黑洞时空中的似正规模研究
|
摘要
黑洞的似正规模一直是黑洞物理学研究的热点,它是指黑洞周围的物质场扰动波包在黑洞时空中演化的中期衰减行为,作为黑洞的“特征声音”被认为是直接探测黑洞存在的有力证据,将在以后几年的引力波实验中被观测到.对各种黑洞似正规模的深入研究和认识,有利于我们通过引力波实验探测黑洞的存在,能帮助我们在观测上进一步认识黑洞的特性.除此以外, AdS黑洞时空的似正规模能够在对偶共形场论中找到很好的解释,被认为是AdS/CFT对应性的探针.更有一些研究试图通过dS时空的似正规模和共形场论的关系来进一步讨论dS/CFT对应性.另外一些研究将平坦空间中似正规模频谱的实部与圈量子理论中的Barbero-Immirzi(BI)系数联系起来,揭示黑洞经典振动和其量子效应之间的联系.最近黑洞的似正规模还被用于提示黑洞相变.本文致力于通过数值方法研究某些特殊黑洞外部场扰动行为,以分析似正规模与背景黑洞特征参数的关系；分析动力学稳定性与热力学稳定性之间的联系；考虑吸积暗能量时周围扰动场的影响,提出观测暗能量的新方法；以及探讨黑洞相变在似正规模中的印迹等问题.
本文分为三部分：第一部分是绪论,介绍似正规模的概念、研究意义和现状,并介绍几种常用的计算似正规模的数值方法；第二部分是本文的核心,采用数值方法研究了某些特殊黑洞的似正规模,包括荷电扁视界Kaluza-Klein黑洞(SqKK黑洞)时空的似正规模及其热力学稳定性、Godel宇宙中荷电扁视界Kaluza-Klein黑洞(SqKK-Godel黑洞)时空中的似正规模现象、动态黑洞的似正规模与暗能量观测的研究,以及黑洞相变及其在似正规模中的印迹的研究；第三部分是本文的总结与展望。
具体来说,在第一章中,简单介绍了似正规模的概念、研究意义和现状,并介绍了常用的几种数值方法。
在第二章中,利用连续分数方法研究了荷电扁视界Kaluza-Klein黑洞(SqKK黑洞)时空无质量标量场扰动的似正规模.发现似正规模频谱能够反映比热容的不连续点。这说明反应动力学扰动的似正规模频谱与反应热力学稳定性的Davies点之间的确存在关联.
在第三章中,研究了Godel时空中荷电压扁视界Klauza-Klein黑洞(SqKK-Godel黑洞)的似正规模.发现由于宇宙的整体旋转,似正规模频谱呈现与以往各向同性、均匀宇宙不同的性质.此外我们也考察了视界被压扁的程度ρ0以及额外维度对似正规模的影响.
在第四章中,考虑被暗能量包裹的黑洞时空,发现由于黑洞吸积暗能量,黑洞自身的质量随时间发生改变.该动态时空的似正规模由于携带有暗能量状态参数w的信息,能帮助我们判断暗能量状态参数是大于还是小于w=-1.随着引力波探测手段的成熟,这将成为我们分析暗能量模型的有力工具.
在第五章中,研究了d维AdS荷电黑洞和AdS孤子外部荷电标量场的扰动.发现与纯标量场扰动时衰减的似正规模不同,荷电标量场的扰动在晚期出现增涨而非衰减的波形,这说明原来荷电AdS黑洞和AdS孤子背景开始变得不稳定.这一现象发生的时刻正好和发现的荷电标量场在该时空背景上凝聚的发生时刻吻合,反映了从原来的时空背景变成带荷电标量场毛的新时空的相变.但是在荷磁AdS孤子时空中我们却发现了持续振荡的波,这说明荷磁AdS孤子时空不同于荷电AdS孤子,在该时空不会发生标量荷凝聚,其似正规模上没有根本变化.
在第六章中我们对本文的工作进行了总结,并对后续研究进行了展望.
Quasinormal mode (QNM) of black holes has been an intriguing subject of discussions for the last three decades. It is believed that QNM is a characteristic sound of black holes which could lead to the direct identification of the black hole existence through gravitational wave observations to be realized in the near future. In addition to its potential astrophysical interest, theoretically QNM was believed as a tool to learn more about black hole and was even argued as a testing ground for fundamental physics. It was found that the QNMs of anti-de Sitter (AdS) black holes have direct interpretation in terms of the dual conformal field theory (CFT). This could serve as a support of the AdS/CFT correspondence. Attempts of using QNMs to investigate the dS/CFT correspondence have also been proposed. It was argued that QNMs might reflect the possible connection between the classical vi-brations of a black hole spacetime and various quantum aspects by relating the real part of the QNM frequencies to the Barbero-Immirzi (BI) parameter, which was introduced by hand in order that loop quantum gravity reproduces correctly the black hole entropy. Recently the quasinormal frequencies was used in studing the black hole phase transition. This thesis is devoted to investigating the perturbation around some special black holes through numerical methods, and trying to find the relationship between thermodynamical and dynamical properties of black holes, to distinguish the dark energy in the time-dependant spacetime, and to find the phase transition.
The thesis consists of three parts. The first part is the brief introduction of QNM'definition, importance and status, and several numerical methods are given. The second part is about the studying of the perturbations around some special black holes, including the QNM of charged Kaluza-Klein black hole with squashed horizon and its thermodynamical stability, the QNM of charged Kaluza-Klein black hole with squashed horizon in Godel universe, the QNM of black holes absorbing dark energy, and the signature of the black hole phase transition in QNM. The summery of the thesis and expectation of future research are given in the last part.
In Section 1, a brief introduction of QNM'definition, importance and status, and several numerical methods mainly used in studying the QNM are given.
In Section 2, the massless scalar perturbation in the background of the charged Kaluza-Klein black holes with squashed horizons was studied by continued fraction method. We find that the position of infinite discontinuities of the heat capacities can be reflected in quasinormal spectrum. This shows the possible non-trivial relation between the thermodynamical and dynamical properties of black holes.
In Section 3, the QNM of scalar perturbation in the background of 5 dimensional charged Kaluza-Klein black holes with squashed horizons immersed in the Godel universe was studied. Besides the influence due to the compactness of the extra dimension, we disclose the cosmological rotational effect in the wave dynamics. The wave behavior affected by the Goodel parameter provides an interesting insight into the Godel universe.
In Section 4, the perturbations of black holes absorbing dark energy was studied. Due to the accretion of dark energy, the black hole mass changes. We observe distinct perturbation behaviors for absorption of different forms of dark energy onto the black holes. This provides the possibility of extracting information whether dark energy lies above or below the cosmological constant boundary w=-1.
