Kerr-Newman黑洞周围电磁场的拟正则模
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摘要
自黑洞在广义相对论中被预言以来,科学家们在理论及观测上做了大量的工作,试图揭示黑洞的本质。但限于黑洞特殊的性质,我们不能对黑洞进行直接的观测。但即便如此,我们仍有一些对黑洞进行间接观测的方式,黑洞的拟正则模便是其中之一。拟正则模是扰动场的一种振动或演化模式,它描述了黑洞在受到外界初始扰动后,扰动场以拟正则模的形式演化并向外传播。其振动的频率以及衰减时间对初始扰动不敏感,而是随着黑洞各特征参数,如:质量,荷电量,角动量的变化而变化。这种方法可以帮助我们确认黑洞的存在以及揭示黑洞的特性。虽然对拟正则模的研究还处在理论阶段,但是对它的探测有望在今后的引力波观测中实现。本文分为四个部分来讨论黑洞以及拟正则模,特别是在Kerr-Newman黑洞周围电磁场的拟正则模。本文的结构安排如下:
在第一部分中我们首先回顾了黑洞的形成及其理论发展,随后介绍了一些黑洞间接观测的方式并分析了它们的局限。
第二部分,我们介绍了拟正则模研究的起源,给出了拟正则模的定义与特点并分析了拟正则模观测的可行性。
在第三部分中我们以两个例子说明了在拟正则模研究中的数学处理方法,随后对目前常用的关于拟正则模的计算方法给予简单的介绍。
第四部分是本文的主体部分。首先我们假设在K-N黑洞附近存在一个电磁场扰动,由电磁场满足的麦克斯韦方程出发,经过推导,得到扰动电磁场满足的波动方程。经过数值计算,我们得到了若干电磁场演化的图像,对其进行讨论并分析了在K-N黑洞时空背景下的电磁场的拟正则模的特点及物理意义。
Since black holes have been predicted in General Relativity, scientists have done a lot of job to reveal the nature of black hole in theory and observation. Because of its characteristics, we can not detect black hole directly. Even so, we still have some indirect methods. Quasi-normal mode (QNM), one of indirect methods, is a kind of oscillation or evolution modes of a perturbed field. The idea of QNM is that after the initial perturbation, there is QNM ringing outside black hole. The frequencies and damping times of the ringing are insensitive to initial perturbation, but varies with different characteristic parameters of black hole such as mass, charge and angular momentum which can help us to identify the existence of black hole and reveal its nature. Although the study on QNM is in the phase of theory, the detection of QNM is expected to be realized through gravitational wave observation. This paper is divided into four sections to discuss the black hole and QNM, especially QNM about electromagnetic field on the background of Kerr-Newman (K-N) black hole. The paper is organized as follows:
In Section One, firstly we review the process of black holes’formation and development of black hole theory and then present some ways about black holes’indirect observations and analyze their disadvantages.
In Section Two, we introduce the origin of QNM study and present QNM’s definition and characteristic and then analyze the feasibility of QNM’s observation.
In Section Three, we show two mathematical methods on QNM’s study with two examples and then briefly introduce two common calculation methods on QNM.
In Section four, the main body of our work, firstly we assume an electromagnetic perturbation around K-N black hole and then starting with Maxwell equation, we derive the wave equation obeyed by electromagnetic perturbation. Using numerical computation, we present several pictures to show the evolution of electromagnetic field and then analyze characteristics of QNM about electromagnetic field on the background of K-N black hole. Of course, the physical meaning is mentioned.
在第一部分中我们首先回顾了黑洞的形成及其理论发展,随后介绍了一些黑洞间接观测的方式并分析了它们的局限。
第二部分,我们介绍了拟正则模研究的起源,给出了拟正则模的定义与特点并分析了拟正则模观测的可行性。
在第三部分中我们以两个例子说明了在拟正则模研究中的数学处理方法,随后对目前常用的关于拟正则模的计算方法给予简单的介绍。
第四部分是本文的主体部分。首先我们假设在K-N黑洞附近存在一个电磁场扰动,由电磁场满足的麦克斯韦方程出发,经过推导,得到扰动电磁场满足的波动方程。经过数值计算,我们得到了若干电磁场演化的图像,对其进行讨论并分析了在K-N黑洞时空背景下的电磁场的拟正则模的特点及物理意义。
Since black holes have been predicted in General Relativity, scientists have done a lot of job to reveal the nature of black hole in theory and observation. Because of its characteristics, we can not detect black hole directly. Even so, we still have some indirect methods. Quasi-normal mode (QNM), one of indirect methods, is a kind of oscillation or evolution modes of a perturbed field. The idea of QNM is that after the initial perturbation, there is QNM ringing outside black hole. The frequencies and damping times of the ringing are insensitive to initial perturbation, but varies with different characteristic parameters of black hole such as mass, charge and angular momentum which can help us to identify the existence of black hole and reveal its nature. Although the study on QNM is in the phase of theory, the detection of QNM is expected to be realized through gravitational wave observation. This paper is divided into four sections to discuss the black hole and QNM, especially QNM about electromagnetic field on the background of Kerr-Newman (K-N) black hole. The paper is organized as follows:
In Section One, firstly we review the process of black holes’formation and development of black hole theory and then present some ways about black holes’indirect observations and analyze their disadvantages.
