静态球对称时空的反作用、统计热力学及其de Broglie-Bohm量子化
摘要
球对称引力场是引力理论中最重要而又最简单的模型之一,它已为人们广泛的研究。本文将重点研究Reissner-Nordstr(?)m时空、de Sitter时空以及Reissner-Nordstr(?)m-de Sitter时空中的一些量子效应及热力学性质.同时也对Brans-Dicke引力理论中的球对称黑洞的量子化做了一些尝试。本文分为三个部分:
第一部分研究黑洞热辐射对时空的反作用问题,考虑了量子效应以后,Hawking发现黑洞具有辐射谱为黑体谱的热辐射。这些存在于黑洞周围的热辐射场必定会反过来引起时空度规发生改变。这就是黑洞的反作用问题。1985年York提出了一种解决黑洞反作用问题的微扰方法,将在背景时空上求得的量子场的真空能动张量作为引起背景时空改变的源,再求解半经典爱因斯坦场方程,从而求得受量子扰动后的度规。York用这种方法计算了共形不变标量场对史瓦西时空的反作用。这种方法的前提是首先要求得在背景时空中的量子场的真空重整化能动张量。
第一部分包括第一章和两个附录(附录A、B)。在第一章里,我们利用黄超光求得的Reissner-Nordstr(?)m(RN)时空中的共形不变标量场的重整化能动张量,求解了共形不变标量场对RN时空的反作用,考察了受扰动后的RN度规的性质。利用求得的度规我们还讨论了RN黑洞的柯西(Cauchy)视界的不稳定性。我们发现,考虑了量子场的反作用后,扰动前的类光柯西视界变成了扰动后的类空超曲面。在附录里,我们简要地介绍了重整化能动张量的概念及其求解的方法。
第二部分采用引力场的欧氏化作用量探讨了黑洞的统计热力学问题。这一部分包括三章(第二、三、四章)。Hawking提出的引力场的路径积分量子化方法是研究黑洞热力学的有力工具,在第二章里我们首先简要介绍了Hawking的路径积分方法,然后介绍了York等人发展的York形式的路径积分方法。York形式的路径积分方法要用一个半径有限的球形空腔把黑洞包在里面,通过在边界上给出不同的边界条件就可以建立不同的热力学系综。这种方法有它固有的优点,并已得到广泛的应用。我们发现如果将球形边界取在宇宙视界里面,则可以将York
The spherically symmetric space-time, which is widely studied, is one of the simple and important models in gravitational theory. The contents of the present thesis are mainly focus on some effects of quantum and thermodynamical properties for Reissner-Nordstrom (RN) space-time, de Sitter space-time and Reissner-Nordstrom-de Sitter (RNdS) space-times. In addition we try to study the quantum effect of the spherically symmetric black hole in Brans-Dicke theory in view of the de Broglie-Bohm (dBB) interpretation. The thesis is composed of three parts as follows.In Part one, we study that the back-reaction of the thermal radiation to the black hole geometry. By consideration of quantum field theory in curved spacetime, Hawking in 1974 found that the black hole should emit a black body radiation. However, the radiation will alter the background geometry according to the semiclassical Einstein equation. In 1985 York introduced a perturbed method to solve the back-reaction problem, which regard the renormalized energy-stress tensoe of a vacuum quantum matter field as the source of gravity and then solve the semiclassical Einstein equation to obtain the perturbed geometry. As an example, York found the back-reaction of the conformally invariant scalar field on the Schwarzschild black hole. The most important thing in this method is obtained a renormalized energy-stress tensor of the vacuum quantum matter field.Part one includes one Chapter and two appendixes (Appendix A and B). In chapter one, by using the renormalized energy-stress tensor of the conformally invariant scalar field given by Huang, we study the back-reaction on RN space-time and then the properties of the perturbed RN spacetime. We also discuss the stability of the Cauchy horizon in RN spacetime and find that it will become spacelike hypersurface in perturbed RN spacetime. In Appendix A and B. we give a brief introduction on the renormalized energy-stress tensor and its calculation.In Part two we study the statistical thermodynamics of the black hole by using the Euclidean action of the gravity. This Part includes three chapters (Chapter two,
Chapter three and Chapter four). In Chapter two, we briefly introduce Hawking's path-integral approach to quantum gravity and York's formalism developed from the Hawking's path-integral approach. In York's formalism, the black hole is surrounded by a cavity of a boundary with a finite radius. By fixing various data on the boundary we can get various thermodynamical ensemble. We find that if the boundary is located inside the Cosmological horizon, the York's formalism can be generalized to the asymptotically de Sitter sapcetime. So using the York's formalism we study the de Sitter spacetime (the first section of the Chapter three) and luckwarm black hole ( the second section of the Chapter four) and then obtain their first law of the thermodynamics.In Second two of the Chapter three, considering the universe is not "empty" we investigate the thermodynamics of the de Sitter universe by using a grand