Quasinormal frequencies of the Dirac field in the massless topological black hole

A. López–Ortega

Centro de Investigación en Ciencia Aplicada y Tecnología Avanzada, Unidad Legaria, Instituto Politécnico Nacional, Calzada Legaria # 694. Colonia Irrigación, Miguel Hidalgo México, D.F., 11500 México.

Recibido el 1 de septiembre de 2009
Aceptado el 1 de diciembre de 2009

Abstract

Motivated by the recent computations of the quasinormal frequencies of higher dimensional black holes, we exactly calculate the quasinormal frequencies of the Dirac field, propagating in D–dimensional (D > 4) massless topological black hole. From the exact values of the quasinormal frequencies for the fermion and boson fields we discuss whether the recently proposed bound on the relaxation time of a perturbed thermodynamical system is satisfied in the D–dimensional massless topological black hole. Also we study the consequences of these results.

Keywords: Quasinormal modes; massless topological black hole; Dirac field; relaxation time.

Resumen

Motivados por el cálculo de las frecuencias cuasinormales de agujeros negros cuyo número de dimensiones D es mayor o igual a cuatro, en el presente artículo calculamos exactamente las frecuencias cuasinormales del campo de Dirac moviéndose en el agujero negro topológico de masa cero con D > 4. Usando los valores exactos de las frecuencias cuasinormales para los fermiones y bosones, discutimos si el límite, recientemente propuesto, sobre el tiempo de relajamiento de un sistema termodinámico perturbado se satisface en el agujero negro topológico de masa cero con D > 4. Adicionalmente estudiamos algunas consecuencias de estos resultados.

Descriptores: Modos cuasinormales; agujero negro topológico de masa cero; campo de Dirac; tiempo de relajamiento.

PACS: 04.70.Bw; 04.50.Gh; 04.70.Dy

DESCARGAR ARTÍCULO EN FORMATO PDF

Acknowledgments

I thank Dr. C. E. Mora Ley and A. Tellez Felipe for their interest in this paper. This work was supported by CONACYT México, SNI México, EDI–IPN, COFAA–IPN, and Research Projects SIP–20090952 and SIP–20091344.

References

1. K.D. Kokkotas and B.G. Schmidt, Living Rev. Rel. 2 (1999) 2. [arXiv:gr–qc/9909058].        [ Links ]

2. E. Berti, V. Cardoso, and A.O. Starinets, Class. Quant. Grav. 26 (2009) 163001. [arXiv:0905.2975 [gr–qc]].        [ Links ]

3. H.P. Nollert, Class. Quant. Grav. 16 (1999) R159.        [ Links ]

4. G.T. Horowitz and V.E. Hubeny, Phys. Rev. D 62 (2000) 024027. [arXiv:hep–th/9909056].        [ Links ]

5. D. Birmingham, I. Sachs, and S.N. Solodukhin, Phys. Rev. Lett. 88 (2002) 151301. [arXiv:hep–th/0112055].        [ Links ]

6. S. Hod, Phys. Rev. Lett. 81 (1998) 4293. [arXiv:gr–qc/9812002].        [ Links ]

7. M. Maggiore, Phys. Rev. Lett. 100 (2008) 141301. [arXiv:0711.3145[gr–qc]].        [ Links ]

8. F. Denef, S.A. Hartnoll, and S. Sachdev, "Quantum oscillations and black hole ringing," arXiv:0908.1788 [hep–th].        [ Links ]

9. F. Denef, S.A. Hartnoll, and S. Sachdev, "Black hole determinants and quasinormal modes," arXiv:0908.2657 [hep–th].        [ Links ]

10. M. Casals, S.R. Dolan, A.C. Ottewill, and B. Wardell, Phys. Rev. D 79 (2009) 124043. [arXiv:0903.0395 [gr–qc]].        [ Links ]

11. V. Cardoso and J.P.S. Lemos, Phys. Rev. D 63 (2001) 124015. [arXiv:gr–qc/0101052].        [ Links ]

12. D. Birmingham, Phys. Rev. D 64 (2001) 064024. [arXiv:hep–th/0101194].        [ Links ]

13. J. Crisostomo, S. Lepe, and J. Saavedra, Class. Quant. Grav. 21 (2004) 2801. [arXiv:hep–th/0402048].        [ Links ]

