稳态轴对称爱因斯坦—麦克斯韦伸缩子黑洞时空中标量场的衰减
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摘要
弦理论是目前唯一能够量子化引力并将引力与电磁、弱和强相互作用统一起来的自治理论。这一理论具有解释宇宙起源与运行以及现代物理中很多难题的潜力。由弦理论得到的伸缩子黑洞,其时空有着与通常广义相对论中的时空不一样的性质,其原因在于伸缩子参数的存在。因此,多年来人们对伸缩子时空的各种研究极为关注。但是,在含伸缩子时空中扰动场的演化研究方面,由于问题的复杂性,人们目前只讨论了静态球对称情况。由于稳态时空具有更普遍的性质,所以研究低能弦理论稳态轴对称解的伸缩子时空中标量场的衰减行为是非常有意义的。本文中我们将研究在该时空背景下标量场的拖尾以及似正模等问题。
利用引入了频谱分解技术的格林函数,我们对稳态轴对称爱因斯坦—麦克斯韦伸缩子黑洞时空中无质量和有质量标量场的晚期拖尾进行了解析的研究。结果表明:无质量扰动场的晚期渐近行为将由负幂律拖尾t~(-(2l+3)主导,有质量标量扰动场的中晚期衰减行为由振荡负幂律拖尾t~(-(l+3/2))sin(μt)主导,而极晚期则由一种衰减更缓慢的振荡负幂律拖尾t~(-(5/6))sin(μt)主导。我们认为,极晚期的振荡负幂律拖尾t~(-(5/6))sin(μt)是任意四维渐近平直旋转时空中有质量标量扰动场晚期衰减的普遍形式。
另一方面,我们利用Leaver最先提出的连续分数法对这一伸缩子时空中无质量标量似正模进行了数值的计算。我们获得了缓慢衰减的似正频率即基频,并且对它们的行为进行研究。结果表明:(1)这些标量似正模受制于伸缩子参数D、旋转参数a、多极矩l和磁量子数m;(2)在a、l和m相同的情况下,似正模基频的实部ωR随着D的减小始终是增大的,但是虚部ωI总会有一个先减小后增大的过程;(3)在D和l相同的情况下,当m≥0时,似正模基频的实部ωR和虚部ωI一般都会随着a的增加而增大,而当m<0时,似正模基频的实部ωR会随着a的增加而减小,虚部ωI则一般出现一个先减小后增大的过程;(4)在D和a相同的情况下,似正模基频的实部ωR只会随着l的增加而增大,除个别情况外,虚部ωI一般也会增大;(5)在D和l相同的情况下,当a=0即对于静态黑洞,m对似正模不存在任何影响,而当a≠0即对于
Superstring theory springing up in recent years is still the only known self-consistent theory which can quantize the gravity and unify the gravity, the electromagnetic interaction, the weak interaction and the strong interaction. This theory can explain the cosmic genesis and evolution, and figure out the problems of modern physics. In terms of the dilaton spacetimes in string theory, it has qualitatively different properties from those appearing in general relativity because of the apear-ance of dilaton. So there have been many investigations concerning the spacetimes of dilaton black hole. But people only discuss the evolution of perturbations in the static, spherically symmetric black hole geometry up to the present because of the complexity. Thus, it is worthwhile to investigate the decay of scalar fields in the background of dilaton black hole being the stationary axisymmetric solution of the so-called low-energy string theory. We will study the scalar late-time tails and quasinormal modes (QNMs) in this considered black hole.We investigate analytically the late-time tails of massless and massive scalar fields in a stationary axisymmetric Einstein-Maxwell dilaton-axion (EMDA) black hole geometry using the black-hole Green's function with the spectral decomposition method. It is shown that the asymptotic behavior of massless perturbations is dominated by the inverse power-law decaying tail t~(-(2l+3)), and the intermediate asymptotic behavior of massive perturbations is dominated by the oscillatory inverse power-law decaying tail t~(-(l+3/2)) sin(μt), but at asymptotically late times by the asymptotic tail t~(-5/6) sin(μt) which may be a quite general feature for the evolution of massive scalar fields in any four-dimensional asymptotically flat rotating black hole geometry.We also calculated numerically the massless scalar QNMs of this EMDA black hole using the continued fraction method first proposed by Leaver. We obtain the slowly damped QNMs (fundamental quasinormal frequencies) and study the peculiar behaviors of them. The main conclusions and their physical implications are the following: (1) The massless scalar fundamental quasinormal frequencies of the EMDA black hole are affected by the dilaton parameter D, the rotational parameter a, the multiple moment / and the azimuthal number m. (2) If a, l and m are fixed,
