探讨Quintessence包围的Schwarzschild黑洞时空
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摘要
本文所探讨的是Quintessence对Schwarzschild黑洞时空的影响。近年来,Quintessence作为暗能量模型中的一种已经成为物理学界所关注的一个热点问题。另一方面黑洞一直以来是现代物理理论中一个具有吸引力的研究对象。探讨Quintessence对Schwarzschild黑洞时空的影响,将有助于我们进一步深入地理解暗能量和黑洞的一些特性。
本文首先考虑了Quintessence和标量场的质量对Schwarzschild黑洞的似正规模的影响。我们用三阶WKB方法计算了Quintessence包围的Schwarzschild黑洞时空的有质量标量场的似正规模。我们发现Quintessence和标量场的质量对似正规模的影响很大,因为Quintessence和标量场质量的引入使得似正规模频谱衰减变慢。除此之外,似正规频谱的实部随着标量场质量的增加而线性地增加,虚部的绝对值随着标量场的质量增加而减少。并且似正规频谱有一个极限值,这些都将有助于我们获得似正规频谱的数据。这样我们就有可能通过观测数据来研究Quintessence和黑洞的性质,从而来推测出暗能量的一些性质。
本文其次考虑了Quintessence对Schwarzschild黑洞熵的影响。我们用薄层brick-wall方法计算了Quintessence包围的Schwarzschild黑洞时空的熵。我们发现由于Quintessence的存在,黑洞的熵变成了两个视界面积的和的四分之一。也就是说对于黑洞的熵的贡献不仅来自于黑洞视界还来自于宇宙视界。并且这个体系的熵是和宇宙中Quintessence的含量有关。这主要是因为Quintessence产生了宇宙视界并改变了原来Schwarzschild黑洞的黑洞视界的大小,这就导致了体系的熵变成了两个视界面积的和的四分之一,这也就再一次揭示了黑洞的熵与视界面积之间的内在联系,也更进一步揭示了brick-wall模型的本质。
The influence of Quintessence to Schwarzschild black hole is discussed in this paper. Recently, Quintessence as one of dark energy candidates has been one of the "hot-spots" in physics field. In modern physics, another attractive object is black hole. The investigation of the influence of Quintessence to Schwarzschild black hole will help us further understand some properties of dark energy and black hole.
In this paper, the contribution of Quintessence and the mass of the scale field to quasinormal modes of Schwarzschild black hole is researched at first. The quasinormal frequencies of the massive scalar field in the background of a Schwarzschild black hole surrounded by Quintessence are evaluated with the third-order WKB method. It is found that the quasinormal modes are greatly influenced by the Quintessence and the mass of scalar filed, because the introduction of the Quintessence and the mass of scalar field leads to less damping of the quasinormal modes. Moreover, the real part of the frequencies linearly increases, while the magnitude of imaginary part linearly decreases as the mass of scalar field increases. And the quasinormal frequencies have a limited value, so it is easier to detect the quasinormal by experiment. It is possible to investigate some properties of Quintessence and black hole by the experiment datum. Consequently, it can be used to conjecture the properties of dark energy.
In this paper, how the Quintessence affects the entropy of black hole is also discussed. The entropy of a Schwarzschild black hole with Quintessence is calculated by using the thin layer brick-wall model. It is found that due to the present of Quintessence, the entropy of the system becomes 1/4 of the sum of the areas of the two events horizons. In other words, the contribution to the entropy of the system is not only from the black hole horizon but also from the cosmological horizon. It is also noted that the entropy of the system is related to the amount of Quintessence contained in the universe. The main reason is that Quintessence generate the cosmological horizon and change the isolated black hole horizon, which induce that the entropy of the system becomes 1/4 of the sum of the areas of the two events horizons. This reconfirms that there is an internal relation between the event horizon and the entropy, and also reveals what is the brick-wall model.
