黑洞视界面面积量子化的研究
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摘要
本论文所探讨的关于黑洞视界面面积的量子化问题是近年来被国际上理论物理学界所关注的一个热点问题。它涉及的内容非常的广泛,涵盖了从广义相对论到热力学统计以及量子场论等许多方面的知识。对它的研究有望成为解开引力量子化困难的突破口。
全文共分为六个章节。第一章以及第二章是关于黑洞背景知识的相关介绍。讨论了黑洞的热力学性质,热辐射性质以及有关黑洞熵的本质问题。第三章主要介绍了黑洞表面积及其熵的量子化的概况。率先研究黑洞量子化问题的是Bekenstein,他基于对非极端黑洞的视界面面积的观察和分析,发现视界面面积类似于经典绝热量。利用Ehrenfest原理,Bekenstein认为,非极端量子黑洞的视界面面积应该就是有分立的本征值的。随后,Hod提出可以利用Bohr's correspondence principle计算黑洞面积谱,同时利用准证规模频率(quasinormal frequencies)的实部确定面积谱的间距。2002年,Kunstatter利用Bekenstein提出的黑洞面积绝热不变量以及Hod提出的其与准证规模频率的联系,证明了黑洞面积及其熵是量子化的,并且也计算出了面积谱及熵谱。2007年,Maggiore寸黑洞的准证规模频率提出了新的物理诠释,认为对Schwarzschild黑洞应该被视为阻尼谐振子,其频率由实部与虚部组成。因此,应该考虑复准证规模频率的绝对值,而不是Hod讨论中频率的实部。并且,Maggiore讨论了快速衰减的准证规模,此时,频率的虚部远远大于实部,其主导地位。根据这一对黑洞准正规模频率的新的物理解释以及利用黑洞绝热不变量的性质,可以发现黑洞的面积谱也是等间距的,并且计算出了相应的面积谱。最近,Ropotenko根据角动量量子化的方法,同样得到了等间距的黑洞面积谱,其结果与Maggiore及Bekenstein的结果相同。虽然这些方法都是半经典的理论,但这些讨论无疑为未来的量子引力理论提供了许多有用的启示。第四章通过两种不同的方法介绍了高维情况下Gauss-Bonnet黑洞视界面的面积及其熵的量子化。第五章则通过角动量量子化的方法讨论了de Sitter时空下宇宙视界面的面积谱。最后一章进行了小结
It has been almost forty years since Bekenstein showed that black hole entropy is proportional to its horizon area as well as that the horizon area is equally quantized. Since then the study of black hole area/entropy quantization has been an intriguing subject of discussions for the last four decades.
The thesis consists of four parts. The first part (Chapter1,2and3) gives a brief introduction of black holes as well as it's thermodynamics properties, and several methods to calculate area/entropy spectrum are given.
In the second part (chapter4), the entropy and area spectrum for D-dimensional Gauss-Bonnet black holes are investigated from two different approaches, which are the quasinormal modes approach developed by Hod, Kunstatter and Mag-giore, and the very recent method proposed by Ropotenko. The two methods give the same results, which show that the entropy spectrum of Gauss-Bonnet black holes is equidistantly quantized with spacing△S=2πh, while the area spectrum is not equally spaced.
In the third part, according to Ropotenko's recent work, we study the area spectrum of the cosmological horizon in d-dimensional pure de Sitter Space-time. In particular, by using the notion of energy E suggested by Padmanabhan, we modify Ropotenko's approach and extend it to de Sitter type universes. The derived area spectrum is evenly spaced and independent of the cosmological constant A. Furthermore, the area quantum of cosmological horizon in de Sit- ter space-time relies on the space-time dimension d and will decrease as the dimension d increases. In the last part (chapter6),a brief summary is given.
