引力和视界热力学及Hawking辐射量子隧穿方法的相关研究
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摘要
引力和视界热力学的关系是理论物理中非常令人感兴趣的问题之一。更为重要的是,近年来联系于视界热力学一些新的性质被发现:Einstein方程在视界上可以作为热力学第一定律的形式出现;在Friedmann-Robertson-Walker(FRW)宇宙表观视界上,时空动力学方程(Friedmann方程)也可以写成一个所谓的统一热力学第一定律的形式;局域定义的FRW宇宙表观视界,也存在着在Kodama矢量观测者看来的Hawking辐射效应。
Hawking辐射是黑洞物理极其重要的一个量子现象,Parikh和Wilczek进一步整理和推广了Keski-Vakkuri和Kraus把辐射解释为粒子经过视界势垒隧穿的思想,称为辐射的Parikh-Wilczek隧穿方法。本文进一步研究了动力学时空FRW宇宙表观视界的Hawking辐射效应,以及动态黑洞模型-Vaidya黑洞和McVittie黑洞表观视界上的热力学,探讨了辐射和视界热力学的关系。
首先,在动力学时空FRW宇宙背景下,发展了超半经典近似的量子隧穿方法,研究了FRW宇宙表观视界的Hawking辐射效应,形式上给出了Hawking温度和熵的高阶量子修正项,Kodama矢量对定义辐射粒子的能量起着关键作用,我们分别用FRW坐标以及类Painlevé坐标给出了详细的分析。
其次,基于静态黑洞时空隧穿方法的一般性研究,那里黑洞的辐射率Γ-exp(ΔS),是视界上热力学第一定律dE_H=TdS+PdV的必然结果,我们在动力学时空FRW宇宙背景下,发展了这种分析。动力学时空与静态时空有着很大不同,我们采用一般性分析方法,从视界上统一热力学第一定律dE_H=TdS+WdV(其中W=(p-P)/2)出发,研究了FRW宇宙表观视界的Hawking辐射效应,形象的表明了在动力学时空下,Kodama矢量观测者意义下的Hawking辐射率Γ-exp(-ΔS),也是其视界热力学第一定律的必然结果。
最后,利用Kodama-Hayward动态黑洞理论,我们研究了两个动态黑洞模型-Vaidya黑洞和McVittie黑洞,分析了表观视界上Hawking辐射。发现对Vaidya黑洞,隧穿方法的一般性分析可以很好的给出辐射率,从而给出辐射谱;对于McVittie黑洞,我们给出了表观视界上的统一热力学第一定律形式,分析发现McVittie黑洞存在内行模式的Hawking辐射效应,这从物理上是难以理解的。我们认为McVittie黑洞的动态特性来源于背景时空FRW宇宙的动态因素,从这个角度也说明了McVittie解并不是严格意义上的动态黑洞解。
The relation between gravity and thermodynamics of horizons is one of the most interesting things in theoretical physics. Recently, and more importantly, several features related to horizon thermodynamics were discovered: Einstein equations can be expressed as a form of the first law of thermodynamics on horizons; on the apparent horizon of the Friedmann-Robertson-Walker(FRW) universe, the dynamical equation of spacetime (Friedmann equation) also can be expressed as a form of the so-called unified first law of thermodynamics; the locally defined apparent horizon of the FRW universe also related with Hawking radiation in the view of the Kodama vector.
Hawking radiation is a very important phenomenon of black hole physics. Parikh and Wilczek further refined and extended the thought of Keski-Vakkuri and Kraus, the radiation was explained as a tunneling process crossing the barrier of horizon, and in literatures the method is called Parikh-Wilczek tunneling method. In our paper, we further studied the effect of Hawking radiation of the apparent horizon of the dynamical spacetime, the FRW universe, and the thermodynamics of the apparent horizon of the dynamical black hole modes, the Vaidya black hole and the McVittie black hole. We also investigate the relations between the radiations and thermodynamics on horizons.
Firstly, in the background of the dynamical spacetime, the FRW universe, we extend the quantum tunneling method beyond semiclassical approximation. And the effect of Hawking radiation of the apparent horizon of the FRW universe is studied. In the form, we give out the high order correction terms of the Hawking temperature and the entropy. The Kodama vector plays a crucial role in the definition of energies of the radiation particles. We present our analysis under the FRW like coordinate system and the much-like to Painleve coordinate system, respectively.
Secondly, based on the general analysis of tunneling method of the static spacetime, where the tunneling rateΓ- exp(△S) arises as a consequence of the first law of thermodynamics for horizons holds the form dE_H = TdS + PdV, we develop the general method in the dynamical spacetime, the FRW universe. However, there are many differences between the static and the dynamical spcetimes. Using the the general analysis, from the unified first law of thermodynamics dE_H = TdS+WdV(where W = (ρ- P)/2) on the apparent horizon, we study the effect of Hawking radiation of the apparent horizon of the FRW universe. In such a dynamical system and in the view of the Kodama vector, we find that the tunneling rateΓ- exp(-△S) of the Hawking radiation also arises as a consequence of the unified first law of thermodynamics on horizons.
In the end, using the Kodama-Hayward dynamical black hole theory, we study two dynamical black hole modes, the Vaidya black hole and the McVittie black hole, and the effects of Hawking radiation are analyzed. For the Vaidya black hole, we find the general analysis of tunneling method follows well, the tunneling rate can be given out, thus the emission spectrum. For the McVittie black hole, we give out the form of the unified first law of thermodynamics on the apparent horizon, and we find that there exists the ingoing mode of Hawking radiation. However as a black hole, the physical meaning of the radiation of the ingoing mode is unclear and even puzzling. We believe that the dynamic characteristics of the McVittie black hole come from the dynamic performances of the background spacetime, the FRW universe. In this sense, the McVittie spacetime may also not actually be viewed as a dynamical black hole, despite its resemblance.
Hawking辐射是黑洞物理极其重要的一个量子现象,Parikh和Wilczek进一步整理和推广了Keski-Vakkuri和Kraus把辐射解释为粒子经过视界势垒隧穿的思想,称为辐射的Parikh-Wilczek隧穿方法。本文进一步研究了动力学时空FRW宇宙表观视界的Hawking辐射效应,以及动态黑洞模型-Vaidya黑洞和McVittie黑洞表观视界上的热力学,探讨了辐射和视界热力学的关系。
首先,在动力学时空FRW宇宙背景下,发展了超半经典近似的量子隧穿方法,研究了FRW宇宙表观视界的Hawking辐射效应,形式上给出了Hawking温度和熵的高阶量子修正项,Kodama矢量对定义辐射粒子的能量起着关键作用,我们分别用FRW坐标以及类Painlevé坐标给出了详细的分析。
其次,基于静态黑洞时空隧穿方法的一般性研究,那里黑洞的辐射率Γ-exp(ΔS),是视界上热力学第一定律dE_H=TdS+PdV的必然结果,我们在动力学时空FRW宇宙背景下,发展了这种分析。动力学时空与静态时空有着很大不同,我们采用一般性分析方法,从视界上统一热力学第一定律dE_H=TdS+WdV(其中W=(p-P)/2)出发,研究了FRW宇宙表观视界的Hawking辐射效应,形象的表明了在动力学时空下,Kodama矢量观测者意义下的Hawking辐射率Γ-exp(-ΔS),也是其视界热力学第一定律的必然结果。
最后,利用Kodama-Hayward动态黑洞理论,我们研究了两个动态黑洞模型-Vaidya黑洞和McVittie黑洞,分析了表观视界上Hawking辐射。发现对Vaidya黑洞,隧穿方法的一般性分析可以很好的给出辐射率,从而给出辐射谱;对于McVittie黑洞,我们给出了表观视界上的统一热力学第一定律形式,分析发现McVittie黑洞存在内行模式的Hawking辐射效应,这从物理上是难以理解的。我们认为McVittie黑洞的动态特性来源于背景时空FRW宇宙的动态因素,从这个角度也说明了McVittie解并不是严格意义上的动态黑洞解。
The relation between gravity and thermodynamics of horizons is one of the most interesting things in theoretical physics. Recently, and more importantly, several features related to horizon thermodynamics were discovered: Einstein equations can be expressed as a form of the first law of thermodynamics on horizons; on the apparent horizon of the Friedmann-Robertson-Walker(FRW) universe, the dynamical equation of spacetime (Friedmann equation) also can be expressed as a form of the so-called unified first law of thermodynamics; the locally defined apparent horizon of the FRW universe also related with Hawking radiation in the view of the Kodama vector.
