黑洞热力学的相关研究及时空的Killing约化
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摘要
本论文总结了作者三方面的工作: (1)讨论动态黑洞的量子热效应; (2)将广义测不准关系用于黑洞熵的讨论; (3)研究Killing对称性时空的约化。全文共分五章,第一章给出了作者工作的背景和相关理论的综述,第二、三、四章是作者自己的工作,第五章是工作总结和对今后工作的展望。
第一方面的工作,即第二章研究了缓变动态黑洞,包括任意加速的旋转黑洞和动态Kerr-Newman黑洞的量子热效应。首先,研究了任意加速的旋转黑洞,对缓变动态黑洞引入了局域热平衡的概念;求出了黑洞的辐射温度,其随时间和视界面上的位置而变化,是一个局域量;给出了Hawking辐射谱公式,其为准黑体谱;得到了黑洞熵,只当取某一特定的截断关系该熵才正比于黑洞的视界面面积。讨论中给出了两种截断关系,其一非常简单且因与静态球对称的截断关系相同而显示出动态黑洞和静态球对称黑洞之间似乎有某种共性,其二比较复杂但更显合理且与一些已知的较简单的动态黑洞或非球对称黑洞一样,其截断也均偏离静态球对称黑洞的截断关系。然后,研究了动态Kerr-Newman黑洞,得到了黑洞的辐射温度,其仍然是一个局域量;给出了Hawking热谱公式,其仍为准黑体谱;求出了黑洞熵,当取某一特定的截断关系时该熵与黑洞视界面面积成正比。本章对这两种动态黑洞的研究表明,改进的Damour-Rulilni方法和薄膜模型能够有效地用到比较复杂的动态黑洞上面去,唯一的条件是这种动态黑洞为缓变的。
第二方面的工作,即第三章讨论了广义测不准关系对黑洞熵计算的影响。首先,将广义测不准关系用于砖墙模型讨论了Schwarzschild黑洞,论证了当两堵砖墙之间的距离足够大时我们从熵表达式中得不到正比于黑洞视界面面积的项。从而表明,当用广义测不准关系讨论黑洞熵时,砖墙模型会失效。然后,将广义测不准关系与薄膜模型结合起来,分别计算了黑洞视界面上的Planck薄层内的Klein-Gordon场和Dirac场的统计熵,既自然地去掉了截断又得到了正比于黑洞视界面面积的黑洞熵。本章的工作进一步说明,将广义测不准关系用于黑洞热力学的讨论是有效的,薄膜模型是合理的。
第三方面的工作,即第四章给出了四维和五维时空的Killing约化。首先,讨论了由四维Hilbert作用量作Killing约化而来的三维作用量,对其变分得到的方程与Geroch
The paper gives a summary of the author' s research work in three aspects: (1) discussing the quantum thermal effect of nonstationary black hole; (2) describing the application of the generalized uncertainty relation to the discussion of black hole entropy; (3) studying the reduction of spacetimes with a Killing symmetry. There are five chapters in the paper. Chapter 1 provides a review about the background and theories related to the work of the author. Chapter 2,3,4 summarize the research work of the author. Chapter 5 gives the summarization and prospects of the research work.
The first work described in chapter 2 is the research on the quantum thermal effect of a radiating rotating arbitrarily accelerating black hole and a nonstationary Kerr-Newman black hole, whose metrics are changing slowly. First, a radiating rotating arbitrarily accelerating black hole is studied. The concept of local thermal equilibrium is introduced into a nonstationary black hole whose metric is changing slowly; the radiation temperature is obtained which depends on the time and the angles on the horizon and is a local quantity; the Hawking thermal spectrum formula is gained which is a quasi-black-body' s thermal spectrum; The black hole entropy is calculated which is proportional to the horizon area only with a appropriate geometrical cutoff relationship. Alternative cutoff relationships are got through discussion. One of them is very simple and it seems that there is something in common between the nonstationary black hole and a spherical and static black hole due to the same of their cutoff relationships. Another of them is more complicated but more reasonable and consistent with the ones of some known more simple nonstationary black hole and non-spherical black hole, whose cutoff relationships also deviate static and spherically symmetric case. Then, a nonstationary Kerr-Newman black hole is researched. The radiation temperature is calculated which is also a local quantity; the Hawking thermal spectrum formula is obtained which is also a quasi-black-body' s thermal spectrum; The black hole entropy is gained and is proportional to the horizon area with a suitable cutoff relationship. The research on the two nonstationary black holes shows that the improved method of Damour-Ruffini and the thin film model
