黑洞热力学和引力理论的全息描述
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摘要
全息原理将反de Sitter空间上的量子引力理论对偶于低维度的共形场论,这一思想被广泛用于研究黑洞热力学和三维引力理论。论文利用全息方法考察了极端Kaluza-Klein黑洞和非球面黑洞的热力学,并研究了与引力理论的Weyl不变扩展有关的变分问题。
绪论部分回顾了黑洞热力学的四条定律和Bekenstein-Hawking面积熵公式,概述了弦理论通过假设D膜束缚态的物理图像对黑洞熵进行微观解释的基本思路。在此基础上介绍了简化的全息方法,后者不依赖于弦理论的具体细节以及与超对称有关的知识。只需要利用与黑洞AdS3几何结构对偶的二维共形场论的基本信息,以及描述渐近态密度的Cardy公式,即可方便地得到与宏观的Bekenstein-Hawking熵公式一致的微观熵。之后,介绍了近几年引起广泛重视的Kerr/CFT对应,它有助于研究与现实黑洞较为类似的各种转动黑洞。最后,概述了用AdS3/CFT2对应研究各种三维质量引力理论的基本方法,后者的单圈配分函数和关联函数等物理信息都可以通过全息方法推导出来。
对于非极端黑洞的情形,Kerr/CFT对应中的隐共形对称性方法指出,在近区域、低频率极限下,与黑洞背景下的波动方程相关的标量Laplace算子可以写成SL(2,R)×SL(2,R)代数的平方Casimir的形式。这种局域共形对称性并不直接依赖于几何结构本身,而是存在于波动方程的解空间上面。对于极端黑洞,则需要引入与非极端情形不同的一组新的共形坐标。在第2章中,将这种新的隐共形对称性方法应用于四维极端Kaluza-Klein黑洞,给出了相应的共形坐标和局域矢量生成元的表达式。此外,从微观角度出发导出了共轭荷和关联函数,发现与引力方面的结果是完全一致的。
在早期文献的基础上,一种改进的方法认为二维Liouville理论可以用来描述一般黑洞的近视界极限。根据能动张量在变换到光锥坐标情况下的渐近行为,并利用Christensen-Fulling关系式就能得到Hawking温度。另外,中心荷和零模式与维数约化过程中的参数之间的一般关系式被找到，再利用Cardy公式即可得到黑洞的微观熵。在第3章中，利用这种新的Liouville方法得到了与约化之后的一般二维度规对应的普遍公式。然后，将其应用于考察五维黑环和四维拓扑黑洞，由此导出的热力学参数与宏观分析相一致。最后，还探讨了与旧的Liouville方法之间的联系，以及改进现有研究思路的某些途径。
引力的Weyl不变扩展是通过把原作用量中的曲率张量替换为Weyl几何中的曲率张量，并引入辅助的标量场和Weyl规范场，使新的作用量在Weyl变换下保持不变。为讨论其物理性质，需要运用变分原理来得到能动张量和Weyl规范场的运动方程，这种过程是非常冗长和复杂的。在第4章中，提出了一种修正的Weyl几何以及相应的简化的变分方法。通过引入协变权重为1/2的新的Weyl协变导数，并把附加Weyl联络视为普通的(1，2)型张量，Riemann张量可以改写为更加紧凑的形式。然后，对于协变导数和变分算子同时存在的情况引入两个新的运算法则，从而导出了Weyl几何中的Palatini等式。由此提供了一种相对简化的方法来进行变分操作以及与Riemann张量有关的运算，这样就为讨论Weyl不变的引力理论的物理内涵带来了便利。
结论部分对相关课题的进一步研究作了展望，其中还简要提到了AdS_3／CFT_2对应的高自旋推广以及平直空间极限。
The holographic principle implies that a quantum theory of gravity in anti-deSitter (AdS) space is dual to a conformal feld theory (CFT) in lower dimensions.It has been extensively employed to investigate the black hole thermodynamicsand three-dimensional gravity theories. This dissertation uses the holographicmethod to explore the thermodynamics of extremal Kaluza-Klein black holeand nonspherical black holes, then studies the variational problem related to theWeyl-invariant extension of gravity theories.