Hawking radiation is a very important phenomenon of black hole physics. Parikh and Wilczek further refined and extended the thought of Keski-Vakkuri and Kraus, the radiation was explained as a tunneling process crossing the barrier of horizon, and in literatures the method is called Parikh-Wilczek tunneling method. In our paper, we further studied the effect of Hawking radiation of the apparent horizon of the dynamical spacetime, the FRW universe, and the thermodynamics of the apparent horizon of the dynamical black hole modes, the Vaidya black hole and the McVittie black hole. We also investigate the relations between the radiations and thermodynamics on horizons.
Firstly, in the background of the dynamical spacetime, the FRW universe, we extend the quantum tunneling method beyond semiclassical approximation. And the effect of Hawking radiation of the apparent horizon of the FRW universe is studied. In the form, we give out the high order correction terms of the Hawking temperature and the entropy. The Kodama vector plays a crucial role in the definition of energies of the radiation particles. We present our analysis under the FRW like coordinate system and the much-like to Painleve coordinate system, respectively.
Secondly, based on the general analysis of tunneling method of the static spacetime, where the tunneling rateΓ- exp(△S) arises as a consequence of the first law of thermodynamics for horizons holds the form dE_H = TdS + PdV, we develop the general method in the dynamical spacetime, the FRW universe. However, there are many differences between the static and the dynamical spcetimes. Using the the general analysis, from the unified first law of thermodynamics dE_H = TdS+WdV(where W = (ρ- P)/2) on the apparent horizon, we study the effect of Hawking radiation of the apparent horizon of the FRW universe. In such a dynamical system and in the view of the Kodama vector, we find that the tunneling rateΓ- exp(-△S) of the Hawking radiation also arises as a consequence of the unified first law of thermodynamics on horizons.
In the end, using the Kodama-Hayward dynamical black hole theory, we study two dynamical black hole modes, the Vaidya black hole and the McVittie black hole, and the effects of Hawking radiation are analyzed. For the Vaidya black hole, we find the general analysis of tunneling method follows well, the tunneling rate can be given out, thus the emission spectrum. For the McVittie black hole, we give out the form of the unified first law of thermodynamics on the apparent horizon, and we find that there exists the ingoing mode of Hawking radiation. However as a black hole, the physical meaning of the radiation of the ingoing mode is unclear and even puzzling. We believe that the dynamic characteristics of the McVittie black hole come from the dynamic performances of the background spacetime, the FRW universe. In this sense, the McVittie spacetime may also not actually be viewed as a dynamical black hole, despite its resemblance.
引文
[1] J. D. Bekenstein, Black holes and the second law, Nuovo Cim.Lett. 4 (1972)737-740.
[2] J. D. Bekenstein, Black Holes and Entropy, Phys. Rev. D 7 (1973) 2333-2346.
[3] J. D. Bekenstein, Generalized second law of thermodynamics in black-hole physics, Phys. Rev. D 9 (1974) 3292-3300.
[4] S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43(1975) 199.
[5] J. M. Bardeen, B. Carter, S. W. Hawking, The four laws of black hole mechanics,Commun. Math. Phys. 31 (1973) 161.
[6] R. M. Wald,The Thermodynamics of Black Holes, Living Rev. Relativity 4(2001)6,http://relativity.livingreviews.org/Articles/lrr-2001-6/index.html
[7] J. D. Bekenstein, Do We Understand Black Hole Entropy ?, gr-qc/9409015.
[8] S. A. Fulling, Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time, Phys. Rev. D 7 (1973) 2850-2862.
[9] P. C. W. Davies, Scalar particle production in Schwarzschild and Rindler metrics, J. Phys. A 8 (1975) 609-616.
[10] U. H. Gerlach, The mechanism of blackbody radiation from an incipient black hole, Phys. Rev. D 14 (1976) 1479-1508.
[11] G. t'Hooft, On the quantum structure of a black hole, Nucl.Phys. B 256 (1985)727.
[12] J. W. York, Jr., Dynamical origin of black-hole radiance, Phys. Rev. D 28 (1983)2929.
[13] W. H. Zurek and K. S. Thorne, Statistical Mechanical Origin of the Entropy of a Rotating, Charged Black Hole, Phys. Rev. Lett. 54 (1985) 2171.
[14] L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373.
[15] R. D. Sorkin, The Statistical Mechanics of Black Hole Thermodynamics, gr-qc/9705006.
[16] T. Jacobson, Thermodynamics of Spacetime: The Einstein Equation of State,Phys. Rev. Lett. 75 (1995) 1260.
[17] E. Verlinde, On the Holographic Principle in a Radiation Dominated Universe,hep-th/0008140.
[18] T Padmanabhan, Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes, Class. Quantum Grav.19(2002) 5387-5408.
[19] D. Kothawala, S. Sarkar, T. Padmanabhan, Einstein's equations as a thermodynamic identity-The cases of stationary axisymmetric horizons and evolving spherically symmetric horizons, Phys. Lett. B 652 (2007) 338-342.
[20] A. Paranjape, S. Sarkar, T. Padmanabhan, Thermodynamic route to field equations in Lanczos-Lovelock gravity, Phys. Rev. D 74,104015 (2006).
[21] T Padmanabhan, Gravity and the thermodynamics of horizons, Phys. Rep. 406(2005)49-125.
[22] T Padmanabhan, A. Paranjape, Entropy of null surfaces and dynamics of spacetime, Phys. Rev. D 75 (2007) 064004.
[23] T Padmanabhan, Is gravity an intrinsically quantum phenomenon ? Dynamics of Gravity from the Entropy of Spacetime and the Principle of Equivalence, Mod.Phys. Lett. A17 (2002) 1147.
[24] T Padmanabhan, Topological interpretation of the horizon temperature, Gen.Rel.Grav. 34 (2002) 2029-2035.
[25] G. W. Gibbons and S. W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15 (1977) 2738.
[26] R. G. Cai, Cardy-Verlinde formula and thermodynamics of black holes in de Sitter spaces, Nucl.Phys. B 628,375 (2002).
[27] R. G. Cai, Cardy-Verlinde formula and asymptotically de Sitter spaces, Phys.Lett.B 525,331 (2002).
[28] M. Banados, C. Teitelboim, and J. Zanelli, Black hole in three-dimensional spacetime,Phys. Rev. Lett. 69 (1992) 1849-1851;M. Banados, M. Henneaux, C. Teitelboim, and J. Zanelli, Geometry of the 2+1 black hole, Phys. Rev. D 48 (1993) 1506-1525.
[29] R. Arnowitt, S. Deser and C. Misner, "The Dynamics of General Relativity"in Gravitation: An Introduction to Current Research, ed. L. Witten (New York,Wiley) 1962.
[30] H. Bondi, M. G. J. van der Burg, and A.W.K.Metzner, Gravitational waves in general relativity.Ⅶ.Waves from axi-symmetric isolated systems, Proc.R.Soc.A 269 (1962) 21;R.K.Sachs,Gravitational waves in general relativity.Ⅷ.Waves in asymptotically flat space-time, Proc. R. Soc. A 270,103 (1962).
[31] C. W. Misner and D. H. Sharp, Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse, Phys. Rev. 136 (1964) B571.
[32] J. D. Brown and J.W. York, Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47, 1407 (1993).
[33] C. C. M. Liu and S. T. Yau, Positivity of Quasilocal Mass, Phys. Rev. Lett. 90(2003)231102.
[34] L. B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in GR-A Review Article, Living Rev. Relativity 7 (2004) 4,http://relativity.livingreviews.org/ArticlesArr-2004-4/.
[35] H.Kodama, Conserved Energy Flux for the Spherically Symmetric System and the Backreaction Problem in the Black Hole Evaporation, Prog. Theor. Phys. 63(1980)1217.
[36] S.A. Hay ward, General laws of black-hole dynamics, Phys. Rev. D 49 (1994)6467.
[37] S. A. Hayward, Gravitational energy in spherical symmetry, Phys. Rev. D 53(1996) 1938.