canonical ensemble, and gives the first law of thermodynamics including the vacuum energy and pressure contributed by the cosmological constant.In Second three of the Chapter three, using the canonical ensemble and anti-Wick rotation we study again the thermodynamics of the de Sitter spacetime. Many years ago, Hawking conjectured that the spacetime foam may be formed from the S2 x S2 bubble. Recently Liu shows that the S2 x S2 bubble can be created really from vacuum fluctuation in one-loop approximation of both steady state universe and closed de Sitter universe. Using this result, we calculate the one-loop correction of the entropy for de Sitter sapcetime in the Section three of the Chapter three.In chapter four we first briefly introduce some special UN spacetime, and then using the York's formlism we study the thermodynamics of the luckwarm black hole which is a RNdS spacetime in thermal equilibrium.In Part three (the Chapter five), using the canonical quantum approach of gravity, an spherically symmetry black hole in BD theory is studied in view of the dBB interpretation. The Wheeler-DeWitt(WD) equation is solved in a minisuperspace model. The dBB interpretation of the wave function is different from the Copenhagen interprata-
tion, the latter is the usual probability interpretation of the wave function. According to the dBB interpretation the wave function is considered to have a " trajectory ", which gives the quantum trajectory and the quantum potential. When the quantum effect can be negligible, the quantum potential will tends to zero and the quantum trajectory will reduced to the classical trajectory. Kenmoku et al investigated the quantum effects of the Schwarzschild black hole by using the canonical quantum approach in view of the dBB interpretation. We generalize their method to the BD theory and study the quantum effect of the Brans type I black hole, which is a non-trivial BD black hole solution different from the usual Schwarzschild solution. In addition we also discuss some thermodynamical properties of the quantum Brans type I.
第一部分研究黑洞热辐射对时空的反作用问题,考虑了量子效应以后,Hawking发现黑洞具有辐射谱为黑体谱的热辐射。这些存在于黑洞周围的热辐射场必定会反过来引起时空度规发生改变。这就是黑洞的反作用问题。1985年York提出了一种解决黑洞反作用问题的微扰方法,将在背景时空上求得的量子场的真空能动张量作为引起背景时空改变的源,再求解半经典爱因斯坦场方程,从而求得受量子扰动后的度规。York用这种方法计算了共形不变标量场对史瓦西时空的反作用。这种方法的前提是首先要求得在背景时空中的量子场的真空重整化能动张量。
第一部分包括第一章和两个附录(附录A、B)。在第一章里,我们利用黄超光求得的Reissner-Nordstr(?)m(RN)时空中的共形不变标量场的重整化能动张量,求解了共形不变标量场对RN时空的反作用,考察了受扰动后的RN度规的性质。利用求得的度规我们还讨论了RN黑洞的柯西(Cauchy)视界的不稳定性。我们发现,考虑了量子场的反作用后,扰动前的类光柯西视界变成了扰动后的类空超曲面。在附录里,我们简要地介绍了重整化能动张量的概念及其求解的方法。
第二部分采用引力场的欧氏化作用量探讨了黑洞的统计热力学问题。这一部分包括三章(第二、三、四章)。Hawking提出的引力场的路径积分量子化方法是研究黑洞热力学的有力工具,在第二章里我们首先简要介绍了Hawking的路径积分方法,然后介绍了York等人发展的York形式的路径积分方法。York形式的路径积分方法要用一个半径有限的球形空腔把黑洞包在里面,通过在边界上给出不同的边界条件就可以建立不同的热力学系综。这种方法有它固有的优点,并已得到广泛的应用。我们发现如果将球形边界取在宇宙视界里面,则可以将York
The spherically symmetric space-time, which is widely studied, is one of the simple and important models in gravitational theory. The contents of the present thesis are mainly focus on some effects of quantum and thermodynamical properties for Reissner-Nordstrom (RN) space-time, de Sitter space-time and Reissner-Nordstrom-de Sitter (RNdS) space-times. In addition we try to study the quantum effect of the spherically symmetric black hole in Brans-Dicke theory in view of the de Broglie-Bohm (dBB) interpretation. The thesis is composed of three parts as follows.In Part one, we study that the back-reaction of the thermal radiation to the black hole geometry. By consideration of quantum field theory in curved spacetime, Hawking in 1974 found that the black hole should emit a black body radiation. However, the radiation will alter the background geometry according to the semiclassical Einstein equation. In 1985 York introduced a perturbed method to solve the back-reaction problem, which regard the renormalized energy-stress tensoe of a vacuum quantum matter field as the source of gravity and then solve the semiclassical Einstein equation to obtain the perturbed geometry. As