can be effectively applied to the study of a complex black hole on the only condition that nonstationary metric is changing slowly.The next work summarized in chapter 3 is the study of how the generalized uncertainty relation influence the calculation of black hole entropy. First, the generalized uncertainty relation and the brick wall model are utilized to investigate Schwarzschild black hole. The conclusion is expounded and proved that we can't obtain the black hole entropy proportional to the horizon area from the expression of entropy when the distance of two brick walls is long enough. So, we can conclude that the brick wall model is ineffectual while the generalized uncertainty relation is applied to the discussion of black hole entropy. Then, the statistical entropies of Klein-Gordon field and Dirac field inside a Planck thin film on the horizon surface are calculated respectively by combining the generalized uncertainty relation with the thin film model. This leads to both removing the cutoff and acquiring a black hole entropy proportional to the horizon area. The work of this chapter testifies further that applying the generalized uncertainty relation to the study of black hole thermodynamics is effective and the thin film model is rational.The final work given in chapter 3 is the investigation on Killing reductions of 4-dimensional and 5-dimensional spacetimes. First, the 3-dimensional action obtained from the Killing reduction of 4-dimensional action is discussed, the variations of which lead to the same field equations as those reduced from the vacuum Einstein equation by Ge-roch; Then, a series of equations parallel to Geroch's are acquired by extending Geroch's approach to a 5-dimensional spacetime with a Killing symmetry, this provides a new version 5-dimensional Kaluza-Klein theory during the spacelike Killing vector field and a vacuum case. Next, the Killing reduction of spacetime is studied from the viewpoint of variation principle. It turns out that the symmetry-reduced 4-dimensional action from the 5-dimensional Hilbert action would give the correct field equations reduced from 5-dimensional vacuum spacetime with a Killing symmetry by variations with respect to a series of suitable field variables. Finally, an alternative special 4-dimensional action is obtained. Its variations with respect to the other series of field variables can also give the
same field equations reduced from 5-dimensional vacuum field equations. The research described in this chapter shows that the final result of " n-dimensional action^=> reduced (n-l)-dimensional action^=> (n-l)-dimensional spacetime field equations obtained by variations " and that of " n-dimensional action^=> n-dimensional spacetime field equations obtained by variations^=> (n-1)-dimensional spacetime field equations obtained by reduction of n-dimensional field equations " are the same and 5-dimensional gravity with a Killing symmetry is equivalent to 4-dimensional gravity coupled to a vector field and a scalar field.
第一方面的工作,即第二章研究了缓变动态黑洞,包括任意加速的旋转黑洞和动态Kerr-Newman黑洞的量子热效应。首先,研究了任意加速的旋转黑洞,对缓变动态黑洞引入了局域热平衡的概念;求出了黑洞的辐射温度,其随时间和视界面上的位置而变化,是一个局域量;给出了Hawking辐射谱公式,其为准黑体谱;得到了黑洞熵,只当取某一特定的截断关系该熵才正比于黑洞的视界面面积。讨论中给出了两种截断关系,其一非常简单且因与静态球对称的截断关系相同而显示出动态黑洞和静态球对称黑洞之间似乎有某种共性,其二比较复杂但更显合理且与一些已知的较简单的动态黑洞或非球对称黑洞一样,其截断也均偏离静态球对称黑洞的截断关系。然后,研究了动态Kerr-Newman黑洞,得到了黑洞的辐射温度,其仍然是一个局域量;给出了Hawking热谱公式,其仍为准黑体谱;求出了黑洞熵,当取某一特定的截断关系时该熵与黑洞视界面面积成正比。本章对这两种动态黑洞的研究表明,改进的Damour-Rulilni方法和薄膜模型能够有效地用到比较复杂的动态黑洞上面去,唯一的条件是这种动态黑洞为缓变的。
第二方面的工作,即第三章讨论了广义测不准关系对黑洞熵计算的影响。首先,将广义测不准关系用于砖墙模型讨论了Schwarzschild黑洞,论证了当两堵砖墙之间的距离足够大时我们从熵表达式中得不到正比于黑洞视界面面积的项。从而表明,当用广义测不准关系讨论黑洞熵时,砖墙模型会失效。然后,将广义测不准关系与薄膜模型结合起来,分别计算了黑洞视界面上的Planck薄层内的Klein-Gordon场和Dirac场的统计熵,既自然地去掉了截断又得到了正比于黑洞视界面面积的黑洞熵。本章的工作进一步说明,将广义测不准关系用于黑洞热力学的讨论是有效的,薄膜模型是合理的。
第三方面的工作,即第四章给出了四维和五维时空的Killing约化。首先,讨论了由四维Hilbert作用量作Killing约化而来的三维作用量,对其变分得到的方程与Geroch
The paper gives a summary of the author' s research work in three aspects: (1) discussing the quantum thermal effect of nonstationary black hole; (2) describing the application of the generalized uncertainty relation to the discussion of black hole entropy; (3) studying the reduction of spacetimes with a Killing symmetry. There are five chapters in the paper. Chapter 1 provides a review about the background and theories related to the work of the author. Chapter 2,3,4 summarize the research work of the author. Chapter 5 gives the summarization and prospects of the research work.