14. S. Fernando, Gen. Rel. Grav. 36 (2004) 71. [arXiv:hep–th/0306214].        [ Links ]

15. A. López–Ortega, Gen. Rel. Grav. 37 (2005) 167.        [ Links ]

16. S. Fernando, Phys. Rev. D 77 (2008) 124005. [arXiv:0802.3321 [hep–th]].        [ Links ]

17. S. Fernando, Phys. Rev. D 79 (2009) 124026. [arXiv:0903.0088 [hep–th]].        [ Links ]

18. R. Becar, S. Lepe, and J. Saavedra, Phys. Rev. D 75 (2007) 084021. [arXiv:gr–qc/0701099].        [ Links ]

19. A. Lopez–Ortega, Int. J. Mod. Phys. D 9 (2009) 1441. [arXiv:0905.0073 [gr–qc]].        [ Links ]

20. D.P. Du, B. Wang, and R.K. Su, Phys. Rev. D 70 (2004) 064024. [arXiv:hep–th/0404047].        [ Links ]

21. J. Natario and R. Schiappa, Adv. Theor. Math. Phys. 8 (2004) 1001. [arXiv:hep–th/0411267].        [ Links ]

22. T.R. Choudhury and T. Padmanabhan, Phys. Rev. D 69 (2004) 064033. [arXiv:gr–qc/0311064].        [ Links ]

23. A. López–Ortega, Gen. Rel. Grav. 38 (2006) 1565. [arXiv:gr–qc/0605027].        [ Links ]

24. A. Zelnikov, JHEP 0807 (2008) 010. [arXiv:0805.4031 [hep–th]].        [ Links ]

25. A. López–Ortega, Gen. Rel. Grav. 38 (2006) 743. [arXiv:gr–qc/0605022].        [ Links ]

26. A. López–Ortega, Gen. Rel. Grav. 39 (2007) 1011. [arXiv:0704.2468 [gr–qc]].        [ Links ]

27. L.H. Liu and B. Wang, Phys. Rev. D 78 (2008) 064001. [arXiv:0803.0455 [hep–th]].        [ Links ]

28. L. Vanzo and S. Zerbini, Phys. Rev. D 70 (2004) 044030. [arXiv:hep–th/0402103].        [ Links ]

29. J. Saavedra, Mod. Phys. Lett. A 21 (2006) 1601. [arXiv:gr–qc/0508040].        [ Links ]

30. L. Vanzo, Phys. Rev. D 56 (1997) 6475. [arXiv:gr–qc/9705004].        [ Links ]

31. D.R. Brill, J. Louko, and P. Peldan, Phys. Rev. D 56 (1997) 3600. [arXiv:gr–qc/9705012].        [ Links ]

32. R.B. Mann, "Topological black holes: Outside looking in" in Internal Structure of Black Holes and Spacetime Singularities, ed. L. Burko and A. Ori. (Technion University Press. Haifa, 1998). [arXiv:gr–qc/9709039].        [ Links ]

33. R.B. Mann, Class. Quant. Grav. 14 (1997) 2927. [arXiv:gr–qc/9705007].        [ Links ]

34. S. Aminneborg, I. Bengtsson, S. Holst, and P. Peldan, Class. Quant. Grav. 13 (1996) 2707. [arXiv:gr–qc/9604005].        [ Links ]

35. D. Birmingham, Class. Quant. Grav. 16 (1999) 1197. [arXiv:hep–th/9808032].        [ Links ]

36. J.P.S. Lemos, Phys. Lett. B 353 (1995) 46. [arXiv:gr–qc/9404041].        [ Links ]

37. D. Birmingham and S. Mokhtari, Phys. Rev. D 74 (2006) 084026. [arXiv:hep–th/0609028].        [ Links ]

38. R. Aros, C. Martinez, R. Troncoso, and J. Zanelli, Phys. Rev. D 67 (2003) 044014. [arXiv:hep–th/0211024].        [ Links ]