In Section 5, the perturbation of the scalar field interacting with the Maxwell field in the background of d-dimensional charged AdS black hole and AdS soliton was studied. Different from the single classical field perturbation, which always has the decay mode in the black hole background, we observe the possible growing mode when the perturbation of the scalar field strongly couples to the Maxwell field. The sudden change of the perturbation to growing mode is also observed in the AdS soliton with electric potential. However in the magnetic charged AdS soliton background, we observe the consistent perturbation behavior when the interaction between scalar field and Maxwell field is considered.
In Section 6, we give a summary and the expection of future research of the thesis.
本文分为三部分：第一部分是绪论,介绍似正规模的概念、研究意义和现状,并介绍几种常用的计算似正规模的数值方法；第二部分是本文的核心,采用数值方法研究了某些特殊黑洞的似正规模,包括荷电扁视界Kaluza-Klein黑洞(SqKK黑洞)时空的似正规模及其热力学稳定性、Godel宇宙中荷电扁视界Kaluza-Klein黑洞(SqKK-Godel黑洞)时空中的似正规模现象、动态黑洞的似正规模与暗能量观测的研究,以及黑洞相变及其在似正规模中的印迹的研究；第三部分是本文的总结与展望。
具体来说,在第一章中,简单介绍了似正规模的概念、研究意义和现状,并介绍了常用的几种数值方法。
在第二章中,利用连续分数方法研究了荷电扁视界Kaluza-Klein黑洞(SqKK黑洞)时空无质量标量场扰动的似正规模.发现似正规模频谱能够反映比热容的不连续点。这说明反应动力学扰动的似正规模频谱与反应热力学稳定性的Davies点之间的确存在关联.
在第三章中,研究了Godel时空中荷电压扁视界Klauza-Klein黑洞(SqKK-Godel黑洞)的似正规模.发现由于宇宙的整体旋转,似正规模频谱呈现与以往各向同性、均匀宇宙不同的性质.此外我们也考察了视界被压扁的程度ρ0以及额外维度对似正规模的影响.
在第四章中,考虑被暗能量包裹的黑洞时空,发现由于黑洞吸积暗能量,黑洞自身的质量随时间发生改变.该动态时空的似正规模由于携带有暗能量状态参数w的信息,能帮助我们判断暗能量状态参数是大于还是小于w=-1.随着引力波探测手段的成熟,这将成为我们分析暗能量模型的有力工具.
在第五章中,研究了d维AdS荷电黑洞和AdS孤子外部荷电标量场的扰动.发现与纯标量场扰动时衰减的似正规模不同,荷电标量场的扰动在晚期出现增涨而非衰减的波形,这说明原来荷电AdS黑洞和AdS孤子背景开始变得不稳定.这一现象发生的时刻正好和发现的荷电标量场在该时空背景上凝聚的发生时刻吻合,反映了从原来的时空背景变成带荷电标量场毛的新时空的相变.但是在荷磁AdS孤子时空中我们却发现了持续振荡的波,这说明荷磁AdS孤子时空不同于荷电AdS孤子,在该时空不会发生标量荷凝聚,其似正规模上没有根本变化.
在第六章中我们对本文的工作进行了总结,并对后续研究进行了展望.
Quasinormal mode (QNM) of black holes has been an intriguing subject of discussions for the last three decades. It is believed that QNM is a characteristic sound of black holes which could lead to the direct identification of the black hole existence through gravitational wave observations to be realized in the near future. In addition to its potential astrophysical interest, theoretically QNM was believed as a tool to learn more about black hole and was even argued as a testing ground for fundamental physics. It was found that the QNMs of anti-de Sitter (AdS) black holes have direct interpretation in terms of the dual conformal field theory (CFT). This could serve as a support of the AdS/CFT correspondence. Attempts of using QNMs to investigate the dS/CFT correspondence have also been proposed. It was argued that QNMs might reflect the possible connection between the classical vi-brations of a black hole spacetime and various quantum aspects by relating the real part of the QNM frequencies to the Barbero-Immirzi (BI) parameter, which was introduced by hand in order that loop quantum gravity reproduces correctly the black hole entropy. Recently the quasinormal frequencies was used in studing the black hole phase transition. This thesis is devoted to investigating the perturbation around some special black holes through numerical methods, and trying to find the relationship between thermodynamical and dynamical properties of black holes, to distinguish the dark energy in the time-dependant spacetime, and to find the phase transition.
The thesis consists of three parts. The first part is the brief introduction of QNM'definition, importance and status, and several numerical methods are given. The second part is about the studying of the perturbations around some special black holes, including the QNM of charged Kaluza-Klein black hole with squashed horizon and its thermodynamical stability, the QNM of charged Kaluza-Klein black hole with squashed horizon in Godel universe, the QNM of black holes absorbing dark energy, and the signature of the black hole phase transition in QNM. The summery of the thesis and expectation of future research are given in the last part.
In Section 1, a brief introduction of QNM'definition, importance and status, and several numerical methods mainly used in studying the QNM are given.
In Section 2, the massless scalar perturbation in the background of the charged Kaluza-Klein black holes with squashed horizons was studied by continued fraction method. We find that the position of infinite discontinuities of the heat capacities can be reflected in quasinormal spectrum. This shows the possible non-trivial relation between the thermodynamical and dynamical properties of black holes.
In Section 3, the QNM of scalar perturbation in the background of 5 dimensional charged Kaluza-Klein black holes with squashed horizons immersed in the Godel universe was studied. Besides the influence due to the compactness of the extra dimension, we disclose the cosmological rotational effect in the wave dynamics. The wave behavior affected by the Goodel parameter provides an interesting insight into the Godel universe.
In Section 4, the perturbations of black holes absorbing dark energy was studied. Due to the accretion of dark energy, the black hole mass changes. We observe distinct perturbation behaviors for absorption of different forms of dark energy onto the black holes. This provides the possibility of extracting information whether dark energy lies above or below the cosmological constant boundary w=-1.
In Section 5, the perturbation of the scalar field interacting with the Maxwell field in the background of d-dimensional charged AdS black hole and AdS soliton was studied. Different from the single classical field perturbation, which always has the decay mode in the black hole background, we observe the possible growing mode when the perturbation of the scalar field strongly couples to the Maxwell field. The sudden change of the perturbation to growing mode is also observed in the AdS soliton with electric potential. However in the magnetic charged AdS soliton background, we observe the consistent perturbation behavior when the interaction between scalar field and Maxwell field is considered.