本文首先考虑了Quintessence和标量场的质量对Schwarzschild黑洞的似正规模的影响。我们用三阶WKB方法计算了Quintessence包围的Schwarzschild黑洞时空的有质量标量场的似正规模。我们发现Quintessence和标量场的质量对似正规模的影响很大,因为Quintessence和标量场质量的引入使得似正规模频谱衰减变慢。除此之外,似正规频谱的实部随着标量场质量的增加而线性地增加,虚部的绝对值随着标量场的质量增加而减少。并且似正规频谱有一个极限值,这些都将有助于我们获得似正规频谱的数据。这样我们就有可能通过观测数据来研究Quintessence和黑洞的性质,从而来推测出暗能量的一些性质。
本文其次考虑了Quintessence对Schwarzschild黑洞熵的影响。我们用薄层brick-wall方法计算了Quintessence包围的Schwarzschild黑洞时空的熵。我们发现由于Quintessence的存在,黑洞的熵变成了两个视界面积的和的四分之一。也就是说对于黑洞的熵的贡献不仅来自于黑洞视界还来自于宇宙视界。并且这个体系的熵是和宇宙中Quintessence的含量有关。这主要是因为Quintessence产生了宇宙视界并改变了原来Schwarzschild黑洞的黑洞视界的大小,这就导致了体系的熵变成了两个视界面积的和的四分之一,这也就再一次揭示了黑洞的熵与视界面积之间的内在联系,也更进一步揭示了brick-wall模型的本质。
The influence of Quintessence to Schwarzschild black hole is discussed in this paper. Recently, Quintessence as one of dark energy candidates has been one of the "hot-spots" in physics field. In modern physics, another attractive object is black hole. The investigation of the influence of Quintessence to Schwarzschild black hole will help us further understand some properties of dark energy and black hole.
In this paper, the contribution of Quintessence and the mass of the scale field to quasinormal modes of Schwarzschild black hole is researched at first. The quasinormal frequencies of the massive scalar field in the background of a Schwarzschild black hole surrounded by Quintessence are evaluated with the third-order WKB method. It is found that the quasinormal modes are greatly influenced by the Quintessence and the mass of scalar filed, because the introduction of the Quintessence and the mass of scalar field leads to less damping of the quasinormal modes. Moreover, the real part of the frequencies linearly increases, while the magnitude of imaginary part linearly decreases as the mass of scalar field increases. And the quasinormal frequencies have a limited value, so it is easier to detect the quasinormal by experiment. It is possible to investigate some properties of Quintessence and black hole by the experiment datum. Consequently, it can be used to conjecture the properties of dark energy.
In this paper, how the Quintessence affects the entropy of black hole is also discussed. The entropy of a Schwarzschild black hole with Quintessence is calculated by using the thin layer brick-wall model. It is found that due to the present of Quintessence, the entropy of the system becomes 1/4 of the sum of the areas of the two events horizons. In other words, the contribution to the entropy of the system is not only from the black hole horizon but also from the cosmological horizon. It is also noted that the entropy of the system is related to the amount of Quintessence contained in the universe. The main reason is that Quintessence generate the cosmological horizon and change the isolated black hole horizon, which induce that the entropy of the system becomes 1/4 of the sum of the areas of the two events horizons. This reconfirms that there is an internal relation between the event horizon and the entropy, and also reveals what is the brick-wall model.
引文
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[52] Plamen P Fiziev. Exact solutions of Regge - Wheeler equation and quasi-normal modes of compact objects. Class. Quantum. Grav. 2006, 23(7):2447-2468.
[53] Songbai Chen and Jiliang Jing. Quasinormal modes of a black hole surrounded by quitessence. Class. Quantum Grav. 2005, 22:4651-4657.
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[2]Oppenheimer J R and Volkoff G M.On Massive Neutron Cores.Phys.Rev.1939,55(4):374-381.
[3]赵铮.黑洞与弯曲的时空.太原:山西科学技术出版社,2001.
[4]Kerr R P.Gravitational field of a spinning mass as an example of algebraically special metrics.Phys.Rev.Lett.1963,11(5):2 37-238.
[5]Newman E.T.Metric of a rotating charged mass.J.Math.Phys.1995,6(6):918-919.
[6]Carter.B.Axisymmetric Black Hole Has Only Two Degrees of Freedom.Phys.Rev.Lett.1971,26(6):331-333.