39. A. Lopez–Ortega, Gen. Rel. Grav.40 (2008) 1379. [arXiv:0706.2933 [gr–qc]].        [ Links ]

40. G. Gibbons and S.A. Hartnoll, Phys. Rev. D 66 (2002) 064024. [arXiv:hep–th/0206202].        [ Links ]

41. D. Birmingham and S. Mokhtari, Phys. Rev. D 76 (2007) 124039. [arXiv: 0709.2388 [hep–th]].        [ Links ]

42. J.S.F. Chan and R.B. Mann, Phys. Rev. D 59 (1999 064025).        [ Links ]

43. G. Koutsoumbas, E. Papantonopoulos, and G. Siopsis, Class. Quant. Grav. 26 (2009) 105004. [arXiv:0806.1452 [hep–th]].        [ Links ]

44. G. Koutsoumbas, S. Musiri, E. Papantonopoulos, and G. Siopsis, JHEP 0610 (2006) 006. [arXiv:hep–th/0606096].        [ Links ]

45. A.S. Miranda, J. Morgan, and V.T. Zanchin, JHEP 0811 (2008) 030. [arXiv:0809.0297 [hep–th]].        [ Links ]

46. A.S. Miranda and V.T. Zanchin, Phys. Rev. D 73 (2006) 064034. [arXiv:gr–qc/0510066].        [ Links ]

47. J. Morgan, V. Cardoso, A.S. Miranda, C. Molina, and V.T. Zanchin, "Gravitational quasinormal modes of AdS black branes in d spacetime dimensions," arXiv:0907.5011 [hep–th].        [ Links ]

48. B. Wang, E. Abdalla and R.B. Mann, Phys. Rev. D 65 (2002) 084006. [arXiv:hep–th/0107243].        [ Links ]

49. W.G. Unruh, Phys. Rev. D 10 (1974) 3194.        [ Links ]

50. W.G. Unruh, Phys. Rev. Lett. 31 (1973) 1265.        [ Links ]

51. M. Martellini and A. Treves, Phys. Rev. D 15 (1977) 3060.        [ Links ]

52. B.R. Iyer and A. Kumar, Phys. Rev. D 18 (1978) 4799.        [ Links ]

53. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, Oxford, 1983).        [ Links ]

54. H.T. Cho, A.S. Cornell, J. Doukas, and W. Naylor, Phys. Rev. D 75 (2007) 104005. [arXiv:hep–th/0701193].        [ Links ]

55. H.T. Cho, A.S. Cornell, J. Doukas, and W. Naylor, Phys. Rev. D 77 (2008) 041502. [arXiv:0710.5267 [hep–th]].        [ Links ]

56. A. Lopez–Ortega, Lat. Am. J. Phys. Educ. 3 (2009) 578. [arXiv:0906.2754 [gr–qc]].        [ Links ]

57. S.K. Chakrabarti, Eur. Phys. J. C 61 (2009) 477. [arXiv:0809.1004 [gr–qc]].        [ Links ]

58. S. Hod, Phys. Rev. D 75 (2007) 064013. [arXiv:gr–qc/0611004].        [ Links ]

59. H. Kodama and A. Ishibashi, Prog. Theor. Phys. 110 (2003) 701. [arXiv:hep–th/0305147].        [ Links ]

60. G.W. Gibbons and A.R. Steif, Phys. Lett. B 314 (1993) 13. [arXiv:gr–qc/9305018].        [ Links ]

61. S.R. Das, G.W. Gibbons, and S.D. Mathur, Phys. Rev. Lett. 78 (1997) 417. [arXiv:hep–th/9609052].        [ Links ]

62. G.W. Gibbons, M. Rogatko, and A. Szyplowska, Phys. Rev. D 11 (2008) 064024. [arXiv:0802.3259 [hep–th]].        [ Links ]

63. G.W. Gibbons and M. Rogatko, Phys. Rev. D 77 (2008) 044034. [arXiv:0801.3130 [hep–th]].        [ Links ]

64. I.I. Cotaescu, Int. J. Mod. Phys. A 19 (2004) 2217. [arXiv:gr–qc/0306127].        [ Links ]

65. I.I. Cotaescu, Mod. Phys. Lett. A 13 (1998) 2991. [arXiv:gr–qc/9808030].        [ Links ]

66. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Graphs, and Mathematical Table, (Dover Publications, New York, 1965).        [ Links ]