In Section 6, we give a summary and the expection of future research of the thesis.
引文
[I]T. Regge and J. A. Wheeler, Phys. Rev.,1957,108(4):1063-1069.
[2]C. V. Vishveshwara, Phys. Rev.D,1970,1:2870-2879.
[3]F. J. Zerilli, Phys. Rev. D,1970,1:2870-2879.
[4]S. Teukolsky, Phys. Rev.Lett,1972,29(16):1114-1118.
[5]S. Teukolsky, AstroPhysical Journal,1973,185:635-647.
[6]S. Chandrasekhar, Oxford University press,1983:135-204.
[7]V. P. Frolov and I. D. Novikov, edited by ALWYN VAN FER MERWE, University of Denver,USA:87-149.
[8]H. P. Nollert, Phys. Rev. D,1993,47:5253-5258.
[9]O. Dreyer, Phys. Rev. Lett,2003,90:081301.
[10]S. Chandrasekhar and S. Detweiler, Royal Society of London Proceedings Series A, 1975,344:441.
[11]S. Chandrasekhar and S. Detweiler, Proc. R. Soc. Lond. A,1984,392:1.
[12]S. Chandrasekhar, Proc. R. Soc. Lond. A,1984,392:1.
[13]D. L. Gunter, Phil. Trans. Roy. Soc. Lond. A,1980,296: 497.
[14]E. W. Leaver, Phys. Rev. D.1990,41:2986; N. Andersson, Royal Society of London Proceedings Series A.1993.442:427; H. Onozawa, T. Mishima, T. Okamura and H. Ishihara, Phys. Rev. D,1996,53:7033; N. Andersson and H. Onozawa, Phys. Rev. D,1996,54:7470; E. Berti, gr-qc/0411025; E. Berti and K. D. Kokkotas, Phys. Rev. D,2003,68:044027.
[15]D. N. Page, Phys. Rev. D,1976,14:1509.
[16]E. Berti and K. D. Kokkotas, Phys. Rev. D,2005,71:124008.
[17]J. L. Jing and Q. Y. Pan, Nucl. Phys. B,2005,728:109.
[18]S. Detweiler, Astrophys. J,1980,239:292; S. Hod and U. Keshet, Class. Quant. Grav,2005,22:L71; U. Keshet and S. Hod, Phys. Rev. D,2007,76:061501; U. Keshet and A. Neitzke, Phys. Rev. D,2008,78:044006; H. C. Kao and D. Tomino, Phys. Rev. D,2008,77:127503.
[19]J. S. F. Chan and R. B. Mann, Phys. Rev. D,1997,55:7546.
[20]G. T. Horowitz and V. E. Hubeny, Phys. Rev. D,2000,62:024027.
[21]B. Wang, C. Y. Lin and E. Abdalla, Phys. Lett. B,2000,481:79.
[22]R. A. Konoplya, Phys. Rev. D,2002,66:084007.
[23]J. L. Jing and Q. Y. Pan, Phys. Rev. D,2005,71:124011; Y. Zhang and J. L. Jing, Int. J. Mod. Phys. D,2006,15:905.
[24]R. A. Konoplya and A. Zhidenko, Phys. Rev. D,2008,78:104017.
[25]E. Berti, V. Cardoso and P. Pani,0903.5311.
[26]J. Natario and R. Schiappa, Adv. Theor. Math. Phys,2004,8:1001; J. D. Brown, Singapore, Singapore:World Scientific,1998,152p.
[27]F. Mellor and I. Moss, Phys. Rev. D,1990,41:403.
[28]P. R. Brady, C. M. Chambers, W. G. Laarakkers and E. Poisson, Phys. Rev. D,1999, 60:064003.
[29]A. Zhidenko, Class. Quant. Grav,2004,21:273:V. Cardoso and J. P. S. Lemos, Phys. Rev. D,2003,67:084020.
[30]B. R. Iyer, S. Iyer and C. V. Vishveshwara, Class. Quant. Grav,1989,6:1627.
[31]R. G. Daghigh, G. Kunstatter and J. Ziprick, Class. Quant. Grav,2007,24:1981.
[32]S. B. Chen and J. L. Jing, Class. Quant. Grav,2005,22:4651:Y. Zhang, Y. X. Gui and F. Li, Gen. Rel. Grav,2007,39:1003; C. Ma, Y. Gui, W. Wang and F. Wang, Central Eur. J. Phys,2008,6:194:Y. Zhang and Y. X. Gui, Class. Quant. Grav, 2006,23:6141; Y. Zhang, Y. X. Gui and F. Yu, Chin. Phys. Lett,2009,26:030401; N. Varghese and V. C. Kuriakose,0802.1397.
[33]C. Gundlach, R. H. Price and J. Pullin, Phys. Rev. D,1994,49:883-889.
[34]C. Gundlach, R. H. Price and J. Pullin, Phys. Rev. D,1994,49:890-899.
[35]G. Z. Wenzel, Phys.,1926,38:518.
[36]H. M. Z. Kramers, Phys.,1926,39:828.
[37]L. Brillouin, Compes Rendus.,1926,183:24; L. Brillouin, J. de Physique et le Rad., 1926,7:353;
[38]V. Ferrari and B. Mashhoon, Phys. Rev. D,1984,30:295-304.
[39]G. Koutsoumbas, S. Musiri, E. Papantonopoulos and G. Siopsis, JHEP,2006,0610: 006; G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, arXiv:0801.4921.
[40]J. Shen, B. Wang, C. Y. Lin, R. G. Cai and R. K. Su, JHEP,2007,0707:037; X. P. Rao, B. Wang and G. H. Yang, Phys. Lett. B,2007,649:472.
[41]Y. S. Myung, arXiv:0801.2434.
[42]J. Jing and Q. Pan, Phys. Lett. B,2008,660:13.
[43]P. C. W. Davies, Proc. Roy. Soc. Lond. A,1977,353:499; P. C. W. Davies, Class. Quant. Grav,1989,6:1909.
[44]E. Berti and V. Cardoso, arXiv:0802.1889.
[45]D. Pavon and J.M. Rube, Phys. Rev. D,1988,37:2052:D. Pavon, Phys. Rev. D, 1991,43:2495; R. G. Cai, R. K. Su and P. K. N. Yu, Phys. Rev. D,1993,48:3473; B. Wang and J. M. Zhu, Mod. Phys. Lett. A,1995,10:1269.
[46]O. Kaburaki, I. Okamoto and J. Katz, Phys. Rev. D,1993,47:2234:J. Katz, I. Okamoto and O. Kaburaki, Class. Quant. Gray,1993,10:1323.
