Back
 JAMP  Vol.6 No.6 , June 2018
BTZ Quasinormal Frequencies as Poles of Green’s Function
Abstract: Based on the well known fact that the quasinormal frequencies are the poles of the frequency domain Green’s function we describe a method that allows us to calculate exactly the quasinormal frequencies of the Klein-Gordon field moving in the three-dimensional rotating Bañados-Teitelboim-Zanelli black hole. These quasinormal frequencies are already published and widely explored in several applications, but we use this example to expound the proposed method of computation. We think that the described procedure can be useful to calculate exactly the quasinormal frequencies of classical fields propagating in other backgrounds. Furthermore, we compare with previous results and discuss some related facts.
Cite this paper: López-Ortega, A. and Mata-Pacheco, D. (2018) BTZ Quasinormal Frequencies as Poles of Green’s Function. Journal of Applied Mathematics and Physics, 6, 1170-1178. doi: 10.4236/jamp.2018.66099.
References

[1]   Kokkotas, K.D. and Schmidt, B.G. (1999) Quasi-Normal Modes of Stars and Black Holes. Living Reviews in Relativity, 2, 2.
https://doi.org/10.12942/lrr-1999-2

[2]   Berti, E., Cardoso, V. and Starinets, A.O. (2009) Quasinormal Modes of Black Holes and Black Branes. Classical and Quantum Gravity, 26, Article ID: 163001.
https://doi.org/10.1088/0264-9381/26/16/163001

[3]   Horowitz, G.T. and Hubeny, V.E. (2000) Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium. Physical Review D, 62, Article ID: 024027.
https://doi.org/10.1103/PhysRevD.62.024027

[4]   Birmingham, D., Sachs, I. and Solodukhin, S.N. (2002) Conformal Field Theory Interpretation of Black Hole Quasi-Normal Modes. Physical Review Letters, 88, Article ID: 151301.

[5]   Cardoso, V. and Lemos, J.P.S. (2001) Quasinormal Modes of Schwarzschild Anti-De Sitter Black Holes: Electromagnetic and Gravitational Perturbations. Physical Review D, 64, Article ID: 084017.
https://doi.org/10.1103/PhysRevD.64.084017

[6]   Cardoso, V. and Lemos, J.P.S. (2001) Quasinormal Modes of Toroidal, Cylindrical and Planar Black Holes in Anti-De Sitter Space-Times. Classical and Quantum Gravity, 18, 5257.
https://doi.org/10.1088/0264-9381/18/23/319

[7]   Cardoso, V., Konoplya, R. and Lemos, J.P.S. (2003) Quasinormal Frequencies of Schwarzschild Black Holes in Anti-De Sitter Space-Times: A Complete Study on the Asymptotic Behavior. Physical Review D, 68, Article ID: 044024.
https://doi.org/10.1103/PhysRevD.68.044024

[8]   Berti, E. and Kokkotas, K.D. (2003) Quasinormal Modes of Reissner-Nordstrom-anti-de Sitter Black Holes: Scalar, Electromagnetic and Gravitational Perturbations. Physical Review D, 67, Article ID: 064020.
https://doi.org/10.1103/PhysRevD.67.064020

[9]   Lopez-Ortega, A. (2006) Electromagnetic Quasinormal Modes of D-Dimensional Black Holes. General Relativity and Gravitation, 38, 1747.
https://doi.org/10.1007/s10714-006-0358-2

[10]   Giammatteo, M. and Jing, J.L. (2005) Dirac Quasinormal Frequencies in Schwarzschild-AdS Space-Time. Physical Review D, 71, Article ID: 024007.
https://doi.org/10.1103/PhysRevD.71.024007

[11]   Jing, J.L. and Pan, Q.Y. (2005) Dirac Quasinormal Frequencies of Reissner-Nordstrom Black Hole in Anti-De Sitter Spacetime. Physical Review D, 71, Article ID: 124011.

[12]   Starinets, A.O. (2002) Quasinormal Modes of near Extremal Black Branes. Physical Review D, 66, Article ID: 124013.
https://doi.org/10.1103/PhysRevD.66.124013

[13]   Nunez, A. and Starinets, A.O. (2003) Ad/CFT Correspondence, Quasinormal Modes, and Thermal Correlators in N = 4 SYM. Physical Review D, 67, Article ID: 124013.
https://doi.org/10.1103/PhysRevD.67.124013

[14]   Kovtun, P.K. and Starinets, A.O. (2005) Quasinormal Modes and Holography. Physical Review D, 72, Article ID: 086009.
https://doi.org/10.1103/PhysRevD.72.086009