an example, York found the back-reaction of the conformally invariant scalar field on the Schwarzschild black hole. The most important thing in this method is obtained a renormalized energy-stress tensor of the vacuum quantum matter field.Part one includes one Chapter and two appendixes (Appendix A and B). In chapter one, by using the renormalized energy-stress tensor of the conformally invariant scalar field given by Huang, we study the back-reaction on RN space-time and then the properties of the perturbed RN spacetime. We also discuss the stability of the Cauchy horizon in RN spacetime and find that it will become spacelike hypersurface in perturbed RN spacetime. In Appendix A and B. we give a brief introduction on the renormalized energy-stress tensor and its calculation.In Part two we study the statistical thermodynamics of the black hole by using the Euclidean action of the gravity. This Part includes three chapters (Chapter two,
Chapter three and Chapter four). In Chapter two, we briefly introduce Hawking's path-integral approach to quantum gravity and York's formalism developed from the Hawking's path-integral approach. In York's formalism, the black hole is surrounded by a cavity of a boundary with a finite radius. By fixing various data on the boundary we can get various thermodynamical ensemble. We find that if the boundary is located inside the Cosmological horizon, the York's formalism can be generalized to the asymptotically de Sitter sapcetime. So using the York's formalism we study the de Sitter spacetime (the first section of the Chapter three) and luckwarm black hole ( the second section of the Chapter four) and then obtain their first law of the thermodynamics.In Second two of the Chapter three, considering the universe is not "empty" we investigate the thermodynamics of the de Sitter universe by using a grand canonical ensemble, and gives the first law of thermodynamics including the vacuum energy and pressure contributed by the cosmological constant.In Second three of the Chapter three, using the canonical ensemble and anti-Wick rotation we study again the thermodynamics of the de Sitter spacetime. Many years ago, Hawking conjectured that the spacetime foam may be formed from the S2 x S2 bubble. Recently Liu shows that the S2 x S2 bubble can be created really from vacuum fluctuation in one-loop approximation of both steady state universe and closed de Sitter universe. Using this result, we calculate the one-loop correction of the entropy for de Sitter sapcetime in the Section three of the Chapter three.In chapter four we first briefly introduce some special UN spacetime, and then using the York's formlism we study the thermodynamics of the luckwarm black hole which is a RNdS spacetime in thermal equilibrium.In Part three (the Chapter five), using the canonical quantum approach of gravity, an spherically symmetry black hole in BD theory is studied in view of the dBB interpretation. The Wheeler-DeWitt(WD) equation is solved in a minisuperspace model. The dBB interpretation of the wave function is different from the Copenhagen interprata-
tion, the latter is the usual probability interpretation of the wave function. According to the dBB interpretation the wave function is considered to have a " trajectory ", which gives the quantum trajectory and the quantum potential. When the quantum effect can be negligible, the quantum potential will tends to zero and the quantum trajectory will reduced to the classical trajectory. Kenmoku et al investigated the quantum effects of the Schwarzschild black hole by using the canonical quantum approach in view of the dBB interpretation. We generalize their method to the BD theory and study the quantum effect of the Brans type I black hole, which is a non-trivial BD black hole solution different from the usual Schwarzschild solution. In addition we also discuss some thermodynamical properties of the quantum Brans type I.
引文
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