[38] S. A. Hayward, Unified first law of black-hole dynamics and relativistic thermodynamics, Class. Quantum Grav. 15 (1998) 3147.
[39] S. A. Hayward, S. Mukohyama, M. C. Ashworth, Dynamic black-hole entropy,Phys. Lett. A 256 (1999) 347.
[40] M. C. Ashworth, S. A. Hayward, Boundary terms and Noether current of spherical black holes, Phys. Rev. D 60 (1999) 084004.
[41] R.-G. Cai, S. P. Kim, First Law of Thermodynamics and Friedmann Equations of Friedmann-Robertson-Walker Universe, JHEP 02 (2005) 050.
[42] R.-G. Cai, L.-M. Cao, Unified first law and the thermodynamics of the apparent horizon in the FRW universe, Phys. Rev. D 75 (2007) 064008.
[43] M. Akbar, R.-G. Cai, Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe, Phys. Rev. D 75 (2007) 084003.
[44] Y. Gong, A. Wang, Friedmann Equations and Thermodynamics of Apparent Horizons, Phys. Rev. Lett. 99 (2007) 211301.
[45] L. Randall, R. Sundrum,Large Mass Hierarchy from a Small Extra Dimension,Phys. Rev. Lett. 83 (1999) 3370.L.Randall, R. Sundrum, An Alternative to Compactification, Phys. Rev. Lett. 83(1999)4690.
[46] G. Dvali, G. Gabadadze, M. Porrati, Large Mass Hierarchy from a Small Extra Dimension, Phys. Lett. B 485 (2000) 208.G. Dvali, G. Gabadadze, Gravity on a brane in infinite-volume extra space, Phys.Rev.D 63 (2001) 065007.
[47] R. G. Cai and L. M. Cao, Thermodynamics of apparent horizon in brane world scenario, Nucl.Phys. B 785 (2007) 135.
[48] A. Sheykhi, B. Wang and R. G. Cai, Thermodynamical properties of apparent horizon in warped DGP braneworld, Nucl. Phys. B 779 (2007) 1.
[49] A. Sheykhi, B. Wang and R. G. Cai, Deep connection between thermodynamics and gravity in Gauss-Bonnet braneworlds, Phys. Rev. D 76 (2007) 023515.
[50] R. G. Cai, L. M. Cao and Y. P. Hu, Corrected Entropy-Area Relation and Modified Friedmann Equations, JHEP 0808 (2008) 090.
[51] R. G. Cai, Thermodynamics of Apparent Horizon in Brane World Scenarios,Prog. Theor. Phys. Suppl.172 (2008) 100.
[52] X. H. Ge, First law of thermodynamics and Friedmann-like equations in braneworld cosmology, Phys. Lett. B 651 (2007) 49.
[53] S. F. Wu, G. H. Yang and P. M. Zhang, Cosmological equations and Thermodynamics on Apparent Horizon in Thick Braneworld, arXiv:0710.5394.
[54] S. F. Wu, B. Wang and G. H. Yang, Thermodynamics on the apparent horizon in generalized gravity theories, Nucl. Phys. B 799 (2008) 330-344.
[55] T. Zhu, J. R. Ren and S. F. Mo, Thermodynamics of Friedmann Equation and Masslike Function in General Braneworld, arXiv:0805.1162.
[56] P. Kraus and F. Wilczek, Selfinteraction correction to black hole radiance, Nucl.Phys.B 433 (1995) 403.
[57] P. Kraus and F. Wilczek, Effect of self-interaction on charged black hole radiance, Nucl.Phys. B 437 (1995) 231-242.
[58] E. Keski-Vakkuri and P. Kraus, Microcanonical D-branes and back reaction,Nucl.Phys.B 491 (1997)249.
[59] M. K. Parikh, A Secret Tunnel Through the Horizon, Gen.Rel.Grav.36 (2004)2419-2422.
[60] M. K. Parikh and F. Wilczek, Hawking Radiation As Tunneling, Phys. Rev. Lett.85 (2000)5042.
[61] M. K. Parikh, New coordinates for de Sitter space and de Sitter radiation, Phys.Lett. B 546 (2002) 189.
[62] M. K. Parikh, A Secret Tunnel Thwugh the Horizon, Int. J. Mod. Phys. D13(2004)2351.
[63] E. C. Vagenas,Semiclassical corrections to the Bekenstein-Hawking entropy of the BTZ black hole via self-gravitation,Phys. Lett. B 533 (2002)302.
[64] A. J. M. Medved, On Hawking Radiation as Tunneling with Back-Reaction, E.C. Vagenas, Mod. Phys. Lett. A 20 (2005) 2449.
[65] M. Arzano, A. J. M. Medved, E. C. Vagenas, Hawking radiation as tunneling through the quantum horizon, JHEP 0509 (2005) 037.
[66] Q.-Q. Jiang, S.-Q.Wu,X.Cai, Hawking radiation as tunneling from the Kerr and Kerr-Newman black holes, Phys. Rev. D 73 (2006) 064003.
[67] Y. Hu, J. Zhang, Z. Zhao, Massive particles' Hawking radiation via tunneling from the G.H Dilaton black hole, Mod. Phys. Lett. A 21 (2006) 2143.
[68] Y. Hu, J. Zhang, Z. Zhao, Massive uncharged and charged particles' tunneling from the Howwitz-Strominger Dilaton black hole, Int. J. Mod. Phys. D 16 (2007)847-856.
[69] Z. Xu, B. Chen, Hawking Radiation from General Kerr-(anti)de Sitter Black Holes, Phys. Rev. D 75 (2007) 024041.
[70] X. Wu, S. Gao, Tunneling Effect Near Weakly Isolated Horizon, Phys. Rev. D 75(2007)044027.
[71] C.-Z. Liu, J.-Y. Zhu, Hawking radiation as tunneling from Gravity's rainbow, Gen. Relat. Grav. 40 (2008) 1899-1911.
[72] R. Kerner, R. B. Mann, Tunnelling, Temperature and Taub-NUT Black Holes, Phys. Rev. D 73 (2006) 104010.
[73] R. Kerner, R. B. Mann, Tunnelling from Godel black holes, Phys. Rev. D 75 (2007)084022.
[74] B. Chatterjee, A. Ghosh, P. Mitra, Tunnelling from black holes and tunnelling into white holes, Phys. Lett. B 661 (2008) 307-311.
[75] J.-R. Ren, R. Li, F.-H. Liu, Hawking Temperature of Dilaton Black Holes from Tunneling, Mod. Phys. Lett. A 23 (2009) 3419-3429.
[76] R. Emparan and H. S. Reall, Generalized Weyl solutions, Phys. Rev. D 65 (2002) 084025.
[77] R. Emparan and H. S. Reall, A Rotating Black Ring Solution in Five Dimensions, Phys. Rev. Lett. 88 (2002) 101101.
[78] R. Emparan, Rotating Circular Strings, and Infinite Non-Uniqueness of Black Rings, JHEP 03 (2004) 064.
[79] H. K. Kunduri and J. Lucietti, Electrically charged dilatonic black rings, Phys. Lett. B 609 (2005) 143-149.
[80] L. Zhao, Tunneling Through Black Rings, Commun. Theor. Phys. 47 (2007) 835.
[81] Q.-Q. Jiang, Dirac particle tunneling from black rings, Phys. Rev. D 78 (2008) 044009.
[82] R. Banerjee, B. R. Majhi, Quantum tunneling and back reaction, Phys. Lett. B 662 (2008) 62-65.
[83] R. Banerjee, B. R. Majhi, Quantum Tunneling Beyond Semiclassical Approximation, JHEP 0806 (2008) 095.
[84] B. R. Majhi, Fermion Tunneling Beyond Semiclassical Approximation, Phys.Rev. D 79 (2009) 044005.
[85] S. K. Modak, Corrected entropy of BTZ black hole in tunneling approach, Phys.Lett.B 671 (2009) 167-173.