[15]   Musiri, S., Ness, S. and Siopsis, G. (2006) Perturbative Calculation of Quasi-Normal Modes of AdS Schwarzschild Black Holes. Physical Review D, 73, Article ID: 064001.
https://doi.org/10.1103/PhysRevD.73.064001

[16]   Miranda, A.S., Morgan, J. and Zanchin, V.T. (2008) Quasinormal Modes of Plane-Symmetric Black Holes According to the AdS/CFT Correspondence. Journal of High Energy Physics, 0811, 030.
https://doi.org/10.1088/1126-6708/2008/11/030

[17]   Cardoso, V. and Lemos, J.P.S. (2001) Scalar, Electromagnetic and Weyl Perturbations of BTZ Black Holes: Quasinormal Modes. Physical Review D, 63, Article ID: 124015.

[18]   Birmingham, D. (2001) Choptuik Scaling and Quasinormal Modes in the AdS/CFT Correspondence. Physical Review D, 64, Article ID: 064024.

[19]   Cordero, R., Lopez-Ortega, A. and Vega-Acevedo, I. (2012) Quasinormal Frequencies of Asymptotically Anti-De Sitter Black Holes in Two Dimensions. General Relativity and Gravitation, 44, 917-940.
https://doi.org/10.1007/s10714-011-1316-1

[20]   Gomez-Navarro, D.V. and Lopez-Ortega, A. (2017) Electromagnetic Quasinormal Modes of Five-Dimensional Topological Black Holes. Revista Mexicana de Física, 63, 541-548.

[21]   Aros, R., Martinez, C., Troncoso, R. and Zanelli, J. (2003) Quasinormal Modes for Massless Topological Black Holes. Physical Review D, 67, Article ID: 044014.
https://doi.org/10.1103/PhysRevD.67.044014

[22]   Birmingham, D. and Mokhtari, S. (2006) Exact Gravitational Quasinormal Frequencies of Topological Black Holes. Physical Review D, 74, Article ID: 084026.

[23]   Gupta, K.S., Juri, T. and Samsarov, A. (2017) Noncommutative Duality and Fermionic Quasinormal Modes of the BTZ Black Hole. Journal of High Energy Physics, 17, 107.
https://doi.org/10.1007/JHEP06(2017)107

[24]   Kandemir, B.S. and Ertem, ü. (2017) Quasinormal Modes of BTZ Black Hole and Hawking-Like Radiation in Graphene. Annalen der Physik, 529, Article ID: 1600330.
https://doi.org/10.1002/andp.201600330

[25]   Lopez-Ortega, A. (2008) Electromagnetic Quasinormal Modes of D-Dimensional Black Holes II. General Relativity and Gravitation, 40, 1379-1401.
https://doi.org/10.1007/s10714-007-0538-8

[26]   Lopez-Ortega, A. (2010) Quasinormal Frequencies of the Dirac Field in the Massless Topological Black Hole. Revista Mexicana de Física, 56, 44-53.

[27]   Lopez-Ortega, A. (2014) Electromagnetic Quasinormal Modes of an Asymptotically Lifshitz Black Hole. General Relativity and Gravitation, 46, 1756.
https://doi.org/10.1007/s10714-014-1756-5

[28]   Oliva, J. and Troncoso, R. (2010) Exact Quasinormal Modes for a Special Class of Black Holes. Physical Review D, 82, Article ID: 027502.
https://doi.org/10.1103/PhysRevD.82.027502

[29]   Banados, M., Teitelboim, C. and Zanelli, J. (1992) The Black Hole in Three-Dimensional Space-Time. Physical Review Letters, 69, 1849-1851.
https://doi.org/10.1103/PhysRevLett.69.1849

[30]   Banados, M., Henneaux, M., Teitelboim, C. and Zanelli, J. (1993) Geometry of the (2+1) Black Hole. Physical Review D, 48, 1506-1525.
https://doi.org/10.1103/PhysRevD.48.1506

[31]   Birmingham, D., Sachs, I. and Sen, S. (2001) Exact Results for the BTZ Black Hole. International Journal of Modern Physics D, 10, 833-857.
https://doi.org/10.1142/S0218271801001207

[32]   Leaver, E.W. (1986) Spectral Decomposition of the Perturbation Response of the Schwarzschild Geometry. Physical Review D, 34, 384-408.
https://doi.org/10.1103/PhysRevD.34.384

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

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

[35]   Olver, F.W.J., Lozier, D.W., Boisvert, R.F. and Clark, C.W. (2010) NIST Handbook of Mathematical Functions. Cambridge University Press, New York.

 
 
Top