全文共分为六个章节。第一章以及第二章是关于黑洞背景知识的相关介绍。讨论了黑洞的热力学性质,热辐射性质以及有关黑洞熵的本质问题。第三章主要介绍了黑洞表面积及其熵的量子化的概况。率先研究黑洞量子化问题的是Bekenstein,他基于对非极端黑洞的视界面面积的观察和分析,发现视界面面积类似于经典绝热量。利用Ehrenfest原理,Bekenstein认为,非极端量子黑洞的视界面面积应该就是有分立的本征值的。随后,Hod提出可以利用Bohr's correspondence principle计算黑洞面积谱,同时利用准证规模频率(quasinormal frequencies)的实部确定面积谱的间距。2002年,Kunstatter利用Bekenstein提出的黑洞面积绝热不变量以及Hod提出的其与准证规模频率的联系,证明了黑洞面积及其熵是量子化的,并且也计算出了面积谱及熵谱。2007年,Maggiore寸黑洞的准证规模频率提出了新的物理诠释,认为对Schwarzschild黑洞应该被视为阻尼谐振子,其频率由实部与虚部组成。因此,应该考虑复准证规模频率的绝对值,而不是Hod讨论中频率的实部。并且,Maggiore讨论了快速衰减的准证规模,此时,频率的虚部远远大于实部,其主导地位。根据这一对黑洞准正规模频率的新的物理解释以及利用黑洞绝热不变量的性质,可以发现黑洞的面积谱也是等间距的,并且计算出了相应的面积谱。最近,Ropotenko根据角动量量子化的方法,同样得到了等间距的黑洞面积谱,其结果与Maggiore及Bekenstein的结果相同。虽然这些方法都是半经典的理论,但这些讨论无疑为未来的量子引力理论提供了许多有用的启示。第四章通过两种不同的方法介绍了高维情况下Gauss-Bonnet黑洞视界面的面积及其熵的量子化。第五章则通过角动量量子化的方法讨论了de Sitter时空下宇宙视界面的面积谱。最后一章进行了小结
It has been almost forty years since Bekenstein showed that black hole entropy is proportional to its horizon area as well as that the horizon area is equally quantized. Since then the study of black hole area/entropy quantization has been an intriguing subject of discussions for the last four decades.
The thesis consists of four parts. The first part (Chapter1,2and3) gives a brief introduction of black holes as well as it's thermodynamics properties, and several methods to calculate area/entropy spectrum are given.
In the second part (chapter4), the entropy and area spectrum for D-dimensional Gauss-Bonnet black holes are investigated from two different approaches, which are the quasinormal modes approach developed by Hod, Kunstatter and Mag-giore, and the very recent method proposed by Ropotenko. The two methods give the same results, which show that the entropy spectrum of Gauss-Bonnet black holes is equidistantly quantized with spacing△S=2πh, while the area spectrum is not equally spaced.
In the third part, according to Ropotenko's recent work, we study the area spectrum of the cosmological horizon in d-dimensional pure de Sitter Space-time. In particular, by using the notion of energy E suggested by Padmanabhan, we modify Ropotenko's approach and extend it to de Sitter type universes. The derived area spectrum is evenly spaced and independent of the cosmological constant A. Furthermore, the area quantum of cosmological horizon in de Sit- ter space-time relies on the space-time dimension d and will decrease as the dimension d increases. In the last part (chapter6),a brief summary is given.
引文
[1]S. W. Hawking, Particle creation by black holes, Communications in Mathematical Physics 43 (1975)199.
[2]S. W. Hawking, Black hole explosions?, Nature 248 (1974) 30-31.
[3]J. D. Bekenstein, Black Holes and Entropy, Physical Review D 7 (Apr.,1973) 2333-2346.
[4]S. F. Ross, Black hole thermodynamics, hep-th/0502195.
[5]P. Townsend, Black holes:Lecture notes, gr-qc/9707012. Lecture notes given as part of the Cambridge University Mathematical Tripos.
[6]王永久,经典黑洞和量子黑洞.北京：科学出版社,2008.
[7]赵峥,黑洞与弯曲的时空.太原：山西科学技术出版社,2001.
[8]刘辽,赵峥,田贵花,张靖仪,黑洞与时间的性质.北京：北京大学出版社,2008.
[9]R. Penrose, Republication of:Conformal treatment of infinity, General Relativity and Gravitation (Nov.,2010) 1-22.
[10]R. Penrose, Zero rest-mass fields including gravitation:Asymptotic behaviour, Proc Roy Soc Lond 284 (1965), no.1397 159-203.
[11]M. Heusler, Stationary Black Holes:Uniqueness and Beyond, Living Reviews in Relativity 1 (May,1998) 6.
[12]W. Israel, Event Horizons in Static Vacuum Space-Times, Physical Review 164 (Dec.,1967) 1776-1779.
[13]G. W. Gibbons, D. Ida, and T. Shiromizu, Uniqueness and Non-Uniqueness of Static Vacuum Black Holesin Higher Dimensions, Progress of Theoretical Physics Supplement 148 (2002) 284-290, [gr-qc/020].
[14]R. Emparan and H. S. Reall, A Rotating Black Ring Solution in Five Dimensions, Physical Review Letters 88 (Mar.,2002) 101101, [hep-th/01].
[15]J. D. Bekenstein, Extraction of energy and charge from a black hole, Phys. Rev. D7 (1973) 949-953.
