (Kluwer Academic, London, 1998)
    [76] P. P. Fiziev, Class. Quantum Grav., 27:135001 (2010)
    [77] S. A. Teukolsky, Phys. Rev. Lett., 29:1114 (1972); S. A. Teukolsky, Astrophys. J., 185:635 (1973); W. H. Press and S. A. Teukolsky, Astrophys. J., 185:649 (1973); S. A. Teukolsky and W. H. Press, Astrophys. J., 193:443 (1974)
    [78] P. P. Fiziev, Phys. Rev. D, 80:124001 (2009)
    [79] S. Chandrasekhar, Proc. R. Soc. London A, 348:39 (1976); Proc. R. Soc. London A, 372:475 (1980)
    [80] S. Hod, arXiv:gr-qc/0307060 (2003)
    [81] E. Berti, V. Cardoso, K. D. Kokkotas, and H. Onozawa, Phys. Rev. D, 68:124018 (2003)
    [82] L. Motl, Ad. Theor. Math. Phys., 6:1135 (2002)
    [83] R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys., 83:793 (2011)
    [84] H. J. Blome and B. Mashhoon, Phys. Lett. A, 110:231 (1984)
    [85] P. Musgrave and K. Lake, Class. Quant. Grav., 13:1885 (1996)
    [86] B. Toshmatov, A. Abdujabbarov, Z. Stuchlik, and B. Ahmedov, Phys. Rev. D, 91:083008 (2015)
    [87] B. Toshmatov, A. Abdujabbarov, J. Schee, and B. Ahmedov, Phys. Rev. D, 93:124017 (2016)
    [88] V. Cardoso, A. S. Miranda, E. Berti, H. Witeck, and V. T. Zanchin, Phys. Rev. D, 79:064016 (2009)
    [89] B. Mashhoon, Phys. Rev. D, 31:290 (1985)
    [90] R. A. Konoplya and Z. Stuchlik, Phys. Lett. B, 771:597 (2017)