利用引入了频谱分解技术的格林函数,我们对稳态轴对称爱因斯坦—麦克斯韦伸缩子黑洞时空中无质量和有质量标量场的晚期拖尾进行了解析的研究。结果表明:无质量扰动场的晚期渐近行为将由负幂律拖尾t~(-(2l+3)主导,有质量标量扰动场的中晚期衰减行为由振荡负幂律拖尾t~(-(l+3/2))sin(μt)主导,而极晚期则由一种衰减更缓慢的振荡负幂律拖尾t~(-(5/6))sin(μt)主导。我们认为,极晚期的振荡负幂律拖尾t~(-(5/6))sin(μt)是任意四维渐近平直旋转时空中有质量标量扰动场晚期衰减的普遍形式。
另一方面,我们利用Leaver最先提出的连续分数法对这一伸缩子时空中无质量标量似正模进行了数值的计算。我们获得了缓慢衰减的似正频率即基频,并且对它们的行为进行研究。结果表明:(1)这些标量似正模受制于伸缩子参数D、旋转参数a、多极矩l和磁量子数m;(2)在a、l和m相同的情况下,似正模基频的实部ωR随着D的减小始终是增大的,但是虚部ωI总会有一个先减小后增大的过程;(3)在D和l相同的情况下,当m≥0时,似正模基频的实部ωR和虚部ωI一般都会随着a的增加而增大,而当m<0时,似正模基频的实部ωR会随着a的增加而减小,虚部ωI则一般出现一个先减小后增大的过程;(4)在D和a相同的情况下,似正模基频的实部ωR只会随着l的增加而增大,除个别情况外,虚部ωI一般也会增大;(5)在D和l相同的情况下,当a=0即对于静态黑洞,m对似正模不存在任何影响,而当a≠0即对于
Superstring theory springing up in recent years is still the only known self-consistent theory which can quantize the gravity and unify the gravity, the electromagnetic interaction, the weak interaction and the strong interaction. This theory can explain the cosmic genesis and evolution, and figure out the problems of modern physics. In terms of the dilaton spacetimes in string theory, it has qualitatively different properties from those appearing in general relativity because of the apear-ance of dilaton. So there have been many investigations concerning the spacetimes of dilaton black hole. But people only discuss the evolution of perturbations in the static, spherically symmetric black hole geometry up to the present because of the complexity. Thus, it is worthwhile to investigate the decay of scalar fields in the background of dilaton black hole being the stationary axisymmetric solution of the so-called low-energy string theory. We will study the scalar late-time tails and quasinormal modes (QNMs) in this considered black hole.We investigate analytically the late-time tails of massless and massive scalar fields in a stationary axisymmetric Einstein-Maxwell dilaton-axion (EMDA) black hole geometry using the black-hole Green's function with the spectral decomposition method. It is shown that the asymptotic behavior of massless perturbations is dominated by the inverse power-law decaying tail t~(-(2l+3)), and the intermediate asymptotic behavior of massive perturbations is dominated by the oscillatory inverse power-law decaying tail t~(-(l+3/2)) sin(μt), but at asymptotically late times by the asymptotic tail t~(-5/6) sin(μt) which may be a quite general feature for the evolution of massive scalar fields in any four-dimensional asymptotically flat rotating black hole geometry.We also calculated numerically the massless scalar QNMs of this EMDA black hole using the continued fraction method first proposed by Leaver. We obtain the slowly damped QNMs (fundamental quasinormal frequencies) and study the peculiar behaviors of them. The main conclusions and their physical implications are the following: (1) The massless scalar fundamental quasinormal frequencies of the EMDA black hole are affected by the dilaton parameter D, the rotational parameter a, the multiple moment / and the azimuthal number m. (2) If a, l and m are fixed,
引文
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[3] R. Ruffoni and J. A. Wheeler, Introducing the black hole [J]. Phys. Today., 1971, 24(1): 30-41.
