旋转黑洞时空中Dirac粒子的晚期拖尾
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摘要
爱因斯坦的广义相对论提出了两个惊人的预言：第一，一些大质量恒星的最终归宿是坍缩成为“黑洞”；第二，时空本身存在奇点。奇点是时空结束或开始的地方，在这些地方，现有的物理定律不再生效，黑洞中就存在这样的奇点。自广义相对论提出以来，黑洞物理学一直是一个最具挑战性，也最让人激动的天文学说。然而黑洞研究的意义远在天文学之外，今天的黑洞物理学已发展成为联系广义相对论与量力力学、弦理论、粒子物理、热力学和统计物理等诸多学科的交叉学科。但到目前为止，黑洞物理本身还有许多问题尚未搞清楚，比如黑洞熵的统计起源、黑洞是否稳定、黑洞信息疑难等等。黑洞微扰理论中的幂率拖尾问题涉及到黑洞“无毛”定理和质量暴涨定律及柯西视界的稳定性问题，因此对这个问题的研究具有非常重要的意义。
本文采用Newman-Penrose形式将弯曲时空中的Dirac方程进行退耦和分离变量，得到径向和角向波动方程。然后用黑洞格林函数(Green'sfunction)法研究在低频近似下，对处于无穷远处的观察者而言，Kerr和Kerr-Newman黑洞附近有质量的Dirac微扰场的晚期幂率拖尾。我们得到如下结论：
1、Kerr时空中有质量的Dirac粒子的中晚期拖尾，不仅与量子数l、m有关，与扰动粒子的质量μ和自旋s有关，而且还与背景黑洞的参量相关：(a)mλ＞0时，黑洞的比角动量a越大，衰减越慢，反之亦然；mλ＜0时，比角动量a越大，衰减越快，反之亦然。(b)λ＞0时，对于相同的轨道角量子数l，磁量子数m越大，衰减越慢；λ＜0时，m越大，衰减越快。
2、对于Kerr-Newman时空中有质量的Dirac粒子的中晚期拖尾，我们发现：(a)有质量的荷电Dirac粒子的中晚期拖尾不仅与量子数l、m有关，而且与seQ的值有关，这里s和e分别是Dirac粒子的自旋参数和带电量，Q是黑洞的电量。seQ＞0时加速衰减，seQ＜0时减缓衰减。(b)对于一定的黑洞电量Q，当mλ＞0时，a越大时衰减越慢，a越小时衰减越快；mλ＜0时，情况刚好相反，即a越大时衰减越快，a越小时衰减越慢。
3、对于极晚期拖尾，我们发现有质量的Dirac粒子在Kerr-Newman黑洞时空中以t(-5／6)sin(μt)衰减。综合以往的结论，这可能是所有有质量粒子在四维黑洞时空中的普适的衰减规律。
4、在旋转黑洞时空中，通过与有质量的scalar粒子的中晚期拖尾作比较，我们发现，有质量的Dirac微扰场的中晚期拖尾具有与之不同的衰减形式，即：(a)有质量Dirac粒子的中晚期拖尾是一个没有振荡的衰减，而有质量scalar粒子的中晚期衰减有振荡的。(b)有质量的Dirac粒子的中晚期拖尾明显慢于有质量scalar粒子。
One of the two exciting predictions in Einstein's general relativity is that there exists a black hole which is formed from the massive collapsing star. The other is the presence of the singularity, where the modern physics is invalid and the spacetime begins or ends. The black hole has this singularity. Indeed, the analysis of black hole has a significance going far beyond astrophysics, and the black hole physics has become an intersectional field of general relativity, quantum mechanics, string theory, particle physics, thermodynamics and statistics. However, there are still many open questions to be addressed, such as the statistical origin of black hole entropy, the stability of black hole, and the information puzzle. The late-time tail in the black hole perturbation theory is associated with the no-hair theorem, the mass-inflation scenario and the stability of the Cauchy horizon. Therefore it is interesting to investigate this topic.