[86] E. T. Akhmedov, V. Akhmedova, D. Singleton,Integrability and higher loops in AdS/dCFT correspondence, Phys. Lett. B 642 (2006) 124;
E. T. Akhmedov, V. Akhmedova, T. Pilling, D. Singleton, Thermal radiation of various gravitational backgrounds, Int. J. Mod. Phys. A 2 (2007) 1705;
T. K. Nakamura, Factor Two Discrepancy of Hawking Radiation Temperature,hep-th/0706.2916;
P. Mitra, Hawking temperature from tunnelling formalism, Phys. Lett. B 648(2007) 240;
E. T. Akhmedov, T. Pilling, D. Singleton, Subtleties in the quasi-classical calculation of Hawking radiation, hep-th/0805.2653;
V. Akhmedova, T. Pilling, A. de Gill, and D. Singleton, Comments on anomaly versus WKB/tunneling methods for calculating Unruh radiation,arXiv:0808.3413;
B. D. Chowdhury, Problems with Tunneling of Thin Shells from Black Holes,Pramana 70(2008) 593-612;
T. Pilling, Quasi-classical Hawking Temperatures and Black Hole Thermodynamics, arXiv:0809.2701;
V. Akhmedova, T. Pilling, A. de Gill, D. Singleton, Temporal contribution to gravitational WKB-like calculations, Phys. Lett. B 666 (2008) 269-271.
[87] G. W. Gibbons and S. W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15 (1977) 2738.
[88] W. G. Unruh, Notes on black-hole evaporation, Phys. Rev. D14(1976) 870. [89] P. Hajicek, Origin of Hawking radiation, Phys. Rev. D 36 (1987) 1065.
[90] M. Visser, Essential and inessential features of Hawking radiation, Int. J. Mod.Phys. D 12 (2003) 649.
[91] S. A. Hayward, R. Di Criscienzo, M. Nadalini, L. Vanzo, S. Zerbini, Local Hawking temperature for dynamical black holes, arXiv:0806.0014.
[92] S. A. Hayward, Dynamics of black holes, arXiv:0810.0923.
[93] S. A. Hayward, R. Di Criscienzo, M. Nadalini, L. Vanzo, S. Zerbini, Local temperature for dynamical black holes, arXiv:0812.2534.
[94] R.-G. Cai, L.-M. Cao, Y.-P. Hu, Hawking Radiation of Apparent Horizon in a FRW Universe, arXiv:0809.1554.
[95] J. W. York Jr., Black hole in thermal equilibrium with a scalar field: The backreaction, Phys. Rev. D 31 (1985) 775.
[96] C. O. Lousto, N. Sanchez, Back reaction effects in black hole spacetimes, Phys.Lett.B 212 (1988) 411-414.
[97] D. V. Fursaev, Temperature and entropy of a quantum black hole and conformal anomaly, Phys. Rev. D 51 (1995) R5352-R5355.
[98] R. B. Mann, S. N. Solodukhin, Universality of quantum entropy for extreme black holes, Nucl.Phys. B 523 (1998) 293.
[99] R. K. Kaul, P. Majumdar, Logarithmic Correction to the Bekenstein-Hawking Entropy, Phys. Rev. Lett. 84 (2000) 5255-5257.
[100] A. Ghosh, P. Mitra, An improved estimate of black hole entropy in the quantum geometry approach, Phys. Lett. B 616 (2005) 114-117.
[101] D. N. Page, Hawking Radiation and Black Hole Thermodynamics, New J. Phys.7 (2005) 203.
[102] R. Banerjee, B. R. Majhi, Quantum Tunneling Beyond Semiclassical Approximation, JHEP 06 (2008) 095.
[103] K. Srinivasan, T. Padmanabhan, Particle production and complex path analysis,Phys. Rev. D 60 (1999) 024007.
[104] S. Sarkar, D. Kothawala, Hawking radiation as tunneling for spherically symmetric black holes-A generalized treatment, Phys. Lett. B 659 (2008) 683-687.
[105] T. Pilling, Black hole thermodynamics and the factor of 2 problem, Phys. Lett.B 660 (2008) 402.
[106] R. Li, J.-R. Ren, D.-F. Shi, Fermions tunneling from apparent horizon of FRW universe, Phys. Lett. B 670 (2009) 446.
[107] P. C. Vaidya, The External Field of a Radiating Star in General Relativity, Curr.Sci. 12,183 (1943), Reprinted: Gen. Relativ. Gravit. 31 (1999) 119-120;P. C. Vaidya, The Gravitational Field of a Radiating Star, Proc.Indian Acad.Sci. A 33 (1951) 264, Reprinted: Gen. Relativ. Gravit. 31 (1999) 121-135;P. C. Vaidya, Newtonian' Time in General Relativity, Nature 171 (1953) 260,Reprinted: Gen. Relativ. Gravit. 31 (1999) 137-138 V. V. Narlikar, P. C. Vaidya, A Spherically Symmetrical Non-Static Electromagnetic Field, Nature 159 (1947) 642.
[108] W. B. Bonnor, P. C. Vaidya, Spherically symmetric radiation of charge in Einstein-Maxwell theory,Gen. Relativ. Gravit.1(1970) 127-130;J. M. Bardeen, Black Holes Do Evaporate Thermally, Phys. Rev. Lett. 46 (1981)382;W. A. Hiscock, Models of evaporating black holes.Ⅰ,Phys. Rev. D 23, 2813(1981);W. A. Hiscock, Models of evaporating black holes.Ⅱ.Effects of the outgoing created radiation, Phys. Rev. D 23, 2823 (1981);W. A. Hiscock, L. G. Williams, and D. M. Eardley, Creation of particles by shell-focusing singularities, Phys. Rev. D 26,751 (1982);J. W. York, Dynamical origin of black-hole radiance, Phys. Rev. D 28 (1983)2929;Y. Kuroda, A Model for Evaporating Black Holes, Prog. Theor. Phys. 71, 100 (1984);
Y. Kuroda, Vaidya spacetime as an evaporating black hole, Prog. Theor. Phys. 71,1422(1984);
R. L. Mallet, Radiating Vaidya metric imbedded in de Sitter space, Phys. Rev. D 31 (1985)416;
J. Bi(c|ˇ)ák and K.V. Kucha(r|ˇ), Null dust in canonical gravity, Phys. Rev. D 56,4878 (1997);
J. Bi(c|ˇ)ák and P. Hájí(c|ˇ)ek, Canonical theory of spherically symmetric spacetimes with cross-streaming null dusts, Phys. Rev. D 68, 104016 (2003);
S. G. Ghosh and N. Dadhich, Naked singularities in higher dimensional Vaidya space-times, Phys. Rev. D 64,047501 (2001);
T. Harko, Gravitational collapse of a Hagedorn fluid in Vaidya geometry, Phys. Rev. D 68,064005 (2003);
I. Brevik and G. Halnes, Cosmic evolution and primordial black hole evaporation, Phys. Rev. D 67,023508 (2003);
F. Girotto and A. Saa, Semianalytical approach for the Vaidya metric in double-null coordinates, Phys. Rev. D 70, 084014 (2004);
J. Podolsky and O. Svitek, Radiative spacetimes approaching the Vaidya metric, Phys. Rev. D 71,124001 (2005);
S. A. Hayward, Formation and Evaporation of Nonsingular Black Holes, Phys. Rev. Lett. 96,031103 (2006).
[109] T. Kobayashi, A Vaidya-type radiating solution in Einstein-Gauss-Bonnet gravity and its application to braneworld, Gen. Relativ. Gravit. 37, 1869-1876 (2005);
H. Maeda, Effects of Gauss-Bonnet term on the final fate of gravitational collapse, Class.Quant.Grav. 23 (2006) 2155;
A. E. Dominguez and E. Gallo, Radiating black hole solutions in Einstein-Gauss-Bonnet gravity, Phys. Rev. D 73,064018 (2006);
S. G. Ghosh and D.W. Deshkar, Horizons of radiating black holes in Einstein-Gauss-Bonnet gravity, Phys. Rev. D 77,047504 (2008).
[110] R.-G. Cai, L.-M. Cao, Y.-P. Hu, S. P. Kim, Generalized Vaidya spacetime in Lovelock gravity and thermodynamics on the apparent horizon, Phys. Rev. D 78(2008)124012.