[47]T. Harmark, V. Niarchos and N. A. Obers, Class. Quant. Grav,2007,24:R1.
[48]S. S. Gubser and I. Mitra, arXiv:hep-th/0009126; S. S. Gubser and I. Mitra, JHEP, 2001,0108:018; H. S. Reall, Phys. Rev. D,2001,64:044005.
[49]H. Ishihara and K. Matsuno, arXiv:hep-th/0510094.
[50]R. G. Cai, L.M. Cao and N. Ohta, Phys. Lett. B,2006,639:354.
[51]M. Kimura, K. Murata, H. Ishihara and J. Soda, arXiv:0712.4202.
[52]H. Ishihara, M. Kimura, R. A. Konoplya, K. Murata and J. Soda, het-th:0802.0655.
[53]Y. S. Myung, arXiv:0801.2434.
[54]J. L. Jing and Q. Y. Pan, Phys. Lett. B,2008,660:13.
[55]X. He, B. Wang, S. B. Chen, R. G. Cai and C. Y. Lin, Phys. Lett. B,2008,665:392.
[56]Y. N. Obukhov, astro-ph/0008106.
[57]K. Godel, Rev. Mod. Phys,1949,21:447.
[58]J. P. Gauntlett, J. B. Gutowski, C. M. Hull, S. Pakis and H. S. Reall, Class. Quant. Grav,2003,20:4587.
[59]E. G. Gimon and A. Hashimoto, Phys. Rev. Lett,2003,91:021601.
[60]C. A. R. Herdeiro, Class. Quant. Grav,2003,20:4891; D. Brecher, U. H. Danielsson, J. P. Gregory and M. E. Olsson, JHEP,2003,0311:033; K. Behrndt and D. Klemm, Class. Quant. Grav,2004,21:4107;S. Q. Wu, arXiv:0709.1749.
[61]S. Tomizawa, H. Ishihara, K. Matsuno and T. Nakagawa, arXiv:0803.3873.
[62]K. Matsuno, H. Ishihara, T. Nakagawa and S. Tomizawa, arXiv:0806.3316;S. Tomizawa and A. Ishibashi. arXiv:0807.1564.
[63]R. A. Konoplya and E. Abdalla, Phys. Rev. D,2005,71:084015.
[64]S. B. Chen, B. Wang, and J. L. Jing, Phys. Rev. D,2008,78:064030.
[65]H. Ishihara, M. Kimura, R. A. Konoplya, K. Murata and J. Soda, Phys. Rev. D, 2008,77:084019.
[66]H. Ishihara and K. Matsuno, Prog. Theor. Phys,2006,116:417.
[67]N. Andersson and H. Onozawa, Phys. Rev. D,1996,54:7470; H. Onozawa, Phys. Rev. D,1996,55:3593; E. Berti and K. D. Kokkotas, Phys. Rev. D,2003,68:044027.
[68]C. Stelea, K. Schleich and D. Witt, arXiv:0807.4338.
[69]A. G. Riess, et al., Astron. J,1998,116:1009; S. Perlmutter, et al., Astrophys. J, 1999,517:565; S. P. de Bernardis, et al., Nature (London),2000,404:955.
[70]S. Perlmutter, et al., Astrophys. J,2003,598:102; M. V. John, Astrophys. J,2004, 614:1; S. Boughn and R. Chrittenden, Nature(London),2004,427:45; S. Cole et al., Mon. Not. R. Astron. Soc,2005,362:505; P. Astier et al., J. Astron. Astrophys, 2006,447:31; V. Springel, C.S. Frenk and S. M. D. White, Nature (London),2006, 440:1137; W. M. Wood-Vasey et al., Astrophys. J,2007,666:694.
[71]D. N. Spergel, et al., Astrophys. J. Suppl,2003,148:175; D. N. Spergel, et. al., Astrophys. J. Suppl,2007,170:377; E. Komatsu, et al., arXiv:0803.0547.
[72]U. Alam, V. Sahni and A. Starobinsky, JCAP,2004,0406:008; Y. G. Gong, Class. Quant. Grav,2005,22:2121; Y. Wang and M. Tegmark, Phys. Rev. D,2005,71: 103513; Y. Wang and P. Mukherjee, Astrophys. J,2004,606:654; R. Daly and S. Djorgovski, Astrophys. J,2004,612:652; U. Alam, V. Sahni, T. Saini and A. Starobinsky, Mon. Not. Roy. Astron. Soc,2004,354:275; T. Choudhury and T. Padmanabhan, Astron. Astrophys,2005,429:807.
[73]E. Babichev, V. Dokuchaev and Y. Eroshenko, Phys. Rev. Lett,2004,93:021102;E. Babichev, V. Dokuchaev and Y. Eroshenko, J. Exp. Theor. Phys,2005,100:528-538; E. Babichev, V. Dokuchaev and Y. Eroshenko, gr-qc/0507119.
[74]L. M. Houghton and A. Kelleher, arXiv:0808.3419.
[75]H. P. Nollert, Class. Quant. Grav,1999,16:R159; K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel,1999,2:2.
[76]R. Ruffini and.J. A. Wheeler, Phys. Today,1971,24(1):30; R. H. Price, Phys. Rev. D,1972,5:2419; S. Hod and T. Piran, Phys. Rev. D,1998,58:024017.
[77]G. T. Horowitz and V. E. Hubeny, Phys. Rev. D,2000,62:024027; B. Wang, C. Y. Lin and E. Abdalla, Phys. Lett. B,2000,481:79; B. Wang, Braz. J. Phys,2005,35: 1029.
[78]B. Wang, C. Molina and E. Abdalla, Phys. Rev. D,2001,63:084001; J. M. Zhu, B. Wang and E. Abdalla, Phys. Rev. D,2001,63:124004; B. Wang, E. Abdalla and R. B. Mann, Phys. Rev. D,2002,65:084006; J. S. F. Chan and R. B. Mann, Phys. Rev. D,1999,59:064025; B. Wang, C.Y. Lin and C. Molina, Phys. Rev. D,2004, 70:064025.
[79]E. Winstanley, Phys. Rev. D,2001,64:104010; J. Crisstomo, S. Lepe and J. Saavedra, Class. Quant. Grav,2004,21:2801-2810; S. Lepe, F. Mendez, J. Saavedra and L. Vergara, Class. Quant. Grav,2003,20:2417-2428; S. Musiri and G. Siopsis, Phys. Lett. B,2003,576:309-313; A. Nunez and A. O. Starinets, Phys. Rev. D,2003,67: 124013.