In this thesis, we focus our attention on the late-time tails of the massive Dirac particles around the slowly rotating black hole, while both the observer and the initial data are situated far away from the black hole. The Dirac equations in the curve spacetime are decoupled and separated into the radial and angular parts by using the Newman-Penrose formalism, and then we use the black hole Green's function with the spectral decomposition method to analyze the late-time behavior. Our main conclusions are as follow:
1. The intermediate late-time tails of massive Dirac particles in the Kerr spacetime depend not only on the multiple number l and m, the particle massμand spin weight s, but also on the black hole parameter: (a) For mλ＞0, the larger the parameter a is, the more slowly it decays; for mλ＜0, the larger the parameter a is, the more quickly it decays. (b) Forλ＞0, with the same multiple number l, m decrease the decay rate; and forλ＜0 case, m increase it.
2. The intermediate late-time tails of massive Dirac particles in the KerrNewman spacetime are studied:(a) The dumping exponent depends not only on the angular quantum number l and m, but also on the product seQ of the spin weight of the Dirac field and the charges of the black hole and the fields, seQ＜0 speeds up the decay but seQ＞0 slows it down.(b) For both positive and negative electric charge Q, the rotating parameter a slows the decay rate down for mλ＞0 but speeds it up for mλ＜0.
3. The decay rate of the asymptotically late-time tails of massive Dirac fields in Kerr-Newmem spacetime is t~(-5/6) sin(μt), which may be a general feature for late-time tails of massive fields in the 4-dimensional black hole background.
4. Compared with the massive scaler fields, the massive Dirac particles in the rotating background have completely different intermediate late-time tails:(a) The intermediate late-time behavior of massive Dirac particles is dominated by a decaying tail without any oscillation, which is different from the oscillatory decaying tail of the scalar fields.(b) The intermediate late-time tails of massive Dirac particles are slower them those of the massive scalar fields.
本文采用Newman-Penrose形式将弯曲时空中的Dirac方程进行退耦和分离变量，得到径向和角向波动方程。然后用黑洞格林函数(Green'sfunction)法研究在低频近似下，对处于无穷远处的观察者而言，Kerr和Kerr-Newman黑洞附近有质量的Dirac微扰场的晚期幂率拖尾。我们得到如下结论：
1、Kerr时空中有质量的Dirac粒子的中晚期拖尾，不仅与量子数l、m有关，与扰动粒子的质量μ和自旋s有关，而且还与背景黑洞的参量相关：(a)mλ＞0时，黑洞的比角动量a越大，衰减越慢，反之亦然；mλ＜0时，比角动量a越大，衰减越快，反之亦然。(b)λ＞0时，对于相同的轨道角量子数l，磁量子数m越大，衰减越慢；λ＜0时，m越大，衰减越快。
2、对于Kerr-Newman时空中有质量的Dirac粒子的中晚期拖尾，我们发现：(a)有质量的荷电Dirac粒子的中晚期拖尾不仅与量子数l、m有关，而且与seQ的值有关，这里s和e分别是Dirac粒子的自旋参数和带电量，Q是黑洞的电量。seQ＞0时加速衰减，seQ＜0时减缓衰减。(b)对于一定的黑洞电量Q，当mλ＞0时，a越大时衰减越慢，a越小时衰减越快；mλ＜0时，情况刚好相反，即a越大时衰减越快，a越小时衰减越慢。
3、对于极晚期拖尾，我们发现有质量的Dirac粒子在Kerr-Newman黑洞时空中以t(-5／6)sin(μt)衰减。综合以往的结论，这可能是所有有质量粒子在四维黑洞时空中的普适的衰减规律。
4、在旋转黑洞时空中，通过与有质量的scalar粒子的中晚期拖尾作比较，我们发现，有质量的Dirac微扰场的中晚期拖尾具有与之不同的衰减形式，即：(a)有质量Dirac粒子的中晚期拖尾是一个没有振荡的衰减，而有质量scalar粒子的中晚期衰减有振荡的。(b)有质量的Dirac粒子的中晚期拖尾明显慢于有质量scalar粒子。
One of the two exciting predictions in Einstein's general relativity is that there exists a black hole which is formed from the massive collapsing star. The other is the presence of the singularity, where the modern physics is invalid and the spacetime begins or ends. The black hole has this singularity. Indeed, the analysis of black hole has a significance going far beyond astrophysics, and the black hole physics has become an intersectional field of general relativity, quantum mechanics, string theory, particle physics, thermodynamics and statistics. However, there are still many open questions to be addressed, such as the statistical origin of black hole entropy, the stability of black hole, and the information puzzle. The late-time tail in the black hole perturbation theory is associated with the no-hair theorem, the mass-inflation scenario and the stability of the Cauchy horizon. Therefore it is interesting to investigate this topic.