[16]J. D. Bekenstein, Quantum black holes as atoms, gr-qc/9710076.
[17]S. Hod, Bohr's correspondence principle and the area spectrum of quantum black holes, Phys.Rev.Lett.81 (1998)4293, [gr-qc/9812002].
[18]S. Hod, Gravitation, the quantum, and Bohr's correspondence principle, Gen.Rel.Grav.31 (1999) 1639, [gr-qc/0002002].
[19]G. Kunstatter, d-dimensional black hole entropy spectrum from quasinormal modes, Phys.Rev.Lett.90(2003) 161301, [gr-qc/0212014].
[20]J. D. Bekenstein and V. F. Mukhanov, Spectroscopy of the quantum black hole, Phys.Lett. B360 (1995) 7-12, [gr-qc/9505012].
[21]M. Maggiore, The Physical interpretation of the spectrum of black hole quasinormal modes, Phys.Rev.Lett.100 (2008) 141301, [arXiv:0711.3145].
[22]E. C. Vagenas, Area spectrum of rotating black holes via the new interpretation of quasinormal modes, JHEP0811 (2008) 073, [arXiv:0804.3264].
[23]A. Medved, On the Kerr Quantum Area Spectrum, Class.Quant.Grav.25 (2008) 205014, [arXiv:0804.4346].
[24]K. Ropotenko, Quantization of the black hole area as quantization of the angular momentum component, Phys.Rev. D80 (2009) 044022, [arXiv:0906.1949].
[25]J. Makela and P. Repo, A quantum mechanical model of the Reissner-Nordstroem black hole, Phys. Rev. D57 (1998) 4899-4916, [gr-qc/9708029].
[26]D. Kothawala, T. Padmanabhan, and S. Sarkar, Is gravitational entropy quantized?, Phys.Rev. D78 (2008) 104018, [arXiv:0807.1481].
[27]S.-W. Wei, R. Li, Y.-X. Liu, and J.-R. Ren, Quantization of Black Hole Entropy from Quasinormal Modes, JHEP 0903 (2009) 076, [arXiv:0901.0587].
[28]W. Li, L. Xu, and J. Lu, Area spectrum of near-extremal sds black holes via the new interpretation of quasinormal modes, Physics Letters B 676 (2009), no.4-5 177-179.
[29]S. Fernando, Spinning Dilaton Black Holes in 2+1 Dimensions:Quasi-normal Modes and the Area Spectrum, Phys. Rev. D79 (2009) 124026, [arXiv:0903.0088].
[30]M. R. Setare, Area spectrum of extremal Reissner-Nordstroem black holes from quasi-normal modes, Phys. Rev. D69 (2004) 044016, [hep-th/0312061].
[31]S. Hod, Quasinormal spectrum and quantization of charged black holes, Class. Quant. Grav.23 (2006) L23-L28, [gr-qc/0511047].
[32]S. Hod, A note on the quantization of a multi-horizon black hole, Class. Quant. Grav.24 (2007) 4871, [arXiv:0709.2041].
[33]Y. Kwon and S. Nam, Area spectra of the rotating BTZ black hole from quasinormal modes, arXiv:1001.5106.
[34]A. Medved, On the 'Universal' Quantum Area Spectrum, Mod.Phys.Lett. A24 (2009) 2601-2609, [arXiv:0906.2641].
[35]A. Lopez-Ortega, Area spectrum of the D-dimensional de Sitter spacetime, Phys.Lett. B682 (2009) 85-88, [arXiv:0910.5779].
[36]D. G. Boulware and S. Deser, String Generated Gravity Models, Phys. Rev. Lett.55 (1985) 2656.
[37]R.-G. Cai, Gauss-Bonnet black holes in AdS spaces, Phys. Rev. D65 (2002) 084014, [hep-th/0109133].
[38]S. K. Chakrabarti and K. S. Gupta, Asymptotic Quasinormal Modes of d-Dimensional Schwarzschild Black Hole with Gauss-Bonnet Correction, Int.J.Mod.Phys. A21 (2006) 3565-3574, [hep-th/0506133].
[39]S. K. Chakrabarti, Quasinormal modes of tensor and vector type perturbation of Gauss Bonnet black hole using third order WKB approach, Gen.Rel. Grav.39 (2007) 567-582, [hep-th/0603123].
[40]R. G. Daghigh, G. Kunstatter, and J. Ziprick, The Mystery of the asymptotic quasinormal modes of Gauss-Bonnet black holes, Class.Quant.Grav.24 (2007) 1981-1992, [gr-qc/0611139].