[80]V. Cardoso and J. P. S. Lemos, Phys. Rev. D,2001,63:124015; V. Cardoso and J. P. S. Lemos, Phys. Rev. D,2001,64:084017; E. Berti and K. D. Kokkotas, Phys. Rev. D,2003,67:064020; V. Cardoso and J. P. S. Lemos, Class. Quant. Grav,2001, 18:5257.
[81]R. Aros, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D,2003,67:044014; D. Birmingham, I. Sachs and S. N. Solodukhin, Phys. Rev. Lett,2002,88:151301; D. Birmingham, Phys. Rev. D,2001,64:064024.
[82]E. Abdalla, B. Wang, A. Lima-Santos and W. G. Qiu, Phys. Lett. B,2002,538: 435; E. Abdalla, K. H. Castello-Branco and A. Lima-Santos, Phys. Rev. D,2002,66: 104018.
[83]G. Koutsoumbas, S. Musiri. E. Papantonopoulos and G. Siopsis, JHEP,2006,0610: 006; G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, JHEP,2008,0805:107; G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, arXiv:0806.1452.
[84]J. Shen, B. Wang, C. Y. Lin, R. G. Cai and R. K. Su, JHEP,2007,0707:037; X. P. Rao, B. Wang and G. H. Yang, Phys. Lett. B,2007,649:472.
[85]J. Frieman, C. T. Hill, A. Stebbins and I. Waga, Phys. Rev. Lett,1995,75:2077.
[86]R. R. Caldwell, Phys. Lett. B,2002,545:23; R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett,2003,91:071301; J. M. Cline, S. Y. Jeon and G. D. Moore, Phys. Rev. D,2004,70:043543.
[87]H. Koyama and A. Tomimatsu, Phys. Rev. D,2001,64:064032; H. Koyama and A. Tomimatsu, Phys. Rev. D,2002,65:084031; L. H. Xue, B. Wang and R. K. Su, Phys. Rev. D,2002,66:024032.
[88]C. G. Shao, B. Wang, E. Abdalla and R. K. Su, Phys. Rev. D,2005,71:044003.
[89]L. H. Xue, Z. X. Shen, B. Wang and R. K. Su, Mod. Phys. Lett. A,2004,19:239; E. Abdalla, C. B. M. Chirenti and A. Saa, Phys. Rev. D,2006,74:084029.
[90]P. C. Vaidya, Nature,1953,171:260.
[91]S. Chen, J. Jing and Q. Y. Pan, arXiv:0809.1152.
[92]B. Wang, E. Abdalla and R. B. Mann, Phys. Rev. D,2002,65:084006.
[93]G. T. Horowitz and V. E. Hubeny, Phys. Rev. D,2000,62:024027.
[94]B. Wang, C. Y. Lin and E. Abdalla, Phys. Lett. B,2000,481:79; B. Wang, C. Molina and E. Abdalla, Phys. Rev. D,2001,63:084001; J. M. Zhu, B. Wang and E. Abdalla, Phys. Rev. D,2001,63:124004:B. Wang, C. Y. Lin and C. Molina, Phys. Rev. D, 2004,70:064025.
[95]V. Cardoso and J. P. S. Lemos, Phys. Rev. D,2001,63:124015; V. Cardoso and J. P. S. Lemos, Phys. Rev. D,2001,64:084017; E. Berti and K. D. Kokkotas, Phys. Rev. D,2003,67:064020; V. Cardoso and J. P. S. Lemos, Class. Quant. Grav,2001, 18:5257; E. Winstanley, Phys. Rev. D,2001,64:104010:J. Crisstomo, S. Lepe and J. Saavedra, Class. Quant. Grav,2004.21:2801;S. Lepe, F. Mendez, J. Saavedra and L. Vergara, Class. Quant. Grav,2003,20:2417.
[96]D. Birmingham, I. Sachs and S. N. Solodukhin, Phys. Rev. Lett,2002,88:151301; D. Birmingham, Phys. Rev. D,2001,64:064024.
[97]B. Wang, E. Abdalla and R. B. Mann, Phys. Rev. D,2002,65: 084006; J. S. F. Chan and R.B. Mann, Phys. Rev. D,1999,59:064025; S. Musiri and G. Siopsis, Phys. Lett. B,2003,576:309; R. Aros, C. Martinez, R. Troncoso and J. Zanelli,
Phys. Rev. D,2003,67:044014; A. Nunez and A. O. Starinets, Phys. Rev. D,2003, 67:124013.
[98]J. Shen, B. Wang, R. K. Su, C.Y. Lin and R. G. Cai, JHEP,2007,0707:037.
[99]X. Rao, B. Wang and G. H. Yang, Phys. Lett. B,2007,649:472.
[100]S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Phys. Rev. Lett,2008,101:031601.
[101]S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, JHEP,2008,0812:015.
[102]G. T. Horowitz and M. M. Roberts, Phys. Rev. D,2008,78:126008.
[103]E. Nakano and Wen-Yu Wen, Phys. Rev. D,2008,78:046004.
[104]I. Amado, M. Kaminski and K. Landsteiner, JHEP,2009,0905:021; K. Maeda, M. Natsuume and T. Okamura, Phys. Rev. D,2009,79:126004.
[105]G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, JHEP,2009,0907:026.
[106]Q. Y. Pan, B. Wang, E. Papantonopoulos, J. Oliveira and A. Pavan, arXiv: 0912.2475.
[107]J. Sonner, Phys. Rev. D,2009,80:084031; R. G. Cai and H. Zhang, arXiv: 0911.4867; R. A. Konoplya, A. Zhidenko, arXiv:0909.2138.
[108]S. A. Hartnoll, arXiv:0903.3246.
[109]C. P. Herzog, J. Phys. A,2009,42:343001.
[110]S. S. Gubser, arXiv:0803.3483.
[111]S. S. Gubser and A. Nellore, JHEP,2009,0904:008.
[112]S. S. Gubser, Phys. Rev. D,2008,78:065034.
[113]F. Denef, S. A. Hartnoll and S. Sachdev, arXiv:0908.1788; S. A. Hartnoll, arXiv: 0909.3553.
[114]G. T. Horowitz and R. C. Myers, Phys. Rev. D,1999,59:026005.
[115]S. Surya, K. Schleich and D. M. Witt, Phys. Rev. Lett,2001,86:5231.
[116]R. G. Cai, S. P. Kim and B. Wang, Phys. Rev. D,2007,76:024011.
[117]T. Nishioka, S. Ryu and T. Takayanagi, arXiv:0911.0962.