In Section Two, we introduce the origin of QNM study and present QNM’s definition and characteristic and then analyze the feasibility of QNM’s observation.
In Section Three, we show two mathematical methods on QNM’s study with two examples and then briefly introduce two common calculation methods on QNM.
In Section four, the main body of our work, firstly we assume an electromagnetic perturbation around K-N black hole and then starting with Maxwell equation, we derive the wave equation obeyed by electromagnetic perturbation. Using numerical computation, we present several pictures to show the evolution of electromagnetic field and then analyze characteristics of QNM about electromagnetic field on the background of K-N black hole. Of course, the physical meaning is mentioned.
引文
[1 ]Vishveshware, C.V, Scattering of gravitational radiation by a Schwarzschild black hole, Nature [J], 1970, 227, 936-938. [2 ]Press, W.H, Long wave trains of gravitational waves from a vibrating black hole, Astrophys [J], 1971, 170, 105-108.
[3]Kerr, R. P, Phys. Rev. Lett [J], 1963, 237-238.
[4]Newman, E.T, Math. Phys [J], 1965, 918-919.
[5]李宗伟. 肖兴华, 《天体物理学》[M], 高等教育出版社, 2007, 147-220.
[6]张镇九, 黑洞物理, 天文学进展 [J], 第二卷, 第四期, 341-354.
[7]俞允强, 《广义相对论引论》(第二版) [M], 北京大学出版社, 1997, 107-117.
[8]P.A.M. Dirac, 《广义相对论》 [M], 朱培豫译, 科学出版社, 1979, 32-37.
[9]王正行, 《近代物理学》[M], 北京大学出版社, 2004, 522-526.
[10]S. Hawking, 《History Of Time》 [M], Hunan Science & Technology Press, 76-106.
[11]Regge and Wheeler, Stability of a Schwarzchild singularity, Phys. Rev [J], 1957, 1063-1069.
[12]F. J. Zerilli, Perturbation analysis for gravitational and electromagnetic radiation in a Reissner-Norstr?m geometry, Phys. Rev [J], 1974, 860-868.
[13]S. Teukolsky, Rotating black holes-separable wave equations for gravitational and electromagnetic perturbations, Phys. Rev. lett [J], 1972, 1114-1118.
[14]王永久, 《黑洞物理学》[M], 湖南师范大学出版社, 2000.
[15]P Xi and L X Zhou, Object picture of quasinormal modes for string black holes, CHIN. PHYS. LETT [J], 2005, 22, 2763-2765.
[16]Kostas D. Kokkotas, Quasinormal mode of stars and black holes, PDF, http://www.astro.auth.gr/`kokktas.
[17]J M Zhu and B Wang and E.Abdalla, Object picture of quasinormal ringing on the background of small Schwarzschild anti-de Sitter black holes, PDF, arXiv:hep-th/010133v2, 2001.
[18]R. Ruffini, Black holes: les Astres Occlus [M], Gordon and Breaach Science Publishers,1973
[19]Schutz B.F and Will C.M, Black hole normal modes: a semi-analytic approach. Astrophys [J], 1985, 291, 33-36.
[20]V. Ferrari and B. Mashoon, Oscillation of a black hole, Phys, Rev, lett [J], 1984, 52,1361-1364.
[21]Iyer. S and Will. C. M, Black-hole normal modes: A WKB approach I foundations and applications of a high WKB analysis of potential-barrier scattering, Phys, Rev, D [J], 1987, 35, 3632-3636.
[22]P Xi and J M Zhu, Object picture of electromagnetic quasinormal ringing on the background of small Schwarzschild Anti-de Sitter black holes, IL Nuovo Cimento 119B [J], 2003, N.4,353-359.
[23] G. T. Horowitz and V. E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, PDF, arXiv:hep-th/9909056v2, 2000.
[24]V. Cardoso, Quasinormal modes and Gravitational radiation in black hole spacetimes, PDF, arXiv:gr-qc/0101093v1, 2004.