[41]E. Berti, V. Cardoso, and A. O. Starinets, Quasinormal modes of black holes and black branes, Class.Quant.Grav.26 (2009) 163001, [arXiv:0905.2975].
[42]V. Olkhovsky and E. Recami, Time as a quantum observable, Int.J.Mod.Phys. A22 (2007) 5063-5087, [quant-ph/0605069].
[43]T. Padmanabhan, Classical and quantum thermodynamics of horizons in spherically symmetric space-times, Class.Quant.Grav.19(2002) 5387-5408, [gr-qc/0204019].
[44]T. Padmanabhan, Thermodynamics and/of horizons:A Comparison of Schwarzschild, Rindler and de Sitter space-times, Mod.Phys.Lett. A17 (2002) 923-942, [gr-qc/0202078].
[45]G. Hayward, Gravitational action for space-times with nonsmooth boundaries, Phys. Rev. D47 (1993)3275-3280.
[46]E. A. Martinez, Quasilocal energy for a Kerr black hole, Phys. Rev. D50 (1994) 4920-4928, [gr-qc/9405033].
[47]R. J. Epp, Angular momentum and an invariant quasilocal energy in general relativity, Phys.Rev. D62 (2000) 124018, [gr-qc/0003035].
[48]L. B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in GR:A Review Article, Living Reviews in Relativity 7 (Mar.,2004) 4.
[2]S. W. Hawking, Black hole explosions?, Nature 248 (1974) 30-31.
[3]J. D. Bekenstein, Black Holes and Entropy, Physical Review D 7 (Apr.,1973) 2333-2346.
[4]S. F. Ross, Black hole thermodynamics, hep-th/0502195.
[5]P. Townsend, Black holes:Lecture notes, gr-qc/9707012. Lecture notes given as part of the Cambridge University Mathematical Tripos.
[6]王永久,经典黑洞和量子黑洞.北京：科学出版社,2008.
[7]赵峥,黑洞与弯曲的时空.太原：山西科学技术出版社,2001.
[8]刘辽,赵峥,田贵花,张靖仪,黑洞与时间的性质.北京：北京大学出版社,2008.
[9]R. Penrose, Republication of:Conformal treatment of infinity, General Relativity and Gravitation (Nov.,2010) 1-22.
[10]R. Penrose, Zero rest-mass fields including gravitation:Asymptotic behaviour, Proc Roy Soc Lond 284 (1965), no.1397 159-203.
[11]M. Heusler, Stationary Black Holes:Uniqueness and Beyond, Living Reviews in Relativity 1 (May,1998) 6.
[12]W. Israel, Event Horizons in Static Vacuum Space-Times, Physical Review 164 (Dec.,1967) 1776-1779.
[13]G. W. Gibbons, D. Ida, and T. Shiromizu, Uniqueness and Non-Uniqueness of Static Vacuum Black Holesin Higher Dimensions, Progress of Theoretical Physics Supplement 148 (2002) 284-290, [gr-qc/020].
[14]R. Emparan and H. S. Reall, A Rotating Black Ring Solution in Five Dimensions, Physical Review Letters 88 (Mar.,2002) 101101, [hep-th/01].
[15]J. D. Bekenstein, Extraction of energy and charge from a black hole, Phys. Rev. D7 (1973) 949-953.
[16]J. D. Bekenstein, Quantum black holes as atoms, gr-qc/9710076.
[17]S. Hod, Bohr's correspondence principle and the area spectrum of quantum black holes, Phys.Rev.Lett.81 (1998)4293, [gr-qc/9812002].
[18]S. Hod, Gravitation, the quantum, and Bohr's correspondence principle, Gen.Rel.Grav.31 (1999) 1639, [gr-qc/0002002].
[19]G. Kunstatter, d-dimensional black hole entropy spectrum from quasinormal modes, Phys.Rev.Lett.90(2003) 161301, [gr-qc/0212014].
[20]J. D. Bekenstein and V. F. Mukhanov, Spectroscopy of the quantum black hole, Phys.Lett. B360 (1995) 7-12, [gr-qc/9505012].
[21]M. Maggiore, The Physical interpretation of the spectrum of black hole quasinormal modes, Phys.Rev.Lett.100 (2008) 141301, [arXiv:0711.3145].
[22]E. C. Vagenas, Area spectrum of rotating black holes via the new interpretation of quasinormal modes, JHEP0811 (2008) 073, [arXiv:0804.3264].
[23]A. Medved, On the Kerr Quantum Area Spectrum, Class.Quant.Grav.25 (2008) 205014, [arXiv:0804.4346].