67. Z.X. Wang and D.R. Guo, Special Functions, (World Scientific Publishing, Singapore, 1989).        [ Links ]

68. R. Aros, C. Martinez, R. Troncoso, and J. Zanelli, JHEP 0205 (2002) 020. [arXiv:hep–th/0204029].        [ Links ]

69. C. Bär, Soc. Math. Fce., Semin. & Cong. 4 (2000) 17.        [ Links ]

70. T. Friedrich, Dirac Operators in Riemannian Geometry (American Mathematical Society, 2000).        [ Links ]

71. R. Camporesi and E. Pedon, Colloq. Math. 87 (2001) 245.        [ Links ]

72. R. Camporesi and A. Higuchi, J. Geom. Phys. 20 (1996) 1. [arXiv:gr–qc/9505009].        [ Links ]

73. N. Ginoux, The Dirac Spectrum, Lecture Notes in Mathematics (Springer, 2009) Vol. 1976.        [ Links ]

74. J. Seade and B. Steer, Topology 26 (1987) 79.        [ Links ]

75. H. Baum, Math. Z. 206 (1991) 409.        [ Links ]

76. S. Hod, Class. Quant. Grav. 24 (2007) 4235. [arXiv:0705.2306 [gr–qc]].        [ Links ]

77. A. Pesci, Int. J. Mod. Phys. D 18 (2009) 831. [arXiv:0807.0300 [gr–qc]].        [ Links ]

78. K. Ropotenko, "Universal Bound on Dynamical Relaxation Time from Condition for Relaxing Quantity to be Classical," arXiv:0705.3625 [gr–qc].        [ Links ]

79. S. Hod, Phys. Rev. D 78 (2008) 084035. [arXiv:0811.3806 [gr–qc]].        [ Links ]

80. R. Dutt, A. Khare, and U.P. Sukhatme, Am. J. Phys. 56 (1988) 163.        [ Links ]

81. A. Lopez–Ortega, Phys. Lett. B 682 (2009) 85. [arXiv:0910.5779 [gr–qc]].        [ Links ] ]]> 1 1999 2 2 26 3 1999 16 R159 4 2000 62 5 2002 88 6 1998 81 4293 7 2008 100 8 9 10 2009 79 11 2001 63 12 2001 64 13 2004 21 2801 14 2004 36 71 15 2005 37 167 16 2008 77 17 2009 79 18 2007 75 19 2009 9 1441 20 2004 70 21 2004 8 1001 22 2004 69 23 2006 38 1565 24 2008 0807 010 25 2006 38 743[arXiv:gr-qc/0605022] 26 2007 39 1011 27 2008 78 28 2004 70 29 2006 21 1601 30 1997 56 6475 31 1997 56 3600[arXiv:gr-qc/9705012] 32 1998 33 1997 14 2927 34 1996 13 2707 35 1999 16 1197 36 1995 353 46 37 2006 74 38 2003 67 39 2008 40 1379 40 2002 66 41 2007 76 42 1999 59 43 2009 26 44 2006 0610 006 45 2008 0811 030 46 2006 73 47 48 2002 65 49 1974 10 3194 50 1973 31 1265 51 1977 15 3060 52 1978 18 4799 53 1983 54 2007 75 55 2008 77 56 2009 3 578 57 2009 61 477 58 2007 75 59 2003 110 701 60 1993 314 13 61 1997 78 417 62 63 2008 77 64 2004 19 2217 65 1998 13 2991 66 1965 67 1989 68 2002 0205 020 69 2000 4 17 70 2000 71 2001 87 245 72 1996 20 1 73 2009 1976 74 1987 26 79 75 1991 206 409 76 2007 24 4235[arXiv:0705.2306 [gr-qc]] 77 2009 18 831[arXiv:0807.0300 [gr-qc]] 78 79 2008 78 80 1988 56 163 81 2009 682 85