[25]R.A.Konoplya, On quasinormal modes of small Schwarzschild anti-de Sitter black hole, PDF, arXiv:hep-th/0205142v2, 2002.
[26]B Wang, Perturbation around black holes, Brazilian Journal of Physics [J], 2005, Vol.35, No.4B, 1029-1037.
[27]曾谨言, 《量子力学》 [M]. 第三版, 科学出版社, 2000, 105-120
[28]R.A.Konoplya, On quasinormal mode of small Schwarzschild Anti-de-Sitter black hole, PDF, arXiv:hep-th/0205142v2, 2002.
[29]奚萍, 黑洞的拟正则模 [D], 上海师范大学, 2005.
[30]傅莉萍, 引力透镜、CFHTLS 宇宙剪切巡天及盘状星系的角动量 [D], 上海师范大学, 2005.
[3]Kerr, R. P, Phys. Rev. Lett [J], 1963, 237-238.
[4]Newman, E.T, Math. Phys [J], 1965, 918-919.
[5]李宗伟. 肖兴华, 《天体物理学》[M], 高等教育出版社, 2007, 147-220.
[6]张镇九, 黑洞物理, 天文学进展 [J], 第二卷, 第四期, 341-354.
[7]俞允强, 《广义相对论引论》(第二版) [M], 北京大学出版社, 1997, 107-117.
[8]P.A.M. Dirac, 《广义相对论》 [M], 朱培豫译, 科学出版社, 1979, 32-37.
[9]王正行, 《近代物理学》[M], 北京大学出版社, 2004, 522-526.
[10]S. Hawking, 《History Of Time》 [M], Hunan Science & Technology Press, 76-106.
[11]Regge and Wheeler, Stability of a Schwarzchild singularity, Phys. Rev [J], 1957, 1063-1069.
[12]F. J. Zerilli, Perturbation analysis for gravitational and electromagnetic radiation in a Reissner-Norstr?m geometry, Phys. Rev [J], 1974, 860-868.
[13]S. Teukolsky, Rotating black holes-separable wave equations for gravitational and electromagnetic perturbations, Phys. Rev. lett [J], 1972, 1114-1118.
[14]王永久, 《黑洞物理学》[M], 湖南师范大学出版社, 2000.
[15]P Xi and L X Zhou, Object picture of quasinormal modes for string black holes, CHIN. PHYS. LETT [J], 2005, 22, 2763-2765.
[16]Kostas D. Kokkotas, Quasinormal mode of stars and black holes, PDF, http://www.astro.auth.gr/`kokktas.
[17]J M Zhu and B Wang and E.Abdalla, Object picture of quasinormal ringing on the background of small Schwarzschild anti-de Sitter black holes, PDF, arXiv:hep-th/010133v2, 2001.
[18]R. Ruffini, Black holes: les Astres Occlus [M], Gordon and Breaach Science Publishers,1973
[19]Schutz B.F and Will C.M, Black hole normal modes: a semi-analytic approach. Astrophys [J], 1985, 291, 33-36.
[20]V. Ferrari and B. Mashoon, Oscillation of a black hole, Phys, Rev, lett [J], 1984, 52,1361-1364.
[21]Iyer. S and Will. C. M, Black-hole normal modes: A WKB approach I foundations and applications of a high WKB analysis of potential-barrier scattering, Phys, Rev, D [J], 1987, 35, 3632-3636.
[22]P Xi and J M Zhu, Object picture of electromagnetic quasinormal ringing on the background of small Schwarzschild Anti-de Sitter black holes, IL Nuovo Cimento 119B [J], 2003, N.4,353-359.
[23] G. T. Horowitz and V. E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, PDF, arXiv:hep-th/9909056v2, 2000.
[24]V. Cardoso, Quasinormal modes and Gravitational radiation in black hole spacetimes, PDF, arXiv:gr-qc/0101093v1, 2004.
[25]R.A.Konoplya, On quasinormal modes of small Schwarzschild anti-de Sitter black hole, PDF, arXiv:hep-th/0205142v2, 2002.
[26]B Wang, Perturbation around black holes, Brazilian Journal of Physics [J], 2005, Vol.35, No.4B, 1029-1037.
[27]曾谨言, 《量子力学》 [M]. 第三版, 科学出版社, 2000, 105-120
[28]R.A.Konoplya, On quasinormal mode of small Schwarzschild Anti-de-Sitter black hole, PDF, arXiv:hep-th/0205142v2, 2002.
[29]奚萍, 黑洞的拟正则模 [D], 上海师范大学, 2005.
[30]傅莉萍, 引力透镜、CFHTLS 宇宙剪切巡天及盘状星系的角动量 [D], 上海师范大学, 2005.
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