[118]J. Y. Shen, B. Wang and R. K. Su, Phys. Rev. D,2006,74:044036.
[2]C. V. Vishveshwara, Phys. Rev.D,1970,1:2870-2879.
[3]F. J. Zerilli, Phys. Rev. D,1970,1:2870-2879.
[4]S. Teukolsky, Phys. Rev.Lett,1972,29(16):1114-1118.
[5]S. Teukolsky, AstroPhysical Journal,1973,185:635-647.
[6]S. Chandrasekhar, Oxford University press,1983:135-204.
[7]V. P. Frolov and I. D. Novikov, edited by ALWYN VAN FER MERWE, University of Denver,USA:87-149.
[8]H. P. Nollert, Phys. Rev. D,1993,47:5253-5258.
[9]O. Dreyer, Phys. Rev. Lett,2003,90:081301.
[10]S. Chandrasekhar and S. Detweiler, Royal Society of London Proceedings Series A, 1975,344:441.
[11]S. Chandrasekhar and S. Detweiler, Proc. R. Soc. Lond. A,1984,392:1.
[12]S. Chandrasekhar, Proc. R. Soc. Lond. A,1984,392:1.
[13]D. L. Gunter, Phil. Trans. Roy. Soc. Lond. A,1980,296: 497.
[14]E. W. Leaver, Phys. Rev. D.1990,41:2986; N. Andersson, Royal Society of London Proceedings Series A.1993.442:427; H. Onozawa, T. Mishima, T. Okamura and H. Ishihara, Phys. Rev. D,1996,53:7033; N. Andersson and H. Onozawa, Phys. Rev. D,1996,54:7470; E. Berti, gr-qc/0411025; E. Berti and K. D. Kokkotas, Phys. Rev. D,2003,68:044027.
[15]D. N. Page, Phys. Rev. D,1976,14:1509.
[16]E. Berti and K. D. Kokkotas, Phys. Rev. D,2005,71:124008.
[17]J. L. Jing and Q. Y. Pan, Nucl. Phys. B,2005,728:109.
[18]S. Detweiler, Astrophys. J,1980,239:292; S. Hod and U. Keshet, Class. Quant. Grav,2005,22:L71; U. Keshet and S. Hod, Phys. Rev. D,2007,76:061501; U. Keshet and A. Neitzke, Phys. Rev. D,2008,78:044006; H. C. Kao and D. Tomino, Phys. Rev. D,2008,77:127503.
[19]J. S. F. Chan and R. B. Mann, Phys. Rev. D,1997,55:7546.
[20]G. T. Horowitz and V. E. Hubeny, Phys. Rev. D,2000,62:024027.
[21]B. Wang, C. Y. Lin and E. Abdalla, Phys. Lett. B,2000,481:79.
[22]R. A. Konoplya, Phys. Rev. D,2002,66:084007.
[23]J. L. Jing and Q. Y. Pan, Phys. Rev. D,2005,71:124011; Y. Zhang and J. L. Jing, Int. J. Mod. Phys. D,2006,15:905.
[24]R. A. Konoplya and A. Zhidenko, Phys. Rev. D,2008,78:104017.
[25]E. Berti, V. Cardoso and P. Pani,0903.5311.
[26]J. Natario and R. Schiappa, Adv. Theor. Math. Phys,2004,8:1001; J. D. Brown, Singapore, Singapore:World Scientific,1998,152p.
[27]F. Mellor and I. Moss, Phys. Rev. D,1990,41:403.
[28]P. R. Brady, C. M. Chambers, W. G. Laarakkers and E. Poisson, Phys. Rev. D,1999, 60:064003.
[29]A. Zhidenko, Class. Quant. Grav,2004,21:273:V. Cardoso and J. P. S. Lemos, Phys. Rev. D,2003,67:084020.
[30]B. R. Iyer, S. Iyer and C. V. Vishveshwara, Class. Quant. Grav,1989,6:1627.
[31]R. G. Daghigh, G. Kunstatter and J. Ziprick, Class. Quant. Grav,2007,24:1981.
[32]S. B. Chen and J. L. Jing, Class. Quant. Grav,2005,22:4651:Y. Zhang, Y. X. Gui and F. Li, Gen. Rel. Grav,2007,39:1003; C. Ma, Y. Gui, W. Wang and F. Wang, Central Eur. J. Phys,2008,6:194:Y. Zhang and Y. X. Gui, Class. Quant. Grav, 2006,23:6141; Y. Zhang, Y. X. Gui and F. Yu, Chin. Phys. Lett,2009,26:030401; N. Varghese and V. C. Kuriakose,0802.1397.
[33]C. Gundlach, R. H. Price and J. Pullin, Phys. Rev. D,1994,49:883-889.
[34]C. Gundlach, R. H. Price and J. Pullin, Phys. Rev. D,1994,49:890-899.
[35]G. Z. Wenzel, Phys.,1926,38:518.
[36]H. M. Z. Kramers, Phys.,1926,39:828.
[37]L. Brillouin, Compes Rendus.,1926,183:24; L. Brillouin, J. de Physique et le Rad., 1926,7:353;
[38]V. Ferrari and B. Mashhoon, Phys. Rev. D,1984,30:295-304.
[39]G. Koutsoumbas, S. Musiri, E. Papantonopoulos and G. Siopsis, JHEP,2006,0610: 006; G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, arXiv:0801.4921.
[40]J. Shen, B. Wang, C. Y. Lin, R. G. Cai and R. K. Su, JHEP,2007,0707:037; X. P. Rao, B. Wang and G. H. Yang, Phys. Lett. B,2007,649:472.
[41]Y. S. Myung, arXiv:0801.2434.
[42]J. Jing and Q. Pan, Phys. Lett. B,2008,660:13.
[43]P. C. W. Davies, Proc. Roy. Soc. Lond. A,1977,353:499; P. C. W. Davies, Class. Quant. Grav,1989,6:1909.
[44]E. Berti and V. Cardoso, arXiv:0802.1889.
[45]D. Pavon and J.M. Rube, Phys. Rev. D,1988,37:2052:D. Pavon, Phys. Rev. D, 1991,43:2495; R. G. Cai, R. K. Su and P. K. N. Yu, Phys. Rev. D,1993,48:3473; B. Wang and J. M. Zhu, Mod. Phys. Lett. A,1995,10:1269.
[46]O. Kaburaki, I. Okamoto and J. Katz, Phys. Rev. D,1993,47:2234:J. Katz, I. Okamoto and O. Kaburaki, Class. Quant. Gray,1993,10:1323.