  • References

    [1] S. Fernando, Phys. Rev. D, 79:124026 (2009)
    [2] I. Sakalli, Int. J. Mod. Phys. A, 26:2263-2269 (2011); Int. J. Mod. Phys. A, 28:1392002 (2013)
    [3] D. Du, B. Wang, and R. Su, Phys. Rev. D, 70:064024 (2004)
    [4] C. Chirenti, Braz. J. Phys., 48(1):102 (2018)
    [5] K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel., 2:2 (1999), arXiv:gr-qc/9909058
    [6] H-P Nollert, Class. Quantum Grav., 16:R159 (1999)
    [7] E. Berti, V. Cardoso, and A. O. Starinets, Class. Quantum Grav., 26:163001 (2009)
    [8] A. Flachi and J. P. S. Lemos, Phys. Rev. D, 87:024034 (2013)
    [9] A. Nagar and L. Rezzolla, Class. Quantum Grav., 22:R167 (2005)
    [10] C. B. M. H. Chirenti and L. Rezzolla, Class. Quantum Grav., 24:4191 (2007)
    [11] B. Toshmatov, C. Bambi, B. Ahmedov, Z. Stuchlik, and J. Schee, Phys. Rev. D, 96:064028 (2017)
    [12] S. Aneesh, S. Bose, and S. Kar, Phys. Rev. D, 97:124004 (2018)
    [13] B. P. Abbott et al (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett., 116:061102 (2016)
    [14] B. P. Abbott et al (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett., 119:161101 (2017)
    [15] A. Krolak and M. Patil, Universe, 3:59 (2017)
    [16] N. Stergioulas, Living Rev Relativ., 1:8 (1998)
    [17] J. Abadie et al (LIGO Scientific Collaboration), Nat. Phys., 7:962 (2011)
    [18] M. Page, J. Qin, J. La Fontaine, C. Zhao, and D. Blair, Phys. Rev. D, 97:124060 (2018)
    [19] P. A. Gonzalez, J. Saavedra, and Y. Vasquez, Int. J. Mod. Phys. D, 21:125005 (2012)
    [20] S. Iyer and C. M. Will, Phys. Rev. D, 35:3621 (1987)
    [21] R. A. Konoplya, Phys. Rev. D, 68:024018 (2003)
    [22] R. Konoplya, Phys. Rev. D, 71:024038 (2005)
    [23] R. A. Konoplya and A. Zhidenko, Phys. Rev. Lett., 103:161101 (2009)
    [24] S. Fernando, Phys. Rev. D, 77:124005 (2008)
    [25] S. Fernando, Gen. Rel. Grav., 36:71 (2004)
    [26] S. Fernando, Int. J. Mod. Phys. D, 24:1550104 (2015)
    [27] N. Breton, T. Clark, and S. Fernando, Int. J. Mod. Phys. D, 26(10):1750112 (2017)
    [28] I. Sakalli, Mod. Phys. Lett. A, 28:1350109 (2013)
    [29] I. Sakalli and S. F. Mirekhtiary, Astrophys.Space Sci., 350:727 (2014)
    [30] I. Sakalli, Eur. Phys. J. C, 75(4):144 (2015)
    [31] B. S. Kandemir and U. Ertem, Annalen Der Physik, 529:1600330 (2017)
    [32] I. Sakalli and G. Tokgoz, Annalen Der Physik, 528:612 (2016)
    [33] A. vgn and K. Jusufi, Annals Phys. 395:138 (2018)
    [34] I. Sakalli, K. Jusufi, and A. vgn, arXiv:1803.10583[gr-qc]
    [35] K. Jusufi, I. Sakalli and A. vgn, Gen. Rel. Grav., 50(1):10 (2018)
    [36] P. A. Gonzalez, A. vgn, J. Saavedra, and Y. Vasquez, Gen. Rel. Grav., 50(6):62 (2018)
    [37] R. Becar, S. Lepe, and J. Saavedra, Phys.Rev. D, 75:084021 (2007)
    [38] J. Saavedra, Mod. Phys. Lett. A, 21:1601 (2006)
    [39] S. Lepe, and J. Saavedra, Phys. Lett. B, 617:174 (2005)
    [40] J. Crisostomo, Samuel Lepe and J. Saavedra, Class. Quant. Grav. 21:2801 (2004)
    [41] P. A. Gonzalez, E. Papantonopoulos, J. Saavedra, and Y.Vasquez, Phys. Rev. D, 95(6):064046 (2017)
    [42] M. Cruz, M. Gonzalez-Espinoza, J. Saavedra, and D. Vargas-Arancibia, Eur. Phys. J. C, 76(2):75 (2016)
    [43] X. Kuang and J. Wu, Phys. Lett. B, 770:117 (2017)
    [44] X. He, B. Wang, S. Wu, and C. Lin, Phys. Lett. B, 673:156 (2009)
    [45] X. Rao, B. Wang, and G. Yang, Phys. Lett. B, 649:472 (2007)
    [46] S. Chen, B. Wang and R. Su, Class. Quantum Grav., 23:7581 (2006)
    [47] X. He, Songbai-Chen, B. Wang, R. Cai, and C. Lin, Phys. Lett. B, 665:392 (2008)
    [48] X. He, B. Wang, and S. Chen, Phys. Rev. D, 79:084005 (2009)
    [49] B. Wang, C. Lin and E. Abdalla, Phys. Lett. B, 481:79 (2000)
    [50] B. Wang, C. Lin, and C. Molina, Phys. Rev. D, 70:064025 (2004)
    [51] A. Jansen, Eur. Phys. J. Plus, 132(12):546 (2017)
    [52] M. Halilsoy and A. Al-Badawi, IL Nuovo Cimento B, 113:761 (1998)
    [53] M. Halilsoy, Gen. Relativ. Gravit., 25(3):275 (1992)
    [54] M. Halilsoy and A. Al-Badawi, Class. Quantum Grav., 12:3013 (1995)
    [55] A. Ovgun, Int. J. Theor. Phys., 55(6):2919 (2016)
    [56] H. Reissner, Annalen der Physik (in German), 50:106 (1916)
    [57] G. Nordstrm, Verhandl. Koninkl. Ned. Akad. Wetenschap., Afdel. Natuurk., Amsterdam., 26:1201 (1918)
    [58] S. Chandrasekhar, The Mathematical Theory of Black Holes (New York:Oxford University Press, 1983)
    [59] R. M. Wald, General Relativity (The University of Chicago Press, Chicago and London, 1984)
    [60] S. Q. Wu and X. Cai, J. Math. Phys. (N.Y.), 44:1084 (2003)
    [61] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965)
    [62] H. S. Vieira, V. B. Bezerra, and G. V. Silva, Ann. Phys. (Amsterdam), 362:576 (2015)
    [63] K. Heun, Math. Ann., 33:161 (1888)
    [64] A. Ronveaux, Heun's differential equations:(New York:Oxford University Press, 1995)
    [65] R. S. Maier, The 192 Solutions of Heun Equation:(Preprint math CA/0408317, 2004)
    [66] P. P. Fiziev, J. Phys. A:Math. Theor., 43:035203 (2010)
    [67] I. Sakalli and M. Halilsoy, Phys. Rev. D, 69:124012 (2004)
    [68] A. Al-Badawi and I. Sakalli, J. Math. Phys., 49:052501 (2008)
    [69] T. Birkandan and M. Hortacsu, EPL, 119(2):20002 (2017)
    [70] I. Sakalli, Phys. Rev. D, 94:084040 (2016)
    [71] G. V. Kraniotis, 2016 Class. Quantum Grav., 33:225011 (2016)
    [72] Here, the computer package, Maple 2017 is used for solving the confluent Heun differential equation
    [73] H. S. Vieira and V. B. Bezerra, Ann. Phys. (NY), 373:28 (2016)
    [74] H. S. Vieira, J. P. Morais Graca, and V. B. Bezerra, Chin. Phys. C, 41:095102 (2017)
    [75] V. Frolov and I. Novikov, Black Hole Physics:Basic Concepts and New Developments. Fundamental Theories of Physics (Kluwer Academic, London, 1998)
    [76] P. P. Fiziev, Class. Quantum Grav., 27:135001 (2010)
    [77] S. A. Teukolsky, Phys. Rev. Lett., 29:1114 (1972); S. A. Teukolsky, Astrophys. J., 185:635 (1973); W. H. Press and S. A. Teukolsky, Astrophys. J., 185:649 (1973); S. A. Teukolsky and W. H. Press, Astrophys. J., 193:443 (1974)
    [78] P. P. Fiziev, Phys. Rev. D, 80:124001 (2009)
    [79] S. Chandrasekhar, Proc. R. Soc. London A, 348:39 (1976); Proc. R. Soc. London A, 372:475 (1980)
    [80] S. Hod, arXiv:gr-qc/0307060 (2003)
    [81] E. Berti, V. Cardoso, K. D. Kokkotas, and H. Onozawa, Phys. Rev. D, 68:124018 (2003)
    [82] L. Motl, Ad. Theor. Math. Phys., 6:1135 (2002)
    [83] R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys., 83:793 (2011)
    [84] H. J. Blome and B. Mashhoon, Phys. Lett. A, 110:231 (1984)
    [85] P. Musgrave and K. Lake, Class. Quant. Grav., 13:1885 (1996)
    [86] B. Toshmatov, A. Abdujabbarov, Z. Stuchlik, and B. Ahmedov, Phys. Rev. D, 91:083008 (2015)
    [87] B. Toshmatov, A. Abdujabbarov, J. Schee, and B. Ahmedov, Phys. Rev. D, 93:124017 (2016)
    [88] V. Cardoso, A. S. Miranda, E. Berti, H. Witeck, and V. T. Zanchin, Phys. Rev. D, 79:064016 (2009)
    [89] B. Mashhoon, Phys. Rev. D, 31:290 (1985)
    [90] R. A. Konoplya and Z. Stuchlik, Phys. Lett. B, 771:597 (2017)