In the Introduction, the four laws of black hole thermodynamics and theBekenstein-Hawking area-entropy formula are reviewed. After that we explain thebasic idea of the microscopic description of black hole entropy using string theory.This approach assumes the physical picture of the bound states of D-branes.Based on these results, a simplifed holographic method is also introduced whichdoes not need the details of string theory and the knowledge of supersymmetry.Starting from the AdS3geometric structure associated with a black hole, oneobtains the basic information of the dual two-dimensional conformal feld theory.Then through the application of Cardy formula which describes the asymptoticdensity of states, the microscopic entropy can be easily found to agree with theBekenstein-Hawking formula on the macroscopic side. Afterwards, we introducethe Kerr/CFT correspondence which has attracted a lot of attention in recentyears. This may help us to study rotating black holes which have more relevanceto the reality. At the end of this chapter, the above AdS3/CF T2correspondenceis employed to research the three-dimensional massive gravities. Through thisholographic method, the one-loop partition functions and correlation functionsof the latter can be derived.
In the case of non-extremal black holes, the hidden conformal symmetryapproach has been proposed in the feld of Kerr/CFT correspondence. It statesthat in the near region, low frequency limit, the scalar Laplacian corresponding tothe wave equation in black hole background could be rewritten as the quadraticCasimir of an SL(2, R)×SL(2, R) algebra. This local conformal symmetry doesnot directly rely on the geometric structure but pertains to the solution spaceof wave equation. As for the extremal black holes, a new set of conformal coor-dinates should be introduced which is diferent from the non-extremal case. InChapter2, this new hidden conformal symmetry method is applied to the four-dimensional extremal Kaluza-Klein black hole, and the corresponding expressionsof the conformal coordinates and local vector generators are given. The conjugatecharges and correlation functions are also derived from the microscopic side, andthe results agree with the gravitational expectation.
Based on earlier studies, an improved method has been proposed to describethe near horizon limit of general black holes using Liouville theory. By inspectingthe asymptotic behavior of the energy-momentum tensor in light-cone coordi-nates, and applying the Christensen-Fulling relation, the Hawking temperaturecan be easily obtained. Besides, the relations between the central charge, zeromode and the parameters in the process of dimensional reduction have been giv-en. Thus the microscopic entropy could be obtained by resorting to the Cardyformula. In Chapter3, this new Liouville formalism is used to derive the generalformulas for any two-dimensional metric resulting from the dimensional reduction.With their help, the fve-dimensional black rings and four-dimensional topologicalblack holes are studied, and the thermodynamic parameters are found to agreewith the macroscopic side. At last, we discuss the relations to the old Liouvilleformalism and propose some ways to improve the current approaches.
The Weyl-invariant extension of a gravity theory involves the replacement ofcurvature tensors in the action by those in Weyl geometry, and the introductionof complementary bosonic felds and the Weyl gauge feld. The resulting actionis required to be invariant under the Weyl transformations. To investigate itsphysical properties, one needs to apply the variational principle to derive theenergy-momentum tensor and the equation of motion for Weyl gauge feld. Thisprocedure may become cumbersome and lengthy. In Chapter4, a modifed Weylgeometry and a corresponding simplifed variational approach are proposed. Byintroducing a new Weyl covariant derivative with covariant weight12, and treatingthe additive Weyl connection as a usual (1,2)-type tensor, the Riemann tensor inWeyl geometry can be rewritten in a more compact form. We also introduce twonew transformation laws when both the covariant derivative and the variationoperator are involved. In this way, the Weyl version of Palatini identity is de-rived. With these discoveries combined together, a simplifed method is obtainedto deal with the variational procedure and the calculations involving Rieman-n tensor. This also facilitates the investigation of the physical implications ofWeyl-invariant gravity theories.