[47]T. Harmark, V. Niarchos and N. A. Obers, Class. Quant. Grav,2007,24:R1.
[48]S. S. Gubser and I. Mitra, arXiv:hep-th/0009126; S. S. Gubser and I. Mitra, JHEP, 2001,0108:018; H. S. Reall, Phys. Rev. D,2001,64:044005.
[49]H. Ishihara and K. Matsuno, arXiv:hep-th/0510094.
[50]R. G. Cai, L.M. Cao and N. Ohta, Phys. Lett. B,2006,639:354.
[51]M. Kimura, K. Murata, H. Ishihara and J. Soda, arXiv:0712.4202.
[52]H. Ishihara, M. Kimura, R. A. Konoplya, K. Murata and J. Soda, het-th:0802.0655.
[53]Y. S. Myung, arXiv:0801.2434.
[54]J. L. Jing and Q. Y. Pan, Phys. Lett. B,2008,660:13.
[55]X. He, B. Wang, S. B. Chen, R. G. Cai and C. Y. Lin, Phys. Lett. B,2008,665:392.
[56]Y. N. Obukhov, astro-ph/0008106.
[57]K. Godel, Rev. Mod. Phys,1949,21:447.
[58]J. P. Gauntlett, J. B. Gutowski, C. M. Hull, S. Pakis and H. S. Reall, Class. Quant. Grav,2003,20:4587.
[59]E. G. Gimon and A. Hashimoto, Phys. Rev. Lett,2003,91:021601.
[60]C. A. R. Herdeiro, Class. Quant. Grav,2003,20:4891; D. Brecher, U. H. Danielsson, J. P. Gregory and M. E. Olsson, JHEP,2003,0311:033; K. Behrndt and D. Klemm, Class. Quant. Grav,2004,21:4107;S. Q. Wu, arXiv:0709.1749.
[61]S. Tomizawa, H. Ishihara, K. Matsuno and T. Nakagawa, arXiv:0803.3873.
[62]K. Matsuno, H. Ishihara, T. Nakagawa and S. Tomizawa, arXiv:0806.3316;S. Tomizawa and A. Ishibashi. arXiv:0807.1564.
[63]R. A. Konoplya and E. Abdalla, Phys. Rev. D,2005,71:084015.
[64]S. B. Chen, B. Wang, and J. L. Jing, Phys. Rev. D,2008,78:064030.
[65]H. Ishihara, M. Kimura, R. A. Konoplya, K. Murata and J. Soda, Phys. Rev. D, 2008,77:084019.
[66]H. Ishihara and K. Matsuno, Prog. Theor. Phys,2006,116:417.
[67]N. Andersson and H. Onozawa, Phys. Rev. D,1996,54:7470; H. Onozawa, Phys. Rev. D,1996,55:3593; E. Berti and K. D. Kokkotas, Phys. Rev. D,2003,68:044027.
[68]C. Stelea, K. Schleich and D. Witt, arXiv:0807.4338.
[69]A. G. Riess, et al., Astron. J,1998,116:1009; S. Perlmutter, et al., Astrophys. J, 1999,517:565; S. P. de Bernardis, et al., Nature (London),2000,404:955.
[70]S. Perlmutter, et al., Astrophys. J,2003,598:102; M. V. John, Astrophys. J,2004, 614:1; S. Boughn and R. Chrittenden, Nature(London),2004,427:45; S. Cole et al., Mon. Not. R. Astron. Soc,2005,362:505; P. Astier et al., J. Astron. Astrophys, 2006,447:31; V. Springel, C.S. Frenk and S. M. D. White, Nature (London),2006, 440:1137; W. M. Wood-Vasey et al., Astrophys. J,2007,666:694.
[71]D. N. Spergel, et al., Astrophys. J. Suppl,2003,148:175; D. N. Spergel, et. al., Astrophys. J. Suppl,2007,170:377; E. Komatsu, et al., arXiv:0803.0547.
[72]U. Alam, V. Sahni and A. Starobinsky, JCAP,2004,0406:008; Y. G. Gong, Class. Quant. Grav,2005,22:2121; Y. Wang and M. Tegmark, Phys. Rev. D,2005,71: 103513; Y. Wang and P. Mukherjee, Astrophys. J,2004,606:654; R. Daly and S. Djorgovski, Astrophys. J,2004,612:652; U. Alam, V. Sahni, T. Saini and A. Starobinsky, Mon. Not. Roy. Astron. Soc,2004,354:275; T. Choudhury and T. Padmanabhan, Astron. Astrophys,2005,429:807.
[73]E. Babichev, V. Dokuchaev and Y. Eroshenko, Phys. Rev. Lett,2004,93:021102;E. Babichev, V. Dokuchaev and Y. Eroshenko, J. Exp. Theor. Phys,2005,100:528-538; E. Babichev, V. Dokuchaev and Y. Eroshenko, gr-qc/0507119.
[74]L. M. Houghton and A. Kelleher, arXiv:0808.3419.
[75]H. P. Nollert, Class. Quant. Grav,1999,16:R159; K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel,1999,2:2.
[76]R. Ruffini and.J. A. Wheeler, Phys. Today,1971,24(1):30; R. H. Price, Phys. Rev. D,1972,5:2419; S. Hod and T. Piran, Phys. Rev. D,1998,58:024017.
[77]G. T. Horowitz and V. E. Hubeny, Phys. Rev. D,2000,62:024027; B. Wang, C. Y. Lin and E. Abdalla, Phys. Lett. B,2000,481:79; B. Wang, Braz. J. Phys,2005,35: 1029.
[78]B. Wang, C. Molina and E. Abdalla, Phys. Rev. D,2001,63:084001; J. M. Zhu, B. Wang and E. Abdalla, Phys. Rev. D,2001,63:124004; B. Wang, E. Abdalla and R. B. Mann, Phys. Rev. D,2002,65:084006; J. S. F. Chan and R. B. Mann, Phys. Rev. D,1999,59:064025; B. Wang, C.Y. Lin and C. Molina, Phys. Rev. D,2004, 70:064025.
[79]E. Winstanley, Phys. Rev. D,2001,64:104010; J. Crisstomo, S. Lepe and J. Saavedra, Class. Quant. Grav,2004,21:2801-2810; S. Lepe, F. Mendez, J. Saavedra and L. Vergara, Class. Quant. Grav,2003,20:2417-2428; S. Musiri and G. Siopsis, Phys. Lett. B,2003,576:309-313; A. Nunez and A. O. Starinets, Phys. Rev. D,2003,67: 124013.