The first work described in chapter 2 is the research on the quantum thermal effect of a radiating rotating arbitrarily accelerating black hole and a nonstationary Kerr-Newman black hole, whose metrics are changing slowly. First, a radiating rotating arbitrarily accelerating black hole is studied. The concept of local thermal equilibrium is introduced into a nonstationary black hole whose metric is changing slowly; the radiation temperature is obtained which depends on the time and the angles on the horizon and is a local quantity; the Hawking thermal spectrum formula is gained which is a quasi-black-body' s thermal spectrum; The black hole entropy is calculated which is proportional to the horizon area only with a appropriate geometrical cutoff relationship. Alternative cutoff relationships are got through discussion. One of them is very simple and it seems that there is something in common between the nonstationary black hole and a spherical and static black hole due to the same of their cutoff relationships. Another of them is more complicated but more reasonable and consistent with the ones of some known more simple nonstationary black hole and non-spherical black hole, whose cutoff relationships also deviate static and spherically symmetric case. Then, a nonstationary Kerr-Newman black hole is researched. The radiation temperature is calculated which is also a local quantity; the Hawking thermal spectrum formula is obtained which is also a quasi-black-body' s thermal spectrum; The black hole entropy is gained and is proportional to the horizon area with a suitable cutoff relationship. The research on the two nonstationary black holes shows that the improved method of Damour-Ruffini and the thin film model
can be effectively applied to the study of a complex black hole on the only condition that nonstationary metric is changing slowly.The next work summarized in chapter 3 is the study of how the generalized uncertainty relation influence the calculation of black hole entropy. First, the generalized uncertainty relation and the brick wall model are utilized to investigate Schwarzschild black hole. The conclusion is expounded and proved that we can't obtain the black hole entropy proportional to the horizon area from the expression of entropy when the distance of two brick walls is long enough. So, we can conclude that the brick wall model is ineffectual while the generalized uncertainty relation is applied to the discussion of black hole entropy. Then, the statistical entropies of Klein-Gordon field and Dirac field inside a Planck thin film on the horizon surface are calculated respectively by combining the generalized uncertainty relation with the thin film model. This leads to both removing the cutoff and acquiring a black hole entropy proportional to the horizon area. The work of this chapter testifies further that applying the generalized uncertainty relation to the study of black hole thermodynamics is effective and the thin film model is rational.The final work given in chapter 3 is the investigation on Killing reductions of 4-dimensional and 5-dimensional spacetimes. First, the 3-dimensional action obtained from the Killing reduction of 4-dimensional action is discussed, the variations of which lead to the same field equations as those reduced from the vacuum Einstein equation by Ge-roch; Then, a series of equations parallel to Geroch's are acquired by extending Geroch's approach to a 5-dimensional spacetime with a Killing symmetry, this provides a new version 5-dimensional Kaluza-Klein theory during the spacelike Killing vector field and a vacuum case. Next, the Killing reduction of spacetime is studied from the viewpoint of variation principle. It turns out that the symmetry-reduced 4-dimensional action from the 5-dimensional Hilbert action would give the correct field equations reduced from 5-dimensional vacuum spacetime with a Killing symmetry by variations with respect to a series of suitable field variables. Finally, an alternative special 4-dimensional action is obtained. Its variations with respect to the other series of field variables can also give the
same field equations reduced from 5-dimensional vacuum field equations. The research described in this chapter shows that the final result of " n-dimensional action^=> reduced (n-l)-dimensional action^=> (n-l)-dimensional spacetime field equations obtained by variations " and that of " n-dimensional action^=> n-dimensional spacetime field equations obtained by variations^=> (n-1)-dimensional spacetime field equations obtained by reduction of n-dimensional field equations " are the same and 5-dimensional gravity with a Killing symmetry is equivalent to 4-dimensional gravity coupled to a vector field and a scalar field.
引文
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