In the Conclusion, further directions and perspectives relevant to this re-search project are discussed. The higher spin extension and the flat space limitof AdS_3/CFT_2correspondence are also briefly mentioned.
绪论部分回顾了黑洞热力学的四条定律和Bekenstein-Hawking面积熵公式,概述了弦理论通过假设D膜束缚态的物理图像对黑洞熵进行微观解释的基本思路。在此基础上介绍了简化的全息方法,后者不依赖于弦理论的具体细节以及与超对称有关的知识。只需要利用与黑洞AdS3几何结构对偶的二维共形场论的基本信息,以及描述渐近态密度的Cardy公式,即可方便地得到与宏观的Bekenstein-Hawking熵公式一致的微观熵。之后,介绍了近几年引起广泛重视的Kerr/CFT对应,它有助于研究与现实黑洞较为类似的各种转动黑洞。最后,概述了用AdS3/CFT2对应研究各种三维质量引力理论的基本方法,后者的单圈配分函数和关联函数等物理信息都可以通过全息方法推导出来。
对于非极端黑洞的情形,Kerr/CFT对应中的隐共形对称性方法指出,在近区域、低频率极限下,与黑洞背景下的波动方程相关的标量Laplace算子可以写成SL(2,R)×SL(2,R)代数的平方Casimir的形式。这种局域共形对称性并不直接依赖于几何结构本身,而是存在于波动方程的解空间上面。对于极端黑洞,则需要引入与非极端情形不同的一组新的共形坐标。在第2章中,将这种新的隐共形对称性方法应用于四维极端Kaluza-Klein黑洞,给出了相应的共形坐标和局域矢量生成元的表达式。此外,从微观角度出发导出了共轭荷和关联函数,发现与引力方面的结果是完全一致的。
在早期文献的基础上,一种改进的方法认为二维Liouville理论可以用来描述一般黑洞的近视界极限。根据能动张量在变换到光锥坐标情况下的渐近行为,并利用Christensen-Fulling关系式就能得到Hawking温度。另外,中心荷和零模式与维数约化过程中的参数之间的一般关系式被找到，再利用Cardy公式即可得到黑洞的微观熵。在第3章中，利用这种新的Liouville方法得到了与约化之后的一般二维度规对应的普遍公式。然后，将其应用于考察五维黑环和四维拓扑黑洞，由此导出的热力学参数与宏观分析相一致。最后，还探讨了与旧的Liouville方法之间的联系，以及改进现有研究思路的某些途径。
引力的Weyl不变扩展是通过把原作用量中的曲率张量替换为Weyl几何中的曲率张量，并引入辅助的标量场和Weyl规范场，使新的作用量在Weyl变换下保持不变。为讨论其物理性质，需要运用变分原理来得到能动张量和Weyl规范场的运动方程，这种过程是非常冗长和复杂的。在第4章中，提出了一种修正的Weyl几何以及相应的简化的变分方法。通过引入协变权重为1/2的新的Weyl协变导数，并把附加Weyl联络视为普通的(1，2)型张量，Riemann张量可以改写为更加紧凑的形式。然后，对于协变导数和变分算子同时存在的情况引入两个新的运算法则，从而导出了Weyl几何中的Palatini等式。由此提供了一种相对简化的方法来进行变分操作以及与Riemann张量有关的运算，这样就为讨论Weyl不变的引力理论的物理内涵带来了便利。
结论部分对相关课题的进一步研究作了展望，其中还简要提到了AdS_3／CFT_2对应的高自旋推广以及平直空间极限。
The holographic principle implies that a quantum theory of gravity in anti-deSitter (AdS) space is dual to a conformal feld theory (CFT) in lower dimensions.It has been extensively employed to investigate the black hole thermodynamicsand three-dimensional gravity theories. This dissertation uses the holographicmethod to explore the thermodynamics of extremal Kaluza-Klein black holeand nonspherical black holes, then studies the variational problem related to theWeyl-invariant extension of gravity theories.