[24]K. Ropotenko, Quantization of the black hole area as quantization of the angular momentum component, Phys.Rev. D80 (2009) 044022, [arXiv:0906.1949].
[25]J. Makela and P. Repo, A quantum mechanical model of the Reissner-Nordstroem black hole, Phys. Rev. D57 (1998) 4899-4916, [gr-qc/9708029].
[26]D. Kothawala, T. Padmanabhan, and S. Sarkar, Is gravitational entropy quantized?, Phys.Rev. D78 (2008) 104018, [arXiv:0807.1481].
[27]S.-W. Wei, R. Li, Y.-X. Liu, and J.-R. Ren, Quantization of Black Hole Entropy from Quasinormal Modes, JHEP 0903 (2009) 076, [arXiv:0901.0587].
[28]W. Li, L. Xu, and J. Lu, Area spectrum of near-extremal sds black holes via the new interpretation of quasinormal modes, Physics Letters B 676 (2009), no.4-5 177-179.
[29]S. Fernando, Spinning Dilaton Black Holes in 2+1 Dimensions:Quasi-normal Modes and the Area Spectrum, Phys. Rev. D79 (2009) 124026, [arXiv:0903.0088].
[30]M. R. Setare, Area spectrum of extremal Reissner-Nordstroem black holes from quasi-normal modes, Phys. Rev. D69 (2004) 044016, [hep-th/0312061].
[31]S. Hod, Quasinormal spectrum and quantization of charged black holes, Class. Quant. Grav.23 (2006) L23-L28, [gr-qc/0511047].
[32]S. Hod, A note on the quantization of a multi-horizon black hole, Class. Quant. Grav.24 (2007) 4871, [arXiv:0709.2041].
[33]Y. Kwon and S. Nam, Area spectra of the rotating BTZ black hole from quasinormal modes, arXiv:1001.5106.
[34]A. Medved, On the 'Universal' Quantum Area Spectrum, Mod.Phys.Lett. A24 (2009) 2601-2609, [arXiv:0906.2641].
[35]A. Lopez-Ortega, Area spectrum of the D-dimensional de Sitter spacetime, Phys.Lett. B682 (2009) 85-88, [arXiv:0910.5779].
[36]D. G. Boulware and S. Deser, String Generated Gravity Models, Phys. Rev. Lett.55 (1985) 2656.
[37]R.-G. Cai, Gauss-Bonnet black holes in AdS spaces, Phys. Rev. D65 (2002) 084014, [hep-th/0109133].
[38]S. K. Chakrabarti and K. S. Gupta, Asymptotic Quasinormal Modes of d-Dimensional Schwarzschild Black Hole with Gauss-Bonnet Correction, Int.J.Mod.Phys. A21 (2006) 3565-3574, [hep-th/0506133].
[39]S. K. Chakrabarti, Quasinormal modes of tensor and vector type perturbation of Gauss Bonnet black hole using third order WKB approach, Gen.Rel. Grav.39 (2007) 567-582, [hep-th/0603123].
[40]R. G. Daghigh, G. Kunstatter, and J. Ziprick, The Mystery of the asymptotic quasinormal modes of Gauss-Bonnet black holes, Class.Quant.Grav.24 (2007) 1981-1992, [gr-qc/0611139].
[41]E. Berti, V. Cardoso, and A. O. Starinets, Quasinormal modes of black holes and black branes, Class.Quant.Grav.26 (2009) 163001, [arXiv:0905.2975].
[42]V. Olkhovsky and E. Recami, Time as a quantum observable, Int.J.Mod.Phys. A22 (2007) 5063-5087, [quant-ph/0605069].
[43]T. Padmanabhan, Classical and quantum thermodynamics of horizons in spherically symmetric space-times, Class.Quant.Grav.19(2002) 5387-5408, [gr-qc/0204019].
[44]T. Padmanabhan, Thermodynamics and/of horizons:A Comparison of Schwarzschild, Rindler and de Sitter space-times, Mod.Phys.Lett. A17 (2002) 923-942, [gr-qc/0202078].
[45]G. Hayward, Gravitational action for space-times with nonsmooth boundaries, Phys. Rev. D47 (1993)3275-3280.
[46]E. A. Martinez, Quasilocal energy for a Kerr black hole, Phys. Rev. D50 (1994) 4920-4928, [gr-qc/9405033].
[47]R. J. Epp, Angular momentum and an invariant quasilocal energy in general relativity, Phys.Rev. D62 (2000) 124018, [gr-qc/0003035].
[48]L. B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in GR:A Review Article, Living Reviews in Relativity 7 (Mar.,2004) 4.