In this thesis, we focus our attention on the late-time tails of the massive Dirac particles around the slowly rotating black hole, while both the observer and the initial data are situated far away from the black hole. The Dirac equations in the curve spacetime are decoupled and separated into the radial and angular parts by using the Newman-Penrose formalism, and then we use the black hole Green's function with the spectral decomposition method to analyze the late-time behavior. Our main conclusions are as follow:
1. The intermediate late-time tails of massive Dirac particles in the Kerr spacetime depend not only on the multiple number l and m, the particle massμand spin weight s, but also on the black hole parameter: (a) For mλ＞0, the larger the parameter a is, the more slowly it decays; for mλ＜0, the larger the parameter a is, the more quickly it decays. (b) Forλ＞0, with the same multiple number l, m decrease the decay rate; and forλ＜0 case, m increase it.
2. The intermediate late-time tails of massive Dirac particles in the KerrNewman spacetime are studied:(a) The dumping exponent depends not only on the angular quantum number l and m, but also on the product seQ of the spin weight of the Dirac field and the charges of the black hole and the fields, seQ＜0 speeds up the decay but seQ＞0 slows it down.(b) For both positive and negative electric charge Q, the rotating parameter a slows the decay rate down for mλ＞0 but speeds it up for mλ＜0.
3. The decay rate of the asymptotically late-time tails of massive Dirac fields in Kerr-Newmem spacetime is t~(-5/6) sin(μt), which may be a general feature for late-time tails of massive fields in the 4-dimensional black hole background.
4. Compared with the massive scaler fields, the massive Dirac particles in the rotating background have completely different intermediate late-time tails:(a) The intermediate late-time behavior of massive Dirac particles is dominated by a decaying tail without any oscillation, which is different from the oscillatory decaying tail of the scalar fields.(b) The intermediate late-time tails of massive Dirac particles are slower them those of the massive scalar fields.
引文
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[2] 王永久，广义相对论和宇宙学[M]．长沙：湖南科学技术出版社，2000：106-468．
[3] J. D. Bekenstein and J. A. Wheeler, Introducing the black hole[J]. Phys. Rev. D, 1973, 7: 2333-2346.
[4] R. M. Wald, General Relativity[M]. Chicago and London: the university of Chicago press, 1984.
[5] S. W. Hawking, The large Scale Structure of Space-time[M]. Cambridge: Cambridge University Press, 1973.
[6] T. Regge and J. A. Wheeler, Stability of a Schwarzschild Singularity[J]. Phys. Rev., 1957, 108(4): 1063-1069.
[7] C. V. Vishveshwara, Stability of the Schwarzschild Metric[J]. Phys. Rev. D, 1970, 1: 2870-2879.
[8] F. J. Zerilli, Perturbation analysis for gravitational and electromagnetic radiation in a Reissner-Nordstrom geometry[J]. Phys. Rev. D, 1970, 1: 2870-2879.
[9] S. Teukolsky, Rotating black holes-seperable wave equations for gravitational and electromagnetic perturbations[J]. Phys. Rev. Lett, 1972, 29(16): 1114-1118.
[10] S. Teukolsky, Perturbations of a rotating black hole I. Fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations[J]. AstroPhysical Journal, 1973, 185: 635-647.
[11] S. Chandrasekhar, THE MATHEMATICAL THEORY OF BLACK HOLES[M]. Oxford University press, 1983: 135-204.
[12] V. P. Frolov and I. D. Novikov, Black Hole Physics: Basic Concepts and New Development [M]. edited by ALWYN VAN FER MERWE, University of Den- ver,USA:87-149.
[13] V. Ferrari and B. Mashhoon, New approach to the quasinormal modes of a black hole[J]. Phys. Rev. D,1984, 30: 295-304.