[7]Robinson D C.Classification of black holes with electromagnetic fields.Phys.Rev.D,1974,10(2):458-460.
[8]Wald R.M.Black hole in a uniform magnetic field.Phys.Rev.D,1974,10:1680-1685.
[9]Jang P.S and Wald R.M.The positive energy conjecture and the cosmic censor hypothesis.J.Math.Phys.1977,18(1):41-44.
[10]Bekenstein J.D.Black holes and energy.Phys.Rev.1973,7(8):2333-2346.
[11]赵峥.四维静态黎曼时空中的Hawking辐射.物理学报.1981,30(11):1508-1519.
[12]R.M.Wald.General Relativity.Chicago and London:the university of Chicago press.1984.
[13]J M Bardeen,B Carter and S W Hawking.The four laws of black hole mechanics.[J].Commun.Math.Phys.1973,31(2):161-170.
[14]S W Hawking.Particle creation by black holes.[J].Commun.Math.Phys.1975,43(3):199-220.
[15]S W Hawking.Black hole explosions.[J].Nature.1974,248(2):30-31.
[16]J.S.Alcaniz.Testing dark energy beyond the cosmological constant barrier.Phys.Rev.D 2004,69:083521(5 pages).
[17]R.R.Caldwell,Rahul Dave and Paul J.Steinhardt.Cosmological Imprint of an Energy Component with General Equation of State.Phys.Rev.Lett.1998,80:1582-1585.
[18]Robert J.Scherrer.Purely Kinetic k Essence as Unified Dark Matter.Phys.Rev.Lett.2004,93:011301(4 pages).
[19]Alexander Vikman.Can dark energy evolve to the phantom? Phys.Rev.D 2005,71:023515(14 pages).
[20]Gong-Bo Zhao,Jun-Qing Xia and Mingzhe Li et al.Perturbations of the quintom models of dark energy and the effects on observations.Phys.Rev.D 2005,72:123515(16 pages).
[21] Zlatev I, Wang L and Steinhardt P J. Quintessence, Cosmic Coincidence, and the Cosmological Constant. Phys. Rev. Lett. 1999, 82(2):869-899.
[22] Ratra B and Peebels P J E. Cosmological Consequences of a Rolling Homogeneous Scalar field. Phys. Rev. D 1988, 37(12):3406-3427.
[23] Brax P and Martin J. Robustness of Quintessence. Phys. Rev. D2000, 61(10): 103502(14).
[24] Varun Sahni and Limin Wang. New cosmological model of quintessence and dark matter. Phys. Rev. D 2000, 62:103517(4).
[25] T. Barreiro, E. J. Copeland and N. J. Nunes. Quintessence arising from exponential potentials. Phys. Rev. D 2000, 61:127301(4).
[26] Kiselev VV. Quintessence and black hole. Class. Quantum. Grav. 2003, 20(6): 1187-1197.
[27] Tullio Regge and John A. Wheeler. Stability of a Schwarzschild Singularity. Phys. Rev. 1957, 108:1063-1069.
[28] Frank J. Zerilli. Effective Potential for Even-Parity Regge-Wheeler Gravitational Perturbation Equations. Phys. Rev. Lett. 1970, 24:737-738.
[29] Vishveshwara C. V. Scattering of gravitational radiation by a Schwarzschild black hole. Nature. 1970, 227:936-938.
[30] Peter Anninos, David Hobill, Edward Seidel et al. Collision of two black holes. Phys. Rev. Lett. 1993, 71:2851-2854.
[31] Reinaldo J. Gleiser, Carlos O. Nicasio, Richard H. Price et al. Colliding Black Holes: How Far Can the Close Approximation Go? Phys. Rev. Lett. 1996, 77:4483-4486.
[32] Gary T. Horowitz and Veronika E. Hubeny Quasinormal modes of AdS black holes and the approach to thermal equilibrium. Phys. Rev. D 2000, 62:024027(11).
[33] E. Witten. Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories. Adv. Theor. Math. Phys. 1998, 2:505-532.