[111] T. P. Singh, Cenalo Vaz, Radiation flux and spectrum in the Vaidya collapse model, Phys. Lett. B 481 (2000) 74-78;S. Q. Wu, X. Cai, Asymmetry of Hawking Radiation of Dirac Particles in a Charged Vaidya - de Sitter Black Hole, Int. J. Theor. Phys. 40 (2001) 1349-1357;S. Q. Wu, X. Cai, Hawking Radiation of Photons in a Vaidya-de Sitter Black Hole, Int. J. Theor. Phys. 41 (2002) 559-567;J. Ren, J. Zhang, Z. Zhao, Tunnelling Effect and Hawking Radiation from a Vaidya Black Hole, Chin. Phys. Lett. 23 (2006) 2019-2022;H. M. Siahaan, Triyanta,Hawking Radiation from a Vaidya Black Hole: A Semi-Classical Approach and Beyond, arXiv:0811.1132.
[112] R. Di Criscienzo, M. Nadalini, L. Vanzo, S. Zerbini, G. Zoccatelli, On the Hawking radiation as tunneling for a class of dynamical black holes, Phys.Lett. B 657 (2007) 107-111;R. Di Criscienzo, L. Vanzo, Fermion tunneling from dynamical horizons,Europhys. Lett. 82 (2008) 60001.
[113] G. C. McVittie, The mass-particle in an expanding universe, Mon. Not. R. Astron. Soc. 93 (1933) 325.
[114] B. C. Nolan, A point mass in an isotropic universe: Existence, uniqueness, and basic properties, Phys. Rev. D 58 (1998) 064006;B. C. Nolan, A point mass in an isotropic universe:Ⅱ.Global properties, Class.Quantum Grav.16 (1999) 1227;B. C. Nolan, A point mass in an isotropic universe:Ⅲ.The region R≤2m,Class. Quantum Grav. 16 (1999) 3183.
[115] C.-J. Gao, S. N. Zhang, Reissner-Nordstrom metric in the Friedman-Robertson-Walker universe, Phys. Lett. B 595 (2004) 28;
C.-J. Gao, Arbitrary-dimensional Schwarzschild-FRW black holes, Class. Quant. Grav. 21 (2004) 4805;
C.-J. Gao, S. N. Zhang, Higher Dimensional Reissner-Nordstr(o|¨)m-FRW metric, Gen. Rel. Grav. 38 (2006) 23-32;
[116] B. Bolen, L. Bombelli, R. Puzio, Expansion-induced contribution to the precession of binary orbits, Class. Quantum Grav. 18 (2001) 1173;
J. Sultana, and C. C. Dyer, Cosmological black holes: A black hole in the Einstein-de Sitter universe, Gen. Relativ. Gravit. 37 (2005) 1347-1370;
V. Kagramanova, J. Kunz, and C. Lammerzahl, Solar system effects in Schwarzschild-de Sitter space-time, Phys. Lett. B 634 (2006) 465-470;
D. J. Shaw, J. D. Barrow, Local effects of cosmological variations in physical "constants" and scalar fields. I. Spherically symmetric spacetimes, Phys. Rev. D 73(2006)123505;
D. J. Shaw, J. D. Barrow, Local effects of cosmological variations in physical "constants" and scalar fields. II. Quasispherical spacetimes, Phys. Rev. D 73 (2006)123506;
E. Hackmann and C. Lammerzahl, Geodesic equation in Schwarzschild-(anti-)de Sitter space-times: Analytical solutions and applications, Phys. Rev. D 78 (2008)024035;
V. Faraoni and A. Jacques, Cosmological expansion and local physics, Phys. Rev. D 76 (2007) 063510;
C. Gao, X. Chen, V. Faraoni, Y. G. Shen, Does the mass of a black hole decrease due to the accretion of phantom energy?, Phys. Rev. D 78 (2008) 024008;
M. Carrera, D. Giulini, On the influence of global cosmological expansion on the dynamics and kinematics of local systems, arXiv:0810.2712;
V. Faraoni, C. Gao, X. Chen, Y.-G. Shen, What is the fate of a black hole embedded in an expanding universe?, Phys. Lett. B 671 (2009)7-9.
[117] M. C. Begelman, Evidence for Black Holes, Science 300 (2003) 1898-1903;
J. Kormendy, D. Richstone, Inward Bound-The Search for Supermassive Black Holes in Galactic Nuclei, Annu. Rev. Astron. Astrophys. 33 (1995) 581-624; M. J. Rees, Black hole models for active galactic nuclei, Annu. Rev. Astron.Astrophys. 22 (1984) 471-506;Zhi-Qiang Shen, K. Y. Lo, M.-C. Liang, Paul T. P. Ho, J.-H. Zhao, A size of -1 AU for the radio source Sgr A~* at the centre of the Milky Way, Nature 438(2005) 62;D. P. Marrone, J. M. Moran, J.-H. Zhao, R. Rao,Interferometric Measurements of Variable 340 GHz Linear Polarization in Sagittarius A~*,Astrophys. J.640(2006)308-318;A. Eckart,R.Schoedel, L. Meyer, S. Trippe, T. Ott, R. Genzel, Polarimetry of near-infrared flares from Sagittarius A~*, Astron. Astrophys. 455 (2006)1.
[118] M. D. Kruskal, Maximal Extension of Schwarzschild Metric, Phys. Rev.119,1743 (1960);P. Szekeres, On the singularities of a Riemannian manifold, Publ.Math. Debrecen, 7 285 (1960).
[119] J. W. Jr. York, What happents to the horizon when a black hole radiation? In: S.Christenses, ed.Quantum theory of gravition: Essays in honour of the sixtieth birthday of B.S.DeWitt,1984,135.
[120] 李淼，引力为什么要量子化?，2006年4月17日，http：//limiao.net/95。
[2] J. D. Bekenstein, Black Holes and Entropy, Phys. Rev. D 7 (1973) 2333-2346.
[3] J. D. Bekenstein, Generalized second law of thermodynamics in black-hole physics, Phys. Rev. D 9 (1974) 3292-3300.
[4] S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43(1975) 199.
[5] J. M. Bardeen, B. Carter, S. W. Hawking, The four laws of black hole mechanics,Commun. Math. Phys. 31 (1973) 161.
[6] R. M. Wald,The Thermodynamics of Black Holes, Living Rev. Relativity 4(2001)6,http://relativity.livingreviews.org/Articles/lrr-2001-6/index.html
[7] J. D. Bekenstein, Do We Understand Black Hole Entropy ?, gr-qc/9409015.
[8] S. A. Fulling, Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time, Phys. Rev. D 7 (1973) 2850-2862.
[9] P. C. W. Davies, Scalar particle production in Schwarzschild and Rindler metrics, J. Phys. A 8 (1975) 609-616.
[10] U. H. Gerlach, The mechanism of blackbody radiation from an incipient black hole, Phys. Rev. D 14 (1976) 1479-1508.
[11] G. t'Hooft, On the quantum structure of a black hole, Nucl.Phys. B 256 (1985)727.
[12] J. W. York, Jr., Dynamical origin of black-hole radiance, Phys. Rev. D 28 (1983)2929.
[13] W. H. Zurek and K. S. Thorne, Statistical Mechanical Origin of the Entropy of a Rotating, Charged Black Hole, Phys. Rev. Lett. 54 (1985) 2171.
[14] L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373.
[15] R. D. Sorkin, The Statistical Mechanics of Black Hole Thermodynamics, gr-qc/9705006.
[16] T. Jacobson, Thermodynamics of Spacetime: The Einstein Equation of State,Phys. Rev. Lett. 75 (1995) 1260.
[17] E. Verlinde, On the Holographic Principle in a Radiation Dominated Universe,hep-th/0008140.
[18] T Padmanabhan, Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes, Class. Quantum Grav.19(2002) 5387-5408.
[19] D. Kothawala, S. Sarkar, T. Padmanabhan, Einstein's equations as a thermodynamic identity-The cases of stationary axisymmetric horizons and evolving spherically symmetric horizons, Phys. Lett. B 652 (2007) 338-342.
[20] A. Paranjape, S. Sarkar, T. Padmanabhan, Thermodynamic route to field equations in Lanczos-Lovelock gravity, Phys. Rev. D 74,104015 (2006).
[21] T Padmanabhan, Gravity and the thermodynamics of horizons, Phys. Rep. 406(2005)49-125.