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[6] F. J. Zerilli, Perturbation analysis for gravitational and electromagnetic radiation in a Reissner-Nordstrom geometry[J]. Phys. Rev. D, 1974, 9(4): 860-868.
[7] S. Teukolsky, ROTATING BLACK HOLES-SEPARABLE WAVE EQUATIONS FOR GRAVITATIONAL AND ELECTROMAGNETIC PERTURBATIONS[J]. Phys. Rev. Lett., 1972, 29(16): 1114-1118.
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[9] Juan M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity [J]. Adv. Theor. Math. Phys., 1998, 2: 231-252; Int. J. Theor. Phys., 1999, 38: 1113-1133; arXiv, 1997, hep-th/9711200.
[10] M. Giammatteo and Jiliang Jing, Dirac quasinormal frequencies in Schwarzschild-AdS space-time[J]. Phys. Rev. D, 2005, 71(2): 024007.
[11] Jiliang Jing, Quasinormal modes of Dirac field perturbation in Schwarzschild-anti-de Sitter black hole[J], arXiv, 2005, gr-qc/0502010, accepted for publication in Phys. Rev. D.
[12] Jiliang Jing and Qiyuan Pan, Dirac quasinormal frequencies of Schwarzschild/Reissner-Nordstrom black hole in Anti-de Sitter spacetime[J], arXiv, 2005, gr-qc/0502011, accepted for publication in Phys. Rev. D.
[13] Pan Qi-Yuan and Jing Ji-Liang, Asymptotic Tails of Massive Scalar Fields in a Stationary Axisymmetric EMDA black hole geometry[J]. Chin. Phys. Lett., 2004, 21(10): 1873-1876; arXiv, 2004, gr-qc/0405129.
[14] Pan Qi-Yuan and Jing Ji-Liang, Late-time tails in a stationary axisymmetric EMDA black hole geometry[J]. Chin. Phys., 2005, 14(2): 268-273.
[15] Pan Qi-Yuan and Jing Ji-Liang, Quasinormal modes of a stationary axisym metric EMDA black hole[J]. 2005,投稿于SCI源刊,在审。
[16] C. Vishveshwara, Scattering of gravitational radiation by a Schwarzschild black-hole[J]. Nature (London), 1970, 227(5261): 936-938.
[17] W. Press, Long Wave Trains of Gravitational Waves from a Vibrating Black Hole[J]. Astrophys. J., 1971, 170(3): L105-L108.
[18] V. Cardoso, Quasinormal Modes and Gravitational Radiation in Black Hole Spacetimes[J]. PhD thesis; arXiv, 2004, gr-qc/0404093.
[19] V. Ferrari and B. Mashhoon, New approach to the quasinormal modes of a black hole[J]. Phys. Rev. D, 1984, 30(2): 295-304.
[20] B. Schutz and C. Will, Black hole normal modes - A semianalytic approach[J]. Astrophys. J., 1985, 291(2): L33-L36.
[21] H. Nollert and B. Schmidt, Quasinormal modes of Schwarzschild black holes: Defined and calculated via Laplace transformation[J]. Phys. Rev. D, 1992, 45(8): 2617-2627.
[22] N. Andersson and S. Linnaeus, Quasinormal modes of a Schwarzschild black hole: Improved phase-integral treatment[J]. Phys. Rev. D, 1992, 46(10): 4179- 4187.
[23] E. Leaver, An analytic representation for quasi-normal modes of Kerr blackholes[J]. Proc. R. Soc. London A, 1985, 402: 285-298.
[24] R. H. Price, Nonspherical Perturbations of Relativistic Gravitational Col- lapse. I. Scalar and Gravitational Perturbations[J]. Phys. Rev. D, 1972, 5(10): 2419-2438.
[25] R. H. Price, Nonspherical Perturbations of Relativistic Gravitational Col lapse. II. Integer-Spin, Zero-Rest-Mass Fields[J]. Phys. Rev. D, 1972, 5(10): 2439-2454.
[26] E. W. Leaver, Spectral decomposition of the perturbation response of the Schwarzschild geometry [J]. Phys. Rev. D, 1986, 34(2): 384-408.
[27] J. Bicak, Gravitational collapse with charge and small asymmetries. I. Scalar perturbations [J]. Gen. Relativ. Gravit., 1972, 3(4): 331-349.
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