[14] S. Iyer and C. M. Will, Black-hole normal modes: A WKB approach. 1. Foundations and application of a higher-order WKB analysis of potential- barrier scattering[J] . Phys. Rev. D, 1987, 35: 3621-3636.
[15] H. Nollert and B. Schmidt, Quasinormal modes of Schwarzschild black holes:Defined and calculated via Laplace transformation[J] . Phys. Rev. D, 1992, 45: 2617-2627.
[16] N. Andersson and S. Linnaeus, Quasinormal modes of Schwarzschild black holes: Improved phase-integral treatment[J]. Phys. Rev. D, 1992, 46: 4179- 4187.
[17] K. D. Kokkotas and B. G. Schmidt, Quasi-normal modes of stars and black holes[J]. Living Rev. Relativ.,1999, 2: 2.
[18] R. A. Konoplya, Quasinormal nehavior of the D-dimensional Schwarzschild black hole and the higher order WKB approach[J]. Phys. Rev. D, 2003, 68: 024018.
[19] V. Cardoso, Quasinormal Modes and Gravitational Radiation in Black Hole Spacetimes[J]. PhD thesis, gr-qc/0404093.
[20] Jiliang Jing, Dirac quasinormal modes of Reissner-Nordstrom de Sitter black hole[J]. Phys. Rev. D, 2004, 69: 084009.
[21] M. Giammatteo and Jiliang Jing, Dirac Quasinormal frequencies in Schwarzschild-Ada space-time[J]. Phys. Rev. D,2005, 71: 024007.
[22] R. Ruffini and J. A. Wheeler, Introducing the black hole [J]. Phys. Today, 1971, 24(1): 30-41.
[23] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation[M]. San Francisco: Freeman, 1973: 872-940.
[24] E. eoisson and W. Israel, Internal structure of black hole [J]. Phys. Rev. D, 1990, 41: 1796-1809.
[25] A. Ori, Oscillatory null singularity inside realistic spinning black holes[J]. Phys. Rev. L, 1999, 83: 5423-5426.
[26] R. Herman and W. A. Hiscock, Strength of the mass inflation singularity[J]. Phys. Rev. D, 1992, 46: 1863-1865.
[27] J. S. F. Chan and R. B. Mann, Mass inflation in (1＋1)-dimensional dilaton gravity[J]. Phys. Rev. D, 1994, 50: 7376-7384.
[28] R. H. Price, Nonspherical Perturbations of Relativistic Gravitational Collapse. I. Scalar and Gravitational Perturbations[J]. Phys. Rev. D, 1972, 5(10): 2419-2438.
[29] R. H. Price, Nonspherical Perturbations of Relativistic Gravitational Collapse. Ⅱ. Integer-Spin, Zero-Rest-Mass Fields[J]. Phys. Rev. D, 1972, 5(10): 2439-2454.
[30] H. Koyama and A. Tomimatsu, Asymptotic tails of massive scalar fields in a Schwarzschild background[J]. Phys. Rev. D, 2001, 64(4): 044014.
[31] H. Koyama and A. Tomimatsu, Asymptotic power-law tails of massive scalar fields in a Reissner-Nordstrom background[J]. Phys. Rev. D, 2001, 63(6): 064032.
[32] Jiliang Jing, Late-time behavior of massive Dirac fields in a Schwarzschild background[J]. Phys. Rev. D, 2004, 70: 065004.
[33] Jiliang Jing, Late-time evolution of charged massive Dirac fields in the Reissner-Nordstrom black-hole background[J]. Phys. Rev. D, 2005, 72: 027501.
[34] J. Bicak, Gravitational collapse with charge and small asymmetries. I. Scalar perturbations[J]. Gen. Relativ. Gravit., 1972, 3: 331.
[35] G. Gundlach, R. H. Price and J. Pullin Late-time behavior of steller collapse and explosions. I. Linearized perturbations[J]. Phys. Rev.D, 1994, 49(2): 883- 889.
[36] G. Gundlach, R. H. Price and J. Pullin Late-time behavior of steller collapse and explosions. II. Nonlinear evolution[J]. Phys. Rev.D, 1994, 49(2): 890-899.
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