[22] T Padmanabhan, A. Paranjape, Entropy of null surfaces and dynamics of spacetime, Phys. Rev. D 75 (2007) 064004.
[23] T Padmanabhan, Is gravity an intrinsically quantum phenomenon ? Dynamics of Gravity from the Entropy of Spacetime and the Principle of Equivalence, Mod.Phys. Lett. A17 (2002) 1147.
[24] T Padmanabhan, Topological interpretation of the horizon temperature, Gen.Rel.Grav. 34 (2002) 2029-2035.
[25] G. W. Gibbons and S. W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15 (1977) 2738.
[26] R. G. Cai, Cardy-Verlinde formula and thermodynamics of black holes in de Sitter spaces, Nucl.Phys. B 628,375 (2002).
[27] R. G. Cai, Cardy-Verlinde formula and asymptotically de Sitter spaces, Phys.Lett.B 525,331 (2002).
[28] M. Banados, C. Teitelboim, and J. Zanelli, Black hole in three-dimensional spacetime,Phys. Rev. Lett. 69 (1992) 1849-1851;M. Banados, M. Henneaux, C. Teitelboim, and J. Zanelli, Geometry of the 2+1 black hole, Phys. Rev. D 48 (1993) 1506-1525.
[29] R. Arnowitt, S. Deser and C. Misner, "The Dynamics of General Relativity"in Gravitation: An Introduction to Current Research, ed. L. Witten (New York,Wiley) 1962.
[30] H. Bondi, M. G. J. van der Burg, and A.W.K.Metzner, Gravitational waves in general relativity.Ⅶ.Waves from axi-symmetric isolated systems, Proc.R.Soc.A 269 (1962) 21;R.K.Sachs,Gravitational waves in general relativity.Ⅷ.Waves in asymptotically flat space-time, Proc. R. Soc. A 270,103 (1962).
[31] C. W. Misner and D. H. Sharp, Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse, Phys. Rev. 136 (1964) B571.
[32] J. D. Brown and J.W. York, Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47, 1407 (1993).
[33] C. C. M. Liu and S. T. Yau, Positivity of Quasilocal Mass, Phys. Rev. Lett. 90(2003)231102.
[34] L. B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in GR-A Review Article, Living Rev. Relativity 7 (2004) 4,http://relativity.livingreviews.org/ArticlesArr-2004-4/.
[35] H.Kodama, Conserved Energy Flux for the Spherically Symmetric System and the Backreaction Problem in the Black Hole Evaporation, Prog. Theor. Phys. 63(1980)1217.
[36] S.A. Hay ward, General laws of black-hole dynamics, Phys. Rev. D 49 (1994)6467.
[37] S. A. Hayward, Gravitational energy in spherical symmetry, Phys. Rev. D 53(1996) 1938.
[38] S. A. Hayward, Unified first law of black-hole dynamics and relativistic thermodynamics, Class. Quantum Grav. 15 (1998) 3147.
[39] S. A. Hayward, S. Mukohyama, M. C. Ashworth, Dynamic black-hole entropy,Phys. Lett. A 256 (1999) 347.
[40] M. C. Ashworth, S. A. Hayward, Boundary terms and Noether current of spherical black holes, Phys. Rev. D 60 (1999) 084004.
[41] R.-G. Cai, S. P. Kim, First Law of Thermodynamics and Friedmann Equations of Friedmann-Robertson-Walker Universe, JHEP 02 (2005) 050.
[42] R.-G. Cai, L.-M. Cao, Unified first law and the thermodynamics of the apparent horizon in the FRW universe, Phys. Rev. D 75 (2007) 064008.
[43] M. Akbar, R.-G. Cai, Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe, Phys. Rev. D 75 (2007) 084003.
[44] Y. Gong, A. Wang, Friedmann Equations and Thermodynamics of Apparent Horizons, Phys. Rev. Lett. 99 (2007) 211301.
[45] L. Randall, R. Sundrum,Large Mass Hierarchy from a Small Extra Dimension,Phys. Rev. Lett. 83 (1999) 3370.L.Randall, R. Sundrum, An Alternative to Compactification, Phys. Rev. Lett. 83(1999)4690.
[46] G. Dvali, G. Gabadadze, M. Porrati, Large Mass Hierarchy from a Small Extra Dimension, Phys. Lett. B 485 (2000) 208.G. Dvali, G. Gabadadze, Gravity on a brane in infinite-volume extra space, Phys.Rev.D 63 (2001) 065007.
[47] R. G. Cai and L. M. Cao, Thermodynamics of apparent horizon in brane world scenario, Nucl.Phys. B 785 (2007) 135.
[48] A. Sheykhi, B. Wang and R. G. Cai, Thermodynamical properties of apparent horizon in warped DGP braneworld, Nucl. Phys. B 779 (2007) 1.
[49] A. Sheykhi, B. Wang and R. G. Cai, Deep connection between thermodynamics and gravity in Gauss-Bonnet braneworlds, Phys. Rev. D 76 (2007) 023515.
[50] R. G. Cai, L. M. Cao and Y. P. Hu, Corrected Entropy-Area Relation and Modified Friedmann Equations, JHEP 0808 (2008) 090.
[51] R. G. Cai, Thermodynamics of Apparent Horizon in Brane World Scenarios,Prog. Theor. Phys. Suppl.172 (2008) 100.
[52] X. H. Ge, First law of thermodynamics and Friedmann-like equations in braneworld cosmology, Phys. Lett. B 651 (2007) 49.
[53] S. F. Wu, G. H. Yang and P. M. Zhang, Cosmological equations and Thermodynamics on Apparent Horizon in Thick Braneworld, arXiv:0710.5394.
[54] S. F. Wu, B. Wang and G. H. Yang, Thermodynamics on the apparent horizon in generalized gravity theories, Nucl. Phys. B 799 (2008) 330-344.
[55] T. Zhu, J. R. Ren and S. F. Mo, Thermodynamics of Friedmann Equation and Masslike Function in General Braneworld, arXiv:0805.1162.
[56] P. Kraus and F. Wilczek, Selfinteraction correction to black hole radiance, Nucl.Phys.B 433 (1995) 403.
[57] P. Kraus and F. Wilczek, Effect of self-interaction on charged black hole radiance, Nucl.Phys. B 437 (1995) 231-242.
[58] E. Keski-Vakkuri and P. Kraus, Microcanonical D-branes and back reaction,Nucl.Phys.B 491 (1997)249.
[59] M. K. Parikh, A Secret Tunnel Through the Horizon, Gen.Rel.Grav.36 (2004)2419-2422.
[60] M. K. Parikh and F. Wilczek, Hawking Radiation As Tunneling, Phys. Rev. Lett.85 (2000)5042.
[61] M. K. Parikh, New coordinates for de Sitter space and de Sitter radiation, Phys.Lett. B 546 (2002) 189.
[62] M. K. Parikh, A Secret Tunnel Thwugh the Horizon, Int. J. Mod. Phys. D13(2004)2351.
[63] E. C. Vagenas,Semiclassical corrections to the Bekenstein-Hawking entropy of the BTZ black hole via self-gravitation,Phys. Lett. B 533 (2002)302.
[64] A. J. M. Medved, On Hawking Radiation as Tunneling with Back-Reaction, E.C. Vagenas, Mod. Phys. Lett. A 20 (2005) 2449.
[65] M. Arzano, A. J. M. Medved, E. C. Vagenas, Hawking radiation as tunneling through the quantum horizon, JHEP 0509 (2005) 037.
[66] Q.-Q. Jiang, S.-Q.Wu,X.Cai, Hawking radiation as tunneling from the Kerr and Kerr-Newman black holes, Phys. Rev. D 73 (2006) 064003.
[67] Y. Hu, J. Zhang, Z. Zhao, Massive particles' Hawking radiation via tunneling from the G.H Dilaton black hole, Mod. Phys. Lett. A 21 (2006) 2143.
[68] Y. Hu, J. Zhang, Z. Zhao, Massive uncharged and charged particles' tunneling from the Howwitz-Strominger Dilaton black hole, Int. J. Mod. Phys. D 16 (2007)847-856.
[69] Z. Xu, B. Chen, Hawking Radiation from General Kerr-(anti)de Sitter Black Holes, Phys. Rev. D 75 (2007) 024041.