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Ali Övgün, Izzet Sakalli and Joel Saavedra. Quasinormal modes of a Schwarzschild black hole immersed in an electromagnetic universe[J]. Chinese Physics C, 2018, 42(10): 105102. doi: 10.1088/1674-1137/42/10/105102
Ali Övgün, Izzet Sakalli and Joel Saavedra. Quasinormal modes of a Schwarzschild black hole immersed in an electromagnetic universe[J]. Chinese Physics C, 2018, 42(10): 105102.  doi: 10.1088/1674-1137/42/10/105102 shu
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Quasinormal modes of a Schwarzschild black hole immersed in an electromagnetic universe

  • 1. Instituto de Fí
  • 2. Physics Department, Arts and Sciences Faculty, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey
  • 3. School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA
  • 4.  Instituto de Fí
  • Received Date: 2018-05-16
  • Available Online: 2018-10-05

Abstract: We study the quasinormal modes (QNMs) of a Schwarzschild black hole immersed in an electromagnetic (EM) universe. The immersed Schwarzschild black hole (ISBH) originates from the metric of colliding EM waves with double polarization[Class. Quantum Grav. 12, 3013 (1995)]. The perturbation equations of the scalar fields for the ISBH geometry are written in the form of separable equations. We show that these equations can be transformed to the confluent Heun's equations, for which we are able to use known techniques to perform analytical quasinormal (QNM) analysis of the solutions. Furthermore, we employ numerical methods (Mashhoon and 6th-order Wentzel-Kramers-Brillouin (WKB)) to derive the QNMs. The results obtained are discussed and depicted with the appropriate plots.

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