[34] Gary T. Horowitz and Veronika E. Hubeny. Quasinormal modes of AdS black holes and the approach to thermal equilibrium. Phys. Rev. D 2000, 62:024027(11 pages).
[35] Gary T Horowitz. Comments on black holes in string theory. Class. Quantum. Grav. 2000, 17(5):1107-1116.
[36] T R Govindarajan and V Suneeta. Quasi-normal modes of AdS black holes: a superpotential approach. Class. Quantum Grav. 2001, 18(2):265-276.
[37] B. Wang, C. Molina and E. Abdalla. Evolution of a massless scalar field in Reissner-Nordstrom anti-de Sitter spacetimes. Phys. Rev. D 2001, 63:084001(8 pages).
[38] Jiong-Ming Zhu, Bin Wang and Elcio Abdalla. Object picture of quasinormal ringing on the background of small Schwarzschild anti - de Sitter black holes. Phys. Rev. D 2001, 63:124004(4 pages).
[39] Shahar Hod. Bohr's Correspondence Principle and the Area Spectrum of Quantum Black Holes. Phys. Rev. Lett. 1998, 81:4293-4296.
[40] J. Fernando Barbero G. Real Ashtekar variables for Lorentzian signature space-times. Phys. Rev. D 1995, 51:5507-5510.
[41] G. Immirzi. Quantum Gravity and Regge Calculus, nucl. phys. proc. suppl. 1997, 57:65-72.
[42] N Andersson. On the asymptotic distribution of quasinorraal-mode frequencies for Schwarzschild black holes. Class. Quantum. Grav. 1993, 10(6):L61-L67.
[43] Hans-Peter Nollert. Quasinormal modes of Schwarzschild black holes: The determination of quasinormal frequencies with very large imaginary parts. Phys. Rev. D 1993, 47: 5253-5258.
[44] Yi Ling and Hongbao Zhang. Quasinormal modes prefer supersymmetry? Phys. Rev. D 2003, 68:101501(3 pages).
[45] Carsten Gundlach, Richard H. Price and Jorge Pullin. Late-time behavior of stellar collapse and explosions. I. Linearized perturbations. Phys. Rev. D 1994,49:883-889.
[46] B. Wang, C. Molina and E. Abdalla. Evolution of a massless scalar field in Reissner-Nordstromanti-de Sitter spacetimes. Phys. Rev. D 2001, 63:084001(8 papges)
[47] B. F. Schutz and C. M. Will. Black hole normal modes - A semianalytic approach. Astrophysical Journal. 1985, 291:L33-L36.
[48] Sai Iyer and Clifford M. Will. Black-hole normal modes: A WKB approach. I. Foundations and application of a higher-order WKB analysis of potential-barrier scattering. Phys. Rev. D 1987, 35:3621-3631.
[49] Sai Iyer. Black-hole normal modes: A WKB approach. II. Schwarzschild black holes. Phys. Rev. D 1987, 35:3632-3636.
[50] Kostas D. Kokkotas and Bernard F. Schutz. Black-hole normal modes: A WKB approach. III. The Reissner-Nordstrom black hole. Phys. Rev. D 1988, 37:3378-3387.
[51] Edward Seidel and Sai Iyer. Black-hole normal modes: A WKB approach. IV. Kerr black holes. Phys. Rev. D 1990, 41:374-382.
[52] Plamen P Fiziev. Exact solutions of Regge - Wheeler equation and quasi-normal modes of compact objects. Class. Quantum. Grav. 2006, 23(7):2447-2468.
[53] Songbai Chen and Jiliang Jing. Quasinormal modes of a black hole surrounded by quitessence. Class. Quantum Grav. 2005, 22:4651-4657.
[54] Jacob D. Bekenstein and Marcelo Schiffer. The many faces of superradiance. Phys. Rev. D 1998, 58:064014(13 pages).
[55] Bekenstein J D. Black hole and entropy. Phys. Rev. D 1973, 7:2333-2346.
[56] Hawking S W. Particle creation by black hole. Commun. Math. Phys. 1975, 43:199-220.
[57] Bardeen J M, Carter B and Hawking S W. The four laws of black hole mechanics. Math Phys. 1973, 31:161-170.
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