[70] X. Wu, S. Gao, Tunneling Effect Near Weakly Isolated Horizon, Phys. Rev. D 75(2007)044027.
[71] C.-Z. Liu, J.-Y. Zhu, Hawking radiation as tunneling from Gravity's rainbow, Gen. Relat. Grav. 40 (2008) 1899-1911.
[72] R. Kerner, R. B. Mann, Tunnelling, Temperature and Taub-NUT Black Holes, Phys. Rev. D 73 (2006) 104010.
[73] R. Kerner, R. B. Mann, Tunnelling from Godel black holes, Phys. Rev. D 75 (2007)084022.
[74] B. Chatterjee, A. Ghosh, P. Mitra, Tunnelling from black holes and tunnelling into white holes, Phys. Lett. B 661 (2008) 307-311.
[75] J.-R. Ren, R. Li, F.-H. Liu, Hawking Temperature of Dilaton Black Holes from Tunneling, Mod. Phys. Lett. A 23 (2009) 3419-3429.
[76] R. Emparan and H. S. Reall, Generalized Weyl solutions, Phys. Rev. D 65 (2002) 084025.
[77] R. Emparan and H. S. Reall, A Rotating Black Ring Solution in Five Dimensions, Phys. Rev. Lett. 88 (2002) 101101.
[78] R. Emparan, Rotating Circular Strings, and Infinite Non-Uniqueness of Black Rings, JHEP 03 (2004) 064.
[79] H. K. Kunduri and J. Lucietti, Electrically charged dilatonic black rings, Phys. Lett. B 609 (2005) 143-149.
[80] L. Zhao, Tunneling Through Black Rings, Commun. Theor. Phys. 47 (2007) 835.
[81] Q.-Q. Jiang, Dirac particle tunneling from black rings, Phys. Rev. D 78 (2008) 044009.
[82] R. Banerjee, B. R. Majhi, Quantum tunneling and back reaction, Phys. Lett. B 662 (2008) 62-65.
[83] R. Banerjee, B. R. Majhi, Quantum Tunneling Beyond Semiclassical Approximation, JHEP 0806 (2008) 095.
[84] B. R. Majhi, Fermion Tunneling Beyond Semiclassical Approximation, Phys.Rev. D 79 (2009) 044005.
[85] S. K. Modak, Corrected entropy of BTZ black hole in tunneling approach, Phys.Lett.B 671 (2009) 167-173.
[86] E. T. Akhmedov, V. Akhmedova, D. Singleton,Integrability and higher loops in AdS/dCFT correspondence, Phys. Lett. B 642 (2006) 124;
E. T. Akhmedov, V. Akhmedova, T. Pilling, D. Singleton, Thermal radiation of various gravitational backgrounds, Int. J. Mod. Phys. A 2 (2007) 1705;
T. K. Nakamura, Factor Two Discrepancy of Hawking Radiation Temperature,hep-th/0706.2916;
P. Mitra, Hawking temperature from tunnelling formalism, Phys. Lett. B 648(2007) 240;
E. T. Akhmedov, T. Pilling, D. Singleton, Subtleties in the quasi-classical calculation of Hawking radiation, hep-th/0805.2653;
V. Akhmedova, T. Pilling, A. de Gill, and D. Singleton, Comments on anomaly versus WKB/tunneling methods for calculating Unruh radiation,arXiv:0808.3413;
B. D. Chowdhury, Problems with Tunneling of Thin Shells from Black Holes,Pramana 70(2008) 593-612;
T. Pilling, Quasi-classical Hawking Temperatures and Black Hole Thermodynamics, arXiv:0809.2701;
V. Akhmedova, T. Pilling, A. de Gill, D. Singleton, Temporal contribution to gravitational WKB-like calculations, Phys. Lett. B 666 (2008) 269-271.
[87] G. W. Gibbons and S. W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15 (1977) 2738.
[88] W. G. Unruh, Notes on black-hole evaporation, Phys. Rev. D14(1976) 870. [89] P. Hajicek, Origin of Hawking radiation, Phys. Rev. D 36 (1987) 1065.
[90] M. Visser, Essential and inessential features of Hawking radiation, Int. J. Mod.Phys. D 12 (2003) 649.
[91] S. A. Hayward, R. Di Criscienzo, M. Nadalini, L. Vanzo, S. Zerbini, Local Hawking temperature for dynamical black holes, arXiv:0806.0014.
[92] S. A. Hayward, Dynamics of black holes, arXiv:0810.0923.
[93] S. A. Hayward, R. Di Criscienzo, M. Nadalini, L. Vanzo, S. Zerbini, Local temperature for dynamical black holes, arXiv:0812.2534.
[94] R.-G. Cai, L.-M. Cao, Y.-P. Hu, Hawking Radiation of Apparent Horizon in a FRW Universe, arXiv:0809.1554.
[95] J. W. York Jr., Black hole in thermal equilibrium with a scalar field: The backreaction, Phys. Rev. D 31 (1985) 775.
[96] C. O. Lousto, N. Sanchez, Back reaction effects in black hole spacetimes, Phys.Lett.B 212 (1988) 411-414.
[97] D. V. Fursaev, Temperature and entropy of a quantum black hole and conformal anomaly, Phys. Rev. D 51 (1995) R5352-R5355.
[98] R. B. Mann, S. N. Solodukhin, Universality of quantum entropy for extreme black holes, Nucl.Phys. B 523 (1998) 293.
[99] R. K. Kaul, P. Majumdar, Logarithmic Correction to the Bekenstein-Hawking Entropy, Phys. Rev. Lett. 84 (2000) 5255-5257.
[100] A. Ghosh, P. Mitra, An improved estimate of black hole entropy in the quantum geometry approach, Phys. Lett. B 616 (2005) 114-117.
[101] D. N. Page, Hawking Radiation and Black Hole Thermodynamics, New J. Phys.7 (2005) 203.
[102] R. Banerjee, B. R. Majhi, Quantum Tunneling Beyond Semiclassical Approximation, JHEP 06 (2008) 095.
[103] K. Srinivasan, T. Padmanabhan, Particle production and complex path analysis,Phys. Rev. D 60 (1999) 024007.
[104] S. Sarkar, D. Kothawala, Hawking radiation as tunneling for spherically symmetric black holes-A generalized treatment, Phys. Lett. B 659 (2008) 683-687.
[105] T. Pilling, Black hole thermodynamics and the factor of 2 problem, Phys. Lett.B 660 (2008) 402.
[106] R. Li, J.-R. Ren, D.-F. Shi, Fermions tunneling from apparent horizon of FRW universe, Phys. Lett. B 670 (2009) 446.
[107] P. C. Vaidya, The External Field of a Radiating Star in General Relativity, Curr.Sci. 12,183 (1943), Reprinted: Gen. Relativ. Gravit. 31 (1999) 119-120;P. C. Vaidya, The Gravitational Field of a Radiating Star, Proc.Indian Acad.Sci. A 33 (1951) 264, Reprinted: Gen. Relativ. Gravit. 31 (1999) 121-135;P. C. Vaidya, Newtonian' Time in General Relativity, Nature 171 (1953) 260,Reprinted: Gen. Relativ. Gravit. 31 (1999) 137-138 V. V. Narlikar, P. C. Vaidya, A Spherically Symmetrical Non-Static Electromagnetic Field, Nature 159 (1947) 642.