In the Introduction, the four laws of black hole thermodynamics and theBekenstein-Hawking area-entropy formula are reviewed. After that we explain thebasic idea of the microscopic description of black hole entropy using string theory.This approach assumes the physical picture of the bound states of D-branes.Based on these results, a simplifed holographic method is also introduced whichdoes not need the details of string theory and the knowledge of supersymmetry.Starting from the AdS3geometric structure associated with a black hole, oneobtains the basic information of the dual two-dimensional conformal feld theory.Then through the application of Cardy formula which describes the asymptoticdensity of states, the microscopic entropy can be easily found to agree with theBekenstein-Hawking formula on the macroscopic side. Afterwards, we introducethe Kerr/CFT correspondence which has attracted a lot of attention in recentyears. This may help us to study rotating black holes which have more relevanceto the reality. At the end of this chapter, the above AdS3/CF T2correspondenceis employed to research the three-dimensional massive gravities. Through thisholographic method, the one-loop partition functions and correlation functionsof the latter can be derived.
In the case of non-extremal black holes, the hidden conformal symmetryapproach has been proposed in the feld of Kerr/CFT correspondence. It statesthat in the near region, low frequency limit, the scalar Laplacian corresponding tothe wave equation in black hole background could be rewritten as the quadraticCasimir of an SL(2, R)×SL(2, R) algebra. This local conformal symmetry doesnot directly rely on the geometric structure but pertains to the solution spaceof wave equation. As for the extremal black holes, a new set of conformal coor-dinates should be introduced which is diferent from the non-extremal case. InChapter2, this new hidden conformal symmetry method is applied to the four-dimensional extremal Kaluza-Klein black hole, and the corresponding expressionsof the conformal coordinates and local vector generators are given. The conjugatecharges and correlation functions are also derived from the microscopic side, andthe results agree with the gravitational expectation.
Based on earlier studies, an improved method has been proposed to describethe near horizon limit of general black holes using Liouville theory. By inspectingthe asymptotic behavior of the energy-momentum tensor in light-cone coordi-nates, and applying the Christensen-Fulling relation, the Hawking temperaturecan be easily obtained. Besides, the relations between the central charge, zeromode and the parameters in the process of dimensional reduction have been giv-en. Thus the microscopic entropy could be obtained by resorting to the Cardyformula. In Chapter3, this new Liouville formalism is used to derive the generalformulas for any two-dimensional metric resulting from the dimensional reduction.With their help, the fve-dimensional black rings and four-dimensional topologicalblack holes are studied, and the thermodynamic parameters are found to agreewith the macroscopic side. At last, we discuss the relations to the old Liouvilleformalism and propose some ways to improve the current approaches.
The Weyl-invariant extension of a gravity theory involves the replacement ofcurvature tensors in the action by those in Weyl geometry, and the introductionof complementary bosonic felds and the Weyl gauge feld. The resulting actionis required to be invariant under the Weyl transformations. To investigate itsphysical properties, one needs to apply the variational principle to derive theenergy-momentum tensor and the equation of motion for Weyl gauge feld. Thisprocedure may become cumbersome and lengthy. In Chapter4, a modifed Weylgeometry and a corresponding simplifed variational approach are proposed. Byintroducing a new Weyl covariant derivative with covariant weight12, and treatingthe additive Weyl connection as a usual (1,2)-type tensor, the Riemann tensor inWeyl geometry can be rewritten in a more compact form. We also introduce twonew transformation laws when both the covariant derivative and the variationoperator are involved. In this way, the Weyl version of Palatini identity is de-rived. With these discoveries combined together, a simplifed method is obtainedto deal with the variational procedure and the calculations involving Rieman-n tensor. This also facilitates the investigation of the physical implications ofWeyl-invariant gravity theories.
In the Conclusion, further directions and perspectives relevant to this re-search project are discussed. The higher spin extension and the flat space limitof AdS_3/CFT_2correspondence are also briefly mentioned.
引文
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