[80]V. Cardoso and J. P. S. Lemos, Phys. Rev. D,2001,63:124015; V. Cardoso and J. P. S. Lemos, Phys. Rev. D,2001,64:084017; E. Berti and K. D. Kokkotas, Phys. Rev. D,2003,67:064020; V. Cardoso and J. P. S. Lemos, Class. Quant. Grav,2001, 18:5257.
[81]R. Aros, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D,2003,67:044014; D. Birmingham, I. Sachs and S. N. Solodukhin, Phys. Rev. Lett,2002,88:151301; D. Birmingham, Phys. Rev. D,2001,64:064024.
[82]E. Abdalla, B. Wang, A. Lima-Santos and W. G. Qiu, Phys. Lett. B,2002,538: 435; E. Abdalla, K. H. Castello-Branco and A. Lima-Santos, Phys. Rev. D,2002,66: 104018.
[83]G. Koutsoumbas, S. Musiri. E. Papantonopoulos and G. Siopsis, JHEP,2006,0610: 006; G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, JHEP,2008,0805:107; G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, arXiv:0806.1452.
[84]J. Shen, B. Wang, C. Y. Lin, R. G. Cai and R. K. Su, JHEP,2007,0707:037; X. P. Rao, B. Wang and G. H. Yang, Phys. Lett. B,2007,649:472.
[85]J. Frieman, C. T. Hill, A. Stebbins and I. Waga, Phys. Rev. Lett,1995,75:2077.
[86]R. R. Caldwell, Phys. Lett. B,2002,545:23; R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett,2003,91:071301; J. M. Cline, S. Y. Jeon and G. D. Moore, Phys. Rev. D,2004,70:043543.
[87]H. Koyama and A. Tomimatsu, Phys. Rev. D,2001,64:064032; H. Koyama and A. Tomimatsu, Phys. Rev. D,2002,65:084031; L. H. Xue, B. Wang and R. K. Su, Phys. Rev. D,2002,66:024032.
[88]C. G. Shao, B. Wang, E. Abdalla and R. K. Su, Phys. Rev. D,2005,71:044003.
[89]L. H. Xue, Z. X. Shen, B. Wang and R. K. Su, Mod. Phys. Lett. A,2004,19:239; E. Abdalla, C. B. M. Chirenti and A. Saa, Phys. Rev. D,2006,74:084029.
[90]P. C. Vaidya, Nature,1953,171:260.
[91]S. Chen, J. Jing and Q. Y. Pan, arXiv:0809.1152.
[92]B. Wang, E. Abdalla and R. B. Mann, Phys. Rev. D,2002,65:084006.
[93]G. T. Horowitz and V. E. Hubeny, Phys. Rev. D,2000,62:024027.
[94]B. Wang, C. Y. Lin and E. Abdalla, Phys. Lett. B,2000,481:79; B. Wang, C. Molina and E. Abdalla, Phys. Rev. D,2001,63:084001; J. M. Zhu, B. Wang and E. Abdalla, Phys. Rev. D,2001,63:124004:B. Wang, C. Y. Lin and C. Molina, Phys. Rev. D, 2004,70:064025.
[95]V. Cardoso and J. P. S. Lemos, Phys. Rev. D,2001,63:124015; V. Cardoso and J. P. S. Lemos, Phys. Rev. D,2001,64:084017; E. Berti and K. D. Kokkotas, Phys. Rev. D,2003,67:064020; V. Cardoso and J. P. S. Lemos, Class. Quant. Grav,2001, 18:5257; E. Winstanley, Phys. Rev. D,2001,64:104010:J. Crisstomo, S. Lepe and J. Saavedra, Class. Quant. Grav,2004.21:2801;S. Lepe, F. Mendez, J. Saavedra and L. Vergara, Class. Quant. Grav,2003,20:2417.
[96]D. Birmingham, I. Sachs and S. N. Solodukhin, Phys. Rev. Lett,2002,88:151301; D. Birmingham, Phys. Rev. D,2001,64:064024.
[97]B. Wang, E. Abdalla and R. B. Mann, Phys. Rev. D,2002,65: 084006; J. S. F. Chan and R.B. Mann, Phys. Rev. D,1999,59:064025; S. Musiri and G. Siopsis, Phys. Lett. B,2003,576:309; R. Aros, C. Martinez, R. Troncoso and J. Zanelli,
Phys. Rev. D,2003,67:044014; A. Nunez and A. O. Starinets, Phys. Rev. D,2003, 67:124013.
[98]J. Shen, B. Wang, R. K. Su, C.Y. Lin and R. G. Cai, JHEP,2007,0707:037.
[99]X. Rao, B. Wang and G. H. Yang, Phys. Lett. B,2007,649:472.
[100]S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Phys. Rev. Lett,2008,101:031601.
[101]S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, JHEP,2008,0812:015.
[102]G. T. Horowitz and M. M. Roberts, Phys. Rev. D,2008,78:126008.
[103]E. Nakano and Wen-Yu Wen, Phys. Rev. D,2008,78:046004.
[104]I. Amado, M. Kaminski and K. Landsteiner, JHEP,2009,0905:021; K. Maeda, M. Natsuume and T. Okamura, Phys. Rev. D,2009,79:126004.
[105]G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, JHEP,2009,0907:026.
[106]Q. Y. Pan, B. Wang, E. Papantonopoulos, J. Oliveira and A. Pavan, arXiv: 0912.2475.
[107]J. Sonner, Phys. Rev. D,2009,80:084031; R. G. Cai and H. Zhang, arXiv: 0911.4867; R. A. Konoplya, A. Zhidenko, arXiv:0909.2138.
[108]S. A. Hartnoll, arXiv:0903.3246.
[109]C. P. Herzog, J. Phys. A,2009,42:343001.
[110]S. S. Gubser, arXiv:0803.3483.
[111]S. S. Gubser and A. Nellore, JHEP,2009,0904:008.
[112]S. S. Gubser, Phys. Rev. D,2008,78:065034.
[113]F. Denef, S. A. Hartnoll and S. Sachdev, arXiv:0908.1788; S. A. Hartnoll, arXiv: 0909.3553.
[114]G. T. Horowitz and R. C. Myers, Phys. Rev. D,1999,59:026005.
[115]S. Surya, K. Schleich and D. M. Witt, Phys. Rev. Lett,2001,86:5231.
[116]R. G. Cai, S. P. Kim and B. Wang, Phys. Rev. D,2007,76:024011.
[117]T. Nishioka, S. Ryu and T. Takayanagi, arXiv:0911.0962.
[118]J. Y. Shen, B. Wang and R. K. Su, Phys. Rev. D,2006,74:044036.