| 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 AdS3/CFT2correspondence are also briefly mentioned. |