[108] W. B. Bonnor, P. C. Vaidya, Spherically symmetric radiation of charge in Einstein-Maxwell theory,Gen. Relativ. Gravit.1(1970) 127-130;J. M. Bardeen, Black Holes Do Evaporate Thermally, Phys. Rev. Lett. 46 (1981)382;W. A. Hiscock, Models of evaporating black holes.Ⅰ,Phys. Rev. D 23, 2813(1981);W. A. Hiscock, Models of evaporating black holes.Ⅱ.Effects of the outgoing created radiation, Phys. Rev. D 23, 2823 (1981);W. A. Hiscock, L. G. Williams, and D. M. Eardley, Creation of particles by shell-focusing singularities, Phys. Rev. D 26,751 (1982);J. W. York, Dynamical origin of black-hole radiance, Phys. Rev. D 28 (1983)2929;Y. Kuroda, A Model for Evaporating Black Holes, Prog. Theor. Phys. 71, 100 (1984);
Y. Kuroda, Vaidya spacetime as an evaporating black hole, Prog. Theor. Phys. 71,1422(1984);
R. L. Mallet, Radiating Vaidya metric imbedded in de Sitter space, Phys. Rev. D 31 (1985)416;
J. Bi(c|ˇ)ák and K.V. Kucha(r|ˇ), Null dust in canonical gravity, Phys. Rev. D 56,4878 (1997);
J. Bi(c|ˇ)ák and P. Hájí(c|ˇ)ek, Canonical theory of spherically symmetric spacetimes with cross-streaming null dusts, Phys. Rev. D 68, 104016 (2003);
S. G. Ghosh and N. Dadhich, Naked singularities in higher dimensional Vaidya space-times, Phys. Rev. D 64,047501 (2001);
T. Harko, Gravitational collapse of a Hagedorn fluid in Vaidya geometry, Phys. Rev. D 68,064005 (2003);
I. Brevik and G. Halnes, Cosmic evolution and primordial black hole evaporation, Phys. Rev. D 67,023508 (2003);
F. Girotto and A. Saa, Semianalytical approach for the Vaidya metric in double-null coordinates, Phys. Rev. D 70, 084014 (2004);
J. Podolsky and O. Svitek, Radiative spacetimes approaching the Vaidya metric, Phys. Rev. D 71,124001 (2005);
S. A. Hayward, Formation and Evaporation of Nonsingular Black Holes, Phys. Rev. Lett. 96,031103 (2006).
[109] T. Kobayashi, A Vaidya-type radiating solution in Einstein-Gauss-Bonnet gravity and its application to braneworld, Gen. Relativ. Gravit. 37, 1869-1876 (2005);
H. Maeda, Effects of Gauss-Bonnet term on the final fate of gravitational collapse, Class.Quant.Grav. 23 (2006) 2155;
A. E. Dominguez and E. Gallo, Radiating black hole solutions in Einstein-Gauss-Bonnet gravity, Phys. Rev. D 73,064018 (2006);
S. G. Ghosh and D.W. Deshkar, Horizons of radiating black holes in Einstein-Gauss-Bonnet gravity, Phys. Rev. D 77,047504 (2008).
[110] R.-G. Cai, L.-M. Cao, Y.-P. Hu, S. P. Kim, Generalized Vaidya spacetime in Lovelock gravity and thermodynamics on the apparent horizon, Phys. Rev. D 78(2008)124012.
[111] T. P. Singh, Cenalo Vaz, Radiation flux and spectrum in the Vaidya collapse model, Phys. Lett. B 481 (2000) 74-78;S. Q. Wu, X. Cai, Asymmetry of Hawking Radiation of Dirac Particles in a Charged Vaidya - de Sitter Black Hole, Int. J. Theor. Phys. 40 (2001) 1349-1357;S. Q. Wu, X. Cai, Hawking Radiation of Photons in a Vaidya-de Sitter Black Hole, Int. J. Theor. Phys. 41 (2002) 559-567;J. Ren, J. Zhang, Z. Zhao, Tunnelling Effect and Hawking Radiation from a Vaidya Black Hole, Chin. Phys. Lett. 23 (2006) 2019-2022;H. M. Siahaan, Triyanta,Hawking Radiation from a Vaidya Black Hole: A Semi-Classical Approach and Beyond, arXiv:0811.1132.
[112] R. Di Criscienzo, M. Nadalini, L. Vanzo, S. Zerbini, G. Zoccatelli, On the Hawking radiation as tunneling for a class of dynamical black holes, Phys.Lett. B 657 (2007) 107-111;R. Di Criscienzo, L. Vanzo, Fermion tunneling from dynamical horizons,Europhys. Lett. 82 (2008) 60001.
[113] G. C. McVittie, The mass-particle in an expanding universe, Mon. Not. R. Astron. Soc. 93 (1933) 325.
[114] B. C. Nolan, A point mass in an isotropic universe: Existence, uniqueness, and basic properties, Phys. Rev. D 58 (1998) 064006;B. C. Nolan, A point mass in an isotropic universe:Ⅱ.Global properties, Class.Quantum Grav.16 (1999) 1227;B. C. Nolan, A point mass in an isotropic universe:Ⅲ.The region R≤2m,Class. Quantum Grav. 16 (1999) 3183.
[115] C.-J. Gao, S. N. Zhang, Reissner-Nordstrom metric in the Friedman-Robertson-Walker universe, Phys. Lett. B 595 (2004) 28;
C.-J. Gao, Arbitrary-dimensional Schwarzschild-FRW black holes, Class. Quant. Grav. 21 (2004) 4805;
C.-J. Gao, S. N. Zhang, Higher Dimensional Reissner-Nordstr(o|¨)m-FRW metric, Gen. Rel. Grav. 38 (2006) 23-32;
[116] B. Bolen, L. Bombelli, R. Puzio, Expansion-induced contribution to the precession of binary orbits, Class. Quantum Grav. 18 (2001) 1173;
J. Sultana, and C. C. Dyer, Cosmological black holes: A black hole in the Einstein-de Sitter universe, Gen. Relativ. Gravit. 37 (2005) 1347-1370;
V. Kagramanova, J. Kunz, and C. Lammerzahl, Solar system effects in Schwarzschild-de Sitter space-time, Phys. Lett. B 634 (2006) 465-470;
D. J. Shaw, J. D. Barrow, Local effects of cosmological variations in physical "constants" and scalar fields. I. Spherically symmetric spacetimes, Phys. Rev. D 73(2006)123505;
D. J. Shaw, J. D. Barrow, Local effects of cosmological variations in physical "constants" and scalar fields. II. Quasispherical spacetimes, Phys. Rev. D 73 (2006)123506;
E. Hackmann and C. Lammerzahl, Geodesic equation in Schwarzschild-(anti-)de Sitter space-times: Analytical solutions and applications, Phys. Rev. D 78 (2008)024035;
V. Faraoni and A. Jacques, Cosmological expansion and local physics, Phys. Rev. D 76 (2007) 063510;
C. Gao, X. Chen, V. Faraoni, Y. G. Shen, Does the mass of a black hole decrease due to the accretion of phantom energy?, Phys. Rev. D 78 (2008) 024008;
M. Carrera, D. Giulini, On the influence of global cosmological expansion on the dynamics and kinematics of local systems, arXiv:0810.2712;
V. Faraoni, C. Gao, X. Chen, Y.-G. Shen, What is the fate of a black hole embedded in an expanding universe?, Phys. Lett. B 671 (2009)7-9.
[117] M. C. Begelman, Evidence for Black Holes, Science 300 (2003) 1898-1903;
J. Kormendy, D. Richstone, Inward Bound-The Search for Supermassive Black Holes in Galactic Nuclei, Annu. Rev. Astron. Astrophys. 33 (1995) 581-624; M. J. Rees, Black hole models for active galactic nuclei, Annu. Rev. Astron.Astrophys. 22 (1984) 471-506;Zhi-Qiang Shen, K. Y. Lo, M.-C. Liang, Paul T. P. Ho, J.-H. Zhao, A size of -1 AU for the radio source Sgr A~* at the centre of the Milky Way, Nature 438(2005) 62;D. P. Marrone, J. M. Moran, J.-H. Zhao, R. Rao,Interferometric Measurements of Variable 340 GHz Linear Polarization in Sagittarius A~*,Astrophys. J.640(2006)308-318;A. Eckart,R.Schoedel, L. Meyer, S. Trippe, T. Ott, R. Genzel, Polarimetry of near-infrared flares from Sagittarius A~*, Astron. Astrophys. 455 (2006)1.
[118] M. D. Kruskal, Maximal Extension of Schwarzschild Metric, Phys. Rev.119,1743 (1960);P. Szekeres, On the singularities of a Riemannian manifold, Publ.Math. Debrecen, 7 285 (1960).
[119] J. W. Jr. York, What happents to the horizon when a black hole radiation? In: S.Christenses, ed.Quantum theory of gravition: Essays in honour of the sixtieth birthday of B.S.DeWitt,1984,135.
[120] 李淼，引力为什么要量子化?，2006年4月17日，http：//limiao.net/95。